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📄 BigFloat.pm package Math::BigFloat; # # Mike grinned. 'Two down, infinity to go' - Mike Nostrus in 'Before and After' # # The following hash values are used internally: # sign : "+", "-", "+inf", "-inf", or "NaN" if not a number # _m : mantissa ($LIB thingy) # _es : sign of _e # _e : exponent ($LIB thingy) # _a : accuracy # _p : precision use 5.006001; use strict; use warnings; use Carp qw< carp croak >; use Math::BigInt (); our $VERSION = '1.999818'; require Exporter; our @ISA = qw/Math::BigInt/; our @EXPORT_OK = qw/bpi/; # $_trap_inf/$_trap_nan are internal and should never be accessed from outside our ($AUTOLOAD, $accuracy, $precision, $div_scale, $round_mode, $rnd_mode, $upgrade, $downgrade, $_trap_nan, $_trap_inf); use overload # overload key: with_assign '+' => sub { $_[0] -> copy() -> badd($_[1]); }, '-' => sub { my $c = $_[0] -> copy(); $_[2] ? $c -> bneg() -> badd($_[1]) : $c -> bsub($_[1]); }, '' => sub { $_[0] -> copy() -> bmul($_[1]); }, '/' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bdiv($_[0]) : $_[0] -> copy() -> bdiv($_[1]); }, '%' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bmod($_[0]) : $_[0] -> copy() -> bmod($_[1]); }, '' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bpow($_[0]) : $_[0] -> copy() -> bpow($_[1]); }, '<<' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> blsft($_[0]) : $_[0] -> copy() -> blsft($_[1]); }, '>>' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> brsft($_[0]) : $_[0] -> copy() -> brsft($_[1]); }, # overload key: assign '+=' => sub { $_[0]->badd($_[1]); }, '-=' => sub { $_[0]->bsub($_[1]); }, '=' => sub { $_[0]->bmul($_[1]); }, '/=' => sub { scalar $_[0]->bdiv($_[1]); }, '%=' => sub { $_[0]->bmod($_[1]); }, '*=' => sub { $_[0]->bpow($_[1]); }, '<<=' => sub { $_[0]->blsft($_[1]); }, '>>=' => sub { $_[0]->brsft($_[1]); }, # 'x=' => sub { }, # '.=' => sub { }, # overload key: num_comparison '<' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> blt($_[0]) : $_[0] -> blt($_[1]); }, '<=' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> ble($_[0]) : $_[0] -> ble($_[1]); }, '>' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bgt($_[0]) : $_[0] -> bgt($_[1]); }, '>=' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bge($_[0]) : $_[0] -> bge($_[1]); }, '==' => sub { $_[0] -> beq($_[1]); }, '!=' => sub { $_[0] -> bne($_[1]); }, # overload key: 3way_comparison '<=>' => sub { my $cmp = $_[0] -> bcmp($_[1]); defined($cmp) && $_[2] ? -$cmp : $cmp; }, 'cmp' => sub { $_[2] ? "$_[1]" cmp $_[0] -> bstr() : $_[0] -> bstr() cmp "$_[1]"; }, # overload key: str_comparison # 'lt' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bstrlt($_[0]) # : $_[0] -> bstrlt($_[1]); }, # # 'le' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bstrle($_[0]) # : $_[0] -> bstrle($_[1]); }, # # 'gt' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bstrgt($_[0]) # : $_[0] -> bstrgt($_[1]); }, # # 'ge' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bstrge($_[0]) # : $_[0] -> bstrge($_[1]); }, # # 'eq' => sub { $_[0] -> bstreq($_[1]); }, # # 'ne' => sub { $_[0] -> bstrne($_[1]); }, # overload key: binary '&' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> band($_[0]) : $_[0] -> copy() -> band($_[1]); }, '&=' => sub { $_[0] -> band($_[1]); }, '\|' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bior($_[0]) : $_[0] -> copy() -> bior($_[1]); }, '\|=' => sub { $_[0] -> bior($_[1]); }, '^' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bxor($_[0]) : $_[0] -> copy() -> bxor($_[1]); }, '^=' => sub { $_[0] -> bxor($_[1]); }, # '&.' => sub { }, # '&.=' => sub { }, # '\|.' => sub { }, # '\|.=' => sub { }, # '^.' => sub { }, # '^.=' => sub { }, # overload key: unary 'neg' => sub { $_[0] -> copy() -> bneg(); }, # '!' => sub { }, '~' => sub { $_[0] -> copy() -> bnot(); }, # '~.' => sub { }, # overload key: mutators '++' => sub { $_[0] -> binc() }, '--' => sub { $_[0] -> bdec() }, # overload key: func 'atan2' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> batan2($_[0]) : $_[0] -> copy() -> batan2($_[1]); }, 'cos' => sub { $_[0] -> copy() -> bcos(); }, 'sin' => sub { $_[0] -> copy() -> bsin(); }, 'exp' => sub { $_[0] -> copy() -> bexp($_[1]); }, 'abs' => sub { $_[0] -> copy() -> babs(); }, 'log' => sub { $_[0] -> copy() -> blog(); }, 'sqrt' => sub { $_[0] -> copy() -> bsqrt(); }, 'int' => sub { $_[0] -> copy() -> bint(); }, # overload key: conversion 'bool' => sub { $_[0] -> is_zero() ? '' : 1; }, '""' => sub { $_[0] -> bstr(); }, '0+' => sub { $_[0] -> numify(); }, '=' => sub { $_[0]->copy(); }, ; ############################################################################## # global constants, flags and assorted stuff # the following are public, but their usage is not recommended. Use the # accessor methods instead. # class constants, use Class->constant_name() to access # one of 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common' $round_mode = 'even'; $accuracy = undef; $precision = undef; $div_scale = 40; $upgrade = undef; $downgrade = undef; # the package we are using for our private parts, defaults to: # Math::BigInt->config('lib') my $LIB = 'Math::BigInt::Calc'; # are NaNs ok? (otherwise it dies when encountering an NaN) set w/ config() $_trap_nan = 0; # the same for infinity $_trap_inf = 0; # constant for easier life my $nan = 'NaN'; my $IMPORT = 0; # was import() called yet? used to make require work # some digits of accuracy for blog(undef, 10); which we use in blog() for speed my $LOG_10 = '2.3025850929940456840179914546843642076011014886287729760333279009675726097'; my $LOG_10_A = length($LOG_10)-1; # ditto for log(2) my $LOG_2 = '0.6931471805599453094172321214581765680755001343602552541206800094933936220'; my $LOG_2_A = length($LOG_2)-1; my $HALF = '0.5'; # made into an object if nec. ############################################################################## # the old code had $rnd_mode, so we need to support it, too sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; } sub FETCH { return $round_mode; } sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); } BEGIN { # when someone sets $rnd_mode, we catch this and check the value to see # whether it is valid or not. $rnd_mode = 'even'; tie $rnd_mode, 'Math::BigFloat'; # we need both of them in this package: as_int = \&as_number; } sub DESTROY { # going through AUTOLOAD for every DESTROY is costly, avoid it by empty sub } sub AUTOLOAD { # make fxxx and bxxx both work by selectively mapping fxxx() to MBF::bxxx() my $name = $AUTOLOAD; $name =~ s/(.):://; # split package my $c = $1 \|\| __PACKAGE__; no strict 'refs'; $c->import() if $IMPORT == 0; if (!_method_alias($name)) { if (!defined $name) { # delayed load of Carp and avoid recursion croak("$c: Can't call a method without name"); } if (!_method_hand_up($name)) { # delayed load of Carp and avoid recursion croak("Can't call $c\-\>$name, not a valid method"); } # try one level up, but subst. bxxx() for fxxx() since MBI only got bxxx() $name =~ s/^f/b/; return &{"Math::BigInt"."::$name"}(@_); } my $bname = $name; $bname =~ s/^f/b/; $c .= "::$name"; {$c} = \&{$bname}; &{$c}; # uses @_ } ############################################################################## { # valid method aliases for AUTOLOAD my %methods = map { $_ => 1 } qw / fadd fsub fmul fdiv fround ffround fsqrt fmod fstr fsstr fpow fnorm fint facmp fcmp fzero fnan finf finc fdec ffac fneg fceil ffloor frsft flsft fone flog froot fexp /; # valid methods that can be handed up (for AUTOLOAD) my %hand_ups = map { $_ => 1 } qw / is_nan is_inf is_negative is_positive is_pos is_neg accuracy precision div_scale round_mode fabs fnot objectify upgrade downgrade bone binf bnan bzero bsub /; sub _method_alias { exists $methods{$_[0]\|\|''}; } sub _method_hand_up { exists $hand_ups{$_[0]\|\|''}; } } sub isa { my ($self, $class) = @_; return if $class =~ /^Math::BigInt/; # we aren't one of these UNIVERSAL::isa($self, $class); } sub config { # return (later set?) configuration data as hash ref my $class = shift \|\| 'Math::BigFloat'; # Getter/accessor. if (@_ == 1 && ref($_[0]) ne 'HASH') { my $param = shift; return $class if $param eq 'class'; return $LIB if $param eq 'with'; return $class->SUPER::config($param); } # Setter. my $cfg = $class->SUPER::config(@_); # now we need only to override the ones that are different from our parent $cfg->{class} = $class; $cfg->{with} = $LIB; $cfg; } ############################################################################### # Constructor methods ############################################################################### sub new { # Create a new Math::BigFloat object from a string or another bigfloat object. # _e: exponent # _m: mantissa # sign => ("+", "-", "+inf", "-inf", or "NaN") my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; my ($wanted, @r) = @_; # avoid numify-calls by not using \|\| on $wanted! unless (defined $wanted) { #carp("Use of uninitialized value in new"); return $self->bzero(@r); } # Using $wanted->isa("Math::BigFloat") here causes a 'Deep recursion on # subroutine "Math::BigFloat::as_number"' in some tests. Fixme! if (UNIVERSAL::isa($wanted, 'Math::BigFloat')) { my $copy = $wanted -> copy(); if ($selfref) { # if new() called as instance method %$self = %$copy; } else { # if new() called as class method $self = $copy; } return $copy; } $class->import() if $IMPORT == 0; # make require work # If called as a class method, initialize a new object. $self = bless {}, $class unless $selfref; # shortcut for bigints and its subclasses if ((ref($wanted)) && $wanted -> can("as_number")) { $self->{_m} = $wanted->as_number()->{value}; # get us a bigint copy $self->{_e} = $LIB->_zero(); $self->{_es} = '+'; $self->{sign} = $wanted->sign(); return $self->bnorm(); } # else: got a string or something masquerading as number (with overload) # Handle Infs. if ($wanted =~ /^\s([+-]?)inf(inity)?\s\z/i) { return $downgrade->new($wanted) if $downgrade; my $sgn = $1 \|\| '+'; $self->{sign} = $sgn . 'inf'; # set a default sign for bstr() return $self->binf($sgn); } # Handle explicit NaNs (not the ones returned due to invalid input). if ($wanted =~ /^\s([+-]?)nan\s\z/i) { return $downgrade->new($wanted) if $downgrade; $self = $class -> bnan(); $self->round(@r) unless @r >= 2 && !defined $r[0] && !defined $r[1]; return $self; } # Handle hexadecimal numbers. We auto-detect hexadecimal numbers if they # have a "0x" or "0X" prefix. if ($wanted =~ /^\s[+-]?0[Xx]/) { $self = $class -> from_hex($wanted); $self->round(@r) unless @r >= 2 && !defined $r[0] && !defined $r[1]; return $self; } # Handle octal numbers. We auto-detect octal numbers if they have a "0" # prefix and a binary exponent. if ($wanted =~ / ^ \s # sign [+-]? # prefix 0 # significand using the octal digits 0..7 [0-7]+ (?: _ [0-7]+ )* (?: \. (?: [0-7]+ (?: _ [0-7]+ )* )? )? # exponent (power of 2) using decimal digits [Pp] [+-]? \d+ (?: _ \d+ )* \s* $ /x) { $self = $class -> from_oct($wanted); $self->round(@r) unless @r >= 2 && !defined $r[0] && !defined $r[1]; return $self; } # Handle binary numbers. We auto-detect binary numbers if they have a "0b" # or "0B" prefix. if ($wanted =~ /^\s[+-]?0[Bb]/) { $self = $class -> from_bin($wanted); $self->round(@r) unless @r >= 2 && !defined $r[0] && !defined $r[1]; return $self; } # Shortcut for simple forms like '12' that have no trailing zeros. if ($wanted =~ /^([+-]?)0([1-9][0-9][1-9])$/) { $self->{_e} = $LIB -> _zero(); $self->{_es} = '+'; $self->{sign} = $1 \|\| '+'; $self->{_m} = $LIB -> _new($2); if (!$downgrade) { $self->round(@r) unless @r >= 2 && !defined $r[0] && !defined $r[1]; return $self; } } my ($mis, $miv, $mfv, $es, $ev) = Math::BigInt::_split($wanted); if (!ref $mis) { if ($_trap_nan) { croak("$wanted is not a number initialized to $class"); } return $downgrade->bnan() if $downgrade; $self->{_e} = $LIB->_zero(); $self->{_es} = '+'; $self->{_m} = $LIB->_zero(); $self->{sign} = $nan; } else { # make integer from mantissa by adjusting exp, then convert to int $self->{_e} = $LIB->_new($$ev); # exponent $self->{_es} = $$es \|\| '+'; my $mantissa = "$$miv$$mfv"; # create mant. $mantissa =~ s/^0+(\d)/$1/; # strip leading zeros $self->{_m} = $LIB->_new($mantissa); # create mant. # 3.123E0 = 3123E-3, and 3.123E-2 => 3123E-5 if (CORE::length($$mfv) != 0) { my $len = $LIB->_new(CORE::length($$mfv)); ($self->{_e}, $self->{_es}) = _e_sub($self->{_e}, $len, $self->{_es}, '+'); } # we can only have trailing zeros on the mantissa if $$mfv eq '' else { # Use a regexp to count the trailing zeros in $$miv instead of # _zeros() because that is faster, especially when _m is not stored # in base 10. my $zeros = 0; $zeros = CORE::length($1) if $$miv =~ /[1-9](0)$/; if ($zeros != 0) { my $z = $LIB->_new($zeros); # turn '120e2' into '12e3' $self->{_m} = $LIB->_rsft($self->{_m}, $z, 10); ($self->{_e}, $self->{_es}) = _e_add($self->{_e}, $z, $self->{_es}, '+'); } } $self->{sign} = $$mis; # for something like 0Ey, set y to 0, and -0 => +0 # Check $$miv for being '0' and $$mfv eq '', because otherwise _m could not # have become 0. That's faster than to call $LIB->_is_zero(). $self->{sign} = '+', $self->{_e} = $LIB->_zero() if $$miv eq '0' and $$mfv eq ''; if (!$downgrade) { $self->round(@r) unless @r >= 2 && !defined $r[0] && !defined $r[1]; return $self; } } # if downgrade, inf, NaN or integers go down if ($downgrade && $self->{_es} eq '+') { if ($LIB->_is_zero($self->{_e})) { return $downgrade->new($$mis . $LIB->_str($self->{_m})); } return $downgrade->new($self->bsstr()); } $self->bnorm(); $self->round(@r) unless @r >= 2 && !defined $r[0] && !defined $r[1]; return $self; } sub from_hex { my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; # Don't modify constant (read-only) objects. return if $selfref && $self->modify('from_hex'); my $str = shift; # If called as a class method, initialize a new object. $self = $class -> bzero() unless $selfref; if ($str =~ s/ ^ \s* # sign ( [+-]? ) # optional "hex marker" (?: 0? x )? # significand using the hex digits 0..9 and a..f ( [0-9a-fA-F]+ (?: _ [0-9a-fA-F]+ )* (?: \. (?: [0-9a-fA-F]+ (?: _ [0-9a-fA-F]+ )* )? )? \| \. [0-9a-fA-F]+ (?: _ [0-9a-fA-F]+ )* ) # exponent (power of 2) using decimal digits (?: [Pp] ( [+-]? ) ( \d+ (?: _ \d+ )* ) )? \s* $ //x) { my $s_sign = $1 \|\| '+'; my $s_value = $2; my $e_sign = $3 \|\| '+'; my $e_value = $4 \|\| '0'; $s_value =~ tr/_//d; $e_value =~ tr/_//d; # The significand must be multiplied by 2 raised to this exponent. my $two_expon = $class -> new($e_value); $two_expon -> bneg() if $e_sign eq '-'; # If there is a dot in the significand, remove it and adjust the # exponent according to the number of digits in the fraction part of # the significand. Since the digits in the significand are in base 16, # but the exponent is only in base 2, multiply the exponent adjustment # value by log(16) / log(2) = 4. my $idx = index($s_value, '.'); if ($idx >= 0) { substr($s_value, $idx, 1) = ''; $two_expon -= $class -> new(CORE::length($s_value)) -> bsub($idx) -> bmul("4"); } $self -> {sign} = $s_sign; $self -> {_m} = $LIB -> _from_hex('0x' . $s_value); if ($two_expon > 0) { my $factor = $class -> new("2") -> bpow($two_expon); $self -> bmul($factor); } elsif ($two_expon < 0) { my $factor = $class -> new("0.5") -> bpow(-$two_expon); $self -> bmul($factor); } return $self; } return $self->bnan(); } sub from_oct { my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; # Don't modify constant (read-only) objects. return if $selfref && $self->modify('from_oct'); my $str = shift; # If called as a class method, initialize a new object. $self = $class -> bzero() unless $selfref; if ($str =~ s/ ^ \s* # sign ( [+-]? ) # significand using the octal digits 0..7 ( [0-7]+ (?: _ [0-7]+ )* (?: \. (?: [0-7]+ (?: _ [0-7]+ )* )? )? \| \. [0-7]+ (?: _ [0-7]+ )* ) # exponent (power of 2) using decimal digits (?: [Pp] ( [+-]? ) ( \d+ (?: _ \d+ )* ) )? \s* $ //x) { my $s_sign = $1 \|\| '+'; my $s_value = $2; my $e_sign = $3 \|\| '+'; my $e_value = $4 \|\| '0'; $s_value =~ tr/_//d; $e_value =~ tr/_//d; # The significand must be multiplied by 2 raised to this exponent. my $two_expon = $class -> new($e_value); $two_expon -> bneg() if $e_sign eq '-'; # If there is a dot in the significand, remove it and adjust the # exponent according to the number of digits in the fraction part of # the significand. Since the digits in the significand are in base 8, # but the exponent is only in base 2, multiply the exponent adjustment # value by log(8) / log(2) = 3. my $idx = index($s_value, '.'); if ($idx >= 0) { substr($s_value, $idx, 1) = ''; $two_expon -= $class -> new(CORE::length($s_value)) -> bsub($idx) -> bmul("3"); } $self -> {sign} = $s_sign; $self -> {_m} = $LIB -> _from_oct($s_value); if ($two_expon > 0) { my $factor = $class -> new("2") -> bpow($two_expon); $self -> bmul($factor); } elsif ($two_expon < 0) { my $factor = $class -> new("0.5") -> bpow(-$two_expon); $self -> bmul($factor); } return $self; } return $self->bnan(); } sub from_bin { my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; # Don't modify constant (read-only) objects. return if $selfref && $self->modify('from_bin'); my $str = shift; # If called as a class method, initialize a new object. $self = $class -> bzero() unless $selfref; if ($str =~ s/ ^ \s* # sign ( [+-]? ) # optional "bin marker" (?: 0? b )? # significand using the binary digits 0 and 1 ( [01]+ (?: _ [01]+ )* (?: \. (?: [01]+ (?: _ [01]+ )* )? )? \| \. [01]+ (?: _ [01]+ )* ) # exponent (power of 2) using decimal digits (?: [Pp] ( [+-]? ) ( \d+ (?: _ \d+ )* ) )? \s* $ //x) { my $s_sign = $1 \|\| '+'; my $s_value = $2; my $e_sign = $3 \|\| '+'; my $e_value = $4 \|\| '0'; $s_value =~ tr/_//d; $e_value =~ tr/_//d; # The significand must be multiplied by 2 raised to this exponent. my $two_expon = $class -> new($e_value); $two_expon -> bneg() if $e_sign eq '-'; # If there is a dot in the significand, remove it and adjust the # exponent according to the number of digits in the fraction part of # the significand. my $idx = index($s_value, '.'); if ($idx >= 0) { substr($s_value, $idx, 1) = ''; $two_expon -= $class -> new(CORE::length($s_value)) -> bsub($idx); } $self -> {sign} = $s_sign; $self -> {_m} = $LIB -> _from_bin('0b' . $s_value); if ($two_expon > 0) { my $factor = $class -> new("2") -> bpow($two_expon); $self -> bmul($factor); } elsif ($two_expon < 0) { my $factor = $class -> new("0.5") -> bpow(-$two_expon); $self -> bmul($factor); } return $self; } return $self->bnan(); } sub from_ieee754 { my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; # Don't modify constant (read-only) objects. return if $selfref && $self->modify('from_ieee754'); my $in = shift; # input string (or raw bytes) my $format = shift; # format ("binary32", "decimal64" etc.) my $enc; # significand encoding (applies only to decimal) my $k; # storage width in bits my $b; # base if ($format =~ /^binary(\d+)\z/) { $k = $1; $b = 2; } elsif ($format =~ /^decimal(\d+)(dpd\|bcd)?\z/) { $k = $1; $b = 10; $enc = $2 \|\| 'dpd'; # default is dencely-packed decimals (DPD) } elsif ($format eq 'half') { $k = 16; $b = 2; } elsif ($format eq 'single') { $k = 32; $b = 2; } elsif ($format eq 'double') { $k = 64; $b = 2; } elsif ($format eq 'quadruple') { $k = 128; $b = 2; } elsif ($format eq 'octuple') { $k = 256; $b = 2; } elsif ($format eq 'sexdecuple') { $k = 512; $b = 2; } if ($b == 2) { # Get the parameters for this format. my $p; # precision (in bits) my $t; # number of bits in significand my $w; # number of bits in exponent if ($k == 16) { # binary16 (half-precision) $p = 11; $t = 10; $w = 5; } elsif ($k == 32) { # binary32 (single-precision) $p = 24; $t = 23; $w = 8; } elsif ($k == 64) { # binary64 (double-precision) $p = 53; $t = 52; $w = 11; } else { # binaryN (quadruple-precision and above) if ($k < 128 \|\| $k != 32 * sprintf('%.0f', $k / 32)) { croak "Number of bits must be 16, 32, 64, or >= 128 and", " a multiple of 32"; } $p = $k - sprintf('%.0f', 4 * log($k) / log(2)) + 13; $t = $p - 1; $w = $k - $t - 1; } # The maximum exponent, minimum exponent, and exponent bias. my $emax = Math::BigInt -> new(2) -> bpow($w - 1) -> bdec(); my $emin = 1 - $emax; my $bias = $emax; # Undefined input. unless (defined $in) { carp("Input is undefined"); return $self -> bzero(); } # Make sure input string is a string of zeros and ones. my $len = CORE::length $in; if (8 * $len == $k) { # bytes $in = unpack "B", $in; } elsif (4 $len == $k) { # hexadecimal if ($in =~ /([^\da-f])/i) { croak "Illegal hexadecimal digit '$1'"; } $in = unpack "B", pack "H", $in; } elsif ($len == $k) { # bits if ($in =~ /([^01])/) { croak "Illegal binary digit '$1'"; } } else { croak "Unknown input -- $in"; } # Split bit string into sign, exponent, and mantissa/significand. my $sign = substr($in, 0, 1) eq '1' ? '-' : '+'; my $expo = $class -> from_bin(substr($in, 1, $w)); my $mant = $class -> from_bin(substr($in, $w + 1)); my $x; $expo -> bsub($bias); # subtract bias if ($expo < $emin) { # zero and subnormals if ($mant == 0) { # zero $x = $class -> bzero(); } else { # subnormals # compute (1/$b)(N) rather than ($b)(-N) $x = $class -> new("0.5"); # 1/$b $x -> bpow($bias + $t - 1) -> bmul($mant); $x -> bneg() if $sign eq '-'; } } elsif ($expo > $emax) { # inf and nan if ($mant == 0) { # inf $x = $class -> binf($sign); } else { # nan $x = $class -> bnan(); } } else { # normals $mant = $class -> new(2) -> bpow($t) -> badd($mant); if ($expo < $t) { # compute (1/$b)(N) rather than ($b)(-N) $x = $class -> new("0.5"); # 1/$b $x -> bpow($t - $expo) -> bmul($mant); } else { $x = $class -> new(2); $x -> bpow($expo - $t) -> bmul($mant); } $x -> bneg() if $sign eq '-'; } if ($selfref) { $self -> {sign} = $x -> {sign}; $self -> {_m} = $x -> {_m}; $self -> {_es} = $x -> {_es}; $self -> {_e} = $x -> {_e}; } else { $self = $x; } return $self; } croak("The format '$format' is not yet supported."); } sub bzero { # create/assign '+0' if (@_ == 0) { #carp("Using bone() as a function is deprecated;", # " use bone() as a method instead"); unshift @_, __PACKAGE__; } my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; $self->import() if $IMPORT == 0; # make require work return if $selfref && $self->modify('bzero'); $self = bless {}, $class unless $selfref; $self -> {sign} = '+'; $self -> {_m} = $LIB -> _zero(); $self -> {_es} = '+'; $self -> {_e} = $LIB -> _zero(); # If rounding parameters are given as arguments, use them. If no rounding # parameters are given, and if called as a class method initialize the new # instance with the class variables. if (@_) { croak "can't specify both accuracy and precision" if @_ >= 2 && defined $_[0] && defined $_[1]; $self->{_a} = $_[0]; $self->{_p} = $_[1]; } else { unless($selfref) { $self->{_a} = $class -> accuracy(); $self->{_p} = $class -> precision(); } } return $self; } sub bone { # Create or assign '+1' (or -1 if given sign '-'). if (@_ == 0 \|\| (defined($_[0]) && ($_[0] eq '+' \|\| $_[0] eq '-'))) { #carp("Using bone() as a function is deprecated;", # " use bone() as a method instead"); unshift @_, __PACKAGE__; } my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; $self->import() if $IMPORT == 0; # make require work return if $selfref && $self->modify('bone'); my $sign = shift; $sign = defined $sign && $sign =~ /^\s-/ ? "-" : "+"; $self = bless {}, $class unless $selfref; $self -> {sign} = $sign; $self -> {_m} = $LIB -> _one(); $self -> {_es} = '+'; $self -> {_e} = $LIB -> _zero(); # If rounding parameters are given as arguments, use them. If no rounding # parameters are given, and if called as a class method initialize the new # instance with the class variables. if (@_) { croak "can't specify both accuracy and precision" if @_ >= 2 && defined $_[0] && defined $_[1]; $self->{_a} = $_[0]; $self->{_p} = $_[1]; } else { unless($selfref) { $self->{_a} = $class -> accuracy(); $self->{_p} = $class -> precision(); } } return $self; } sub binf { # create/assign a '+inf' or '-inf' if (@_ == 0 \|\| (defined($_[0]) && !ref($_[0]) && $_[0] =~ /^\s[+-](inf(inity)?)?\s$/)) { #carp("Using binf() as a function is deprecated;", # " use binf() as a method instead"); unshift @_, __PACKAGE__; } my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; { no strict 'refs'; if (${"${class}::_trap_inf"}) { croak("Tried to create +-inf in $class->binf()"); } } $self->import() if $IMPORT == 0; # make require work return if $selfref && $self->modify('binf'); my $sign = shift; $sign = defined $sign && $sign =~ /^\s-/ ? "-" : "+"; $self = bless {}, $class unless $selfref; $self -> {sign} = $sign . 'inf'; $self -> {_m} = $LIB -> _zero(); $self -> {_es} = '+'; $self -> {_e} = $LIB -> _zero(); # If rounding parameters are given as arguments, use them. If no rounding # parameters are given, and if called as a class method initialize the new # instance with the class variables. if (@_) { croak "can't specify both accuracy and precision" if @_ >= 2 && defined $_[0] && defined $_[1]; $self->{_a} = $_[0]; $self->{_p} = $_[1]; } else { unless($selfref) { $self->{_a} = $class -> accuracy(); $self->{_p} = $class -> precision(); } } return $self; } sub bnan { # create/assign a 'NaN' if (@_ == 0) { #carp("Using bnan() as a function is deprecated;", # " use bnan() as a method instead"); unshift @_, __PACKAGE__; } my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; { no strict 'refs'; if (${"${class}::_trap_nan"}) { croak("Tried to create NaN in $class->bnan()"); } } $self->import() if $IMPORT == 0; # make require work return if $selfref && $self->modify('bnan'); $self = bless {}, $class unless $selfref; $self -> {sign} = $nan; $self -> {_m} = $LIB -> _zero(); $self -> {_es} = '+'; $self -> {_e} = $LIB -> _zero(); # If rounding parameters are given as arguments, use them. If no rounding # parameters are given, and if called as a class method initialize the new # instance with the class variables. if (@_) { croak "can't specify both accuracy and precision" if @_ >= 2 && defined $_[0] && defined $_[1]; $self->{_a} = $_[0]; $self->{_p} = $_[1]; } else { unless($selfref) { $self->{_a} = $class -> accuracy(); $self->{_p} = $class -> precision(); } } return $self; } sub bpi { # Called as Argument list # --------- ------------- # Math::BigFloat->bpi() ("Math::BigFloat") # Math::BigFloat->bpi(10) ("Math::BigFloat", 10) # $x->bpi() ($x) # $x->bpi(10) ($x, 10) # Math::BigFloat::bpi() () # Math::BigFloat::bpi(10) (10) # # In ambiguous cases, we favour the OO-style, so the following case # # $n = Math::BigFloat->new("10"); # $x = Math::BigFloat->bpi($n); # # which gives an argument list with the single element $n, is resolved as # # $n->bpi(); my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; my @r; # rounding paramters # If bpi() is called as a function ... # # This cludge is necessary because we still support bpi() as a function. If # bpi() is called with either no argument or one argument, and that one # argument is either undefined or a scalar that looks like a number, then # we assume bpi() is called as a function. if (@_ == 0 && (defined($self) && !ref($self) && $self =~ /^\s[+-]?\d/i) \|\| !defined($self)) { $r[0] = $self; $class = __PACKAGE__; $self = $class -> bzero(@r); # initialize } # ... or if bpi() is called as a method ... else { @r = @_; if ($selfref) { # bpi() called as instance method return $self if $self -> modify('bpi'); } else { # bpi() called as class method $self = $class -> bzero(@r); # initialize } } ($self, @r) = $self -> _find_round_parameters(@r); # The accuracy, i.e., the number of digits. Pi has one digit before the # dot, so a precision of 4 digits is equivalent to an accuracy of 5 digits. my $n = defined $r[0] ? $r[0] : defined $r[1] ? 1 - $r[1] : $self -> div_scale(); my $rmode = defined $r[2] ? $r[2] : $self -> round_mode(); my $pi; if ($n <= 1000) { # 75 x 14 = 1050 digits my $all_digits = <<EOF; 314159265358979323846264338327950288419716939937510582097494459230781640628 620899862803482534211706798214808651328230664709384460955058223172535940812 848111745028410270193852110555964462294895493038196442881097566593344612847 564823378678316527120190914564856692346034861045432664821339360726024914127 372458700660631558817488152092096282925409171536436789259036001133053054882 046652138414695194151160943305727036575959195309218611738193261179310511854 807446237996274956735188575272489122793818301194912983367336244065664308602 139494639522473719070217986094370277053921717629317675238467481846766940513 200056812714526356082778577134275778960917363717872146844090122495343014654 958537105079227968925892354201995611212902196086403441815981362977477130996 051870721134999999837297804995105973173281609631859502445945534690830264252 230825334468503526193118817101000313783875288658753320838142061717766914730 359825349042875546873115956286388235378759375195778185778053217122680661300 192787661119590921642019893809525720106548586327886593615338182796823030195 EOF # Should we round up? my $round_up; # From the string above, we need to extract the number of digits we # want plus extra characters for the newlines. my $nchrs = $n + int($n / 75); # Extract the digits we want. my $digits = substr($all_digits, 0, $nchrs); # Find out whether we should round up or down. Rounding is easy, since # pi is trancendental. With directed rounding, it doesn't matter what # the following digits are. With rounding to nearest, we only have to # look at one extra digit. if ($rmode eq 'trunc') { $round_up = 0; } else { my $next_digit = substr($all_digits, $nchrs, 1); $round_up = $next_digit lt '5' ? 0 : 1; } # Remove the newlines. $digits =~ tr/0-9//cd; # Now do the rounding. We could easily make the regex substitution # handle all cases, but we avoid using the regex engine when it is # simple to avoid it. if ($round_up) { my $last_digit = substr($digits, -1, 1); if ($last_digit lt '9') { substr($digits, -1, 1) = ++$last_digit; } else { $digits =~ s/([0-8])(9+)$/ ($1 + 1) . ("0" x CORE::length($2)) /e; } } # Append the exponent and convert to an object. $pi = Math::BigFloat -> new($digits . 'e-' . ($n - 1)); } else { # For large accuracy, the arctan formulas become very inefficient with # Math::BigFloat, so use Brent-Salamin (aka AGM or Gauss-Legendre). # Use a few more digits in the intermediate computations. $n += 8; $HALF = $class -> new($HALF) unless ref($HALF); my ($an, $bn, $tn, $pn) = ($class -> bone, $HALF -> copy() -> bsqrt($n), $HALF -> copy() -> bmul($HALF), $class -> bone); while ($pn < $n) { my $prev_an = $an -> copy(); $an -> badd($bn) -> bmul($HALF, $n); $bn -> bmul($prev_an) -> bsqrt($n); $prev_an -> bsub($an); $tn -> bsub($pn $prev_an * $prev_an); $pn -> badd($pn); } $an -> badd($bn); $an -> bmul($an, $n) -> bdiv(4 * $tn, $n); $an -> round(@r); $pi = $an; } if (defined $r[0]) { $pi -> accuracy($r[0]); } elsif (defined $r[1]) { $pi -> precision($r[1]); } for my $key (qw/ sign _m _es _e _a _p /) { $self -> {$key} = $pi -> {$key}; } return $self; } sub copy { my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; # If called as a class method, the object to copy is the next argument. $self = shift() unless $selfref; my $copy = bless {}, $class; $copy->{sign} = $self->{sign}; $copy->{_es} = $self->{_es}; $copy->{_m} = $LIB->_copy($self->{_m}); $copy->{_e} = $LIB->_copy($self->{_e}); $copy->{_a} = $self->{_a} if exists $self->{_a}; $copy->{_p} = $self->{_p} if exists $self->{_p}; return $copy; } sub as_number { # return copy as a bigint representation of this Math::BigFloat number my ($class, $x) = ref($_[0]) ? (ref($_[0]), $_[0]) : objectify(1, @_); return $x if $x->modify('as_number'); if (!$x->isa('Math::BigFloat')) { # if the object can as_number(), use it return $x->as_number() if $x->can('as_number'); # otherwise, get us a float and then a number $x = $x->can('as_float') ? $x->as_float() : $class->new(0+"$x"); } return Math::BigInt->binf($x->sign()) if $x->is_inf(); return Math::BigInt->bnan() if $x->is_nan(); my $z = $LIB->_copy($x->{_m}); if ($x->{_es} eq '-') { # < 0 $z = $LIB->_rsft($z, $x->{_e}, 10); } elsif (! $LIB->_is_zero($x->{_e})) { # > 0 $z = $LIB->_lsft($z, $x->{_e}, 10); } $z = Math::BigInt->new($x->{sign} . $LIB->_str($z)); $z; } ############################################################################### # Boolean methods ############################################################################### sub is_zero { # return true if arg (BFLOAT or num_str) is zero my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); ($x->{sign} eq '+' && $LIB->_is_zero($x->{_m})) ? 1 : 0; } sub is_one { # return true if arg (BFLOAT or num_str) is +1 or -1 if signis given my ($class, $x, $sign) = ref($_[0]) ? (undef, @_) : objectify(1, @_); $sign = '+' if !defined $sign \|\| $sign ne '-'; ($x->{sign} eq $sign && $LIB->_is_zero($x->{_e}) && $LIB->_is_one($x->{_m})) ? 1 : 0; } sub is_odd { # return true if arg (BFLOAT or num_str) is odd or false if even my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); (($x->{sign} =~ /^[+-]$/) && # NaN & +-inf aren't ($LIB->_is_zero($x->{_e})) && ($LIB->_is_odd($x->{_m}))) ? 1 : 0; } sub is_even { # return true if arg (BINT or num_str) is even or false if odd my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); (($x->{sign} =~ /^[+-]$/) && # NaN & +-inf aren't ($x->{_es} eq '+') && # 123.45 isn't ($LIB->_is_even($x->{_m}))) ? 1 : 0; # but 1200 is } sub is_int { # return true if arg (BFLOAT or num_str) is an integer my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); (($x->{sign} =~ /^[+-]$/) && # NaN and +-inf aren't ($x->{_es} eq '+')) ? 1 : 0; # 1e-1 => no integer } ############################################################################### # Comparison methods ############################################################################### sub bcmp { # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort) # set up parameters my ($class, $x, $y) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y) = objectify(2, @_); } return $upgrade->bcmp($x, $y) if defined $upgrade && ((!$x->isa($class)) \|\| (!$y->isa($class))); # Handle all 'nan' cases. return undef if ($x->{sign} eq $nan) \|\| ($y->{sign} eq $nan); # Handle all '+inf' and '-inf' cases. return 0 if ($x->{sign} eq '+inf' && $y->{sign} eq '+inf' \|\| $x->{sign} eq '-inf' && $y->{sign} eq '-inf'); return +1 if $x->{sign} eq '+inf'; # x = +inf and y < +inf return -1 if $x->{sign} eq '-inf'; # x = -inf and y > -inf return -1 if $y->{sign} eq '+inf'; # x < +inf and y = +inf return +1 if $y->{sign} eq '-inf'; # x > -inf and y = -inf # Handle all cases with opposite signs. return +1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # also does 0 <=> -y return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # also does -x <=> 0 # Handle all remaining zero cases. my $xz = $x->is_zero(); my $yz = $y->is_zero(); return 0 if $xz && $yz; # 0 <=> 0 return -1 if $xz && $y->{sign} eq '+'; # 0 <=> +y return +1 if $yz && $x->{sign} eq '+'; # +x <=> 0 # Both arguments are now finite, non-zero numbers with the same sign. my $cmp; # The next step is to compare the exponents, but since each mantissa is an # integer of arbitrary value, the exponents must be normalized by the length # of the mantissas before we can compare them. my $mxl = $LIB->_len($x->{_m}); my $myl = $LIB->_len($y->{_m}); # If the mantissas have the same length, there is no point in normalizing the # exponents by the length of the mantissas, so treat that as a special case. if ($mxl == $myl) { # First handle the two cases where the exponents have different signs. if ($x->{_es} eq '+' && $y->{_es} eq '-') { $cmp = +1; } elsif ($x->{_es} eq '-' && $y->{_es} eq '+') { $cmp = -1; } # Then handle the case where the exponents have the same sign. else { $cmp = $LIB->_acmp($x->{_e}, $y->{_e}); $cmp = -$cmp if $x->{_es} eq '-'; } # Adjust for the sign, which is the same for x and y, and bail out if # we're done. $cmp = -$cmp if $x->{sign} eq '-'; # 124 > 123, but -124 < -123 return $cmp if $cmp; } # We must normalize each exponent by the length of the corresponding # mantissa. Life is a lot easier if we first make both exponents # non-negative. We do this by adding the same positive value to both # exponent. This is safe, because when comparing the exponents, only the # relative difference is important. my $ex; my $ey; if ($x->{_es} eq '+') { # If the exponent of x is >= 0 and the exponent of y is >= 0, there is no # need to do anything special. if ($y->{_es} eq '+') { $ex = $LIB->_copy($x->{_e}); $ey = $LIB->_copy($y->{_e}); } # If the exponent of x is >= 0 and the exponent of y is < 0, add the # absolute value of the exponent of y to both. else { $ex = $LIB->_copy($x->{_e}); $ex = $LIB->_add($ex, $y->{_e}); # ex + \|ey\| $ey = $LIB->_zero(); # -ex + \|ey\| = 0 } } else { # If the exponent of x is < 0 and the exponent of y is >= 0, add the # absolute value of the exponent of x to both. if ($y->{_es} eq '+') { $ex = $LIB->_zero(); # -ex + \|ex\| = 0 $ey = $LIB->_copy($y->{_e}); $ey = $LIB->_add($ey, $x->{_e}); # ey + \|ex\| } # If the exponent of x is < 0 and the exponent of y is < 0, add the # absolute values of both exponents to both exponents. else { $ex = $LIB->_copy($y->{_e}); # -ex + \|ey\| + \|ex\| = \|ey\| $ey = $LIB->_copy($x->{_e}); # -ey + \|ex\| + \|ey\| = \|ex\| } } # Now we can normalize the exponents by adding lengths of the mantissas. $ex = $LIB->_add($ex, $LIB->_new($mxl)); $ey = $LIB->_add($ey, $LIB->_new($myl)); # We're done if the exponents are different. $cmp = $LIB->_acmp($ex, $ey); $cmp = -$cmp if $x->{sign} eq '-'; # 124 > 123, but -124 < -123 return $cmp if $cmp; # Compare the mantissas, but first normalize them by padding the shorter # mantissa with zeros (shift left) until it has the same length as the longer # mantissa. my $mx = $x->{_m}; my $my = $y->{_m}; if ($mxl > $myl) { $my = $LIB->_lsft($LIB->_copy($my), $LIB->_new($mxl - $myl), 10); } elsif ($mxl < $myl) { $mx = $LIB->_lsft($LIB->_copy($mx), $LIB->_new($myl - $mxl), 10); } $cmp = $LIB->_acmp($mx, $my); $cmp = -$cmp if $x->{sign} eq '-'; # 124 > 123, but -124 < -123 return $cmp; } sub bacmp { # Compares 2 values, ignoring their signs. # Returns one of undef, <0, =0, >0. (suitable for sort) # set up parameters my ($class, $x, $y) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y) = objectify(2, @_); } return $upgrade->bacmp($x, $y) if defined $upgrade && ((!$x->isa($class)) \|\| (!$y->isa($class))); # handle +-inf and NaN's if ($x->{sign} !~ /^[+-]$/ \|\| $y->{sign} !~ /^[+-]$/) { return undef if (($x->{sign} eq $nan) \|\| ($y->{sign} eq $nan)); return 0 if ($x->is_inf() && $y->is_inf()); return 1 if ($x->is_inf() && !$y->is_inf()); return -1; } # shortcut my $xz = $x->is_zero(); my $yz = $y->is_zero(); return 0 if $xz && $yz; # 0 <=> 0 return -1 if $xz && !$yz; # 0 <=> +y return 1 if $yz && !$xz; # +x <=> 0 # adjust so that exponents are equal my $lxm = $LIB->_len($x->{_m}); my $lym = $LIB->_len($y->{_m}); my ($xes, $yes) = (1, 1); $xes = -1 if $x->{_es} ne '+'; $yes = -1 if $y->{_es} ne '+'; # the numify somewhat limits our length, but makes it much faster my $lx = $lxm + $xes * $LIB->_num($x->{_e}); my $ly = $lym + $yes * $LIB->_num($y->{_e}); my $l = $lx - $ly; return $l <=> 0 if $l != 0; # lengths (corrected by exponent) are equal # so make mantissa equal-length by padding with zero (shift left) my $diff = $lxm - $lym; my $xm = $x->{_m}; # not yet copy it my $ym = $y->{_m}; if ($diff > 0) { $ym = $LIB->_copy($y->{_m}); $ym = $LIB->_lsft($ym, $LIB->_new($diff), 10); } elsif ($diff < 0) { $xm = $LIB->_copy($x->{_m}); $xm = $LIB->_lsft($xm, $LIB->_new(-$diff), 10); } $LIB->_acmp($xm, $ym); } ############################################################################### # Arithmetic methods ############################################################################### sub bneg { # (BINT or num_str) return BINT # negate number or make a negated number from string my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); return $x if $x->modify('bneg'); # for +0 do not negate (to have always normalized +0). Does nothing for 'NaN' $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $LIB->_is_zero($x->{_m})); $x; } sub bnorm { # adjust m and e so that m is smallest possible my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); return $x if $x->{sign} !~ /^[+-]$/; # inf, nan etc my $zeros = $LIB->_zeros($x->{_m}); # correct for trailing zeros if ($zeros != 0) { my $z = $LIB->_new($zeros); $x->{_m} = $LIB->_rsft($x->{_m}, $z, 10); if ($x->{_es} eq '-') { if ($LIB->_acmp($x->{_e}, $z) >= 0) { $x->{_e} = $LIB->_sub($x->{_e}, $z); $x->{_es} = '+' if $LIB->_is_zero($x->{_e}); } else { $x->{_e} = $LIB->_sub($LIB->_copy($z), $x->{_e}); $x->{_es} = '+'; } } else { $x->{_e} = $LIB->_add($x->{_e}, $z); } } else { # $x can only be 0Ey if there are no trailing zeros ('0' has 0 trailing # zeros). So, for something like 0Ey, set y to 1, and -0 => +0 $x->{sign} = '+', $x->{_es} = '+', $x->{_e} = $LIB->_one() if $LIB->_is_zero($x->{_m}); } $x; } sub binc { # increment arg by one my ($class, $x, @r) = ref($_[0]) ? (ref($_[0]), @_) : objectify(1, @_); return $x if $x->modify('binc'); if ($x->{_es} eq '-') { return $x->badd($class->bone(), @r); # digits after dot } if (!$LIB->_is_zero($x->{_e})) # _e == 0 for NaN, inf, -inf { # 1e2 => 100, so after the shift below _m has a '0' as last digit $x->{_m} = $LIB->_lsft($x->{_m}, $x->{_e}, 10); # 1e2 => 100 $x->{_e} = $LIB->_zero(); # normalize $x->{_es} = '+'; # we know that the last digit of $x will be '1' or '9', depending on the # sign } # now $x->{_e} == 0 if ($x->{sign} eq '+') { $x->{_m} = $LIB->_inc($x->{_m}); return $x->bnorm()->bround(@r); } elsif ($x->{sign} eq '-') { $x->{_m} = $LIB->_dec($x->{_m}); $x->{sign} = '+' if $LIB->_is_zero($x->{_m}); # -1 +1 => -0 => +0 return $x->bnorm()->bround(@r); } # inf, nan handling etc $x->badd($class->bone(), @r); # badd() does round } sub bdec { # decrement arg by one my ($class, $x, @r) = ref($_[0]) ? (ref($_[0]), @_) : objectify(1, @_); return $x if $x->modify('bdec'); if ($x->{_es} eq '-') { return $x->badd($class->bone('-'), @r); # digits after dot } if (!$LIB->_is_zero($x->{_e})) { $x->{_m} = $LIB->_lsft($x->{_m}, $x->{_e}, 10); # 1e2 => 100 $x->{_e} = $LIB->_zero(); # normalize $x->{_es} = '+'; } # now $x->{_e} == 0 my $zero = $x->is_zero(); # <= 0 if (($x->{sign} eq '-') \|\| $zero) { $x->{_m} = $LIB->_inc($x->{_m}); $x->{sign} = '-' if $zero; # 0 => 1 => -1 $x->{sign} = '+' if $LIB->_is_zero($x->{_m}); # -1 +1 => -0 => +0 return $x->bnorm()->round(@r); } # > 0 elsif ($x->{sign} eq '+') { $x->{_m} = $LIB->_dec($x->{_m}); return $x->bnorm()->round(@r); } # inf, nan handling etc $x->badd($class->bone('-'), @r); # does round } sub badd { # add second arg (BFLOAT or string) to first (BFLOAT) (modifies first) # return result as BFLOAT # set up parameters my ($class, $x, $y, @r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, @r) = objectify(2, @_); } return $x if $x->modify('badd'); # inf and NaN handling if (($x->{sign} !~ /^[+-]$/) \|\| ($y->{sign} !~ /^[+-]$/)) { # NaN first return $x->bnan() if (($x->{sign} eq $nan) \|\| ($y->{sign} eq $nan)); # inf handling if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/)) { # +inf++inf or -inf+-inf => same, rest is NaN return $x if $x->{sign} eq $y->{sign}; return $x->bnan(); } # +-inf + something => +inf; something +-inf => +-inf $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/; return $x; } return $upgrade->badd($x, $y, @r) if defined $upgrade && ((!$x->isa($class)) \|\| (!$y->isa($class))); $r[3] = $y; # no push! # speed: no add for 0+y or x+0 return $x->bround(@r) if $y->is_zero(); # x+0 if ($x->is_zero()) # 0+y { # make copy, clobbering up x (modify in place!) $x->{_e} = $LIB->_copy($y->{_e}); $x->{_es} = $y->{_es}; $x->{_m} = $LIB->_copy($y->{_m}); $x->{sign} = $y->{sign} \|\| $nan; return $x->round(@r); } # take lower of the two e's and adapt m1 to it to match m2 my $e = $y->{_e}; $e = $LIB->_zero() if !defined $e; # if no BFLOAT? $e = $LIB->_copy($e); # make copy (didn't do it yet) my $es; ($e, $es) = _e_sub($e, $x->{_e}, $y->{_es} \|\| '+', $x->{_es}); my $add = $LIB->_copy($y->{_m}); if ($es eq '-') # < 0 { $x->{_m} = $LIB->_lsft($x->{_m}, $e, 10); ($x->{_e}, $x->{_es}) = _e_add($x->{_e}, $e, $x->{_es}, $es); } elsif (!$LIB->_is_zero($e)) # > 0 { $add = $LIB->_lsft($add, $e, 10); } # else: both e are the same, so just leave them if ($x->{sign} eq $y->{sign}) { # add $x->{_m} = $LIB->_add($x->{_m}, $add); } else { ($x->{_m}, $x->{sign}) = _e_add($x->{_m}, $add, $x->{sign}, $y->{sign}); } # delete trailing zeros, then round $x->bnorm()->round(@r); } sub bsub { # (BINT or num_str, BINT or num_str) return BINT # subtract second arg from first, modify first # set up parameters my ($class, $x, $y, @r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, @r) = objectify(2, @_); } return $x if $x -> modify('bsub'); return $upgrade -> new($x) -> bsub($upgrade -> new($y), @r) if defined $upgrade && (!$x -> isa($class) \|\| !$y -> isa($class)); return $x -> round(@r) if $y -> is_zero(); # To correctly handle the lone special case $x -> bsub($x), we note the # sign of $x, then flip the sign from $y, and if the sign of $x did change, # too, then we caught the special case: my $xsign = $x -> {sign}; $y -> {sign} =~ tr/+-/-+/; # does nothing for NaN if ($xsign ne $x -> {sign}) { # special case of $x -> bsub($x) results in 0 return $x -> bzero(@r) if $xsign =~ /^[+-]$/; return $x -> bnan(); # NaN, -inf, +inf } $x -> badd($y, @r); # badd does not leave internal zeros $y -> {sign} =~ tr/+-/-+/; # refix $y (does nothing for NaN) $x; # already rounded by badd() or no rounding } sub bmul { # multiply two numbers # set up parameters my ($class, $x, $y, @r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, @r) = objectify(2, @_); } return $x if $x->modify('bmul'); return $x->bnan() if (($x->{sign} eq $nan) \|\| ($y->{sign} eq $nan)); # inf handling if (($x->{sign} =~ /^[+-]inf$/) \|\| ($y->{sign} =~ /^[+-]inf$/)) { return $x->bnan() if $x->is_zero() \|\| $y->is_zero(); # result will always be +-inf: # +inf * +/+inf => +inf, -inf * -/-inf => +inf # +inf * -/-inf => -inf, -inf * +/+inf => -inf return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/); return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/); return $x->binf('-'); } return $upgrade->bmul($x, $y, @r) if defined $upgrade && ((!$x->isa($class)) \|\| (!$y->isa($class))); # aEb * cEd = (ac)E(b+d) $x->{_m} = $LIB->_mul($x->{_m}, $y->{_m}); ($x->{_e}, $x->{_es}) = _e_add($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es}); $r[3] = $y; # no push! # adjust sign: $x->{sign} = $x->{sign} ne $y->{sign} ? '-' : '+'; $x->bnorm->round(@r); } sub bmuladd { # multiply two numbers and add the third to the result # set up parameters my ($class, $x, $y, $z, @r) = objectify(3, @_); return $x if $x->modify('bmuladd'); return $x->bnan() if (($x->{sign} eq $nan) \|\| ($y->{sign} eq $nan) \|\| ($z->{sign} eq $nan)); # inf handling if (($x->{sign} =~ /^[+-]inf$/) \|\| ($y->{sign} =~ /^[+-]inf$/)) { return $x->bnan() if $x->is_zero() \|\| $y->is_zero(); # result will always be +-inf: # +inf +/+inf => +inf, -inf * -/-inf => +inf # +inf * -/-inf => -inf, -inf * +/+inf => -inf return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/); return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/); return $x->binf('-'); } return $upgrade->bmul($x, $y, @r) if defined $upgrade && ((!$x->isa($class)) \|\| (!$y->isa($class))); # aEb * cEd = (ac)E(b+d) $x->{_m} = $LIB->_mul($x->{_m}, $y->{_m}); ($x->{_e}, $x->{_es}) = _e_add($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es}); $r[3] = $y; # no push! # adjust sign: $x->{sign} = $x->{sign} ne $y->{sign} ? '-' : '+'; # z=inf handling (z=NaN handled above) $x->{sign} = $z->{sign}, return $x if $z->{sign} =~ /^[+-]inf$/; # take lower of the two e's and adapt m1 to it to match m2 my $e = $z->{_e}; $e = $LIB->_zero() if !defined $e; # if no BFLOAT? $e = $LIB->_copy($e); # make copy (didn't do it yet) my $es; ($e, $es) = _e_sub($e, $x->{_e}, $z->{_es} \|\| '+', $x->{_es}); my $add = $LIB->_copy($z->{_m}); if ($es eq '-') # < 0 { $x->{_m} = $LIB->_lsft($x->{_m}, $e, 10); ($x->{_e}, $x->{_es}) = _e_add($x->{_e}, $e, $x->{_es}, $es); } elsif (!$LIB->_is_zero($e)) # > 0 { $add = $LIB->_lsft($add, $e, 10); } # else: both e are the same, so just leave them if ($x->{sign} eq $z->{sign}) { # add $x->{_m} = $LIB->_add($x->{_m}, $add); } else { ($x->{_m}, $x->{sign}) = _e_add($x->{_m}, $add, $x->{sign}, $z->{sign}); } # delete trailing zeros, then round $x->bnorm()->round(@r); } sub bdiv { # (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return # (BFLOAT, BFLOAT) (quo, rem) or BFLOAT (only quo) # set up parameters my ($class, $x, $y, $a, $p, $r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, $a, $p, $r) = objectify(2, @_); } return $x if $x->modify('bdiv'); my $wantarray = wantarray; # call only once # At least one argument is NaN. This is handled the same way as in # Math::BigInt -> bdiv(). if ($x -> is_nan() \|\| $y -> is_nan()) { return $wantarray ? ($x -> bnan(), $class -> bnan()) : $x -> bnan(); } # Divide by zero and modulo zero. This is handled the same way as in # Math::BigInt -> bdiv(). See the comment in the code for Math::BigInt -> # bdiv() for further details. if ($y -> is_zero()) { my ($quo, $rem); if ($wantarray) { $rem = $x -> copy(); } if ($x -> is_zero()) { $quo = $x -> bnan(); } else { $quo = $x -> binf($x -> {sign}); } return $wantarray ? ($quo, $rem) : $quo; } # Numerator (dividend) is +/-inf. This is handled the same way as in # Math::BigInt -> bdiv(). See the comment in the code for Math::BigInt -> # bdiv() for further details. if ($x -> is_inf()) { my ($quo, $rem); $rem = $class -> bnan() if $wantarray; if ($y -> is_inf()) { $quo = $x -> bnan(); } else { my $sign = $x -> bcmp(0) == $y -> bcmp(0) ? '+' : '-'; $quo = $x -> binf($sign); } return $wantarray ? ($quo, $rem) : $quo; } # Denominator (divisor) is +/-inf. This is handled the same way as in # Math::BigInt -> bdiv(), with one exception: In scalar context, # Math::BigFloat does true division (although rounded), not floored division # (F-division), so a finite number divided by +/-inf is always zero. See the # comment in the code for Math::BigInt -> bdiv() for further details. if ($y -> is_inf()) { my ($quo, $rem); if ($wantarray) { if ($x -> is_zero() \|\| $x -> bcmp(0) == $y -> bcmp(0)) { $rem = $x -> copy(); $quo = $x -> bzero(); } else { $rem = $class -> binf($y -> {sign}); $quo = $x -> bone('-'); } return ($quo, $rem); } else { if ($y -> is_inf()) { if ($x -> is_nan() \|\| $x -> is_inf()) { return $x -> bnan(); } else { return $x -> bzero(); } } } } # At this point, both the numerator and denominator are finite numbers, and # the denominator (divisor) is non-zero. # x == 0? return wantarray ? ($x, $class->bzero()) : $x if $x->is_zero(); # upgrade ? return $upgrade->bdiv($upgrade->new($x), $y, $a, $p, $r) if defined $upgrade; # we need to limit the accuracy to protect against overflow my $fallback = 0; my (@params, $scale); ($x, @params) = $x->_find_round_parameters($a, $p, $r, $y); return $x if $x->is_nan(); # error in _find_round_parameters? # no rounding at all, so must use fallback if (scalar @params == 0) { # simulate old behaviour $params[0] = $class->div_scale(); # and round to it as accuracy $scale = $params[0]+4; # at least four more for proper round $params[2] = $r; # round mode by caller or undef $fallback = 1; # to clear a/p afterwards } else { # the 4 below is empirical, and there might be cases where it is not # enough... $scale = abs($params[0] \|\| $params[1]) + 4; # take whatever is defined } my $rem; $rem = $class -> bzero() if wantarray; $y = $class->new($y) unless $y->isa('Math::BigFloat'); my $lx = $LIB -> _len($x->{_m}); my $ly = $LIB -> _len($y->{_m}); $scale = $lx if $lx > $scale; $scale = $ly if $ly > $scale; my $diff = $ly - $lx; $scale += $diff if $diff > 0; # if lx << ly, but not if ly << lx! # check that $y is not 1 nor -1 and cache the result: my $y_not_one = !($LIB->_is_zero($y->{_e}) && $LIB->_is_one($y->{_m})); # flipping the sign of $y will also flip the sign of $x for the special # case of $x->bsub($x); so we can catch it below: my $xsign = $x->{sign}; $y->{sign} =~ tr/+-/-+/; if ($xsign ne $x->{sign}) { # special case of $x /= $x results in 1 $x->bone(); # "fixes" also sign of $y, since $x is $y } else { # correct $y's sign again $y->{sign} =~ tr/+-/-+/; # continue with normal div code: # make copy of $x in case of list context for later remainder calculation if (wantarray && $y_not_one) { $rem = $x->copy(); } $x->{sign} = $x->{sign} ne $y->sign() ? '-' : '+'; # check for / +-1 (+/- 1E0) if ($y_not_one) { # promote BigInts and it's subclasses (except when already a Math::BigFloat) $y = $class->new($y) unless $y->isa('Math::BigFloat'); # calculate the result to $scale digits and then round it # a 10 ** b / c * 10 ** d => a/c * 10 (b-d) $x->{_m} = $LIB->_lsft($x->{_m}, $LIB->_new($scale), 10); $x->{_m} = $LIB->_div($x->{_m}, $y->{_m}); # a/c # correct exponent of $x ($x->{_e}, $x->{_es}) = _e_sub($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es}); # correct for 10scale ($x->{_e}, $x->{_es}) = _e_sub($x->{_e}, $LIB->_new($scale), $x->{_es}, '+'); $x->bnorm(); # remove trailing 0's } } # end else $x != $y # shortcut to not run through _find_round_parameters again if (defined $params[0]) { delete $x->{_a}; # clear before round $x->bround($params[0], $params[2]); # then round accordingly } else { delete $x->{_p}; # clear before round $x->bfround($params[1], $params[2]); # then round accordingly } if ($fallback) { # clear a/p after round, since user did not request it delete $x->{_a}; delete $x->{_p}; } if (wantarray) { if ($y_not_one) { $x -> bfloor(); $rem->bmod($y, @params); # copy already done } if ($fallback) { # clear a/p after round, since user did not request it delete $rem->{_a}; delete $rem->{_p}; } return ($x, $rem); } $x; } sub bmod { # (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return remainder # set up parameters my ($class, $x, $y, $a, $p, $r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, $a, $p, $r) = objectify(2, @_); } return $x if $x->modify('bmod'); # At least one argument is NaN. This is handled the same way as in # Math::BigInt -> bmod(). if ($x -> is_nan() \|\| $y -> is_nan()) { return $x -> bnan(); } # Modulo zero. This is handled the same way as in Math::BigInt -> bmod(). if ($y -> is_zero()) { return $x; } # Numerator (dividend) is +/-inf. This is handled the same way as in # Math::BigInt -> bmod(). if ($x -> is_inf()) { return $x -> bnan(); } # Denominator (divisor) is +/-inf. This is handled the same way as in # Math::BigInt -> bmod(). if ($y -> is_inf()) { if ($x -> is_zero() \|\| $x -> bcmp(0) == $y -> bcmp(0)) { return $x; } else { return $x -> binf($y -> sign()); } } return $x->bzero() if $x->is_zero() \|\| ($x->is_int() && # check that $y == +1 or $y == -1: ($LIB->_is_zero($y->{_e}) && $LIB->_is_one($y->{_m}))); my $cmp = $x->bacmp($y); # equal or $x < $y? if ($cmp == 0) { # $x == $y => result 0 return $x -> bzero($a, $p); } # only $y of the operands negative? my $neg = $x->{sign} ne $y->{sign} ? 1 : 0; $x->{sign} = $y->{sign}; # calc sign first if ($cmp < 0 && $neg == 0) { # $x < $y => result $x return $x -> round($a, $p, $r); } my $ym = $LIB->_copy($y->{_m}); # 2e1 => 20 $ym = $LIB->_lsft($ym, $y->{_e}, 10) if $y->{_es} eq '+' && !$LIB->_is_zero($y->{_e}); # if $y has digits after dot my $shifty = 0; # correct _e of $x by this if ($y->{_es} eq '-') # has digits after dot { # 123 % 2.5 => 1230 % 25 => 5 => 0.5 $shifty = $LIB->_num($y->{_e}); # no more digits after dot $x->{_m} = $LIB->_lsft($x->{_m}, $y->{_e}, 10); # 123 => 1230, $y->{_m} is already 25 } # $ym is now mantissa of $y based on exponent 0 my $shiftx = 0; # correct _e of $x by this if ($x->{_es} eq '-') # has digits after dot { # 123.4 % 20 => 1234 % 200 $shiftx = $LIB->_num($x->{_e}); # no more digits after dot $ym = $LIB->_lsft($ym, $x->{_e}, 10); # 123 => 1230 } # 123e1 % 20 => 1230 % 20 if ($x->{_es} eq '+' && !$LIB->_is_zero($x->{_e})) { $x->{_m} = $LIB->_lsft($x->{_m}, $x->{_e}, 10); # es => '+' here } $x->{_e} = $LIB->_new($shiftx); $x->{_es} = '+'; $x->{_es} = '-' if $shiftx != 0 \|\| $shifty != 0; $x->{_e} = $LIB->_add($x->{_e}, $LIB->_new($shifty)) if $shifty != 0; # now mantissas are equalized, exponent of $x is adjusted, so calc result $x->{_m} = $LIB->_mod($x->{_m}, $ym); $x->{sign} = '+' if $LIB->_is_zero($x->{_m}); # fix sign for -0 $x->bnorm(); if ($neg != 0 && ! $x -> is_zero()) # one of them negative => correct in place { my $r = $y - $x; $x->{_m} = $r->{_m}; $x->{_e} = $r->{_e}; $x->{_es} = $r->{_es}; $x->{sign} = '+' if $LIB->_is_zero($x->{_m}); # fix sign for -0 $x->bnorm(); } $x->round($a, $p, $r, $y); # round and return } sub bmodpow { # takes a very large number to a very large exponent in a given very # large modulus, quickly, thanks to binary exponentiation. Supports # negative exponents. my ($class, $num, $exp, $mod) = objectify(3, @_); return $num if $num->modify('bmodpow'); # check modulus for valid values return $num->bnan() if ($mod->{sign} ne '+' # NaN, -, -inf, +inf \|\| $mod->is_zero()); # check exponent for valid values if ($exp->{sign} =~ /\w/) { # i.e., if it's NaN, +inf, or -inf... return $num->bnan(); } $num->bmodinv ($mod) if ($exp->{sign} eq '-'); # check num for valid values (also NaN if there was no inverse but $exp < 0) return $num->bnan() if $num->{sign} !~ /^[+-]$/; # $mod is positive, sign on $exp is ignored, result also positive # XXX TODO: speed it up when all three numbers are integers $num->bpow($exp)->bmod($mod); } sub bpow { # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT # compute power of two numbers, second arg is used as integer # modifies first argument # set up parameters my ($class, $x, $y, $a, $p, $r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, $a, $p, $r) = objectify(2, @_); } return $x if $x->modify('bpow'); # $x and/or $y is a NaN return $x->bnan() if $x->is_nan() \|\| $y->is_nan(); # $x and/or $y is a +/-Inf if ($x->is_inf("-")) { return $x->bzero() if $y->is_negative(); return $x->bnan() if $y->is_zero(); return $x if $y->is_odd(); return $x->bneg(); } elsif ($x->is_inf("+")) { return $x->bzero() if $y->is_negative(); return $x->bnan() if $y->is_zero(); return $x; } elsif ($y->is_inf("-")) { return $x->bnan() if $x -> is_one("-"); return $x->binf("+") if $x > -1 && $x < 1; return $x->bone() if $x -> is_one("+"); return $x->bzero(); } elsif ($y->is_inf("+")) { return $x->bnan() if $x -> is_one("-"); return $x->bzero() if $x > -1 && $x < 1; return $x->bone() if $x -> is_one("+"); return $x->binf("+"); } # we don't support complex numbers, so return NaN return $x->bnan() if $x->is_negative() && !$y->is_int(); # cache the result of is_zero my $y_is_zero = $y->is_zero(); return $x->bone() if $y_is_zero; return $x if $x->is_one() \|\| $y->is_one(); my $x_is_zero = $x->is_zero(); return $x->_pow($y, $a, $p, $r) if !$x_is_zero && !$y->is_int(); my $y1 = $y->as_number()->{value}; # make MBI part if ($x->is_one("-")) { # if $x == -1 and odd/even y => +1/-1 because +-1 ^ (+-1) => +-1 return $LIB->_is_odd($y1) ? $x : $x->babs(1); } if ($x_is_zero) { return $x if $y->{sign} eq '+'; # 0y => 0 (if not y <= 0) # 0 -y => 1 / (0 y) => 1 / 0! (1 / 0 => +inf) return $x->binf(); } my $new_sign = '+'; $new_sign = $LIB->_is_odd($y1) ? '-' : '+' if $x->{sign} ne '+'; # calculate $x->{_m} $y and $x->{_e} * $y separately (faster) $x->{_m} = $LIB->_pow($x->{_m}, $y1); $x->{_e} = $LIB->_mul ($x->{_e}, $y1); $x->{sign} = $new_sign; $x->bnorm(); if ($y->{sign} eq '-') { # modify $x in place! my $z = $x->copy(); $x->bone(); return scalar $x->bdiv($z, $a, $p, $r); # round in one go (might ignore y's A!) } $x->round($a, $p, $r, $y); } sub blog { # Return the logarithm of the operand. If a second operand is defined, that # value is used as the base, otherwise the base is assumed to be Euler's # constant. my ($class, $x, $base, $a, $p, $r); # Don't objectify the base, since an undefined base, as in $x->blog() or # $x->blog(undef) signals that the base is Euler's number. if (!ref($_[0]) && $_[0] =~ /^[A-Za-z]\|::/) { # E.g., Math::BigFloat->blog(256, 2) ($class, $x, $base, $a, $p, $r) = defined $_[2] ? objectify(2, @_) : objectify(1, @_); } else { # E.g., Math::BigFloat::blog(256, 2) or $x->blog(2) ($class, $x, $base, $a, $p, $r) = defined $_[1] ? objectify(2, @_) : objectify(1, @_); } return $x if $x->modify('blog'); return $x -> bnan() if $x -> is_nan(); # we need to limit the accuracy to protect against overflow my $fallback = 0; my ($scale, @params); ($x, @params) = $x->_find_round_parameters($a, $p, $r); # no rounding at all, so must use fallback if (scalar @params == 0) { # simulate old behaviour $params[0] = $class->div_scale(); # and round to it as accuracy $params[1] = undef; # P = undef $scale = $params[0]+4; # at least four more for proper round $params[2] = $r; # round mode by caller or undef $fallback = 1; # to clear a/p afterwards } else { # the 4 below is empirical, and there might be cases where it is not # enough... $scale = abs($params[0] \|\| $params[1]) + 4; # take whatever is defined } my $done = 0; if (defined $base) { $base = $class -> new($base) unless ref $base; if ($base -> is_nan() \|\| $base -> is_one()) { $x -> bnan(); $done = 1; } elsif ($base -> is_inf() \|\| $base -> is_zero()) { if ($x -> is_inf() \|\| $x -> is_zero()) { $x -> bnan(); } else { $x -> bzero(@params); } $done = 1; } elsif ($base -> is_negative()) { # -inf < base < 0 if ($x -> is_one()) { # x = 1 $x -> bzero(@params); } elsif ($x == $base) { $x -> bone('+', @params); # x = base } else { $x -> bnan(); # otherwise } $done = 1; } elsif ($x == $base) { $x -> bone('+', @params); # 0 < base && 0 < x < inf $done = 1; } } # We now know that the base is either undefined or positive and finite. unless ($done) { if ($x -> is_inf()) { # x = +/-inf my $sign = defined $base && $base < 1 ? '-' : '+'; $x -> binf($sign); $done = 1; } elsif ($x -> is_neg()) { # -inf < x < 0 $x -> bnan(); $done = 1; } elsif ($x -> is_one()) { # x = 1 $x -> bzero(@params); $done = 1; } elsif ($x -> is_zero()) { # x = 0 my $sign = defined $base && $base < 1 ? '+' : '-'; $x -> binf($sign); $done = 1; } } if ($done) { if ($fallback) { # clear a/p after round, since user did not request it delete $x->{_a}; delete $x->{_p}; } return $x; } # when user set globals, they would interfere with our calculation, so # disable them and later re-enable them no strict 'refs'; my $abr = "$class\::accuracy"; my $ab = $$abr; $$abr = undef; my $pbr = "$class\::precision"; my $pb = $$pbr; $$pbr = undef; # we also need to disable any set A or P on $x (_find_round_parameters took # them already into account), since these would interfere, too delete $x->{_a}; delete $x->{_p}; # need to disable $upgrade in BigInt, to avoid deep recursion local $Math::BigInt::upgrade = undef; local $Math::BigFloat::downgrade = undef; # upgrade $x if $x is not a Math::BigFloat (handle BigInt input) # XXX TODO: rebless! if (!$x->isa('Math::BigFloat')) { $x = Math::BigFloat->new($x); $class = ref($x); } $done = 0; # If the base is defined and an integer, try to calculate integer result # first. This is very fast, and in case the real result was found, we can # stop right here. if (defined $base && $base->is_int() && $x->is_int()) { my $xint = Math::BigInt -> new($x -> bdstr()); my $bint = Math::BigInt -> new($base -> bdstr()); $xint->blog($bint); # if we found the exact result, we're done if ($bint -> bpow($xint) == $x) { my $xflt = Math::BigFloat -> new($xint -> bdstr()); $x->{sign} = $xflt->{sign}; $x->{_m} = $xflt->{_m}; $x->{_es} = $xflt->{_es}; $x->{_e} = $xflt->{_e}; $done = 1; } } if ($done == 0) { # First calculate the log to base e (using reduction by 10 and possibly # also by 2): $x->_log_10($scale); # and if a different base was requested, convert it if (defined $base) { $base = Math::BigFloat->new($base) unless $base->isa('Math::BigFloat'); # log_b(x) = ln(x) / ln(b), so compute ln(b) my $base_log_e = $base->copy()->_log_10($scale); $x->bdiv($base_log_e, $scale); } } # shortcut to not run through _find_round_parameters again if (defined $params[0]) { $x->bround($params[0], $params[2]); # then round accordingly } else { $x->bfround($params[1], $params[2]); # then round accordingly } if ($fallback) { # clear a/p after round, since user did not request it delete $x->{_a}; delete $x->{_p}; } # restore globals $$abr = $ab; $$pbr = $pb; $x; } sub bexp { # Calculate e ** X (Euler's number to the power of X) my ($class, $x, $a, $p, $r) = ref($_[0]) ? (ref($_[0]), @_) : objectify(1, @_); return $x if $x->modify('bexp'); return $x->binf() if $x->{sign} eq '+inf'; return $x->bzero() if $x->{sign} eq '-inf'; # we need to limit the accuracy to protect against overflow my $fallback = 0; my ($scale, @params); ($x, @params) = $x->_find_round_parameters($a, $p, $r); # also takes care of the "error in _find_round_parameters?" case return $x if $x->{sign} eq 'NaN'; # no rounding at all, so must use fallback if (scalar @params == 0) { # simulate old behaviour $params[0] = $class->div_scale(); # and round to it as accuracy $params[1] = undef; # P = undef $scale = $params[0]+4; # at least four more for proper round $params[2] = $r; # round mode by caller or undef $fallback = 1; # to clear a/p afterwards } else { # the 4 below is empirical, and there might be cases where it's not # enough ... $scale = abs($params[0] \|\| $params[1]) + 4; # take whatever is defined } return $x->bone(@params) if $x->is_zero(); if (!$x->isa('Math::BigFloat')) { $x = Math::BigFloat->new($x); $class = ref($x); } # when user set globals, they would interfere with our calculation, so # disable them and later re-enable them no strict 'refs'; my $abr = "$class\::accuracy"; my $ab = $$abr; $$abr = undef; my $pbr = "$class\::precision"; my $pb = $$pbr; $$pbr = undef; # we also need to disable any set A or P on $x (_find_round_parameters took # them already into account), since these would interfere, too delete $x->{_a}; delete $x->{_p}; # need to disable $upgrade in BigInt, to avoid deep recursion local $Math::BigInt::upgrade = undef; local $Math::BigFloat::downgrade = undef; my $x_org = $x->copy(); # We use the following Taylor series: # x x^2 x^3 x^4 # e = 1 + --- + --- + --- + --- ... # 1! 2! 3! 4! # The difference for each term is X and N, which would result in: # 2 copy, 2 mul, 2 add, 1 inc, 1 div operations per term # But it is faster to compute exp(1) and then raising it to the # given power, esp. if $x is really big and an integer because: # * The numerator is always 1, making the computation faster # * the series converges faster in the case of x == 1 # * We can also easily check when we have reached our limit: when the # term to be added is smaller than "1E$scale", we can stop - f.i. # scale == 5, and we have 1/40320, then we stop since 1/40320 < 1E-5. # * we can compute the exact result by simulating bigrat math: # 1 1 gcd(3, 4) = 1 124 + 16 5 # - + - = ---------- = -- # 6 24 624 24 # We do not compute the gcd() here, but simple do: # 1 1 124 + 16 30 # - + - = --------- = -- # 6 24 624 144 # In general: # a c ad + cb and note that c is always 1 and d = (bf) # - + - = --------- # b d bd # This leads to: which can be reduced by b to: # a 1 abf + b af + 1 # - + - = --------- = ------- # b bf bbf bf # The first terms in the series are: # 1 1 1 1 1 1 1 1 13700 # -- + -- + -- + -- + -- + --- + --- + ---- = ----- # 1 1 2 6 24 120 720 5040 5040 # Note that we cannot simply reduce 13700/5040 to 685/252, but must keep # the numerator and the denominator! if ($scale <= 75) { # set $x directly from a cached string form $x->{_m} = $LIB->_new("2718281828459045235360287471352662497757" . "2470936999595749669676277240766303535476"); $x->{sign} = '+'; $x->{_es} = '-'; $x->{_e} = $LIB->_new(79); } else { # compute A and B so that e = A / B. # After some terms we end up with this, so we use it as a starting point: my $A = $LIB->_new("9093339520860578540197197" . "0164779391644753259799242"); my $F = $LIB->_new(42); my $step = 42; # Compute how many steps we need to take to get $A and $B sufficiently big my $steps = _len_to_steps($scale - 4); # print STDERR "# Doing $steps steps for ", $scale-4, " digits\n"; while ($step++ <= $steps) { # calculate $a $f + 1 $A = $LIB->_mul($A, $F); $A = $LIB->_inc($A); # increment f $F = $LIB->_inc($F); } # compute $B as factorial of $steps (this is faster than doing it manually) my $B = $LIB->_fac($LIB->_new($steps)); # print "A ", $LIB->_str($A), "\nB ", $LIB->_str($B), "\n"; # compute A/B with $scale digits in the result (truncate, not round) $A = $LIB->_lsft($A, $LIB->_new($scale), 10); $A = $LIB->_div($A, $B); $x->{_m} = $A; $x->{sign} = '+'; $x->{_es} = '-'; $x->{_e} = $LIB->_new($scale); } # $x contains now an estimate of e, with some surplus digits, so we can round if (!$x_org->is_one()) { # Reduce size of fractional part, followup with integer power of two. my $lshift = 0; while ($lshift < 30 && $x_org->bacmp(2 << $lshift) > 0) { $lshift++; } # Raise $x to the wanted power and round it. if ($lshift == 0) { $x->bpow($x_org, @params); } else { my($mul, $rescale) = (1 << $lshift, $scale+1+$lshift); $x->bpow(scalar $x_org->bdiv($mul, $rescale), $rescale)->bpow($mul, @params); } } else { # else just round the already computed result delete $x->{_a}; delete $x->{_p}; # shortcut to not run through _find_round_parameters again if (defined $params[0]) { $x->bround($params[0], $params[2]); # then round accordingly } else { $x->bfround($params[1], $params[2]); # then round accordingly } } if ($fallback) { # clear a/p after round, since user did not request it delete $x->{_a}; delete $x->{_p}; } # restore globals $$abr = $ab; $$pbr = $pb; $x; # return modified $x } sub bnok { # Calculate n over k (binomial coefficient or "choose" function) as integer. # set up parameters my ($class, $x, $y, @r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, @r) = objectify(2, @_); } return $x if $x->modify('bnok'); return $x->bnan() if $x->is_nan() \|\| $y->is_nan(); return $x->bnan() if (($x->is_finite() && !$x->is_int()) \|\| ($y->is_finite() && !$y->is_int())); my $xint = Math::BigInt -> new($x -> bsstr()); my $yint = Math::BigInt -> new($y -> bsstr()); $xint -> bnok($yint); my $xflt = Math::BigFloat -> new($xint); $x->{_m} = $xflt->{_m}; $x->{_e} = $xflt->{_e}; $x->{_es} = $xflt->{_es}; $x->{sign} = $xflt->{sign}; return $x; } sub bsin { # Calculate a sinus of x. my ($class, $x, @r) = ref($_[0]) ? (ref($_[0]), @_) : objectify(1, @_); # taylor: x^3 x^5 x^7 x^9 # sin = x - --- + --- - --- + --- ... # 3! 5! 7! 9! # we need to limit the accuracy to protect against overflow my $fallback = 0; my ($scale, @params); ($x, @params) = $x->_find_round_parameters(@r); # constant object or error in _find_round_parameters? return $x if $x->modify('bsin') \|\| $x->is_nan(); return $x->bzero(@r) if $x->is_zero(); # no rounding at all, so must use fallback if (scalar @params == 0) { # simulate old behaviour $params[0] = $class->div_scale(); # and round to it as accuracy $params[1] = undef; # disable P $scale = $params[0]+4; # at least four more for proper round $params[2] = $r[2]; # round mode by caller or undef $fallback = 1; # to clear a/p afterwards } else { # the 4 below is empirical, and there might be cases where it is not # enough... $scale = abs($params[0] \|\| $params[1]) + 4; # take whatever is defined } # when user set globals, they would interfere with our calculation, so # disable them and later re-enable them no strict 'refs'; my $abr = "$class\::accuracy"; my $ab = $$abr; $$abr = undef; my $pbr = "$class\::precision"; my $pb = $$pbr; $$pbr = undef; # we also need to disable any set A or P on $x (_find_round_parameters took # them already into account), since these would interfere, too delete $x->{_a}; delete $x->{_p}; # need to disable $upgrade in BigInt, to avoid deep recursion local $Math::BigInt::upgrade = undef; my $over = $x * $x; # X ^ 2 my $x2 = $over->copy(); # X ^ 2; difference between terms $over->bmul($x); # X ^ 3 as starting value my $sign = 1; # start with -= my $below = $class->new(6); my $factorial = $class->new(4); delete $x->{_a}; delete $x->{_p}; my $limit = $class->new("1E-". ($scale-1)); #my $steps = 0; while (3 < 5) { # we calculate the next term, and add it to the last # when the next term is below our limit, it won't affect the outcome # anymore, so we stop: my $next = $over->copy()->bdiv($below, $scale); last if $next->bacmp($limit) <= 0; if ($sign == 0) { $x->badd($next); } else { $x->bsub($next); } $sign = 1-$sign; # alternate # calculate things for the next term $over->bmul($x2); # $x$x $below->bmul($factorial); $factorial->binc(); # n(n+1) $below->bmul($factorial); $factorial->binc(); # n(n+1) } # shortcut to not run through _find_round_parameters again if (defined $params[0]) { $x->bround($params[0], $params[2]); # then round accordingly } else { $x->bfround($params[1], $params[2]); # then round accordingly } if ($fallback) { # clear a/p after round, since user did not request it delete $x->{_a}; delete $x->{_p}; } # restore globals $$abr = $ab; $$pbr = $pb; $x; } sub bcos { # Calculate a cosinus of x. my ($class, $x, @r) = ref($_[0]) ? (ref($_[0]), @_) : objectify(1, @_); # Taylor: x^2 x^4 x^6 x^8 # cos = 1 - --- + --- - --- + --- ... # 2! 4! 6! 8! # we need to limit the accuracy to protect against overflow my $fallback = 0; my ($scale, @params); ($x, @params) = $x->_find_round_parameters(@r); # constant object or error in _find_round_parameters? return $x if $x->modify('bcos') \|\| $x->is_nan(); return $x->bone(@r) if $x->is_zero(); # no rounding at all, so must use fallback if (scalar @params == 0) { # simulate old behaviour $params[0] = $class->div_scale(); # and round to it as accuracy $params[1] = undef; # disable P $scale = $params[0]+4; # at least four more for proper round $params[2] = $r[2]; # round mode by caller or undef $fallback = 1; # to clear a/p afterwards } else { # the 4 below is empirical, and there might be cases where it is not # enough... $scale = abs($params[0] \|\| $params[1]) + 4; # take whatever is defined } # when user set globals, they would interfere with our calculation, so # disable them and later re-enable them no strict 'refs'; my $abr = "$class\::accuracy"; my $ab = $$abr; $$abr = undef; my $pbr = "$class\::precision"; my $pb = $$pbr; $$pbr = undef; # we also need to disable any set A or P on $x (_find_round_parameters took # them already into account), since these would interfere, too delete $x->{_a}; delete $x->{_p}; # need to disable $upgrade in BigInt, to avoid deep recursion local $Math::BigInt::upgrade = undef; my $over = $x $x; # X ^ 2 my $x2 = $over->copy(); # X ^ 2; difference between terms my $sign = 1; # start with -= my $below = $class->new(2); my $factorial = $class->new(3); $x->bone(); delete $x->{_a}; delete $x->{_p}; my $limit = $class->new("1E-". ($scale-1)); #my $steps = 0; while (3 < 5) { # we calculate the next term, and add it to the last # when the next term is below our limit, it won't affect the outcome # anymore, so we stop: my $next = $over->copy()->bdiv($below, $scale); last if $next->bacmp($limit) <= 0; if ($sign == 0) { $x->badd($next); } else { $x->bsub($next); } $sign = 1-$sign; # alternate # calculate things for the next term $over->bmul($x2); # $x$x $below->bmul($factorial); $factorial->binc(); # n(n+1) $below->bmul($factorial); $factorial->binc(); # n(n+1) } # shortcut to not run through _find_round_parameters again if (defined $params[0]) { $x->bround($params[0], $params[2]); # then round accordingly } else { $x->bfround($params[1], $params[2]); # then round accordingly } if ($fallback) { # clear a/p after round, since user did not request it delete $x->{_a}; delete $x->{_p}; } # restore globals $$abr = $ab; $$pbr = $pb; $x; } sub batan { # Calculate a arcus tangens of x. my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; my (@r) = @_; # taylor: x^3 x^5 x^7 x^9 # atan = x - --- + --- - --- + --- ... # 3 5 7 9 # We need to limit the accuracy to protect against overflow. my $fallback = 0; my ($scale, @params); ($self, @params) = $self->_find_round_parameters(@r); # Constant object or error in _find_round_parameters? return $self if $self->modify('batan') \|\| $self->is_nan(); if ($self->{sign} =~ /^[+-]inf\z/) { # +inf result is PI/2 # -inf result is -PI/2 # calculate PI/2 my $pi = $class->bpi(@r); # modify $self in place $self->{_m} = $pi->{_m}; $self->{_e} = $pi->{_e}; $self->{_es} = $pi->{_es}; # -y => -PI/2, +y => PI/2 $self->{sign} = substr($self->{sign}, 0, 1); # "+inf" => "+" $self -> {_m} = $LIB->_div($self->{_m}, $LIB->_new(2)); return $self; } return $self->bzero(@r) if $self->is_zero(); # no rounding at all, so must use fallback if (scalar @params == 0) { # simulate old behaviour $params[0] = $class->div_scale(); # and round to it as accuracy $params[1] = undef; # disable P $scale = $params[0]+4; # at least four more for proper round $params[2] = $r[2]; # round mode by caller or undef $fallback = 1; # to clear a/p afterwards } else { # the 4 below is empirical, and there might be cases where it is not # enough... $scale = abs($params[0] \|\| $params[1]) + 4; # take whatever is defined } # 1 or -1 => PI/4 # inlined is_one() && is_one('-') if ($LIB->_is_one($self->{_m}) && $LIB->_is_zero($self->{_e})) { my $pi = $class->bpi($scale - 3); # modify $self in place $self->{_m} = $pi->{_m}; $self->{_e} = $pi->{_e}; $self->{_es} = $pi->{_es}; # leave the sign of $self alone (+1 => +PI/4, -1 => -PI/4) $self->{_m} = $LIB->_div($self->{_m}, $LIB->_new(4)); return $self; } # This series is only valid if -1 < x < 1, so for other x we need to # calculate PI/2 - atan(1/x): my $pi = undef; if ($self->bacmp($self->copy()->bone) >= 0) { # calculate PI/2 $pi = $class->bpi($scale - 3); $pi->{_m} = $LIB->_div($pi->{_m}, $LIB->_new(2)); # calculate 1/$self: my $self_copy = $self->copy(); # modify $self in place $self->bone(); $self->bdiv($self_copy, $scale); } my $fmul = 1; foreach (0 .. int($scale / 20)) { $fmul = 2; $self->bdiv($self->copy()->bmul($self)->binc->bsqrt($scale + 4)->binc, $scale + 4); } # When user set globals, they would interfere with our calculation, so # disable them and later re-enable them. no strict 'refs'; my $abr = "$class\::accuracy"; my $ab = $$abr; $$abr = undef; my $pbr = "$class\::precision"; my $pb = $$pbr; $$pbr = undef; # We also need to disable any set A or P on $self (_find_round_parameters # took them already into account), since these would interfere, too delete $self->{_a}; delete $self->{_p}; # Need to disable $upgrade in BigInt, to avoid deep recursion. local $Math::BigInt::upgrade = undef; my $over = $self * $self; # X ^ 2 my $self2 = $over->copy(); # X ^ 2; difference between terms $over->bmul($self); # X ^ 3 as starting value my $sign = 1; # start with -= my $below = $class->new(3); my $two = $class->new(2); delete $self->{_a}; delete $self->{_p}; my $limit = $class->new("1E-". ($scale-1)); #my $steps = 0; while (1) { # We calculate the next term, and add it to the last. When the next # term is below our limit, it won't affect the outcome anymore, so we # stop: my $next = $over->copy()->bdiv($below, $scale); last if $next->bacmp($limit) <= 0; if ($sign == 0) { $self->badd($next); } else { $self->bsub($next); } $sign = 1-$sign; # alternatex # calculate things for the next term $over->bmul($self2); # $self$self $below->badd($two); # n += 2 } $self->bmul($fmul); if (defined $pi) { my $self_copy = $self->copy(); # modify $self in place $self->{_m} = $pi->{_m}; $self->{_e} = $pi->{_e}; $self->{_es} = $pi->{_es}; # PI/2 - $self $self->bsub($self_copy); } # Shortcut to not run through _find_round_parameters again. if (defined $params[0]) { $self->bround($params[0], $params[2]); # then round accordingly } else { $self->bfround($params[1], $params[2]); # then round accordingly } if ($fallback) { # Clear a/p after round, since user did not request it. delete $self->{_a}; delete $self->{_p}; } # restore globals $$abr = $ab; $$pbr = $pb; $self; } sub batan2 { # $y -> batan2($x) returns the arcus tangens of $y / $x. # Set up parameters. my ($class, $y, $x, @r) = (ref($_[0]), @_); # Objectify is costly, so avoid it if we can. if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $y, $x, @r) = objectify(2, @_); } # Quick exit if $y is read-only. return $y if $y -> modify('batan2'); # Handle all NaN cases. return $y -> bnan() if $x->{sign} eq $nan \|\| $y->{sign} eq $nan; # We need to limit the accuracy to protect against overflow. my $fallback = 0; my ($scale, @params); ($y, @params) = $y -> _find_round_parameters(@r); # Error in _find_round_parameters? return $y if $y->is_nan(); # No rounding at all, so must use fallback. if (scalar @params == 0) { # Simulate old behaviour $params[0] = $class -> div_scale(); # and round to it as accuracy $params[1] = undef; # disable P $scale = $params[0] + 4; # at least four more for proper round $params[2] = $r[2]; # round mode by caller or undef $fallback = 1; # to clear a/p afterwards } else { # The 4 below is empirical, and there might be cases where it is not # enough ... $scale = abs($params[0] \|\| $params[1]) + 4; # take whatever is defined } if ($x -> is_inf("+")) { # x = inf if ($y -> is_inf("+")) { # y = inf $y -> bpi($scale) -> bmul("0.25"); # pi/4 } elsif ($y -> is_inf("-")) { # y = -inf $y -> bpi($scale) -> bmul("-0.25"); # -pi/4 } else { # -inf < y < inf return $y -> bzero(@r); # 0 } } elsif ($x -> is_inf("-")) { # x = -inf if ($y -> is_inf("+")) { # y = inf $y -> bpi($scale) -> bmul("0.75"); # 3/4 pi } elsif ($y -> is_inf("-")) { # y = -inf $y -> bpi($scale) -> bmul("-0.75"); # -3/4 pi } elsif ($y >= 0) { # y >= 0 $y -> bpi($scale); # pi } else { # y < 0 $y -> bpi($scale) -> bneg(); # -pi } } elsif ($x > 0) { # 0 < x < inf if ($y -> is_inf("+")) { # y = inf $y -> bpi($scale) -> bmul("0.5"); # pi/2 } elsif ($y -> is_inf("-")) { # y = -inf $y -> bpi($scale) -> bmul("-0.5"); # -pi/2 } else { # -inf < y < inf $y -> bdiv($x, $scale) -> batan($scale); # atan(y/x) } } elsif ($x < 0) { # -inf < x < 0 my $pi = $class -> bpi($scale); if ($y >= 0) { # y >= 0 $y -> bdiv($x, $scale) -> batan() # atan(y/x) + pi -> badd($pi); } else { # y < 0 $y -> bdiv($x, $scale) -> batan() # atan(y/x) - pi -> bsub($pi); } } else { # x = 0 if ($y > 0) { # y > 0 $y -> bpi($scale) -> bmul("0.5"); # pi/2 } elsif ($y < 0) { # y < 0 $y -> bpi($scale) -> bmul("-0.5"); # -pi/2 } else { # y = 0 return $y -> bzero(@r); # 0 } } $y -> round(@r); if ($fallback) { delete $y->{_a}; delete $y->{_p}; } return $y; } ############################################################################## sub bsqrt { # calculate square root my ($class, $x, $a, $p, $r) = ref($_[0]) ? (ref($_[0]), @_) : objectify(1, @_); return $x if $x->modify('bsqrt'); return $x->bnan() if $x->{sign} !~ /^\+/; # NaN, -inf or < 0 return $x if $x->{sign} eq '+inf'; # sqrt(inf) == inf return $x->round($a, $p, $r) if $x->is_zero() \|\| $x->is_one(); # we need to limit the accuracy to protect against overflow my $fallback = 0; my (@params, $scale); ($x, @params) = $x->_find_round_parameters($a, $p, $r); return $x if $x->is_nan(); # error in _find_round_parameters? # no rounding at all, so must use fallback if (scalar @params == 0) { # simulate old behaviour $params[0] = $class->div_scale(); # and round to it as accuracy $scale = $params[0]+4; # at least four more for proper round $params[2] = $r; # round mode by caller or undef $fallback = 1; # to clear a/p afterwards } else { # the 4 below is empirical, and there might be cases where it is not # enough... $scale = abs($params[0] \|\| $params[1]) + 4; # take whatever is defined } # when user set globals, they would interfere with our calculation, so # disable them and later re-enable them no strict 'refs'; my $abr = "$class\::accuracy"; my $ab = $$abr; $$abr = undef; my $pbr = "$class\::precision"; my $pb = $$pbr; $$pbr = undef; # we also need to disable any set A or P on $x (_find_round_parameters took # them already into account), since these would interfere, too delete $x->{_a}; delete $x->{_p}; # need to disable $upgrade in BigInt, to avoid deep recursion local $Math::BigInt::upgrade = undef; # should be really parent class vs MBI my $i = $LIB->_copy($x->{_m}); $i = $LIB->_lsft($i, $x->{_e}, 10) unless $LIB->_is_zero($x->{_e}); my $xas = Math::BigInt->bzero(); $xas->{value} = $i; my $gs = $xas->copy()->bsqrt(); # some guess if (($x->{_es} ne '-') # guess can't be accurate if there are # digits after the dot && ($xas->bacmp($gs $gs) == 0)) # guess hit the nail on the head? { # exact result, copy result over to keep $x $x->{_m} = $gs->{value}; $x->{_e} = $LIB->_zero(); $x->{_es} = '+'; $x->bnorm(); # shortcut to not run through _find_round_parameters again if (defined $params[0]) { $x->bround($params[0], $params[2]); # then round accordingly } else { $x->bfround($params[1], $params[2]); # then round accordingly } if ($fallback) { # clear a/p after round, since user did not request it delete $x->{_a}; delete $x->{_p}; } # re-enable A and P, upgrade is taken care of by "local" ${"$class\::accuracy"} = $ab; ${"$class\::precision"} = $pb; return $x; } # sqrt(2) = 1.4 because sqrt(2100) = 1.410; so we can increase the accuracy # of the result by multiplying the input by 100 and then divide the integer # result of sqrt(input) by 10. Rounding afterwards returns the real result. # The following steps will transform 123.456 (in $x) into 123456 (in $y1) my $y1 = $LIB->_copy($x->{_m}); my $length = $LIB->_len($y1); # Now calculate how many digits the result of sqrt(y1) would have my $digits = int($length / 2); # But we need at least $scale digits, so calculate how many are missing my $shift = $scale - $digits; # This happens if the input had enough digits # (we take care of integer guesses above) $shift = 0 if $shift < 0; # Multiply in steps of 100, by shifting left two times the "missing" digits my $s2 = $shift * 2; # We now make sure that $y1 has the same odd or even number of digits than # $x had. So when _e of $x is odd, we must shift $y1 by one digit left, # because we always must multiply by steps of 100 (sqrt(100) is 10) and not # steps of 10. The length of $x does not count, since an even or odd number # of digits before the dot is not changed by adding an even number of digits # after the dot (the result is still odd or even digits long). $s2++ if $LIB->_is_odd($x->{_e}); $y1 = $LIB->_lsft($y1, $LIB->_new($s2), 10); # now take the square root and truncate to integer $y1 = $LIB->_sqrt($y1); # By "shifting" $y1 right (by creating a negative _e) we calculate the final # result, which is than later rounded to the desired scale. # calculate how many zeros $x had after the '.' (or before it, depending # on sign of $dat, the result should have half as many: my $dat = $LIB->_num($x->{_e}); $dat = -$dat if $x->{_es} eq '-'; $dat += $length; if ($dat > 0) { # no zeros after the dot (e.g. 1.23, 0.49 etc) # preserve half as many digits before the dot than the input had # (but round this "up") $dat = int(($dat+1)/2); } else { $dat = int(($dat)/2); } $dat -= $LIB->_len($y1); if ($dat < 0) { $dat = abs($dat); $x->{_e} = $LIB->_new($dat); $x->{_es} = '-'; } else { $x->{_e} = $LIB->_new($dat); $x->{_es} = '+'; } $x->{_m} = $y1; $x->bnorm(); # shortcut to not run through _find_round_parameters again if (defined $params[0]) { $x->bround($params[0], $params[2]); # then round accordingly } else { $x->bfround($params[1], $params[2]); # then round accordingly } if ($fallback) { # clear a/p after round, since user did not request it delete $x->{_a}; delete $x->{_p}; } # restore globals $$abr = $ab; $$pbr = $pb; $x; } sub broot { # calculate $y'th root of $x # set up parameters my ($class, $x, $y, $a, $p, $r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, $a, $p, $r) = objectify(2, @_); } return $x if $x->modify('broot'); # NaN handling: $x 1/0, x or y NaN, or y inf/-inf or y == 0 return $x->bnan() if $x->{sign} !~ /^\+/ \|\| $y->is_zero() \|\| $y->{sign} !~ /^\+$/; return $x if $x->is_zero() \|\| $x->is_one() \|\| $x->is_inf() \|\| $y->is_one(); # we need to limit the accuracy to protect against overflow my $fallback = 0; my (@params, $scale); ($x, @params) = $x->_find_round_parameters($a, $p, $r); return $x if $x->is_nan(); # error in _find_round_parameters? # no rounding at all, so must use fallback if (scalar @params == 0) { # simulate old behaviour $params[0] = $class->div_scale(); # and round to it as accuracy $scale = $params[0]+4; # at least four more for proper round $params[2] = $r; # round mode by caller or undef $fallback = 1; # to clear a/p afterwards } else { # the 4 below is empirical, and there might be cases where it is not # enough... $scale = abs($params[0] \|\| $params[1]) + 4; # take whatever is defined } # when user set globals, they would interfere with our calculation, so # disable them and later re-enable them no strict 'refs'; my $abr = "$class\::accuracy"; my $ab = $$abr; $$abr = undef; my $pbr = "$class\::precision"; my $pb = $$pbr; $$pbr = undef; # we also need to disable any set A or P on $x (_find_round_parameters took # them already into account), since these would interfere, too delete $x->{_a}; delete $x->{_p}; # need to disable $upgrade in BigInt, to avoid deep recursion local $Math::BigInt::upgrade = undef; # should be really parent class vs MBI # remember sign and make $x positive, since -4 (1/2) => -2 my $sign = 0; $sign = 1 if $x->{sign} eq '-'; $x->{sign} = '+'; my $is_two = 0; if ($y->isa('Math::BigFloat')) { $is_two = ($y->{sign} eq '+' && $LIB->_is_two($y->{_m}) && $LIB->_is_zero($y->{_e})); } else { $is_two = ($y == 2); } # normal square root if $y == 2: if ($is_two) { $x->bsqrt($scale+4); } elsif ($y->is_one('-')) { # $x -1 => 1/$x my $u = $class->bone()->bdiv($x, $scale); # copy private parts over $x->{_m} = $u->{_m}; $x->{_e} = $u->{_e}; $x->{_es} = $u->{_es}; } else { # calculate the broot() as integer result first, and if it fits, return # it rightaway (but only if $x and $y are integer): my $done = 0; # not yet if ($y->is_int() && $x->is_int()) { my $i = $LIB->_copy($x->{_m}); $i = $LIB->_lsft($i, $x->{_e}, 10) unless $LIB->_is_zero($x->{_e}); my $int = Math::BigInt->bzero(); $int->{value} = $i; $int->broot($y->as_number()); # if ($exact) if ($int->copy()->bpow($y) == $x) { # found result, return it $x->{_m} = $int->{value}; $x->{_e} = $LIB->_zero(); $x->{_es} = '+'; $x->bnorm(); $done = 1; } } if ($done == 0) { my $u = $class->bone()->bdiv($y, $scale+4); delete $u->{_a}; delete $u->{_p}; # otherwise it conflicts $x->bpow($u, $scale+4); # el cheapo } } $x->bneg() if $sign == 1; # shortcut to not run through _find_round_parameters again if (defined $params[0]) { $x->bround($params[0], $params[2]); # then round accordingly } else { $x->bfround($params[1], $params[2]); # then round accordingly } if ($fallback) { # clear a/p after round, since user did not request it delete $x->{_a}; delete $x->{_p}; } # restore globals $$abr = $ab; $$pbr = $pb; $x; } sub bfac { # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT # compute factorial number, modifies first argument # set up parameters my ($class, $x, @r) = (ref($_[0]), @_); # objectify is costly, so avoid it ($class, $x, @r) = objectify(1, @_) if !ref($x); # inf => inf return $x if $x->modify('bfac') \|\| $x->{sign} eq '+inf'; return $x->bnan() if (($x->{sign} ne '+') \|\| # inf, NaN, <0 etc => NaN ($x->{_es} ne '+')); # digits after dot? if (! $LIB->_is_zero($x->{_e})) { $x->{_m} = $LIB->_lsft($x->{_m}, $x->{_e}, 10); # change 12e1 to 120e0 $x->{_e} = $LIB->_zero(); # normalize $x->{_es} = '+'; } $x->{_m} = $LIB->_fac($x->{_m}); # calculate factorial $x->bnorm()->round(@r); # norm again and round result } sub bdfac { # compute double factorial # set up parameters my ($class, $x, @r) = (ref($_[0]), @_); # objectify is costly, so avoid it ($class, $x, @r) = objectify(1, @_) if !ref($x); # inf => inf return $x if $x->modify('bfac') \|\| $x->{sign} eq '+inf'; return $x->bnan() if (($x->{sign} ne '+') \|\| # inf, NaN, <0 etc => NaN ($x->{_es} ne '+')); # digits after dot? croak("bdfac() requires a newer version of the $LIB library.") unless $LIB->can('_dfac'); if (! $LIB->_is_zero($x->{_e})) { $x->{_m} = $LIB->_lsft($x->{_m}, $x->{_e}, 10); # change 12e1 to 120e0 $x->{_e} = $LIB->_zero(); # normalize $x->{_es} = '+'; } $x->{_m} = $LIB->_dfac($x->{_m}); # calculate factorial $x->bnorm()->round(@r); # norm again and round result } sub blsft { # shift left by $y (multiply by $b $y) # set up parameters my ($class, $x, $y, $b, $a, $p, $r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, $b, $a, $p, $r) = objectify(2, @_); } return $x if $x -> modify('blsft'); return $x if $x -> {sign} !~ /^[+-]$/; # nan, +inf, -inf $b = 2 if !defined $b; $b = $class -> new($b) unless ref($b) && $b -> isa($class); return $x -> bnan() if $x -> is_nan() \|\| $y -> is_nan() \|\| $b -> is_nan(); # shift by a negative amount? return $x -> brsft($y -> copy() -> babs(), $b) if $y -> {sign} =~ /^-/; $x -> bmul($b -> bpow($y), $a, $p, $r, $y); } sub brsft { # shift right by $y (divide $b ** $y) # set up parameters my ($class, $x, $y, $b, $a, $p, $r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, $b, $a, $p, $r) = objectify(2, @_); } return $x if $x -> modify('brsft'); return $x if $x -> {sign} !~ /^[+-]$/; # nan, +inf, -inf $b = 2 if !defined $b; $b = $class -> new($b) unless ref($b) && $b -> isa($class); return $x -> bnan() if $x -> is_nan() \|\| $y -> is_nan() \|\| $b -> is_nan(); # shift by a negative amount? return $x -> blsft($y -> copy() -> babs(), $b) if $y -> {sign} =~ /^-/; # the following call to bdiv() will return either quotient (scalar context) # or quotient and remainder (list context). $x -> bdiv($b -> bpow($y), $a, $p, $r, $y); } ############################################################################### # Bitwise methods ############################################################################### sub band { my $x = shift; my $xref = ref($x); my $class = $xref \|\| $x; croak 'band() is an instance method, not a class method' unless $xref; croak 'Not enough arguments for band()' if @_ < 1; return if $x -> modify('band'); my $y = shift; $y = $class -> new($y) unless ref($y); my @r = @_; my $xtmp = Math::BigInt -> new($x -> bint()); # to Math::BigInt $xtmp -> band($y); $xtmp = $class -> new($xtmp); # back to Math::BigFloat $x -> {sign} = $xtmp -> {sign}; $x -> {_m} = $xtmp -> {_m}; $x -> {_es} = $xtmp -> {_es}; $x -> {_e} = $xtmp -> {_e}; return $x -> round(@r); } sub bior { my $x = shift; my $xref = ref($x); my $class = $xref \|\| $x; croak 'bior() is an instance method, not a class method' unless $xref; croak 'Not enough arguments for bior()' if @_ < 1; return if $x -> modify('bior'); my $y = shift; $y = $class -> new($y) unless ref($y); my @r = @_; my $xtmp = Math::BigInt -> new($x -> bint()); # to Math::BigInt $xtmp -> bior($y); $xtmp = $class -> new($xtmp); # back to Math::BigFloat $x -> {sign} = $xtmp -> {sign}; $x -> {_m} = $xtmp -> {_m}; $x -> {_es} = $xtmp -> {_es}; $x -> {_e} = $xtmp -> {_e}; return $x -> round(@r); } sub bxor { my $x = shift; my $xref = ref($x); my $class = $xref \|\| $x; croak 'bxor() is an instance method, not a class method' unless $xref; croak 'Not enough arguments for bxor()' if @_ < 1; return if $x -> modify('bxor'); my $y = shift; $y = $class -> new($y) unless ref($y); my @r = @_; my $xtmp = Math::BigInt -> new($x -> bint()); # to Math::BigInt $xtmp -> bxor($y); $xtmp = $class -> new($xtmp); # back to Math::BigFloat $x -> {sign} = $xtmp -> {sign}; $x -> {_m} = $xtmp -> {_m}; $x -> {_es} = $xtmp -> {_es}; $x -> {_e} = $xtmp -> {_e}; return $x -> round(@r); } sub bnot { my $x = shift; my $xref = ref($x); my $class = $xref \|\| $x; croak 'bnot() is an instance method, not a class method' unless $xref; return if $x -> modify('bnot'); my @r = @_; my $xtmp = Math::BigInt -> new($x -> bint()); # to Math::BigInt $xtmp -> bnot(); $xtmp = $class -> new($xtmp); # back to Math::BigFloat $x -> {sign} = $xtmp -> {sign}; $x -> {_m} = $xtmp -> {_m}; $x -> {_es} = $xtmp -> {_es}; $x -> {_e} = $xtmp -> {_e}; return $x -> round(@r); } ############################################################################### # Rounding methods ############################################################################### sub bround { # accuracy: preserve $N digits, and overwrite the rest with 0's my $x = shift; my $class = ref($x) \|\| $x; $x = $class->new(shift) if !ref($x); if (($_[0] \|\| 0) < 0) { croak('bround() needs positive accuracy'); } my ($scale, $mode) = $x->_scale_a(@_); return $x if !defined $scale \|\| $x->modify('bround'); # no-op # scale is now either $x->{_a}, $accuracy, or the user parameter # test whether $x already has lower accuracy, do nothing in this case # but do round if the accuracy is the same, since a math operation might # want to round a number with A=5 to 5 digits afterwards again return $x if defined $x->{_a} && $x->{_a} < $scale; # scale < 0 makes no sense # scale == 0 => keep all digits # never round a +-inf, NaN return $x if ($scale <= 0) \|\| $x->{sign} !~ /^[+-]$/; # 1: never round a 0 # 2: if we should keep more digits than the mantissa has, do nothing if ($x->is_zero() \|\| $LIB->_len($x->{_m}) <= $scale) { $x->{_a} = $scale if !defined $x->{_a} \|\| $x->{_a} > $scale; return $x; } # pass sign to bround for '+inf' and '-inf' rounding modes my $m = bless { sign => $x->{sign}, value => $x->{_m} }, 'Math::BigInt'; $m->bround($scale, $mode); # round mantissa $x->{_m} = $m->{value}; # get our mantissa back $x->{_a} = $scale; # remember rounding delete $x->{_p}; # and clear P $x->bnorm(); # del trailing zeros gen. by bround() } sub bfround { # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.' # $n == 0 means round to integer # expects and returns normalized numbers! my $x = shift; my $class = ref($x) \|\| $x; $x = $class->new(shift) if !ref($x); my ($scale, $mode) = $x->_scale_p(@_); return $x if !defined $scale \|\| $x->modify('bfround'); # no-op # never round a 0, +-inf, NaN if ($x->is_zero()) { $x->{_p} = $scale if !defined $x->{_p} \|\| $x->{_p} < $scale; # -3 < -2 return $x; } return $x if $x->{sign} !~ /^[+-]$/; # don't round if x already has lower precision return $x if (defined $x->{_p} && $x->{_p} < 0 && $scale < $x->{_p}); $x->{_p} = $scale; # remember round in any case delete $x->{_a}; # and clear A if ($scale < 0) { # round right from the '.' return $x if $x->{_es} eq '+'; # e >= 0 => nothing to round $scale = -$scale; # positive for simplicity my $len = $LIB->_len($x->{_m}); # length of mantissa # the following poses a restriction on _e, but if _e is bigger than a # scalar, you got other problems (memory etc) anyway my $dad = -(0+ ($x->{_es}.$LIB->_num($x->{_e}))); # digits after dot my $zad = 0; # zeros after dot $zad = $dad - $len if (-$dad < -$len); # for 0.00..00xxx style # print "scale $scale dad $dad zad $zad len $len\n"; # number bsstr len zad dad # 0.123 123e-3 3 0 3 # 0.0123 123e-4 3 1 4 # 0.001 1e-3 1 2 3 # 1.23 123e-2 3 0 2 # 1.2345 12345e-4 5 0 4 # do not round after/right of the $dad return $x if $scale > $dad; # 0.123, scale >= 3 => exit # round to zero if rounding inside the $zad, but not for last zero like: # 0.0065, scale -2, round last '0' with following '65' (scale == zad case) return $x->bzero() if $scale < $zad; if ($scale == $zad) # for 0.006, scale -3 and trunc { $scale = -$len; } else { # adjust round-point to be inside mantissa if ($zad != 0) { $scale = $scale-$zad; } else { my $dbd = $len - $dad; $dbd = 0 if $dbd < 0; # digits before dot $scale = $dbd+$scale; } } } else { # round left from the '.' # 123 => 100 means length(123) = 3 - $scale (2) => 1 my $dbt = $LIB->_len($x->{_m}); # digits before dot my $dbd = $dbt + ($x->{_es} . $LIB->_num($x->{_e})); # should be the same, so treat it as this $scale = 1 if $scale == 0; # shortcut if already integer return $x if $scale == 1 && $dbt <= $dbd; # maximum digits before dot ++$dbd; if ($scale > $dbd) { # not enough digits before dot, so round to zero return $x->bzero; } elsif ($scale == $dbd) { # maximum $scale = -$dbt; } else { $scale = $dbd - $scale; } } # pass sign to bround for rounding modes '+inf' and '-inf' my $m = bless { sign => $x->{sign}, value => $x->{_m} }, 'Math::BigInt'; $m->bround($scale, $mode); $x->{_m} = $m->{value}; # get our mantissa back $x->bnorm(); } sub bfloor { # round towards minus infinity my ($class, $x, $a, $p, $r) = ref($_[0]) ? (ref($_[0]), @_) : objectify(1, @_); return $x if $x->modify('bfloor'); return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf # if $x has digits after dot if ($x->{_es} eq '-') { $x->{_m} = $LIB->_rsft($x->{_m}, $x->{_e}, 10); # cut off digits after dot $x->{_e} = $LIB->_zero(); # trunc/norm $x->{_es} = '+'; # abs e $x->{_m} = $LIB->_inc($x->{_m}) if $x->{sign} eq '-'; # increment if negative } $x->round($a, $p, $r); } sub bceil { # round towards plus infinity my ($class, $x, $a, $p, $r) = ref($_[0]) ? (ref($_[0]), @_) : objectify(1, @_); return $x if $x->modify('bceil'); return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf # if $x has digits after dot if ($x->{_es} eq '-') { $x->{_m} = $LIB->_rsft($x->{_m}, $x->{_e}, 10); # cut off digits after dot $x->{_e} = $LIB->_zero(); # trunc/norm $x->{_es} = '+'; # abs e if ($x->{sign} eq '+') { $x->{_m} = $LIB->_inc($x->{_m}); # increment if positive } else { $x->{sign} = '+' if $LIB->_is_zero($x->{_m}); # avoid -0 } } $x->round($a, $p, $r); } sub bint { # round towards zero my ($class, $x, $a, $p, $r) = ref($_[0]) ? (ref($_[0]), @_) : objectify(1, @_); return $x if $x->modify('bint'); return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf # if $x has digits after the decimal point if ($x->{_es} eq '-') { $x->{_m} = $LIB->_rsft($x->{_m}, $x->{_e}, 10); # cut off digits after dot $x->{_e} = $LIB->_zero(); # truncate/normalize $x->{_es} = '+'; # abs e $x->{sign} = '+' if $LIB->_is_zero($x->{_m}); # avoid -0 } $x->round($a, $p, $r); } ############################################################################### # Other mathematical methods ############################################################################### sub bgcd { # (BINT or num_str, BINT or num_str) return BINT # does not modify arguments, but returns new object unshift @_, __PACKAGE__ unless ref($_[0]) \|\| $_[0] =~ /^[a-z]\w(?:::[a-z]\w)$/i; my ($class, @args) = objectify(0, @_); my $x = shift @args; $x = ref($x) && $x -> isa($class) ? $x -> copy() : $class -> new($x); return $class->bnan() unless $x -> is_int(); while (@args) { my $y = shift @args; $y = $class->new($y) unless ref($y) && $y -> isa($class); return $class->bnan() unless $y -> is_int(); # greatest common divisor while (! $y->is_zero()) { ($x, $y) = ($y->copy(), $x->copy()->bmod($y)); } last if $x -> is_one(); } return $x -> babs(); } sub blcm { # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT # does not modify arguments, but returns new object # Least Common Multiple unshift @_, __PACKAGE__ unless ref($_[0]) \|\| $_[0] =~ /^[a-z]\w(?:::[a-z]\w)$/i; my ($class, @args) = objectify(0, @_); my $x = shift @args; $x = ref($x) && $x -> isa($class) ? $x -> copy() : $class -> new($x); return $class->bnan() if $x->{sign} !~ /^[+-]$/; # x NaN? while (@args) { my $y = shift @args; $y = $class -> new($y) unless ref($y) && $y -> isa($class); return $x->bnan() unless $y -> is_int(); my $gcd = $x -> bgcd($y); $x -> bdiv($gcd) -> bmul($y); } return $x -> babs(); } ############################################################################### # Object property methods ############################################################################### sub length { my $x = shift; my $class = ref($x) \|\| $x; $x = $class->new(shift) unless ref($x); return 1 if $LIB->_is_zero($x->{_m}); my $len = $LIB->_len($x->{_m}); $len += $LIB->_num($x->{_e}) if $x->{_es} eq '+'; if (wantarray()) { my $t = 0; $t = $LIB->_num($x->{_e}) if $x->{_es} eq '-'; return ($len, $t); } $len; } sub mantissa { # return a copy of the mantissa my ($class, $x) = ref($_[0]) ? (ref($_[0]), $_[0]) : objectify(1, @_); if ($x->{sign} !~ /^[+-]$/) { my $s = $x->{sign}; $s =~ s/^\+//; return Math::BigInt->new($s, undef, undef); # -inf, +inf => +inf } my $m = Math::BigInt->new($LIB->_str($x->{_m}), undef, undef); $m->bneg() if $x->{sign} eq '-'; $m; } sub exponent { # return a copy of the exponent my ($class, $x) = ref($_[0]) ? (ref($_[0]), $_[0]) : objectify(1, @_); if ($x->{sign} !~ /^[+-]$/) { my $s = $x->{sign}; $s =~ s/^[+-]//; return Math::BigInt->new($s, undef, undef); # -inf, +inf => +inf } Math::BigInt->new($x->{_es} . $LIB->_str($x->{_e}), undef, undef); } sub parts { # return a copy of both the exponent and the mantissa my ($class, $x) = ref($_[0]) ? (ref($_[0]), $_[0]) : objectify(1, @_); if ($x->{sign} !~ /^[+-]$/) { my $s = $x->{sign}; $s =~ s/^\+//; my $se = $s; $se =~ s/^-//; return ($class->new($s), $class->new($se)); # +inf => inf and -inf, +inf => inf } my $m = Math::BigInt->bzero(); $m->{value} = $LIB->_copy($x->{_m}); $m->bneg() if $x->{sign} eq '-'; ($m, Math::BigInt->new($x->{_es} . $LIB->_num($x->{_e}))); } sub sparts { my $self = shift; my $class = ref $self; croak("sparts() is an instance method, not a class method") unless $class; # Not-a-number. if ($self -> is_nan()) { my $mant = $self -> copy(); # mantissa return $mant unless wantarray; # scalar context my $expo = $class -> bnan(); # exponent return ($mant, $expo); # list context } # Infinity. if ($self -> is_inf()) { my $mant = $self -> copy(); # mantissa return $mant unless wantarray; # scalar context my $expo = $class -> binf('+'); # exponent return ($mant, $expo); # list context } # Finite number. my $mant = $class -> bzero(); $mant -> {sign} = $self -> {sign}; $mant -> {_m} = $LIB->_copy($self -> {_m}); return $mant unless wantarray; my $expo = $class -> bzero(); $expo -> {sign} = $self -> {_es}; $expo -> {_m} = $LIB->_copy($self -> {_e}); return ($mant, $expo); } sub nparts { my $self = shift; my $class = ref $self; croak("nparts() is an instance method, not a class method") unless $class; # Not-a-number. if ($self -> is_nan()) { my $mant = $self -> copy(); # mantissa return $mant unless wantarray; # scalar context my $expo = $class -> bnan(); # exponent return ($mant, $expo); # list context } # Infinity. if ($self -> is_inf()) { my $mant = $self -> copy(); # mantissa return $mant unless wantarray; # scalar context my $expo = $class -> binf('+'); # exponent return ($mant, $expo); # list context } # Finite number. my ($mant, $expo) = $self -> sparts(); if ($mant -> bcmp(0)) { my ($ndigtot, $ndigfrac) = $mant -> length(); my $expo10adj = $ndigtot - $ndigfrac - 1; if ($expo10adj != 0) { my $factor = "1e" . -$expo10adj; $mant -> bmul($factor); return $mant unless wantarray; $expo -> badd($expo10adj); return ($mant, $expo); } } return $mant unless wantarray; return ($mant, $expo); } sub eparts { my $self = shift; my $class = ref $self; croak("eparts() is an instance method, not a class method") unless $class; # Not-a-number and Infinity. return $self -> sparts() if $self -> is_nan() \|\| $self -> is_inf(); # Finite number. my ($mant, $expo) = $self -> nparts(); my $c = $expo -> copy() -> bmod(3); $mant -> blsft($c, 10); return $mant unless wantarray; $expo -> bsub($c); return ($mant, $expo); } sub dparts { my $self = shift; my $class = ref $self; croak("dparts() is an instance method, not a class method") unless $class; # Not-a-number and Infinity. if ($self -> is_nan() \|\| $self -> is_inf()) { my $int = $self -> copy(); return $int unless wantarray; my $frc = $class -> bzero(); return ($int, $frc); } my $int = $self -> copy(); my $frc = $class -> bzero(); # If the input has a fraction part. if ($int->{_es} eq '-') { $int->{_m} = $LIB -> _rsft($int->{_m}, $int->{_e}, 10); $int->{_e} = $LIB -> _zero(); $int->{_es} = '+'; $int->{sign} = '+' if $LIB->_is_zero($int->{_m}); # avoid -0 return $int unless wantarray; $frc = $self -> copy() -> bsub($int); return ($int, $frc); } return $int unless wantarray; return ($int, $frc); } ############################################################################### # String conversion methods ############################################################################### sub bstr { # (ref to BFLOAT or num_str) return num_str # Convert number from internal format to (non-scientific) string format. # internal format is always normalized (no leading zeros, "-0" => "+0") my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); if ($x->{sign} !~ /^[+-]$/) { return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN return 'inf'; # +inf } my $es = '0'; my $len = 1; my $cad = 0; my $dot = '.'; # $x is zero? my $not_zero = !($x->{sign} eq '+' && $LIB->_is_zero($x->{_m})); if ($not_zero) { $es = $LIB->_str($x->{_m}); $len = CORE::length($es); my $e = $LIB->_num($x->{_e}); $e = -$e if $x->{_es} eq '-'; if ($e < 0) { $dot = ''; # if _e is bigger than a scalar, the following will blow your memory if ($e <= -$len) { my $r = abs($e) - $len; $es = '0.'. ('0' x $r) . $es; $cad = -($len+$r); } else { substr($es, $e, 0) = '.'; $cad = $LIB->_num($x->{_e}); $cad = -$cad if $x->{_es} eq '-'; } } elsif ($e > 0) { # expand with zeros $es .= '0' x $e; $len += $e; $cad = 0; } } # if not zero $es = '-'.$es if $x->{sign} eq '-'; # if set accuracy or precision, pad with zeros on the right side if ((defined $x->{_a}) && ($not_zero)) { # 123400 => 6, 0.1234 => 4, 0.001234 => 4 my $zeros = $x->{_a} - $cad; # cad == 0 => 12340 $zeros = $x->{_a} - $len if $cad != $len; $es .= $dot.'0' x $zeros if $zeros > 0; } elsif ((($x->{_p} \|\| 0) < 0)) { # 123400 => 6, 0.1234 => 4, 0.001234 => 6 my $zeros = -$x->{_p} + $cad; $es .= $dot.'0' x $zeros if $zeros > 0; } $es; } # Decimal notation, e.g., "12345.6789". sub bdstr { my $x = shift; if ($x->{sign} ne '+' && $x->{sign} ne '-') { return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN return 'inf'; # +inf } my $mant = $LIB->_str($x->{_m}); my $expo = $x -> exponent(); my $str = $mant; if ($expo >= 0) { $str .= "0" x $expo; } else { my $mantlen = CORE::length($mant); my $c = $mantlen + $expo; $str = "0" x (1 - $c) . $str if $c <= 0; substr($str, $expo, 0) = '.'; } return $x->{sign} eq '-' ? "-$str" : $str; } # Scientific notation with significand/mantissa as an integer, e.g., "12345.6789" # is written as "123456789e-4". sub bsstr { my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); if ($x->{sign} ne '+' && $x->{sign} ne '-') { return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN return 'inf'; # +inf } my $str = $LIB->_str($x->{_m}) . 'e' . $x->{_es}. $LIB->_str($x->{_e}); return $x->{sign} eq '-' ? "-$str" : $str; } # Normalized notation, e.g., "12345.6789" is written as "1.23456789e+4". sub bnstr { my $x = shift; if ($x->{sign} ne '+' && $x->{sign} ne '-') { return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN return 'inf'; # +inf } my ($mant, $expo) = $x -> nparts(); my $esgn = $expo < 0 ? '-' : '+'; my $eabs = $expo -> babs() -> bfround(0) -> bstr(); #$eabs = '0' . $eabs if length($eabs) < 2; return $mant . 'e' . $esgn . $eabs; } # Engineering notation, e.g., "12345.6789" is written as "12.3456789e+3". sub bestr { my $x = shift; if ($x->{sign} ne '+' && $x->{sign} ne '-') { return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN return 'inf'; # +inf } my ($mant, $expo) = $x -> eparts(); my $esgn = $expo < 0 ? '-' : '+'; my $eabs = $expo -> babs() -> bfround(0) -> bstr(); #$eabs = '0' . $eabs if length($eabs) < 2; return $mant . 'e' . $esgn . $eabs; } sub to_hex { # return number as hexadecimal string (only for integers defined) my ($class, $x) = ref($_[0]) ? (ref($_[0]), $_[0]) : objectify(1, @_); return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc return '0' if $x->is_zero(); return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex? my $z = $LIB->_copy($x->{_m}); if (! $LIB->_is_zero($x->{_e})) { # > 0 $z = $LIB->_lsft($z, $x->{_e}, 10); } my $str = $LIB->_to_hex($z); return $x->{sign} eq '-' ? "-$str" : $str; } sub to_oct { # return number as octal digit string (only for integers defined) my ($class, $x) = ref($_[0]) ? (ref($_[0]), $_[0]) : objectify(1, @_); return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc return '0' if $x->is_zero(); return $nan if $x->{_es} ne '+'; # how to do 1e-1 in octal? my $z = $LIB->_copy($x->{_m}); if (! $LIB->_is_zero($x->{_e})) { # > 0 $z = $LIB->_lsft($z, $x->{_e}, 10); } my $str = $LIB->_to_oct($z); return $x->{sign} eq '-' ? "-$str" : $str; } sub to_bin { # return number as binary digit string (only for integers defined) my ($class, $x) = ref($_[0]) ? (ref($_[0]), $_[0]) : objectify(1, @_); return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc return '0' if $x->is_zero(); return $nan if $x->{_es} ne '+'; # how to do 1e-1 in binary? my $z = $LIB->_copy($x->{_m}); if (! $LIB->_is_zero($x->{_e})) { # > 0 $z = $LIB->_lsft($z, $x->{_e}, 10); } my $str = $LIB->_to_bin($z); return $x->{sign} eq '-' ? "-$str" : $str; } sub to_ieee754 { my $x = shift; my $format = shift; my $class = ref $x; my $enc; # significand encoding (applies only to decimal) my $k; # storage width in bits my $b; # base if ($format =~ /^binary(\d+)\z/) { $k = $1; $b = 2; } elsif ($format =~ /^decimal(\d+)(dpd\|bcd)?\z/) { $k = $1; $b = 10; $enc = $2 \|\| 'dpd'; # default is dencely-packed decimals (DPD) } elsif ($format eq 'half') { $k = 16; $b = 2; } elsif ($format eq 'single') { $k = 32; $b = 2; } elsif ($format eq 'double') { $k = 64; $b = 2; } elsif ($format eq 'quadruple') { $k = 128; $b = 2; } elsif ($format eq 'octuple') { $k = 256; $b = 2; } elsif ($format eq 'sexdecuple') { $k = 512; $b = 2; } if ($b == 2) { # Get the parameters for this format. my $p; # precision (in bits) my $t; # number of bits in significand my $w; # number of bits in exponent if ($k == 16) { # binary16 (half-precision) $p = 11; $t = 10; $w = 5; } elsif ($k == 32) { # binary32 (single-precision) $p = 24; $t = 23; $w = 8; } elsif ($k == 64) { # binary64 (double-precision) $p = 53; $t = 52; $w = 11; } else { # binaryN (quadruple-precition and above) if ($k < 128 \|\| $k != 32 * sprintf('%.0f', $k / 32)) { croak "Number of bits must be 16, 32, 64, or >= 128 and", " a multiple of 32"; } $p = $k - sprintf('%.0f', 4 * log($k) / log(2)) + 13; $t = $p - 1; $w = $k - $t - 1; } # The maximum exponent, minimum exponent, and exponent bias. my $emax = $class -> new(2) -> bpow($w - 1) -> bdec(); my $emin = 1 - $emax; my $bias = $emax; # Get numerical sign, exponent, and mantissa/significand for bit # string. my $sign = 0; my $expo; my $mant; if ($x -> is_nan()) { # nan $sign = 1; $expo = $emax -> copy() -> binc(); $mant = $class -> new(2) -> bpow($t - 1); } elsif ($x -> is_inf()) { # inf $sign = 1 if $x -> is_neg(); $expo = $emax -> copy() -> binc(); $mant = $class -> bzero(); } elsif ($x -> is_zero()) { # zero $expo = $emin -> copy() -> bdec(); $mant = $class -> bzero(); } else { # normal and subnormal $sign = 1 if $x -> is_neg(); # Now we need to compute the mantissa and exponent in base $b. my $binv = $class -> new("0.5"); my $b = $class -> new(2); my $one = $class -> bone(); # We start off by initializing the exponent to zero and the # mantissa to the input value. Then we increase the mantissa and # decrease the exponent, or vice versa, until the mantissa is in # the desired range or we hit one of the limits for the exponent. $mant = $x -> copy() -> babs(); # We need to find the base 2 exponent. First make an estimate of # the base 2 exponent, before adjusting it below. We could skip # this estimation and go straight to the while-loops below, but the # loops are slow, especially when the final exponent is far from # zero and even more so if the number of digits is large. This # initial estimation speeds up the computation dramatically. # # log2($m * 10$e) = log10($m + 10$e) * log(10)/log(2) # = (log10($m) + $e) * log(10)/log(2) # = (log($m)/log(10) + $e) * log(10)/log(2) my ($m, $e) = $x -> nparts(); my $ms = $m -> numify(); my $es = $e -> numify(); my $expo_est = (log(abs($ms))/log(10) + $es) * log(10)/log(2); $expo_est = int($expo_est); # Limit the exponent. if ($expo_est > $emax) { $expo_est = $emax; } elsif ($expo_est < $emin) { $expo_est = $emin; } # Don't multiply by a number raised to a negative exponent. This # will cause a division, whose result is truncated to some fixed # number of digits. Instead, multiply by the inverse number raised # to a positive exponent. $expo = $class -> new($expo_est); if ($expo_est > 0) { $mant -> bmul($binv -> copy() -> bpow($expo)); } elsif ($expo_est < 0) { my $expo_abs = $expo -> copy() -> bneg(); $mant -> bmul($b -> copy() -> bpow($expo_abs)); } # Final adjustment. while ($mant >= $b && $expo <= $emax) { $mant -> bmul($binv); $expo -> binc(); } while ($mant < $one && $expo >= $emin) { $mant -> bmul($b); $expo -> bdec(); } # Encode as infinity, normal number or subnormal number? if ($expo > $emax) { # overflow => infinity $expo = $emax -> copy() -> binc(); $mant = $class -> bzero(); } elsif ($expo < $emin) { # subnormal number my $const = $class -> new(2) -> bpow($t - 1); $mant -> bmul($const); $mant -> bfround(0); } else { # normal number $mant -> bdec(); # remove implicit leading bit my $const = $class -> new(2) -> bpow($t); $mant -> bmul($const) -> bfround(0); } } $expo -> badd($bias); # add bias my $signbit = "$sign"; my $mantbits = $mant -> to_bin(); $mantbits = ("0" x ($t - CORE::length($mantbits))) . $mantbits; my $expobits = $expo -> to_bin(); $expobits = ("0" x ($w - CORE::length($expobits))) . $expobits; my $bin = $signbit . $expobits . $mantbits; return pack "B", $bin; } croak("The format '$format' is not yet supported."); } sub as_hex { # return number as hexadecimal string (only for integers defined) my ($class, $x) = ref($_[0]) ? (ref($_[0]), $_[0]) : objectify(1, @_); return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc return '0x0' if $x->is_zero(); return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex? my $z = $LIB->_copy($x->{_m}); if (! $LIB->_is_zero($x->{_e})) { # > 0 $z = $LIB->_lsft($z, $x->{_e}, 10); } my $str = $LIB->_as_hex($z); return $x->{sign} eq '-' ? "-$str" : $str; } sub as_oct { # return number as octal digit string (only for integers defined) my ($class, $x) = ref($_[0]) ? (ref($_[0]), $_[0]) : objectify(1, @_); return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc return '00' if $x->is_zero(); return $nan if $x->{_es} ne '+'; # how to do 1e-1 in octal? my $z = $LIB->_copy($x->{_m}); if (! $LIB->_is_zero($x->{_e})) { # > 0 $z = $LIB->_lsft($z, $x->{_e}, 10); } my $str = $LIB->_as_oct($z); return $x->{sign} eq '-' ? "-$str" : $str; } sub as_bin { # return number as binary digit string (only for integers defined) my ($class, $x) = ref($_[0]) ? (ref($_[0]), $_[0]) : objectify(1, @_); return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc return '0b0' if $x->is_zero(); return $nan if $x->{_es} ne '+'; # how to do 1e-1 in binary? my $z = $LIB->_copy($x->{_m}); if (! $LIB->_is_zero($x->{_e})) { # > 0 $z = $LIB->_lsft($z, $x->{_e}, 10); } my $str = $LIB->_as_bin($z); return $x->{sign} eq '-' ? "-$str" : $str; } sub numify { # Make a Perl scalar number from a Math::BigFloat object. my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); if ($x -> is_nan()) { require Math::Complex; my $inf = Math::Complex::Inf(); return $inf - $inf; } if ($x -> is_inf()) { require Math::Complex; my $inf = Math::Complex::Inf(); return $x -> is_negative() ? -$inf : $inf; } # Create a string and let Perl's atoi()/atof() handle the rest. return 0 + $x -> bnstr(); } ############################################################################### # Private methods and functions. ############################################################################### sub import { my $class = shift; my $l = scalar @_; my $lib = ''; my @a; my $lib_kind = 'try'; $IMPORT=1; for (my $i = 0; $i < $l ; $i++) { if ($_[$i] eq ':constant') { # This causes overlord er load to step in. 'binary' and 'integer' # are handled by BigInt. overload::constant float => sub { $class->new(shift); }; } elsif ($_[$i] eq 'upgrade') { # this causes upgrading $upgrade = $_[$i+1]; # or undef to disable $i++; } elsif ($_[$i] eq 'downgrade') { # this causes downgrading $downgrade = $_[$i+1]; # or undef to disable $i++; } elsif ($_[$i] =~ /^(lib\|try\|only)\z/) { # alternative library $lib = $_[$i+1] \|\| ''; # default Calc $lib_kind = $1; # lib, try or only $i++; } elsif ($_[$i] eq 'with') { # alternative class for our private parts() # XXX: no longer supported # $LIB = $_[$i+1] \|\| 'Math::BigInt'; $i++; } else { push @a, $_[$i]; } } $lib =~ tr/a-zA-Z0-9,://cd; # restrict to sane characters # let use Math::BigInt lib => 'GMP'; use Math::BigFloat; still work my $mbilib = eval { Math::BigInt->config('lib') }; if ((defined $mbilib) && ($LIB eq 'Math::BigInt::Calc')) { # $LIB already loaded Math::BigInt->import($lib_kind, "$lib, $mbilib", 'objectify'); } else { # $LIB not loaded, or with ne "Math::BigInt::Calc" $lib .= ",$mbilib" if defined $mbilib; $lib =~ s/^,//; # don't leave empty # replacement library can handle lib statement, but also could ignore it # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is # used in the same script, or eval inside import(). So we require MBI: require Math::BigInt; Math::BigInt->import($lib_kind => $lib, 'objectify'); } if ($@) { croak("Couldn't load $lib: $! $@"); } # find out which one was actually loaded $LIB = Math::BigInt->config('lib'); # register us with MBI to get notified of future lib changes Math::BigInt::_register_callback($class, sub { $LIB = $_[0]; }); $class->export_to_level(1, $class, @a); # export wanted functions } sub _len_to_steps { # Given D (digits in decimal), compute N so that N! (N factorial) is # at least D digits long. D should be at least 50. my $d = shift; # two constants for the Ramanujan estimate of ln(N!) my $lg2 = log(2 3.14159265) / 2; my $lg10 = log(10); # D = 50 => N => 42, so L = 40 and R = 50 my $l = 40; my $r = $d; # Otherwise this does not work under -Mbignum and we do not yet have "no bignum;" :( $l = $l->numify if ref($l); $r = $r->numify if ref($r); $lg2 = $lg2->numify if ref($lg2); $lg10 = $lg10->numify if ref($lg10); # binary search for the right value (could this be written as the reverse of lg(n!)?) while ($r - $l > 1) { my $n = int(($r - $l) / 2) + $l; my $ramanujan = int(($n * log($n) - $n + log($n * (1 + 4$n(1+2$n))) / 6 + $lg2) / $lg10); $ramanujan > $d ? $r = $n : $l = $n; } $l; } sub _log { # internal log function to calculate ln() based on Taylor series. # Modifies $x in place. my ($x, $scale) = @_; my $class = ref $x; # in case of $x == 1, result is 0 return $x->bzero() if $x->is_one(); # XXX TODO: rewrite this in a similar manner to bexp() # http://www.efunda.com/math/taylor_series/logarithmic.cfm?search_string=log # u = x-1, v = x+1 # _ _ # Taylor: \| u 1 u^3 1 u^5 \| # ln (x) = 2 \| --- + - --- + - * --- + ... \| x > 0 # \|_ v 3 v^3 5 v^5 _\| # This takes much more steps to calculate the result and is thus not used # u = x-1 # _ _ # Taylor: \| u 1 u^2 1 u^3 \| # ln (x) = 2 \| --- + - * --- + - * --- + ... \| x > 1/2 # \|_ x 2 x^2 3 x^3 _\| my ($limit, $v, $u, $below, $factor, $next, $over, $f); $v = $x->copy(); $v->binc(); # v = x+1 $x->bdec(); $u = $x->copy(); # u = x-1; x = x-1 $x->bdiv($v, $scale); # first term: u/v $below = $v->copy(); $over = $u->copy(); $u = $u; $v = $v; # u^2, v^2 $below->bmul($v); # u^3, v^3 $over->bmul($u); $factor = $class->new(3); $f = $class->new(2); $limit = $class->new("1E-". ($scale-1)); while (3 < 5) { # we calculate the next term, and add it to the last # when the next term is below our limit, it won't affect the outcome # anymore, so we stop # calculating the next term simple from over/below will result in quite # a time hog if the input has many digits, since over and below will # accumulate more and more digits, and the result will also have many # digits, but in the end it is rounded to $scale digits anyway. So if we # round $over and $below first, we save a lot of time for the division # (not with log(1.2345), but try log (123*123) to see what I mean. This # can introduce a rounding error if the division result would be f.i. # 0.1234500000001 and we round it to 5 digits it would become 0.12346, but # if we truncated $over and $below we might get 0.12345. Does this matter # for the end result? So we give $over and $below 4 more digits to be # on the safe side (unscientific error handling as usual... :+D $next = $over->copy()->bround($scale+4) ->bdiv($below->copy()->bmul($factor)->bround($scale+4), $scale); ## old version: ## $next = $over->copy()->bdiv($below->copy()->bmul($factor), $scale); last if $next->bacmp($limit) <= 0; delete $next->{_a}; delete $next->{_p}; $x->badd($next); # calculate things for the next term $over = $u; $below = $v; $factor->badd($f); } $x->bmul($f); # $x = 2 } sub _log_10 { # Internal log function based on reducing input to the range of 0.1 .. 9.99 # and then "correcting" the result to the proper one. Modifies $x in place. my ($x, $scale) = @_; my $class = ref $x; # Taking blog() from numbers greater than 10 takes a very long time, so we # break the computation down into parts based on the observation that: # blog(XY) = blog(X) + blog(Y) # We set Y here to multiples of 10 so that $x becomes below 1 - the smaller # $x is the faster it gets. Since 2$x takes about 10 times as # long, we make it faster by about a factor of 100 by dividing $x by 10. # The same observation is valid for numbers smaller than 0.1, e.g. computing # log(1) is fastest, and the further away we get from 1, the longer it takes. # So we also 'break' this down by multiplying $x with 10 and subtract the # log(10) afterwards to get the correct result. # To get $x even closer to 1, we also divide by 2 and then use log(2) to # correct for this. For instance if $x is 2.4, we use the formula: # blog(2.4 * 2) == blog(1.2) + blog(2) # and thus calculate only blog(1.2) and blog(2), which is faster in total # than calculating blog(2.4). # In addition, the values for blog(2) and blog(10) are cached. # Calculate nr of digits before dot. x = 123, dbd = 3; x = 1.23, dbd = 1; # x = 0.0123, dbd = -1; x = 0.000123, dbd = -3, etc. my $dbd = $LIB->_num($x->{_e}); $dbd = -$dbd if $x->{_es} eq '-'; $dbd += $LIB->_len($x->{_m}); # more than one digit (e.g. at least 10), but not exactly 10 to avoid # infinite recursion my $calc = 1; # do some calculation? # disable the shortcut for 10, since we need log(10) and this would recurse # infinitely deep if ($x->{_es} eq '+' && # $x == 10 ($LIB->_is_one($x->{_e}) && $LIB->_is_one($x->{_m}))) { $dbd = 0; # disable shortcut # we can use the cached value in these cases if ($scale <= $LOG_10_A) { $x->bzero(); $x->badd($LOG_10); # modify $x in place $calc = 0; # no need to calc, but round } # if we can't use the shortcut, we continue normally } else { # disable the shortcut for 2, since we maybe have it cached if (($LIB->_is_zero($x->{_e}) && # $x == 2 $LIB->_is_two($x->{_m}))) { $dbd = 0; # disable shortcut # we can use the cached value in these cases if ($scale <= $LOG_2_A) { $x->bzero(); $x->badd($LOG_2); # modify $x in place $calc = 0; # no need to calc, but round } # if we can't use the shortcut, we continue normally } } # if $x = 0.1, we know the result must be 0-log(10) if ($calc != 0 && ($x->{_es} eq '-' && # $x == 0.1 ($LIB->_is_one($x->{_e}) && $LIB->_is_one($x->{_m})))) { $dbd = 0; # disable shortcut # we can use the cached value in these cases if ($scale <= $LOG_10_A) { $x->bzero(); $x->bsub($LOG_10); $calc = 0; # no need to calc, but round } } return $x if $calc == 0; # already have the result # default: these correction factors are undef and thus not used my $l_10; # value of ln(10) to A of $scale my $l_2; # value of ln(2) to A of $scale my $two = $class->new(2); # $x == 2 => 1, $x == 13 => 2, $x == 0.1 => 0, $x == 0.01 => -1 # so don't do this shortcut for 1 or 0 if (($dbd > 1) \|\| ($dbd < 0)) { # convert our cached value to an object if not already (avoid doing this # at import() time, since not everybody needs this) $LOG_10 = $class->new($LOG_10, undef, undef) unless ref $LOG_10; #print "x = $x, dbd = $dbd, calc = $calc\n"; # got more than one digit before the dot, or more than one zero after the # dot, so do: # log(123) == log(1.23) + log(10) * 2 # log(0.0123) == log(1.23) - log(10) * 2 if ($scale <= $LOG_10_A) { # use cached value $l_10 = $LOG_10->copy(); # copy for mul } else { # else: slower, compute and cache result # also disable downgrade for this code path local $Math::BigFloat::downgrade = undef; # shorten the time to calculate log(10) based on the following: # log(1.25 * 8) = log(1.25) + log(8) # = log(1.25) + log(2) + log(2) + log(2) # first get $l_2 (and possible compute and cache log(2)) $LOG_2 = $class->new($LOG_2, undef, undef) unless ref $LOG_2; if ($scale <= $LOG_2_A) { # use cached value $l_2 = $LOG_2->copy(); # copy() for the mul below } else { # else: slower, compute and cache result $l_2 = $two->copy(); $l_2->_log($scale); # scale+4, actually $LOG_2 = $l_2->copy(); # cache the result for later # the copy() is for mul below $LOG_2_A = $scale; } # now calculate log(1.25): $l_10 = $class->new('1.25'); $l_10->_log($scale); # scale+4, actually # log(1.25) + log(2) + log(2) + log(2): $l_10->badd($l_2); $l_10->badd($l_2); $l_10->badd($l_2); $LOG_10 = $l_10->copy(); # cache the result for later # the copy() is for mul below $LOG_10_A = $scale; } $dbd-- if ($dbd > 1); # 20 => dbd=2, so make it dbd=1 $l_10->bmul($class->new($dbd)); # log(10) * (digits_before_dot-1) my $dbd_sign = '+'; if ($dbd < 0) { $dbd = -$dbd; $dbd_sign = '-'; } ($x->{_e}, $x->{_es}) = _e_sub($x->{_e}, $LIB->_new($dbd), $x->{_es}, $dbd_sign); # 123 => 1.23 } # Now: 0.1 <= $x < 10 (and possible correction in l_10) ### Since $x in the range 0.5 .. 1.5 is MUCH faster, we do a repeated div ### or mul by 2 (maximum times 3, since x < 10 and x > 0.1) $HALF = $class->new($HALF) unless ref($HALF); my $twos = 0; # default: none (0 times) while ($x->bacmp($HALF) <= 0) { # X <= 0.5 $twos--; $x->bmul($two); } while ($x->bacmp($two) >= 0) { # X >= 2 $twos++; $x->bdiv($two, $scale+4); # keep all digits } $x->bround($scale+4); # $twos > 0 => did mul 2, < 0 => did div 2 (but we never did both) # So calculate correction factor based on ln(2): if ($twos != 0) { $LOG_2 = $class->new($LOG_2, undef, undef) unless ref $LOG_2; if ($scale <= $LOG_2_A) { # use cached value $l_2 = $LOG_2->copy(); # copy() for the mul below } else { # else: slower, compute and cache result # also disable downgrade for this code path local $Math::BigFloat::downgrade = undef; $l_2 = $two->copy(); $l_2->_log($scale); # scale+4, actually $LOG_2 = $l_2->copy(); # cache the result for later # the copy() is for mul below $LOG_2_A = $scale; } $l_2->bmul($twos); # * -2 => subtract, * 2 => add } else { undef $l_2; } $x->_log($scale); # need to do the "normal" way $x->badd($l_10) if defined $l_10; # correct it by ln(10) $x->badd($l_2) if defined $l_2; # and maybe by ln(2) # all done, $x contains now the result $x; } sub _e_add { # Internal helper sub to take two positive integers and their signs and # then add them. Input ($LIB, $LIB, ('+'\|'-'), ('+'\|'-')), output # ($LIB, ('+'\|'-')). my ($x, $y, $xs, $ys) = @_; # if the signs are equal we can add them (-5 + -3 => -(5 + 3) => -8) if ($xs eq $ys) { $x = $LIB->_add($x, $y); # +a + +b or -a + -b } else { my $a = $LIB->_acmp($x, $y); if ($a == 0) { # This does NOT modify $x in-place. TODO: Fix this? $x = $LIB->_zero(); # result is 0 $xs = '+'; return ($x, $xs); } if ($a > 0) { $x = $LIB->_sub($x, $y); # abs sub } else { # a < 0 $x = $LIB->_sub ($y, $x, 1); # abs sub $xs = $ys; } } $xs = '+' if $xs eq '-' && $LIB->_is_zero($x); # no "-0" return ($x, $xs); } sub _e_sub { # Internal helper sub to take two positive integers and their signs and # then subtract them. Input ($LIB, $LIB, ('+'\|'-'), ('+'\|'-')), # output ($LIB, ('+'\|'-')) my ($x, $y, $xs, $ys) = @_; # flip sign $ys = $ys eq '+' ? '-' : '+'; # swap sign of second operand ... _e_add($x, $y, $xs, $ys); # ... and let _e_add() do the job } sub _pow { # Calculate a power where $y is a non-integer, like 2 0.3 my ($x, $y, @r) = @_; my $class = ref($x); # if $y == 0.5, it is sqrt($x) $HALF = $class->new($HALF) unless ref($HALF); return $x->bsqrt(@r, $y) if $y->bcmp($HALF) == 0; # Using: # a x == e ** (x * ln a) # u = y * ln x # _ _ # Taylor: \| u u^2 u^3 \| # x ** y = 1 + \| --- + --- + ----- + ... \| # \|_ 1 12 123 _\| # we need to limit the accuracy to protect against overflow my $fallback = 0; my ($scale, @params); ($x, @params) = $x->_find_round_parameters(@r); return $x if $x->is_nan(); # error in _find_round_parameters? # no rounding at all, so must use fallback if (scalar @params == 0) { # simulate old behaviour $params[0] = $class->div_scale(); # and round to it as accuracy $params[1] = undef; # disable P $scale = $params[0]+4; # at least four more for proper round $params[2] = $r[2]; # round mode by caller or undef $fallback = 1; # to clear a/p afterwards } else { # the 4 below is empirical, and there might be cases where it is not # enough... $scale = abs($params[0] \|\| $params[1]) + 4; # take whatever is defined } # when user set globals, they would interfere with our calculation, so # disable them and later re-enable them no strict 'refs'; my $abr = "$class\::accuracy"; my $ab = $$abr; $$abr = undef; my $pbr = "$class\::precision"; my $pb = $$pbr; $$pbr = undef; # we also need to disable any set A or P on $x (_find_round_parameters took # them already into account), since these would interfere, too delete $x->{_a}; delete $x->{_p}; # need to disable $upgrade in BigInt, to avoid deep recursion local $Math::BigInt::upgrade = undef; my ($limit, $v, $u, $below, $factor, $next, $over); $u = $x->copy()->blog(undef, $scale)->bmul($y); my $do_invert = ($u->{sign} eq '-'); $u->bneg() if $do_invert; $v = $class->bone(); # 1 $factor = $class->new(2); # 2 $x->bone(); # first term: 1 $below = $v->copy(); $over = $u->copy(); $limit = $class->new("1E-". ($scale-1)); while (3 < 5) { # we calculate the next term, and add it to the last # when the next term is below our limit, it won't affect the outcome # anymore, so we stop: $next = $over->copy()->bdiv($below, $scale); last if $next->bacmp($limit) <= 0; $x->badd($next); # calculate things for the next term $over = $u; $below = $factor; $factor->binc(); last if $x->{sign} !~ /^[-+]$/; } if ($do_invert) { my $x_copy = $x->copy(); $x->bone->bdiv($x_copy, $scale); } # shortcut to not run through _find_round_parameters again if (defined $params[0]) { $x->bround($params[0], $params[2]); # then round accordingly } else { $x->bfround($params[1], $params[2]); # then round accordingly } if ($fallback) { # clear a/p after round, since user did not request it delete $x->{_a}; delete $x->{_p}; } # restore globals $$abr = $ab; $$pbr = $pb; $x; } 1; __END__ =pod =head1 NAME Math::BigFloat - Arbitrary size floating point math package =head1 SYNOPSIS use Math::BigFloat; # Configuration methods (may be used as class methods and instance methods) Math::BigFloat->accuracy(); # get class accuracy Math::BigFloat->accuracy($n); # set class accuracy Math::BigFloat->precision(); # get class precision Math::BigFloat->precision($n); # set class precision Math::BigFloat->round_mode(); # get class rounding mode Math::BigFloat->round_mode($m); # set global round mode, must be one of # 'even', 'odd', '+inf', '-inf', 'zero', # 'trunc', or 'common' Math::BigFloat->config("lib"); # name of backend math library # Constructor methods (when the class methods below are used as instance # methods, the value is assigned the invocand) $x = Math::BigFloat->new($str); # defaults to 0 $x = Math::BigFloat->new('0x123'); # from hexadecimal $x = Math::BigFloat->new('0b101'); # from binary $x = Math::BigFloat->from_hex('0xc.afep+3'); # from hex $x = Math::BigFloat->from_hex('cafe'); # ditto $x = Math::BigFloat->from_oct('1.3267p-4'); # from octal $x = Math::BigFloat->from_oct('0377'); # ditto $x = Math::BigFloat->from_bin('0b1.1001p-4'); # from binary $x = Math::BigFloat->from_bin('0101'); # ditto $x = Math::BigFloat->from_ieee754($b, "binary64"); # from IEEE-754 bytes $x = Math::BigFloat->bzero(); # create a +0 $x = Math::BigFloat->bone(); # create a +1 $x = Math::BigFloat->bone('-'); # create a -1 $x = Math::BigFloat->binf(); # create a +inf $x = Math::BigFloat->binf('-'); # create a -inf $x = Math::BigFloat->bnan(); # create a Not-A-Number $x = Math::BigFloat->bpi(); # returns pi $y = $x->copy(); # make a copy (unlike $y = $x) $y = $x->as_int(); # return as BigInt # Boolean methods (these don't modify the invocand) $x->is_zero(); # if $x is 0 $x->is_one(); # if $x is +1 $x->is_one("+"); # ditto $x->is_one("-"); # if $x is -1 $x->is_inf(); # if $x is +inf or -inf $x->is_inf("+"); # if $x is +inf $x->is_inf("-"); # if $x is -inf $x->is_nan(); # if $x is NaN $x->is_positive(); # if $x > 0 $x->is_pos(); # ditto $x->is_negative(); # if $x < 0 $x->is_neg(); # ditto $x->is_odd(); # if $x is odd $x->is_even(); # if $x is even $x->is_int(); # if $x is an integer # Comparison methods $x->bcmp($y); # compare numbers (undef, < 0, == 0, > 0) $x->bacmp($y); # compare absolutely (undef, < 0, == 0, > 0) $x->beq($y); # true if and only if $x == $y $x->bne($y); # true if and only if $x != $y $x->blt($y); # true if and only if $x < $y $x->ble($y); # true if and only if $x <= $y $x->bgt($y); # true if and only if $x > $y $x->bge($y); # true if and only if $x >= $y # Arithmetic methods $x->bneg(); # negation $x->babs(); # absolute value $x->bsgn(); # sign function (-1, 0, 1, or NaN) $x->bnorm(); # normalize (no-op) $x->binc(); # increment $x by 1 $x->bdec(); # decrement $x by 1 $x->badd($y); # addition (add $y to $x) $x->bsub($y); # subtraction (subtract $y from $x) $x->bmul($y); # multiplication (multiply $x by $y) $x->bmuladd($y,$z); # $x = $x $y + $z $x->bdiv($y); # division (floored), set $x to quotient # return (quo,rem) or quo if scalar $x->btdiv($y); # division (truncated), set $x to quotient # return (quo,rem) or quo if scalar $x->bmod($y); # modulus (x % y) $x->btmod($y); # modulus (truncated) $x->bmodinv($mod); # modular multiplicative inverse $x->bmodpow($y,$mod); # modular exponentiation (($x $y) % $mod) $x->bpow($y); # power of arguments (x y) $x->blog(); # logarithm of $x to base e (Euler's number) $x->blog($base); # logarithm of $x to base $base (e.g., base 2) $x->bexp(); # calculate e ** $x where e is Euler's number $x->bnok($y); # x over y (binomial coefficient n over k) $x->bsin(); # sine $x->bcos(); # cosine $x->batan(); # inverse tangent $x->batan2($y); # two-argument inverse tangent $x->bsqrt(); # calculate square root $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root) $x->bfac(); # factorial of $x (1234..$x) $x->blsft($n); # left shift $n places in base 2 $x->blsft($n,$b); # left shift $n places in base $b # returns (quo,rem) or quo (scalar context) $x->brsft($n); # right shift $n places in base 2 $x->brsft($n,$b); # right shift $n places in base $b # returns (quo,rem) or quo (scalar context) # Bitwise methods $x->band($y); # bitwise and $x->bior($y); # bitwise inclusive or $x->bxor($y); # bitwise exclusive or $x->bnot(); # bitwise not (two's complement) # Rounding methods $x->round($A,$P,$mode); # round to accuracy or precision using # rounding mode $mode $x->bround($n); # accuracy: preserve $n digits $x->bfround($n); # $n > 0: round to $nth digit left of dec. point # $n < 0: round to $nth digit right of dec. point $x->bfloor(); # round towards minus infinity $x->bceil(); # round towards plus infinity $x->bint(); # round towards zero # Other mathematical methods $x->bgcd($y); # greatest common divisor $x->blcm($y); # least common multiple # Object property methods (do not modify the invocand) $x->sign(); # the sign, either +, - or NaN $x->digit($n); # the nth digit, counting from the right $x->digit(-$n); # the nth digit, counting from the left $x->length(); # return number of digits in number ($xl,$f) = $x->length(); # length of number and length of fraction # part, latter is always 0 digits long # for Math::BigInt objects $x->mantissa(); # return (signed) mantissa as BigInt $x->exponent(); # return exponent as BigInt $x->parts(); # return (mantissa,exponent) as BigInt $x->sparts(); # mantissa and exponent (as integers) $x->nparts(); # mantissa and exponent (normalised) $x->eparts(); # mantissa and exponent (engineering notation) $x->dparts(); # integer and fraction part # Conversion methods (do not modify the invocand) $x->bstr(); # decimal notation, possibly zero padded $x->bsstr(); # string in scientific notation with integers $x->bnstr(); # string in normalized notation $x->bestr(); # string in engineering notation $x->bdstr(); # string in decimal notation $x->as_hex(); # as signed hexadecimal string with prefixed 0x $x->as_bin(); # as signed binary string with prefixed 0b $x->as_oct(); # as signed octal string with prefixed 0 $x->to_ieee754($format); # to bytes encoded according to IEEE 754-2008 # Other conversion methods $x->numify(); # return as scalar (might overflow or underflow) =head1 DESCRIPTION Math::BigFloat provides support for arbitrary precision floating point. Overloading is also provided for Perl operators. All operators (including basic math operations) are overloaded if you declare your big floating point numbers as $x = Math::BigFloat -> new('12_3.456_789_123_456_789E-2'); Operations with overloaded operators preserve the arguments, which is exactly what you expect. =head2 Input Input values to these routines may be any scalar number or string that looks like a number and represents a floating point number. =over =item * Leading and trailing whitespace is ignored. =item * Leading and trailing zeros are ignored. =item * If the string has a "0x" prefix, it is interpreted as a hexadecimal number. =item * If the string has a "0b" prefix, it is interpreted as a binary number. =item * For hexadecimal and binary numbers, the exponent must be separated from the significand (mantissa) by the letter "p" or "P", not "e" or "E" as with decimal numbers. =item * One underline is allowed between any two digits, including hexadecimal and binary digits. =item * If the string can not be interpreted, NaN is returned. =back Octal numbers are typically prefixed by "0", but since leading zeros are stripped, these methods can not automatically recognize octal numbers, so use the constructor from_oct() to interpret octal strings. Some examples of valid string input Input string Resulting value 123 123 1.23e2 123 12300e-2 123 0xcafe 51966 0b1101 13 67_538_754 67538754 -4_5_6.7_8_9e+0_1_0 -4567890000000 0x1.921fb5p+1 3.14159262180328369140625e+0 0b1.1001p-4 9.765625e-2 =head2 Output Output values are usually Math::BigFloat objects. Boolean operators C<is_zero()>, C<is_one()>, C<is_inf()>, etc. return true or false. Comparison operators C<bcmp()> and C<bacmp()>) return -1, 0, 1, or undef. =head1 METHODS Math::BigFloat supports all methods that Math::BigInt supports, except it calculates non-integer results when possible. Please see L<Math::BigInt> for a full description of each method. Below are just the most important differences: =head2 Configuration methods =over =item accuracy() $x->accuracy(5); # local for $x CLASS->accuracy(5); # global for all members of CLASS # Note: This also applies to new()! $A = $x->accuracy(); # read out accuracy that affects $x $A = CLASS->accuracy(); # read out global accuracy Set or get the global or local accuracy, aka how many significant digits the results have. If you set a global accuracy, then this also applies to new()! Warning! The accuracy I<sticks>, e.g. once you created a number under the influence of C<< CLASS->accuracy($A) >>, all results from math operations with that number will also be rounded. In most cases, you should probably round the results explicitly using one of L<Math::BigInt/round()>, L<Math::BigInt/bround()> or L<Math::BigInt/bfround()> or by passing the desired accuracy to the math operation as additional parameter: my $x = Math::BigInt->new(30000); my $y = Math::BigInt->new(7); print scalar $x->copy()->bdiv($y, 2); # print 4300 print scalar $x->copy()->bdiv($y)->bround(2); # print 4300 =item precision() $x->precision(-2); # local for $x, round at the second # digit right of the dot $x->precision(2); # ditto, round at the second digit # left of the dot CLASS->precision(5); # Global for all members of CLASS # This also applies to new()! CLASS->precision(-5); # ditto $P = CLASS->precision(); # read out global precision $P = $x->precision(); # read out precision that affects $x Note: You probably want to use L</accuracy()> instead. With L</accuracy()> you set the number of digits each result should have, with L</precision()> you set the place where to round! =back =head2 Constructor methods =over =item from_hex() $x -> from_hex("0x1.921fb54442d18p+1"); $x = Math::BigFloat -> from_hex("0x1.921fb54442d18p+1"); Interpret input as a hexadecimal string.A prefix ("0x", "x", ignoring case) is optional. A single underscore character ("_") may be placed between any two digits. If the input is invalid, a NaN is returned. The exponent is in base 2 using decimal digits. If called as an instance method, the value is assigned to the invocand. =item from_oct() $x -> from_oct("1.3267p-4"); $x = Math::BigFloat -> from_oct("1.3267p-4"); Interpret input as an octal string. A single underscore character ("_") may be placed between any two digits. If the input is invalid, a NaN is returned. The exponent is in base 2 using decimal digits. If called as an instance method, the value is assigned to the invocand. =item from_bin() $x -> from_bin("0b1.1001p-4"); $x = Math::BigFloat -> from_bin("0b1.1001p-4"); Interpret input as a hexadecimal string. A prefix ("0b" or "b", ignoring case) is optional. A single underscore character ("_") may be placed between any two digits. If the input is invalid, a NaN is returned. The exponent is in base 2 using decimal digits. If called as an instance method, the value is assigned to the invocand. =item from_ieee754() Interpret the input as a value encoded as described in IEEE754-2008. The input can be given as a byte string, hex string or binary string. The input is assumed to be in big-endian byte-order. # both $dbl and $mbf are 3.141592... $bytes = "\x40\x09\x21\xfb\x54\x44\x2d\x18"; $dbl = unpack "d>", $bytes; $mbf = Math::BigFloat -> from_ieee754($bytes, "binary64"); =item bpi() print Math::BigFloat->bpi(100), "\n"; Calculate PI to N digits (including the 3 before the dot). The result is rounded according to the current rounding mode, which defaults to "even". This method was added in v1.87 of Math::BigInt (June 2007). =back =head2 Arithmetic methods =over =item bmuladd() $x->bmuladd($y,$z); Multiply $x by $y, and then add $z to the result. This method was added in v1.87 of Math::BigInt (June 2007). =item bdiv() $q = $x->bdiv($y); ($q, $r) = $x->bdiv($y); In scalar context, divides $x by $y and returns the result to the given or default accuracy/precision. In list context, does floored division (F-division), returning an integer $q and a remainder $r so that $x = $q * $y + $r. The remainer (modulo) is equal to what is returned by C<< $x->bmod($y) >>. =item bmod() $x->bmod($y); Returns $x modulo $y. When $x is finite, and $y is finite and non-zero, the result is identical to the remainder after floored division (F-division). If, in addition, both $x and $y are integers, the result is identical to the result from Perl's % operator. =item bexp() $x->bexp($accuracy); # calculate e X Calculates the expression C<e $x> where C<e> is Euler's number. This method was added in v1.82 of Math::BigInt (April 2007). =item bnok() $x->bnok($y); # x over y (binomial coefficient n over k) Calculates the binomial coefficient n over k, also called the "choose" function. The result is equivalent to: ( n ) n! \| - \| = ------- ( k ) k!(n-k)! This method was added in v1.84 of Math::BigInt (April 2007). =item bsin() my $x = Math::BigFloat->new(1); print $x->bsin(100), "\n"; Calculate the sinus of $x, modifying $x in place. This method was added in v1.87 of Math::BigInt (June 2007). =item bcos() my $x = Math::BigFloat->new(1); print $x->bcos(100), "\n"; Calculate the cosinus of $x, modifying $x in place. This method was added in v1.87 of Math::BigInt (June 2007). =item batan() my $x = Math::BigFloat->new(1); print $x->batan(100), "\n"; Calculate the arcus tanges of $x, modifying $x in place. See also L</batan2()>. This method was added in v1.87 of Math::BigInt (June 2007). =item batan2() my $y = Math::BigFloat->new(2); my $x = Math::BigFloat->new(3); print $y->batan2($x), "\n"; Calculate the arcus tanges of C<$y> divided by C<$x>, modifying $y in place. See also L</batan()>. This method was added in v1.87 of Math::BigInt (June 2007). =item as_float() This method is called when Math::BigFloat encounters an object it doesn't know how to handle. For instance, assume $x is a Math::BigFloat, or subclass thereof, and $y is defined, but not a Math::BigFloat, or subclass thereof. If you do $x -> badd($y); $y needs to be converted into an object that $x can deal with. This is done by first checking if $y is something that $x might be upgraded to. If that is the case, no further attempts are made. The next is to see if $y supports the method C<as_float()>. The method C<as_float()> is expected to return either an object that has the same class as $x, a subclass thereof, or a string that C<ref($x)-E<gt>new()> can parse to create an object. In Math::BigFloat, C<as_float()> has the same effect as C<copy()>. =item to_ieee754() Encodes the invocand as a byte string in the given format as specified in IEEE 754-2008. Note that the encoded value is the nearest possible representation of the value. This value might not be exactly the same as the value in the invocand. # $x = 3.1415926535897932385 $x = Math::BigFloat -> bpi(30); $b = $x -> to_ieee754("binary64"); # encode as 8 bytes $h = unpack "H", $b; # "400921fb54442d18" # 3.141592653589793115997963... $y = Math::BigFloat -> from_ieee754($h, "binary64"); All binary formats in IEEE 754-2008 are accepted. For convenience, som aliases are recognized: "half" for "binary16", "single" for "binary32", "double" for "binary64", "quadruple" for "binary128", "octuple" for "binary256", and "sexdecuple" for "binary512". See also L<https://en.wikipedia.org/wiki/IEEE_754>. =back =head2 ACCURACY AND PRECISION See also: L<Rounding\|/Rounding>. Math::BigFloat supports both precision (rounding to a certain place before or after the dot) and accuracy (rounding to a certain number of digits). For a full documentation, examples and tips on these topics please see the large section about rounding in L<Math::BigInt>. Since things like C<sqrt(2)> or C<1 / 3> must presented with a limited accuracy lest a operation consumes all resources, each operation produces no more than the requested number of digits. If there is no global precision or accuracy set, B<and> the operation in question was not called with a requested precision or accuracy, B<and> the input $x has no accuracy or precision set, then a fallback parameter will be used. For historical reasons, it is called C<div_scale> and can be accessed via: $d = Math::BigFloat->div_scale(); # query Math::BigFloat->div_scale($n); # set to $n digits The default value for C<div_scale> is 40. In case the result of one operation has more digits than specified, it is rounded. The rounding mode taken is either the default mode, or the one supplied to the operation after the I<scale>: $x = Math::BigFloat->new(2); Math::BigFloat->accuracy(5); # 5 digits max $y = $x->copy()->bdiv(3); # gives 0.66667 $y = $x->copy()->bdiv(3,6); # gives 0.666667 $y = $x->copy()->bdiv(3,6,undef,'odd'); # gives 0.666667 Math::BigFloat->round_mode('zero'); $y = $x->copy()->bdiv(3,6); # will also give 0.666667 Note that C<< Math::BigFloat->accuracy() >> and C<< Math::BigFloat->precision() >> set the global variables, and thus B<any> newly created number will be subject to the global rounding B<immediately>. This means that in the examples above, the C<3> as argument to C<bdiv()> will also get an accuracy of B<5>. It is less confusing to either calculate the result fully, and afterwards round it explicitly, or use the additional parameters to the math functions like so: use Math::BigFloat; $x = Math::BigFloat->new(2); $y = $x->copy()->bdiv(3); print $y->bround(5),"\n"; # gives 0.66667 or use Math::BigFloat; $x = Math::BigFloat->new(2); $y = $x->copy()->bdiv(3,5); # gives 0.66667 print "$y\n"; =head2 Rounding =over =item bfround ( +$scale ) Rounds to the $scale'th place left from the '.', counting from the dot. The first digit is numbered 1. =item bfround ( -$scale ) Rounds to the $scale'th place right from the '.', counting from the dot. =item bfround ( 0 ) Rounds to an integer. =item bround ( +$scale ) Preserves accuracy to $scale digits from the left (aka significant digits) and pads the rest with zeros. If the number is between 1 and -1, the significant digits count from the first non-zero after the '.' =item bround ( -$scale ) and bround ( 0 ) These are effectively no-ops. =back All rounding functions take as a second parameter a rounding mode from one of the following: 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'. The default rounding mode is 'even'. By using C<< Math::BigFloat->round_mode($round_mode); >> you can get and set the default mode for subsequent rounding. The usage of C<$Math::BigFloat::$round_mode> is no longer supported. The second parameter to the round functions then overrides the default temporarily. The C<as_number()> function returns a BigInt from a Math::BigFloat. It uses 'trunc' as rounding mode to make it equivalent to: $x = 2.5; $y = int($x) + 2; You can override this by passing the desired rounding mode as parameter to C<as_number()>: $x = Math::BigFloat->new(2.5); $y = $x->as_number('odd'); # $y = 3 =head1 Autocreating constants After C<use Math::BigFloat ':constant'> all the floating point constants in the given scope are converted to C<Math::BigFloat>. This conversion happens at compile time. In particular perl -MMath::BigFloat=:constant -e 'print 2E-100,"\n"' prints the value of C<2E-100>. Note that without conversion of constants the expression 2E-100 will be calculated as normal floating point number. Please note that ':constant' does not affect integer constants, nor binary nor hexadecimal constants. Use L<bignum> or L<Math::BigInt> to get this to work. =head2 Math library Math with the numbers is done (by default) by a module called Math::BigInt::Calc. This is equivalent to saying: use Math::BigFloat lib => 'Calc'; You can change this by using: use Math::BigFloat lib => 'GMP'; B<Note>: General purpose packages should not be explicit about the library to use; let the script author decide which is best. Note: The keyword 'lib' will warn when the requested library could not be loaded. To suppress the warning use 'try' instead: use Math::BigFloat try => 'GMP'; If your script works with huge numbers and Calc is too slow for them, you can also for the loading of one of these libraries and if none of them can be used, the code will die: use Math::BigFloat only => 'GMP,Pari'; The following would first try to find Math::BigInt::Foo, then Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc: use Math::BigFloat lib => 'Foo,Math::BigInt::Bar'; See the respective low-level library documentation for further details. Please note that Math::BigFloat does B<not> use the denoted library itself, but it merely passes the lib argument to Math::BigInt. So, instead of the need to do: use Math::BigInt lib => 'GMP'; use Math::BigFloat; you can roll it all into one line: use Math::BigFloat lib => 'GMP'; It is also possible to just require Math::BigFloat: require Math::BigFloat; This will load the necessary things (like BigInt) when they are needed, and automatically. See L<Math::BigInt> for more details than you ever wanted to know about using a different low-level library. =head2 Using Math::BigInt::Lite For backwards compatibility reasons it is still possible to request a different storage class for use with Math::BigFloat: use Math::BigFloat with => 'Math::BigInt::Lite'; However, this request is ignored, as the current code now uses the low-level math library for directly storing the number parts. =head1 EXPORTS C<Math::BigFloat> exports nothing by default, but can export the C<bpi()> method: use Math::BigFloat qw/bpi/; print bpi(10), "\n"; =head1 CAVEATS Do not try to be clever to insert some operations in between switching libraries: require Math::BigFloat; my $matter = Math::BigFloat->bone() + 4; # load BigInt and Calc Math::BigFloat->import( lib => 'Pari' ); # load Pari, too my $anti_matter = Math::BigFloat->bone()+4; # now use Pari This will create objects with numbers stored in two different backend libraries, and B<VERY BAD THINGS> will happen when you use these together: my $flash_and_bang = $matter + $anti_matter; # Don't do this! =over =item stringify, bstr() Both stringify and bstr() now drop the leading '+'. The old code would return '+1.23', the new returns '1.23'. See the documentation in L<Math::BigInt> for reasoning and details. =item brsft() The following will probably not print what you expect: my $c = Math::BigFloat->new('3.14159'); print $c->brsft(3,10),"\n"; # prints 0.00314153.1415 It prints both quotient and remainder, since print calls C<brsft()> in list context. Also, C<< $c->brsft() >> will modify $c, so be careful. You probably want to use print scalar $c->copy()->brsft(3,10),"\n"; # or if you really want to modify $c print scalar $c->brsft(3,10),"\n"; instead. =item Modifying and = Beware of: $x = Math::BigFloat->new(5); $y = $x; It will not do what you think, e.g. making a copy of $x. Instead it just makes a second reference to the B<same> object and stores it in $y. Thus anything that modifies $x will modify $y (except overloaded math operators), and vice versa. See L<Math::BigInt> for details and how to avoid that. =item precision() vs. accuracy() A common pitfall is to use L</precision()> when you want to round a result to a certain number of digits: use Math::BigFloat; Math::BigFloat->precision(4); # does not do what you # think it does my $x = Math::BigFloat->new(12345); # rounds $x to "12000"! print "$x\n"; # print "12000" my $y = Math::BigFloat->new(3); # rounds $y to "0"! print "$y\n"; # print "0" $z = $x / $y; # 12000 / 0 => NaN! print "$z\n"; print $z->precision(),"\n"; # 4 Replacing L</precision()> with L</accuracy()> is probably not what you want, either: use Math::BigFloat; Math::BigFloat->accuracy(4); # enables global rounding: my $x = Math::BigFloat->new(123456); # rounded immediately # to "12350" print "$x\n"; # print "123500" my $y = Math::BigFloat->new(3); # rounded to "3 print "$y\n"; # print "3" print $z = $x->copy()->bdiv($y),"\n"; # 41170 print $z->accuracy(),"\n"; # 4 What you want to use instead is: use Math::BigFloat; my $x = Math::BigFloat->new(123456); # no rounding print "$x\n"; # print "123456" my $y = Math::BigFloat->new(3); # no rounding print "$y\n"; # print "3" print $z = $x->copy()->bdiv($y,4),"\n"; # 41150 print $z->accuracy(),"\n"; # undef In addition to computing what you expected, the last example also does B<not> "taint" the result with an accuracy or precision setting, which would influence any further operation. =back =head1 BUGS Please report any bugs or feature requests to C<bug-math-bigint at rt.cpan.org>, or through the web interface at L<https://rt.cpan.org/Ticket/Create.html?Queue=Math-BigInt> (requires login). We will be notified, and then you'll automatically be notified of progress on your bug as I make changes. =head1 SUPPORT You can find documentation for this module with the perldoc command. perldoc Math::BigFloat You can also look for information at: =over 4 =item RT: CPAN's request tracker L<https://rt.cpan.org/Public/Dist/Display.html?Name=Math-BigInt> =item * AnnoCPAN: Annotated CPAN documentation L<http://annocpan.org/dist/Math-BigInt> =item * CPAN Ratings L<https://cpanratings.perl.org/dist/Math-BigInt> =item * MetaCPAN L<https://metacpan.org/release/Math-BigInt> =item * CPAN Testers Matrix L<http://matrix.cpantesters.org/?dist=Math-BigInt> =item * The Bignum mailing list =over 4 =item * Post to mailing list C<bignum at lists.scsys.co.uk> =item * View mailing list L<http://lists.scsys.co.uk/pipermail/bignum/> =item * Subscribe/Unsubscribe L<http://lists.scsys.co.uk/cgi-bin/mailman/listinfo/bignum> =back =back =head1 LICENSE This program is free software; you may redistribute it and/or modify it under the same terms as Perl itself. =head1 SEE ALSO L<Math::BigFloat> and L<Math::BigInt> as well as the backends L<Math::BigInt::FastCalc>, L<Math::BigInt::GMP>, and L<Math::BigInt::Pari>. The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest because they solve the autoupgrading/downgrading issue, at least partly. =head1 AUTHORS =over 4 =item * Mark Biggar, overloaded interface by Ilya Zakharevich, 1996-2001. =item * Completely rewritten by Tels L<http://bloodgate.com> in 2001-2008. =item * Florian Ragwitz E<lt>flora@cpan.orgE<gt>, 2010. =item * Peter John Acklam E<lt>pjacklam@online.noE<gt>, 2011-. =back =cut	194.67 KB	2019-10-20 12:42
📁 BigInt	-	2025-05-25 12:29
📄 BigInt.pm # -- coding: utf-8-unix -- package Math::BigInt; # # "Mike had an infinite amount to do and a negative amount of time in which # to do it." - Before and After # # The following hash values are used: # value: unsigned int with actual value (as a Math::BigInt::Calc or similar) # sign : +, -, NaN, +inf, -inf # _a : accuracy # _p : precision # Remember not to take shortcuts ala $xs = $x->{value}; $LIB->foo($xs); since # underlying lib might change the reference! use 5.006001; use strict; use warnings; use Carp qw< carp croak >; our $VERSION = '1.999818'; require Exporter; our @ISA = qw(Exporter); our @EXPORT_OK = qw(objectify bgcd blcm); # Inside overload, the first arg is always an object. If the original code had # it reversed (like $x = 2 * $y), then the third parameter is true. # In some cases (like add, $x = $x + 2 is the same as $x = 2 + $x) this makes # no difference, but in some cases it does. # For overloaded ops with only one argument we simple use $_[0]->copy() to # preserve the argument. # Thus inheritance of overload operators becomes possible and transparent for # our subclasses without the need to repeat the entire overload section there. use overload # overload key: with_assign '+' => sub { $_[0] -> copy() -> badd($_[1]); }, '-' => sub { my $c = $_[0] -> copy; $_[2] ? $c -> bneg() -> badd($_[1]) : $c -> bsub($_[1]); }, '' => sub { $_[0] -> copy() -> bmul($_[1]); }, '/' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bdiv($_[0]) : $_[0] -> copy -> bdiv($_[1]); }, '%' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bmod($_[0]) : $_[0] -> copy -> bmod($_[1]); }, '' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bpow($_[0]) : $_[0] -> copy -> bpow($_[1]); }, '<<' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> blsft($_[0]) : $_[0] -> copy -> blsft($_[1]); }, '>>' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> brsft($_[0]) : $_[0] -> copy -> brsft($_[1]); }, # overload key: assign '+=' => sub { $_[0]->badd($_[1]); }, '-=' => sub { $_[0]->bsub($_[1]); }, '=' => sub { $_[0]->bmul($_[1]); }, '/=' => sub { scalar $_[0]->bdiv($_[1]); }, '%=' => sub { $_[0]->bmod($_[1]); }, '*=' => sub { $_[0]->bpow($_[1]); }, '<<=' => sub { $_[0]->blsft($_[1]); }, '>>=' => sub { $_[0]->brsft($_[1]); }, # 'x=' => sub { }, # '.=' => sub { }, # overload key: num_comparison '<' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> blt($_[0]) : $_[0] -> blt($_[1]); }, '<=' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> ble($_[0]) : $_[0] -> ble($_[1]); }, '>' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bgt($_[0]) : $_[0] -> bgt($_[1]); }, '>=' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bge($_[0]) : $_[0] -> bge($_[1]); }, '==' => sub { $_[0] -> beq($_[1]); }, '!=' => sub { $_[0] -> bne($_[1]); }, # overload key: 3way_comparison '<=>' => sub { my $cmp = $_[0] -> bcmp($_[1]); defined($cmp) && $_[2] ? -$cmp : $cmp; }, 'cmp' => sub { $_[2] ? "$_[1]" cmp $_[0] -> bstr() : $_[0] -> bstr() cmp "$_[1]"; }, # overload key: str_comparison # 'lt' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bstrlt($_[0]) # : $_[0] -> bstrlt($_[1]); }, # # 'le' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bstrle($_[0]) # : $_[0] -> bstrle($_[1]); }, # # 'gt' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bstrgt($_[0]) # : $_[0] -> bstrgt($_[1]); }, # # 'ge' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bstrge($_[0]) # : $_[0] -> bstrge($_[1]); }, # # 'eq' => sub { $_[0] -> bstreq($_[1]); }, # # 'ne' => sub { $_[0] -> bstrne($_[1]); }, # overload key: binary '&' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> band($_[0]) : $_[0] -> copy -> band($_[1]); }, '&=' => sub { $_[0] -> band($_[1]); }, '\|' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bior($_[0]) : $_[0] -> copy -> bior($_[1]); }, '\|=' => sub { $_[0] -> bior($_[1]); }, '^' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bxor($_[0]) : $_[0] -> copy -> bxor($_[1]); }, '^=' => sub { $_[0] -> bxor($_[1]); }, # '&.' => sub { }, # '&.=' => sub { }, # '\|.' => sub { }, # '\|.=' => sub { }, # '^.' => sub { }, # '^.=' => sub { }, # overload key: unary 'neg' => sub { $_[0] -> copy() -> bneg(); }, # '!' => sub { }, '~' => sub { $_[0] -> copy() -> bnot(); }, # '~.' => sub { }, # overload key: mutators '++' => sub { $_[0] -> binc() }, '--' => sub { $_[0] -> bdec() }, # overload key: func 'atan2' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> batan2($_[0]) : $_[0] -> copy() -> batan2($_[1]); }, 'cos' => sub { $_[0] -> copy -> bcos(); }, 'sin' => sub { $_[0] -> copy -> bsin(); }, 'exp' => sub { $_[0] -> copy() -> bexp($_[1]); }, 'abs' => sub { $_[0] -> copy() -> babs(); }, 'log' => sub { $_[0] -> copy() -> blog(); }, 'sqrt' => sub { $_[0] -> copy() -> bsqrt(); }, 'int' => sub { $_[0] -> copy() -> bint(); }, # overload key: conversion 'bool' => sub { $_[0] -> is_zero() ? '' : 1; }, '""' => sub { $_[0] -> bstr(); }, '0+' => sub { $_[0] -> numify(); }, '=' => sub { $_[0]->copy(); }, ; ############################################################################## # global constants, flags and accessory # These vars are public, but their direct usage is not recommended, use the # accessor methods instead our $round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common' our $accuracy = undef; our $precision = undef; our $div_scale = 40; our $upgrade = undef; # default is no upgrade our $downgrade = undef; # default is no downgrade # These are internally, and not to be used from the outside at all our $_trap_nan = 0; # are NaNs ok? set w/ config() our $_trap_inf = 0; # are infs ok? set w/ config() my $nan = 'NaN'; # constants for easier life my $LIB = 'Math::BigInt::Calc'; # module to do the low level math # default is Calc.pm my $IMPORT = 0; # was import() called yet? # used to make require work my %CALLBACKS; # callbacks to notify on lib loads ############################################################################## # the old code had $rnd_mode, so we need to support it, too our $rnd_mode = 'even'; sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; } sub FETCH { return $round_mode; } sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); } BEGIN { # tie to enable $rnd_mode to work transparently tie $rnd_mode, 'Math::BigInt'; # set up some handy alias names as_int = \&as_number; is_pos = \&is_positive; is_neg = \&is_negative; } ############################################################################### # Configuration methods ############################################################################### sub round_mode { no strict 'refs'; # make Class->round_mode() work my $self = shift; my $class = ref($self) \|\| $self \|\| __PACKAGE__; if (defined $_[0]) { my $m = shift; if ($m !~ /^(even\|odd\|\+inf\|\-inf\|zero\|trunc\|common)$/) { croak("Unknown round mode '$m'"); } return ${"${class}::round_mode"} = $m; } ${"${class}::round_mode"}; } sub upgrade { no strict 'refs'; # make Class->upgrade() work my $self = shift; my $class = ref($self) \|\| $self \|\| __PACKAGE__; # need to set new value? if (@_ > 0) { return ${"${class}::upgrade"} = $_[0]; } ${"${class}::upgrade"}; } sub downgrade { no strict 'refs'; # make Class->downgrade() work my $self = shift; my $class = ref($self) \|\| $self \|\| __PACKAGE__; # need to set new value? if (@_ > 0) { return ${"${class}::downgrade"} = $_[0]; } ${"${class}::downgrade"}; } sub div_scale { no strict 'refs'; # make Class->div_scale() work my $self = shift; my $class = ref($self) \|\| $self \|\| __PACKAGE__; if (defined $_[0]) { if ($_[0] < 0) { croak('div_scale must be greater than zero'); } ${"${class}::div_scale"} = $_[0]; } ${"${class}::div_scale"}; } sub accuracy { # $x->accuracy($a); ref($x) $a # $x->accuracy(); ref($x) # Class->accuracy(); class # Class->accuracy($a); class $a my $x = shift; my $class = ref($x) \|\| $x \|\| __PACKAGE__; no strict 'refs'; if (@_ > 0) { my $a = shift; if (defined $a) { $a = $a->numify() if ref($a) && $a->can('numify'); # also croak on non-numerical if (!$a \|\| $a <= 0) { croak('Argument to accuracy must be greater than zero'); } if (int($a) != $a) { croak('Argument to accuracy must be an integer'); } } if (ref($x)) { # Set instance variable. $x->bround($a) if $a; # not for undef, 0 $x->{_a} = $a; # set/overwrite, even if not rounded delete $x->{_p}; # clear P # Why return class variable here? Fixme! $a = ${"${class}::accuracy"} unless defined $a; # proper return value } else { # Set class variable. ${"${class}::accuracy"} = $a; # set global A ${"${class}::precision"} = undef; # clear global P } return $a; # shortcut } # Return instance variable. return $x->{_a} if ref($x) && (defined $x->{_a} \|\| defined $x->{_p}); # Return class variable. return ${"${class}::accuracy"}; } sub precision { # $x->precision($p); ref($x) $p # $x->precision(); ref($x) # Class->precision(); class # Class->precision($p); class $p my $x = shift; my $class = ref($x) \|\| $x \|\| __PACKAGE__; no strict 'refs'; if (@_ > 0) { my $p = shift; if (defined $p) { $p = $p->numify() if ref($p) && $p->can('numify'); if ($p != int $p) { croak('Argument to precision must be an integer'); } } if (ref($x)) { # Set instance variable. $x->bfround($p) if $p; # not for undef, 0 $x->{_p} = $p; # set/overwrite, even if not rounded delete $x->{_a}; # clear A # Why return class variable here? Fixme! $p = ${"${class}::precision"} unless defined $p; # proper return value } else { # Set class variable. ${"${class}::precision"} = $p; # set global P ${"${class}::accuracy"} = undef; # clear global A } return $p; # shortcut } # Return instance variable. return $x->{_p} if ref($x) && (defined $x->{_a} \|\| defined $x->{_p}); # Return class variable. return ${"${class}::precision"}; } sub config { # return (or set) configuration data. my $class = shift \|\| __PACKAGE__; no strict 'refs'; if (@_ > 1 \|\| (@_ == 1 && (ref($_[0]) eq 'HASH'))) { # try to set given options as arguments from hash my $args = $_[0]; if (ref($args) ne 'HASH') { $args = { @_ }; } # these values can be "set" my $set_args = {}; foreach my $key (qw/ accuracy precision round_mode div_scale upgrade downgrade trap_inf trap_nan /) { $set_args->{$key} = $args->{$key} if exists $args->{$key}; delete $args->{$key}; } if (keys %$args > 0) { croak("Illegal key(s) '", join("', '", keys %$args), "' passed to $class\->config()"); } foreach my $key (keys %$set_args) { if ($key =~ /^trap_(inf\|nan)\z/) { ${"${class}::_trap_$1"} = ($set_args->{"trap_$1"} ? 1 : 0); next; } # use a call instead of just setting the $variable to check argument $class->$key($set_args->{$key}); } } # now return actual configuration my $cfg = { lib => $LIB, lib_version => ${"${LIB}::VERSION"}, class => $class, trap_nan => ${"${class}::_trap_nan"}, trap_inf => ${"${class}::_trap_inf"}, version => ${"${class}::VERSION"}, }; foreach my $key (qw/ accuracy precision round_mode div_scale upgrade downgrade /) { $cfg->{$key} = ${"${class}::$key"}; } if (@_ == 1 && (ref($_[0]) ne 'HASH')) { # calls of the style config('lib') return just this value return $cfg->{$_[0]}; } $cfg; } sub _scale_a { # select accuracy parameter based on precedence, # used by bround() and bfround(), may return undef for scale (means no op) my ($x, $scale, $mode) = @_; $scale = $x->{_a} unless defined $scale; no strict 'refs'; my $class = ref($x); $scale = ${ $class . '::accuracy' } unless defined $scale; $mode = ${ $class . '::round_mode' } unless defined $mode; if (defined $scale) { $scale = $scale->can('numify') ? $scale->numify() : "$scale" if ref($scale); $scale = int($scale); } ($scale, $mode); } sub _scale_p { # select precision parameter based on precedence, # used by bround() and bfround(), may return undef for scale (means no op) my ($x, $scale, $mode) = @_; $scale = $x->{_p} unless defined $scale; no strict 'refs'; my $class = ref($x); $scale = ${ $class . '::precision' } unless defined $scale; $mode = ${ $class . '::round_mode' } unless defined $mode; if (defined $scale) { $scale = $scale->can('numify') ? $scale->numify() : "$scale" if ref($scale); $scale = int($scale); } ($scale, $mode); } ############################################################################### # Constructor methods ############################################################################### sub new { # Create a new Math::BigInt object from a string or another Math::BigInt # object. See hash keys documented at top. # The argument could be an object, so avoid \|\|, && etc. on it. This would # cause costly overloaded code to be called. The only allowed ops are ref() # and defined. my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; # The POD says: # # "Currently, Math::BigInt->new() defaults to 0, while Math::BigInt->new('') # results in 'NaN'. This might change in the future, so use always the # following explicit forms to get a zero or NaN: # $zero = Math::BigInt->bzero(); # $nan = Math::BigInt->bnan(); # # But although this use has been discouraged for more than 10 years, people # apparently still use it, so we still support it. return $self->bzero() unless @_; my ($wanted, $a, $p, $r) = @_; # Always return a new object, so if called as an instance method, copy the # invocand, and if called as a class method, initialize a new object. $self = $selfref ? $self -> copy() : bless {}, $class; unless (defined $wanted) { #carp("Use of uninitialized value in new()"); return $self->bzero($a, $p, $r); } if (ref($wanted) && $wanted->isa($class)) { # MBI or subclass # Using "$copy = $wanted -> copy()" here fails some tests. Fixme! my $copy = $class -> copy($wanted); if ($selfref) { %$self = %$copy; } else { $self = $copy; } return $self; } $class->import() if $IMPORT == 0; # make require work # Shortcut for non-zero scalar integers with no non-zero exponent. if (!ref($wanted) && $wanted =~ / ^ ([+-]?) # optional sign ([1-9][0-9]) # non-zero significand (\.0)? # ... with optional zero fraction ([Ee][+-]?0+)? # optional zero exponent \z /x) { my $sgn = $1; my $abs = $2; $self->{sign} = $sgn \|\| '+'; $self->{value} = $LIB->_new($abs); no strict 'refs'; if (defined($a) \|\| defined($p) \|\| defined(${"${class}::precision"}) \|\| defined(${"${class}::accuracy"})) { $self->round($a, $p, $r) unless @_ >= 3 && !defined $a && !defined $p; } return $self; } # Handle Infs. if ($wanted =~ /^\s([+-]?)inf(inity)?\s\z/i) { my $sgn = $1 \|\| '+'; $self->{sign} = $sgn . 'inf'; # set a default sign for bstr() return $class->binf($sgn); } # Handle explicit NaNs (not the ones returned due to invalid input). if ($wanted =~ /^\s([+-]?)nan\s\z/i) { $self = $class -> bnan(); $self->round($a, $p, $r) unless @_ >= 3 && !defined $a && !defined $p; return $self; } # Handle hexadecimal numbers. if ($wanted =~ /^\s[+-]?0[Xx]/) { $self = $class -> from_hex($wanted); $self->round($a, $p, $r) unless @_ >= 3 && !defined $a && !defined $p; return $self; } # Handle binary numbers. if ($wanted =~ /^\s[+-]?0[Bb]/) { $self = $class -> from_bin($wanted); $self->round($a, $p, $r) unless @_ >= 3 && !defined $a && !defined $p; return $self; } # Split string into mantissa, exponent, integer, fraction, value, and sign. my ($mis, $miv, $mfv, $es, $ev) = _split($wanted); if (!ref $mis) { if ($_trap_nan) { croak("$wanted is not a number in $class"); } $self->{value} = $LIB->_zero(); $self->{sign} = $nan; return $self; } if (!ref $miv) { # _from_hex or _from_bin $self->{value} = $mis->{value}; $self->{sign} = $mis->{sign}; return $self; # throw away $mis } # Make integer from mantissa by adjusting exponent, then convert to a # Math::BigInt. $self->{sign} = $$mis; # store sign $self->{value} = $LIB->_zero(); # for all the NaN cases my $e = int("$$es$$ev"); # exponent (avoid recursion) if ($e > 0) { my $diff = $e - CORE::length($$mfv); if ($diff < 0) { # Not integer if ($_trap_nan) { croak("$wanted not an integer in $class"); } #print "NOI 1\n"; return $upgrade->new($wanted, $a, $p, $r) if defined $upgrade; $self->{sign} = $nan; } else { # diff >= 0 # adjust fraction and add it to value #print "diff > 0 $$miv\n"; $$miv = $$miv . ($$mfv . '0' x $diff); } } else { if ($$mfv ne '') { # e <= 0 # fraction and negative/zero E => NOI if ($_trap_nan) { croak("$wanted not an integer in $class"); } #print "NOI 2 \$\$mfv '$$mfv'\n"; return $upgrade->new($wanted, $a, $p, $r) if defined $upgrade; $self->{sign} = $nan; } elsif ($e < 0) { # xE-y, and empty mfv # Split the mantissa at the decimal point. E.g., if # $$miv = 12345 and $e = -2, then $frac = 45 and $$miv = 123. my $frac = substr($$miv, $e); # $frac is fraction part substr($$miv, $e) = ""; # $$miv is now integer part if ($frac =~ /[^0]/) { if ($_trap_nan) { croak("$wanted not an integer in $class"); } #print "NOI 3\n"; return $upgrade->new($wanted, $a, $p, $r) if defined $upgrade; $self->{sign} = $nan; } } } unless ($self->{sign} eq $nan) { $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0 $self->{value} = $LIB->_new($$miv) if $self->{sign} =~ /^[+-]$/; } # If any of the globals are set, use them to round, and store them inside # $self. Do not round for new($x, undef, undef) since that is used by MBF # to signal no rounding. $self->round($a, $p, $r) unless @_ >= 3 && !defined $a && !defined $p; $self; } # Create a Math::BigInt from a hexadecimal string. sub from_hex { my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; # Don't modify constant (read-only) objects. return if $selfref && $self->modify('from_hex'); my $str = shift; # If called as a class method, initialize a new object. $self = $class -> bzero() unless $selfref; if ($str =~ s/ ^ \s* ( [+-]? ) (0?x)? ( [0-9a-fA-F]* ( _ [0-9a-fA-F]+ )* ) \s* $ //x) { # Get a "clean" version of the string, i.e., non-emtpy and with no # underscores or invalid characters. my $sign = $1; my $chrs = $3; $chrs =~ tr/_//d; $chrs = '0' unless CORE::length $chrs; # The library method requires a prefix. $self->{value} = $LIB->_from_hex('0x' . $chrs); # Place the sign. $self->{sign} = $sign eq '-' && ! $LIB->_is_zero($self->{value}) ? '-' : '+'; return $self; } # CORE::hex() parses as much as it can, and ignores any trailing garbage. # For backwards compatibility, we return NaN. return $self->bnan(); } # Create a Math::BigInt from an octal string. sub from_oct { my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; # Don't modify constant (read-only) objects. return if $selfref && $self->modify('from_oct'); my $str = shift; # If called as a class method, initialize a new object. $self = $class -> bzero() unless $selfref; if ($str =~ s/ ^ \s* ( [+-]? ) ( [0-7]* ( _ [0-7]+ )* ) \s* $ //x) { # Get a "clean" version of the string, i.e., non-emtpy and with no # underscores or invalid characters. my $sign = $1; my $chrs = $2; $chrs =~ tr/_//d; $chrs = '0' unless CORE::length $chrs; # The library method requires a prefix. $self->{value} = $LIB->_from_oct('0' . $chrs); # Place the sign. $self->{sign} = $sign eq '-' && ! $LIB->_is_zero($self->{value}) ? '-' : '+'; return $self; } # CORE::oct() parses as much as it can, and ignores any trailing garbage. # For backwards compatibility, we return NaN. return $self->bnan(); } # Create a Math::BigInt from a binary string. sub from_bin { my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; # Don't modify constant (read-only) objects. return if $selfref && $self->modify('from_bin'); my $str = shift; # If called as a class method, initialize a new object. $self = $class -> bzero() unless $selfref; if ($str =~ s/ ^ \s* ( [+-]? ) (0?b)? ( [01]* ( _ [01]+ )* ) \s* $ //x) { # Get a "clean" version of the string, i.e., non-emtpy and with no # underscores or invalid characters. my $sign = $1; my $chrs = $3; $chrs =~ tr/_//d; $chrs = '0' unless CORE::length $chrs; # The library method requires a prefix. $self->{value} = $LIB->_from_bin('0b' . $chrs); # Place the sign. $self->{sign} = $sign eq '-' && ! $LIB->_is_zero($self->{value}) ? '-' : '+'; return $self; } # For consistency with from_hex() and from_oct(), we return NaN when the # input is invalid. return $self->bnan(); } # Create a Math::BigInt from a byte string. sub from_bytes { my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; # Don't modify constant (read-only) objects. return if $selfref && $self->modify('from_bytes'); croak("from_bytes() requires a newer version of the $LIB library.") unless $LIB->can('_from_bytes'); my $str = shift; # If called as a class method, initialize a new object. $self = $class -> bzero() unless $selfref; $self -> {sign} = '+'; $self -> {value} = $LIB -> _from_bytes($str); return $self; } sub from_base { my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; # Don't modify constant (read-only) objects. return if $selfref && $self->modify('from_base'); my $str = shift; my $base = shift; $base = $class->new($base) unless ref($base); croak("the base must be a finite integer >= 2") if $base < 2 \|\| ! $base -> is_int(); # If called as a class method, initialize a new object. $self = $class -> bzero() unless $selfref; # If no collating sequence is given, pass some of the conversions to # methods optimized for those cases. if (! @_) { return $self -> from_bin($str) if $base == 2; return $self -> from_oct($str) if $base == 8; return $self -> from_hex($str) if $base == 16; if ($base == 10) { my $tmp = $class -> new($str); $self -> {value} = $tmp -> {value}; $self -> {sign} = '+'; } } croak("from_base() requires a newer version of the $LIB library.") unless $LIB->can('_from_base'); $self -> {sign} = '+'; $self -> {value} = $LIB->_from_base($str, $base -> {value}, @_ ? shift() : ()); return $self } sub bzero { # create/assign '+0' if (@_ == 0) { #carp("Using bzero() as a function is deprecated;", # " use bzero() as a method instead"); unshift @_, __PACKAGE__; } my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; $self->import() if $IMPORT == 0; # make require work # Don't modify constant (read-only) objects. return if $selfref && $self->modify('bzero'); $self = bless {}, $class unless $selfref; $self->{sign} = '+'; $self->{value} = $LIB->_zero(); # If rounding parameters are given as arguments, use them. If no rounding # parameters are given, and if called as a class method initialize the new # instance with the class variables. if (@_) { croak "can't specify both accuracy and precision" if @_ >= 2 && defined $_[0] && defined $_[1]; $self->{_a} = $_[0]; $self->{_p} = $_[1]; } else { unless($selfref) { $self->{_a} = $class -> accuracy(); $self->{_p} = $class -> precision(); } } return $self; } sub bone { # Create or assign '+1' (or -1 if given sign '-'). if (@_ == 0 \|\| (defined($_[0]) && ($_[0] eq '+' \|\| $_[0] eq '-'))) { #carp("Using bone() as a function is deprecated;", # " use bone() as a method instead"); unshift @_, __PACKAGE__; } my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; $self->import() if $IMPORT == 0; # make require work # Don't modify constant (read-only) objects. return if $selfref && $self->modify('bone'); my $sign = '+'; # default if (@_) { $sign = shift; $sign = $sign =~ /^\s-/ ? "-" : "+"; } $self = bless {}, $class unless $selfref; $self->{sign} = $sign; $self->{value} = $LIB->_one(); # If rounding parameters are given as arguments, use them. If no rounding # parameters are given, and if called as a class method initialize the new # instance with the class variables. if (@_) { croak "can't specify both accuracy and precision" if @_ >= 2 && defined $_[0] && defined $_[1]; $self->{_a} = $_[0]; $self->{_p} = $_[1]; } else { unless($selfref) { $self->{_a} = $class -> accuracy(); $self->{_p} = $class -> precision(); } } return $self; } sub binf { # create/assign a '+inf' or '-inf' if (@_ == 0 \|\| (defined($_[0]) && !ref($_[0]) && $_[0] =~ /^\s[+-](inf(inity)?)?\s$/)) { #carp("Using binf() as a function is deprecated;", # " use binf() as a method instead"); unshift @_, __PACKAGE__; } my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; { no strict 'refs'; if (${"${class}::_trap_inf"}) { croak("Tried to create +-inf in $class->binf()"); } } $self->import() if $IMPORT == 0; # make require work # Don't modify constant (read-only) objects. return if $selfref && $self->modify('binf'); my $sign = shift; $sign = defined $sign && $sign =~ /^\s-/ ? "-" : "+"; $self = bless {}, $class unless $selfref; $self -> {sign} = $sign . 'inf'; $self -> {value} = $LIB -> _zero(); # If rounding parameters are given as arguments, use them. If no rounding # parameters are given, and if called as a class method initialize the new # instance with the class variables. if (@_) { croak "can't specify both accuracy and precision" if @_ >= 2 && defined $_[0] && defined $_[1]; $self->{_a} = $_[0]; $self->{_p} = $_[1]; } else { unless($selfref) { $self->{_a} = $class -> accuracy(); $self->{_p} = $class -> precision(); } } return $self; } sub bnan { # create/assign a 'NaN' if (@_ == 0) { #carp("Using bnan() as a function is deprecated;", # " use bnan() as a method instead"); unshift @_, __PACKAGE__; } my $self = shift; my $selfref = ref($self); my $class = $selfref \|\| $self; { no strict 'refs'; if (${"${class}::_trap_nan"}) { croak("Tried to create NaN in $class->bnan()"); } } $self->import() if $IMPORT == 0; # make require work # Don't modify constant (read-only) objects. return if $selfref && $self->modify('bnan'); $self = bless {}, $class unless $selfref; $self -> {sign} = $nan; $self -> {value} = $LIB -> _zero(); return $self; } sub bpi { # Calculate PI to N digits. Unless upgrading is in effect, returns the # result truncated to an integer, that is, always returns '3'. my ($self, $n) = @_; if (@_ == 1) { # called like Math::BigInt::bpi(10); $n = $self; $self = __PACKAGE__; } $self = ref($self) if ref($self); return $upgrade->new($n) if defined $upgrade; # hard-wired to "3" $self->new(3); } sub copy { my $self = shift; my $selfref = ref $self; my $class = $selfref \|\| $self; # If called as a class method, the object to copy is the next argument. $self = shift() unless $selfref; my $copy = bless {}, $class; $copy->{sign} = $self->{sign}; $copy->{value} = $LIB->_copy($self->{value}); $copy->{_a} = $self->{_a} if exists $self->{_a}; $copy->{_p} = $self->{_p} if exists $self->{_p}; return $copy; } sub as_number { # An object might be asked to return itself as bigint on certain overloaded # operations. This does exactly this, so that sub classes can simple inherit # it or override with their own integer conversion routine. $_[0]->copy(); } ############################################################################### # Boolean methods ############################################################################### sub is_zero { # return true if arg (BINT or num_str) is zero (array '+', '0') my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't $LIB->_is_zero($x->{value}); } sub is_one { # return true if arg (BINT or num_str) is +1, or -1 if sign is given my ($class, $x, $sign) = ref($_[0]) ? (undef, @_) : objectify(1, @_); $sign = '+' if !defined $sign \|\| $sign ne '-'; return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either $LIB->_is_one($x->{value}); } sub is_finite { my $x = shift; return $x->{sign} eq '+' \|\| $x->{sign} eq '-'; } sub is_inf { # return true if arg (BINT or num_str) is +-inf my ($class, $x, $sign) = ref($_[0]) ? (undef, @_) : objectify(1, @_); if (defined $sign) { $sign = '[+-]inf' if $sign eq ''; # +- doesn't matter, only that's inf $sign = "[$1]inf" if $sign =~ /^([+-])(inf)?$/; # extract '+' or '-' return $x->{sign} =~ /^$sign$/ ? 1 : 0; } $x->{sign} =~ /^[+-]inf$/ ? 1 : 0; # only +-inf is infinity } sub is_nan { # return true if arg (BINT or num_str) is NaN my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); $x->{sign} eq $nan ? 1 : 0; } sub is_positive { # return true when arg (BINT or num_str) is positive (> 0) my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); return 1 if $x->{sign} eq '+inf'; # +inf is positive # 0+ is neither positive nor negative ($x->{sign} eq '+' && !$x->is_zero()) ? 1 : 0; } sub is_negative { # return true when arg (BINT or num_str) is negative (< 0) my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); $x->{sign} =~ /^-/ ? 1 : 0; # -inf is negative, but NaN is not } sub is_non_negative { # Return true if argument is non-negative (>= 0). my ($class, $x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); return 1 if $x->{sign} =~ /^\+/; return 1 if $x -> is_zero(); return 0; } sub is_non_positive { # Return true if argument is non-positive (<= 0). my ($class, $x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); return 1 if $x->{sign} =~ /^\-/; return 1 if $x -> is_zero(); return 0; } sub is_odd { # return true when arg (BINT or num_str) is odd, false for even my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't $LIB->_is_odd($x->{value}); } sub is_even { # return true when arg (BINT or num_str) is even, false for odd my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't $LIB->_is_even($x->{value}); } sub is_int { # return true when arg (BINT or num_str) is an integer # always true for Math::BigInt, but different for Math::BigFloat objects my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't } ############################################################################### # Comparison methods ############################################################################### sub bcmp { # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort) # (BINT or num_str, BINT or num_str) return cond_code # set up parameters my ($class, $x, $y) = ref($_[0]) && ref($_[0]) eq ref($_[1]) ? (ref($_[0]), @_) : objectify(2, @_); return $upgrade->bcmp($x, $y) if defined $upgrade && ((!$x->isa($class)) \|\| (!$y->isa($class))); if (($x->{sign} !~ /^[+-]$/) \|\| ($y->{sign} !~ /^[+-]$/)) { # handle +-inf and NaN return undef if (($x->{sign} eq $nan) \|\| ($y->{sign} eq $nan)); return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/; return +1 if $x->{sign} eq '+inf'; return -1 if $x->{sign} eq '-inf'; return -1 if $y->{sign} eq '+inf'; return +1; } # check sign for speed first return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0 # have same sign, so compare absolute values. Don't make tests for zero # here because it's actually slower than testing in Calc (especially w/ Pari # et al) # post-normalized compare for internal use (honors signs) if ($x->{sign} eq '+') { # $x and $y both > 0 return $LIB->_acmp($x->{value}, $y->{value}); } # $x && $y both < 0 $LIB->_acmp($y->{value}, $x->{value}); # swapped acmp (lib returns 0, 1, -1) } sub bacmp { # Compares 2 values, ignoring their signs. # Returns one of undef, <0, =0, >0. (suitable for sort) # (BINT, BINT) return cond_code # set up parameters my ($class, $x, $y) = ref($_[0]) && ref($_[0]) eq ref($_[1]) ? (ref($_[0]), @_) : objectify(2, @_); return $upgrade->bacmp($x, $y) if defined $upgrade && ((!$x->isa($class)) \|\| (!$y->isa($class))); if (($x->{sign} !~ /^[+-]$/) \|\| ($y->{sign} !~ /^[+-]$/)) { # handle +-inf and NaN return undef if (($x->{sign} eq $nan) \|\| ($y->{sign} eq $nan)); return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/; return 1 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} !~ /^[+-]inf$/; return -1; } $LIB->_acmp($x->{value}, $y->{value}); # lib does only 0, 1, -1 } sub beq { my $self = shift; my $selfref = ref $self; croak 'beq() is an instance method, not a class method' unless $selfref; croak 'Wrong number of arguments for beq()' unless @_ == 1; my $cmp = $self -> bcmp(shift); return defined($cmp) && ! $cmp; } sub bne { my $self = shift; my $selfref = ref $self; croak 'bne() is an instance method, not a class method' unless $selfref; croak 'Wrong number of arguments for bne()' unless @_ == 1; my $cmp = $self -> bcmp(shift); return defined($cmp) && ! $cmp ? '' : 1; } sub blt { my $self = shift; my $selfref = ref $self; croak 'blt() is an instance method, not a class method' unless $selfref; croak 'Wrong number of arguments for blt()' unless @_ == 1; my $cmp = $self -> bcmp(shift); return defined($cmp) && $cmp < 0; } sub ble { my $self = shift; my $selfref = ref $self; croak 'ble() is an instance method, not a class method' unless $selfref; croak 'Wrong number of arguments for ble()' unless @_ == 1; my $cmp = $self -> bcmp(shift); return defined($cmp) && $cmp <= 0; } sub bgt { my $self = shift; my $selfref = ref $self; croak 'bgt() is an instance method, not a class method' unless $selfref; croak 'Wrong number of arguments for bgt()' unless @_ == 1; my $cmp = $self -> bcmp(shift); return defined($cmp) && $cmp > 0; } sub bge { my $self = shift; my $selfref = ref $self; croak 'bge() is an instance method, not a class method' unless $selfref; croak 'Wrong number of arguments for bge()' unless @_ == 1; my $cmp = $self -> bcmp(shift); return defined($cmp) && $cmp >= 0; } ############################################################################### # Arithmetic methods ############################################################################### sub bneg { # (BINT or num_str) return BINT # negate number or make a negated number from string my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); return $x if $x->modify('bneg'); # for +0 do not negate (to have always normalized +0). Does nothing for 'NaN' $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $LIB->_is_zero($x->{value})); $x; } sub babs { # (BINT or num_str) return BINT # make number absolute, or return absolute BINT from string my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); return $x if $x->modify('babs'); # post-normalized abs for internal use (does nothing for NaN) $x->{sign} =~ s/^-/+/; $x; } sub bsgn { # Signum function. my $self = shift; return $self if $self->modify('bsgn'); return $self -> bone("+") if $self -> is_pos(); return $self -> bone("-") if $self -> is_neg(); return $self; # zero or NaN } sub bnorm { # (numstr or BINT) return BINT # Normalize number -- no-op here my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); $x; } sub binc { # increment arg by one my ($class, $x, $a, $p, $r) = ref($_[0]) ? (ref($_[0]), @_) : objectify(1, @_); return $x if $x->modify('binc'); if ($x->{sign} eq '+') { $x->{value} = $LIB->_inc($x->{value}); return $x->round($a, $p, $r); } elsif ($x->{sign} eq '-') { $x->{value} = $LIB->_dec($x->{value}); $x->{sign} = '+' if $LIB->_is_zero($x->{value}); # -1 +1 => -0 => +0 return $x->round($a, $p, $r); } # inf, nan handling etc $x->badd($class->bone(), $a, $p, $r); # badd does round } sub bdec { # decrement arg by one my ($class, $x, @r) = ref($_[0]) ? (ref($_[0]), @_) : objectify(1, @_); return $x if $x->modify('bdec'); if ($x->{sign} eq '-') { # x already < 0 $x->{value} = $LIB->_inc($x->{value}); } else { return $x->badd($class->bone('-'), @r) unless $x->{sign} eq '+'; # inf or NaN # >= 0 if ($LIB->_is_zero($x->{value})) { # == 0 $x->{value} = $LIB->_one(); $x->{sign} = '-'; # 0 => -1 } else { # > 0 $x->{value} = $LIB->_dec($x->{value}); } } $x->round(@r); } #sub bstrcmp { # my $self = shift; # my $selfref = ref $self; # my $class = $selfref \|\| $self; # # croak 'bstrcmp() is an instance method, not a class method' # unless $selfref; # croak 'Wrong number of arguments for bstrcmp()' unless @_ == 1; # # return $self -> bstr() CORE::cmp shift; #} # #sub bstreq { # my $self = shift; # my $selfref = ref $self; # my $class = $selfref \|\| $self; # # croak 'bstreq() is an instance method, not a class method' # unless $selfref; # croak 'Wrong number of arguments for bstreq()' unless @_ == 1; # # my $cmp = $self -> bstrcmp(shift); # return defined($cmp) && ! $cmp; #} # #sub bstrne { # my $self = shift; # my $selfref = ref $self; # my $class = $selfref \|\| $self; # # croak 'bstrne() is an instance method, not a class method' # unless $selfref; # croak 'Wrong number of arguments for bstrne()' unless @_ == 1; # # my $cmp = $self -> bstrcmp(shift); # return defined($cmp) && ! $cmp ? '' : 1; #} # #sub bstrlt { # my $self = shift; # my $selfref = ref $self; # my $class = $selfref \|\| $self; # # croak 'bstrlt() is an instance method, not a class method' # unless $selfref; # croak 'Wrong number of arguments for bstrlt()' unless @_ == 1; # # my $cmp = $self -> bstrcmp(shift); # return defined($cmp) && $cmp < 0; #} # #sub bstrle { # my $self = shift; # my $selfref = ref $self; # my $class = $selfref \|\| $self; # # croak 'bstrle() is an instance method, not a class method' # unless $selfref; # croak 'Wrong number of arguments for bstrle()' unless @_ == 1; # # my $cmp = $self -> bstrcmp(shift); # return defined($cmp) && $cmp <= 0; #} # #sub bstrgt { # my $self = shift; # my $selfref = ref $self; # my $class = $selfref \|\| $self; # # croak 'bstrgt() is an instance method, not a class method' # unless $selfref; # croak 'Wrong number of arguments for bstrgt()' unless @_ == 1; # # my $cmp = $self -> bstrcmp(shift); # return defined($cmp) && $cmp > 0; #} # #sub bstrge { # my $self = shift; # my $selfref = ref $self; # my $class = $selfref \|\| $self; # # croak 'bstrge() is an instance method, not a class method' # unless $selfref; # croak 'Wrong number of arguments for bstrge()' unless @_ == 1; # # my $cmp = $self -> bstrcmp(shift); # return defined($cmp) && $cmp >= 0; #} sub badd { # add second arg (BINT or string) to first (BINT) (modifies first) # return result as BINT # set up parameters my ($class, $x, $y, @r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, @r) = objectify(2, @_); } return $x if $x->modify('badd'); return $upgrade->badd($upgrade->new($x), $upgrade->new($y), @r) if defined $upgrade && ((!$x->isa($class)) \|\| (!$y->isa($class))); $r[3] = $y; # no push! # inf and NaN handling if ($x->{sign} !~ /^[+-]$/ \|\| $y->{sign} !~ /^[+-]$/) { # NaN first return $x->bnan() if (($x->{sign} eq $nan) \|\| ($y->{sign} eq $nan)); # inf handling if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/)) { # +inf++inf or -inf+-inf => same, rest is NaN return $x if $x->{sign} eq $y->{sign}; return $x->bnan(); } # +-inf + something => +inf # something +-inf => +-inf $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/; return $x; } my ($sx, $sy) = ($x->{sign}, $y->{sign}); # get signs if ($sx eq $sy) { $x->{value} = $LIB->_add($x->{value}, $y->{value}); # same sign, abs add } else { my $a = $LIB->_acmp ($y->{value}, $x->{value}); # absolute compare if ($a > 0) { $x->{value} = $LIB->_sub($y->{value}, $x->{value}, 1); # abs sub w/ swap $x->{sign} = $sy; } elsif ($a == 0) { # speedup, if equal, set result to 0 $x->{value} = $LIB->_zero(); $x->{sign} = '+'; } else # a < 0 { $x->{value} = $LIB->_sub($x->{value}, $y->{value}); # abs sub } } $x->round(@r); } sub bsub { # (BINT or num_str, BINT or num_str) return BINT # subtract second arg from first, modify first # set up parameters my ($class, $x, $y, @r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, @r) = objectify(2, @_); } return $x if $x -> modify('bsub'); return $upgrade -> new($x) -> bsub($upgrade -> new($y), @r) if defined $upgrade && (!$x -> isa($class) \|\| !$y -> isa($class)); return $x -> round(@r) if $y -> is_zero(); # To correctly handle the lone special case $x -> bsub($x), we note the # sign of $x, then flip the sign from $y, and if the sign of $x did change, # too, then we caught the special case: my $xsign = $x -> {sign}; $y -> {sign} =~ tr/+-/-+/; # does nothing for NaN if ($xsign ne $x -> {sign}) { # special case of $x -> bsub($x) results in 0 return $x -> bzero(@r) if $xsign =~ /^[+-]$/; return $x -> bnan(); # NaN, -inf, +inf } $x -> badd($y, @r); # badd does not leave internal zeros $y -> {sign} =~ tr/+-/-+/; # refix $y (does nothing for NaN) $x; # already rounded by badd() or no rounding } sub bmul { # multiply the first number by the second number # (BINT or num_str, BINT or num_str) return BINT # set up parameters my ($class, $x, $y, @r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, @r) = objectify(2, @_); } return $x if $x->modify('bmul'); return $x->bnan() if (($x->{sign} eq $nan) \|\| ($y->{sign} eq $nan)); # inf handling if (($x->{sign} =~ /^[+-]inf$/) \|\| ($y->{sign} =~ /^[+-]inf$/)) { return $x->bnan() if $x->is_zero() \|\| $y->is_zero(); # result will always be +-inf: # +inf * +/+inf => +inf, -inf * -/-inf => +inf # +inf * -/-inf => -inf, -inf * +/+inf => -inf return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/); return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/); return $x->binf('-'); } return $upgrade->bmul($x, $upgrade->new($y), @r) if defined $upgrade && !$y->isa($class); $r[3] = $y; # no push here $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => + $x->{value} = $LIB->_mul($x->{value}, $y->{value}); # do actual math $x->{sign} = '+' if $LIB->_is_zero($x->{value}); # no -0 $x->round(@r); } sub bmuladd { # multiply two numbers and then add the third to the result # (BINT or num_str, BINT or num_str, BINT or num_str) return BINT # set up parameters my ($class, $x, $y, $z, @r) = objectify(3, @_); return $x if $x->modify('bmuladd'); return $x->bnan() if (($x->{sign} eq $nan) \|\| ($y->{sign} eq $nan) \|\| ($z->{sign} eq $nan)); # inf handling of x and y if (($x->{sign} =~ /^[+-]inf$/) \|\| ($y->{sign} =~ /^[+-]inf$/)) { return $x->bnan() if $x->is_zero() \|\| $y->is_zero(); # result will always be +-inf: # +inf * +/+inf => +inf, -inf * -/-inf => +inf # +inf * -/-inf => -inf, -inf * +/+inf => -inf return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/); return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/); return $x->binf('-'); } # inf handling xy and z if (($z->{sign} =~ /^[+-]inf$/)) { # something +-inf => +-inf $x->{sign} = $z->{sign}, return $x if $z->{sign} =~ /^[+-]inf$/; } return $upgrade->bmuladd($x, $upgrade->new($y), $upgrade->new($z), @r) if defined $upgrade && (!$y->isa($class) \|\| !$z->isa($class) \|\| !$x->isa($class)); # TODO: what if $y and $z have A or P set? $r[3] = $z; # no push here $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 +1 or -1 * -1 => + $x->{value} = $LIB->_mul($x->{value}, $y->{value}); # do actual math $x->{sign} = '+' if $LIB->_is_zero($x->{value}); # no -0 my ($sx, $sz) = ( $x->{sign}, $z->{sign} ); # get signs if ($sx eq $sz) { $x->{value} = $LIB->_add($x->{value}, $z->{value}); # same sign, abs add } else { my $a = $LIB->_acmp ($z->{value}, $x->{value}); # absolute compare if ($a > 0) { $x->{value} = $LIB->_sub($z->{value}, $x->{value}, 1); # abs sub w/ swap $x->{sign} = $sz; } elsif ($a == 0) { # speedup, if equal, set result to 0 $x->{value} = $LIB->_zero(); $x->{sign} = '+'; } else # a < 0 { $x->{value} = $LIB->_sub($x->{value}, $z->{value}); # abs sub } } $x->round(@r); } sub bdiv { # This does floored division, where the quotient is floored, i.e., rounded # towards negative infinity. As a consequence, the remainder has the same # sign as the divisor. # Set up parameters. my ($class, $x, $y, @r) = (ref($_[0]), @_); # objectify() is costly, so avoid it if we can. if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, @r) = objectify(2, @_); } return $x if $x -> modify('bdiv'); my $wantarray = wantarray; # call only once # At least one argument is NaN. Return NaN for both quotient and the # modulo/remainder. if ($x -> is_nan() \|\| $y -> is_nan()) { return $wantarray ? ($x -> bnan(), $class -> bnan()) : $x -> bnan(); } # Divide by zero and modulo zero. # # Division: Use the common convention that x / 0 is inf with the same sign # as x, except when x = 0, where we return NaN. This is also what earlier # versions did. # # Modulo: In modular arithmetic, the congruence relation z = x (mod y) # means that there is some integer k such that z - x = k y. If y = 0, we # get z - x = 0 or z = x. This is also what earlier versions did, except # that 0 % 0 returned NaN. # # inf / 0 = inf inf % 0 = inf # 5 / 0 = inf 5 % 0 = 5 # 0 / 0 = NaN 0 % 0 = 0 # -5 / 0 = -inf -5 % 0 = -5 # -inf / 0 = -inf -inf % 0 = -inf if ($y -> is_zero()) { my $rem; if ($wantarray) { $rem = $x -> copy(); } if ($x -> is_zero()) { $x -> bnan(); } else { $x -> binf($x -> {sign}); } return $wantarray ? ($x, $rem) : $x; } # Numerator (dividend) is +/-inf, and denominator is finite and non-zero. # The divide by zero cases are covered above. In all of the cases listed # below we return the same as core Perl. # # inf / -inf = NaN inf % -inf = NaN # inf / -5 = -inf inf % -5 = NaN # inf / 5 = inf inf % 5 = NaN # inf / inf = NaN inf % inf = NaN # # -inf / -inf = NaN -inf % -inf = NaN # -inf / -5 = inf -inf % -5 = NaN # -inf / 5 = -inf -inf % 5 = NaN # -inf / inf = NaN -inf % inf = NaN if ($x -> is_inf()) { my $rem; $rem = $class -> bnan() if $wantarray; if ($y -> is_inf()) { $x -> bnan(); } else { my $sign = $x -> bcmp(0) == $y -> bcmp(0) ? '+' : '-'; $x -> binf($sign); } return $wantarray ? ($x, $rem) : $x; } # Denominator (divisor) is +/-inf. The cases when the numerator is +/-inf # are covered above. In the modulo cases (in the right column) we return # the same as core Perl, which does floored division, so for consistency we # also do floored division in the division cases (in the left column). # # -5 / inf = -1 -5 % inf = inf # 0 / inf = 0 0 % inf = 0 # 5 / inf = 0 5 % inf = 5 # # -5 / -inf = 0 -5 % -inf = -5 # 0 / -inf = 0 0 % -inf = 0 # 5 / -inf = -1 5 % -inf = -inf if ($y -> is_inf()) { my $rem; if ($x -> is_zero() \|\| $x -> bcmp(0) == $y -> bcmp(0)) { $rem = $x -> copy() if $wantarray; $x -> bzero(); } else { $rem = $class -> binf($y -> {sign}) if $wantarray; $x -> bone('-'); } return $wantarray ? ($x, $rem) : $x; } # At this point, both the numerator and denominator are finite numbers, and # the denominator (divisor) is non-zero. return $upgrade -> bdiv($upgrade -> new($x), $upgrade -> new($y), @r) if defined $upgrade; $r[3] = $y; # no push! # Inialize remainder. my $rem = $class -> bzero(); # Are both operands the same object, i.e., like $x -> bdiv($x)? If so, # flipping the sign of $y also flips the sign of $x. my $xsign = $x -> {sign}; my $ysign = $y -> {sign}; $y -> {sign} =~ tr/+-/-+/; # Flip the sign of $y, and see ... my $same = $xsign ne $x -> {sign}; # ... if that changed the sign of $x. $y -> {sign} = $ysign; # Re-insert the original sign. if ($same) { $x -> bone(); } else { ($x -> {value}, $rem -> {value}) = $LIB -> _div($x -> {value}, $y -> {value}); if ($LIB -> _is_zero($rem -> {value})) { if ($xsign eq $ysign \|\| $LIB -> _is_zero($x -> {value})) { $x -> {sign} = '+'; } else { $x -> {sign} = '-'; } } else { if ($xsign eq $ysign) { $x -> {sign} = '+'; } else { if ($xsign eq '+') { $x -> badd(1); } else { $x -> bsub(1); } $x -> {sign} = '-'; } } } $x -> round(@r); if ($wantarray) { unless ($LIB -> _is_zero($rem -> {value})) { if ($xsign ne $ysign) { $rem = $y -> copy() -> babs() -> bsub($rem); } $rem -> {sign} = $ysign; } $rem -> {_a} = $x -> {_a}; $rem -> {_p} = $x -> {_p}; $rem -> round(@r); return ($x, $rem); } return $x; } sub btdiv { # This does truncated division, where the quotient is truncted, i.e., # rounded towards zero. # # ($q, $r) = $x -> btdiv($y) returns $q and $r so that $q is int($x / $y) # and $q * $y + $r = $x. # Set up parameters my ($class, $x, $y, @r) = (ref($_[0]), @_); # objectify is costly, so avoid it if we can. if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, @r) = objectify(2, @_); } return $x if $x -> modify('btdiv'); my $wantarray = wantarray; # call only once # At least one argument is NaN. Return NaN for both quotient and the # modulo/remainder. if ($x -> is_nan() \|\| $y -> is_nan()) { return $wantarray ? ($x -> bnan(), $class -> bnan()) : $x -> bnan(); } # Divide by zero and modulo zero. # # Division: Use the common convention that x / 0 is inf with the same sign # as x, except when x = 0, where we return NaN. This is also what earlier # versions did. # # Modulo: In modular arithmetic, the congruence relation z = x (mod y) # means that there is some integer k such that z - x = k y. If y = 0, we # get z - x = 0 or z = x. This is also what earlier versions did, except # that 0 % 0 returned NaN. # # inf / 0 = inf inf % 0 = inf # 5 / 0 = inf 5 % 0 = 5 # 0 / 0 = NaN 0 % 0 = 0 # -5 / 0 = -inf -5 % 0 = -5 # -inf / 0 = -inf -inf % 0 = -inf if ($y -> is_zero()) { my $rem; if ($wantarray) { $rem = $x -> copy(); } if ($x -> is_zero()) { $x -> bnan(); } else { $x -> binf($x -> {sign}); } return $wantarray ? ($x, $rem) : $x; } # Numerator (dividend) is +/-inf, and denominator is finite and non-zero. # The divide by zero cases are covered above. In all of the cases listed # below we return the same as core Perl. # # inf / -inf = NaN inf % -inf = NaN # inf / -5 = -inf inf % -5 = NaN # inf / 5 = inf inf % 5 = NaN # inf / inf = NaN inf % inf = NaN # # -inf / -inf = NaN -inf % -inf = NaN # -inf / -5 = inf -inf % -5 = NaN # -inf / 5 = -inf -inf % 5 = NaN # -inf / inf = NaN -inf % inf = NaN if ($x -> is_inf()) { my $rem; $rem = $class -> bnan() if $wantarray; if ($y -> is_inf()) { $x -> bnan(); } else { my $sign = $x -> bcmp(0) == $y -> bcmp(0) ? '+' : '-'; $x -> binf($sign); } return $wantarray ? ($x, $rem) : $x; } # Denominator (divisor) is +/-inf. The cases when the numerator is +/-inf # are covered above. In the modulo cases (in the right column) we return # the same as core Perl, which does floored division, so for consistency we # also do floored division in the division cases (in the left column). # # -5 / inf = 0 -5 % inf = -5 # 0 / inf = 0 0 % inf = 0 # 5 / inf = 0 5 % inf = 5 # # -5 / -inf = 0 -5 % -inf = -5 # 0 / -inf = 0 0 % -inf = 0 # 5 / -inf = 0 5 % -inf = 5 if ($y -> is_inf()) { my $rem; $rem = $x -> copy() if $wantarray; $x -> bzero(); return $wantarray ? ($x, $rem) : $x; } return $upgrade -> btdiv($upgrade -> new($x), $upgrade -> new($y), @r) if defined $upgrade; $r[3] = $y; # no push! # Inialize remainder. my $rem = $class -> bzero(); # Are both operands the same object, i.e., like $x -> bdiv($x)? If so, # flipping the sign of $y also flips the sign of $x. my $xsign = $x -> {sign}; my $ysign = $y -> {sign}; $y -> {sign} =~ tr/+-/-+/; # Flip the sign of $y, and see ... my $same = $xsign ne $x -> {sign}; # ... if that changed the sign of $x. $y -> {sign} = $ysign; # Re-insert the original sign. if ($same) { $x -> bone(); } else { ($x -> {value}, $rem -> {value}) = $LIB -> _div($x -> {value}, $y -> {value}); $x -> {sign} = $xsign eq $ysign ? '+' : '-'; $x -> {sign} = '+' if $LIB -> _is_zero($x -> {value}); $x -> round(@r); } if (wantarray) { $rem -> {sign} = $xsign; $rem -> {sign} = '+' if $LIB -> _is_zero($rem -> {value}); $rem -> {_a} = $x -> {_a}; $rem -> {_p} = $x -> {_p}; $rem -> round(@r); return ($x, $rem); } return $x; } sub bmod { # This is the remainder after floored division. # Set up parameters. my ($class, $x, $y, @r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, @r) = objectify(2, @_); } return $x if $x -> modify('bmod'); $r[3] = $y; # no push! # At least one argument is NaN. if ($x -> is_nan() \|\| $y -> is_nan()) { return $x -> bnan(); } # Modulo zero. See documentation for bdiv(). if ($y -> is_zero()) { return $x; } # Numerator (dividend) is +/-inf. if ($x -> is_inf()) { return $x -> bnan(); } # Denominator (divisor) is +/-inf. if ($y -> is_inf()) { if ($x -> is_zero() \|\| $x -> bcmp(0) == $y -> bcmp(0)) { return $x; } else { return $x -> binf($y -> sign()); } } # Calc new sign and in case $y == +/- 1, return $x. $x -> {value} = $LIB -> _mod($x -> {value}, $y -> {value}); if ($LIB -> _is_zero($x -> {value})) { $x -> {sign} = '+'; # do not leave -0 } else { $x -> {value} = $LIB -> _sub($y -> {value}, $x -> {value}, 1) # $y-$x if ($x -> {sign} ne $y -> {sign}); $x -> {sign} = $y -> {sign}; } $x -> round(@r); } sub btmod { # Remainder after truncated division. # set up parameters my ($class, $x, $y, @r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, @r) = objectify(2, @_); } return $x if $x -> modify('btmod'); # At least one argument is NaN. if ($x -> is_nan() \|\| $y -> is_nan()) { return $x -> bnan(); } # Modulo zero. See documentation for btdiv(). if ($y -> is_zero()) { return $x; } # Numerator (dividend) is +/-inf. if ($x -> is_inf()) { return $x -> bnan(); } # Denominator (divisor) is +/-inf. if ($y -> is_inf()) { return $x; } return $upgrade -> btmod($upgrade -> new($x), $upgrade -> new($y), @r) if defined $upgrade; $r[3] = $y; # no push! my $xsign = $x -> {sign}; $x -> {value} = $LIB -> _mod($x -> {value}, $y -> {value}); $x -> {sign} = $xsign; $x -> {sign} = '+' if $LIB -> _is_zero($x -> {value}); $x -> round(@r); return $x; } sub bmodinv { # Return modular multiplicative inverse: # # z is the modular inverse of x (mod y) if and only if # # xz ≡ 1 (mod y) # # If the modulus y is larger than one, x and z are relative primes (i.e., # their greatest common divisor is one). # # If no modular multiplicative inverse exists, NaN is returned. # set up parameters my ($class, $x, $y, @r) = (undef, @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, @r) = objectify(2, @_); } return $x if $x->modify('bmodinv'); # Return NaN if one or both arguments is +inf, -inf, or nan. return $x->bnan() if ($y->{sign} !~ /^[+-]$/ \|\| $x->{sign} !~ /^[+-]$/); # Return NaN if $y is zero; 1 % 0 makes no sense. return $x->bnan() if $y->is_zero(); # Return 0 in the trivial case. $x % 1 or $x % -1 is zero for all finite # integers $x. return $x->bzero() if ($y->is_one() \|\| $y->is_one('-')); # Return NaN if $x = 0, or $x modulo $y is zero. The only valid case when # $x = 0 is when $y = 1 or $y = -1, but that was covered above. # # Note that computing $x modulo $y here affects the value we'll feed to # $LIB->_modinv() below when $x and $y have opposite signs. E.g., if $x = # 5 and $y = 7, those two values are fed to _modinv(), but if $x = -5 and # $y = 7, the values fed to _modinv() are $x = 2 (= -5 % 7) and $y = 7. # The value if $x is affected only when $x and $y have opposite signs. $x->bmod($y); return $x->bnan() if $x->is_zero(); # Compute the modular multiplicative inverse of the absolute values. We'll # correct for the signs of $x and $y later. Return NaN if no GCD is found. ($x->{value}, $x->{sign}) = $LIB->_modinv($x->{value}, $y->{value}); return $x->bnan() if !defined $x->{value}; # Library inconsistency workaround: _modinv() in Math::BigInt::GMP versions # <= 1.32 return undef rather than a "+" for the sign. $x->{sign} = '+' unless defined $x->{sign}; # When one or both arguments are negative, we have the following # relations. If x and y are positive: # # modinv(-x, -y) = -modinv(x, y) # modinv(-x, y) = y - modinv(x, y) = -modinv(x, y) (mod y) # modinv( x, -y) = modinv(x, y) - y = modinv(x, y) (mod -y) # We must swap the sign of the result if the original $x is negative. # However, we must compensate for ignoring the signs when computing the # inverse modulo. The net effect is that we must swap the sign of the # result if $y is negative. $x -> bneg() if $y->{sign} eq '-'; # Compute $x modulo $y again after correcting the sign. $x -> bmod($y) if $x->{sign} ne $y->{sign}; return $x; } sub bmodpow { # Modular exponentiation. Raises a very large number to a very large exponent # in a given very large modulus quickly, thanks to binary exponentiation. # Supports negative exponents. my ($class, $num, $exp, $mod, @r) = objectify(3, @_); return $num if $num->modify('bmodpow'); # When the exponent 'e' is negative, use the following relation, which is # based on finding the multiplicative inverse 'd' of 'b' modulo 'm': # # b^(-e) (mod m) = d^e (mod m) where bd = 1 (mod m) $num->bmodinv($mod) if ($exp->{sign} eq '-'); # Check for valid input. All operands must be finite, and the modulus must be # non-zero. return $num->bnan() if ($num->{sign} =~ /NaN\|inf/ \|\| # NaN, -inf, +inf $exp->{sign} =~ /NaN\|inf/ \|\| # NaN, -inf, +inf $mod->{sign} =~ /NaN\|inf/); # NaN, -inf, +inf # Modulo zero. See documentation for Math::BigInt's bmod() method. if ($mod -> is_zero()) { if ($num -> is_zero()) { return $class -> bnan(); } else { return $num -> copy(); } } # Compute 'a (mod m)', ignoring the signs on 'a' and 'm'. If the resulting # value is zero, the output is also zero, regardless of the signs on 'a' and # 'm'. my $value = $LIB->_modpow($num->{value}, $exp->{value}, $mod->{value}); my $sign = '+'; # If the resulting value is non-zero, we have four special cases, depending # on the signs on 'a' and 'm'. unless ($LIB->_is_zero($value)) { # There is a negative sign on 'a' (= $num$exp) only if the number we # are exponentiating ($num) is negative and the exponent ($exp) is odd. if ($num->{sign} eq '-' && $exp->is_odd()) { # When both the number 'a' and the modulus 'm' have a negative sign, # use this relation: # # -a (mod -m) = -(a (mod m)) if ($mod->{sign} eq '-') { $sign = '-'; } # When only the number 'a' has a negative sign, use this relation: # # -a (mod m) = m - (a (mod m)) else { # Use copy of $mod since _sub() modifies the first argument. my $mod = $LIB->_copy($mod->{value}); $value = $LIB->_sub($mod, $value); $sign = '+'; } } else { # When only the modulus 'm' has a negative sign, use this relation: # # a (mod -m) = (a (mod m)) - m # = -(m - (a (mod m))) if ($mod->{sign} eq '-') { # Use copy of $mod since _sub() modifies the first argument. my $mod = $LIB->_copy($mod->{value}); $value = $LIB->_sub($mod, $value); $sign = '-'; } # When neither the number 'a' nor the modulus 'm' have a negative # sign, directly return the already computed value. # # (a (mod m)) } } $num->{value} = $value; $num->{sign} = $sign; return $num -> round(@r); } sub bpow { # (BINT or num_str, BINT or num_str) return BINT # compute power of two numbers -- stolen from Knuth Vol 2 pg 233 # modifies first argument # set up parameters my ($class, $x, $y, @r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, @r) = objectify(2, @_); } return $x if $x->modify('bpow'); # $x and/or $y is a NaN return $x->bnan() if $x->is_nan() \|\| $y->is_nan(); # $x and/or $y is a +/-Inf if ($x->is_inf("-")) { return $x->bzero() if $y->is_negative(); return $x->bnan() if $y->is_zero(); return $x if $y->is_odd(); return $x->bneg(); } elsif ($x->is_inf("+")) { return $x->bzero() if $y->is_negative(); return $x->bnan() if $y->is_zero(); return $x; } elsif ($y->is_inf("-")) { return $x->bnan() if $x -> is_one("-"); return $x->binf("+") if $x -> is_zero(); return $x->bone() if $x -> is_one("+"); return $x->bzero(); } elsif ($y->is_inf("+")) { return $x->bnan() if $x -> is_one("-"); return $x->bzero() if $x -> is_zero(); return $x->bone() if $x -> is_one("+"); return $x->binf("+"); } return $upgrade->bpow($upgrade->new($x), $y, @r) if defined $upgrade && (!$y->isa($class) \|\| $y->{sign} eq '-'); $r[3] = $y; # no push! # 0 -y => ( 1 / (0 y)) => 1 / 0 => +inf return $x->binf() if $y->is_negative() && $x->is_zero(); # 1 -y => 1 / (1 \|y\|) return $x->bzero() if $y->is_negative() && !$LIB->_is_one($x->{value}); $x->{value} = $LIB->_pow($x->{value}, $y->{value}); $x->{sign} = $x->is_negative() && $y->is_odd() ? '-' : '+'; $x->round(@r); } sub blog { # Return the logarithm of the operand. If a second operand is defined, that # value is used as the base, otherwise the base is assumed to be Euler's # constant. my ($class, $x, $base, @r); # Don't objectify the base, since an undefined base, as in $x->blog() or # $x->blog(undef) signals that the base is Euler's number. if (!ref($_[0]) && $_[0] =~ /^[A-Za-z]\|::/) { # E.g., Math::BigInt->blog(256, 2) ($class, $x, $base, @r) = defined $_[2] ? objectify(2, @_) : objectify(1, @_); } else { # E.g., Math::BigInt::blog(256, 2) or $x->blog(2) ($class, $x, $base, @r) = defined $_[1] ? objectify(2, @_) : objectify(1, @_); } return $x if $x->modify('blog'); # Handle all exception cases and all trivial cases. I have used Wolfram # Alpha (http://www.wolframalpha.com) as the reference for these cases. return $x -> bnan() if $x -> is_nan(); if (defined $base) { $base = $class -> new($base) unless ref $base; if ($base -> is_nan() \|\| $base -> is_one()) { return $x -> bnan(); } elsif ($base -> is_inf() \|\| $base -> is_zero()) { return $x -> bnan() if $x -> is_inf() \|\| $x -> is_zero(); return $x -> bzero(); } elsif ($base -> is_negative()) { # -inf < base < 0 return $x -> bzero() if $x -> is_one(); # x = 1 return $x -> bone() if $x == $base; # x = base return $x -> bnan(); # otherwise } return $x -> bone() if $x == $base; # 0 < base && 0 < x < inf } # We now know that the base is either undefined or >= 2 and finite. return $x -> binf('+') if $x -> is_inf(); # x = +/-inf return $x -> bnan() if $x -> is_neg(); # -inf < x < 0 return $x -> bzero() if $x -> is_one(); # x = 1 return $x -> binf('-') if $x -> is_zero(); # x = 0 # At this point we are done handling all exception cases and trivial cases. return $upgrade -> blog($upgrade -> new($x), $base, @r) if defined $upgrade; # fix for bug #24969: # the default base is e (Euler's number) which is not an integer if (!defined $base) { require Math::BigFloat; my $u = Math::BigFloat->blog(Math::BigFloat->new($x))->as_int(); # modify $x in place $x->{value} = $u->{value}; $x->{sign} = $u->{sign}; return $x; } my ($rc) = $LIB->_log_int($x->{value}, $base->{value}); return $x->bnan() unless defined $rc; # not possible to take log? $x->{value} = $rc; $x->round(@r); } sub bexp { # Calculate e $x (Euler's number to the power of X), truncated to # an integer value. my ($class, $x, @r) = ref($_[0]) ? (ref($_[0]), @_) : objectify(1, @_); return $x if $x->modify('bexp'); # inf, -inf, NaN, <0 => NaN return $x->bnan() if $x->{sign} eq 'NaN'; return $x->bone() if $x->is_zero(); return $x if $x->{sign} eq '+inf'; return $x->bzero() if $x->{sign} eq '-inf'; my $u; { # run through Math::BigFloat unless told otherwise require Math::BigFloat unless defined $upgrade; local $upgrade = 'Math::BigFloat' unless defined $upgrade; # calculate result, truncate it to integer $u = $upgrade->bexp($upgrade->new($x), @r); } if (defined $upgrade) { $x = $u; } else { $u = $u->as_int(); # modify $x in place $x->{value} = $u->{value}; $x->round(@r); } } sub bnok { # Calculate n over k (binomial coefficient or "choose" function) as # integer. # Set up parameters. my ($self, $n, $k, @r) = (ref($_[0]), @_); # Objectify is costly, so avoid it. if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($self, $n, $k, @r) = objectify(2, @_); } return $n if $n->modify('bnok'); # All cases where at least one argument is NaN. return $n->bnan() if $n->{sign} eq 'NaN' \|\| $k->{sign} eq 'NaN'; # All cases where at least one argument is +/-inf. if ($n -> is_inf()) { if ($k -> is_inf()) { # bnok(+/-inf,+/-inf) return $n -> bnan(); } elsif ($k -> is_neg()) { # bnok(+/-inf,k), k < 0 return $n -> bzero(); } elsif ($k -> is_zero()) { # bnok(+/-inf,k), k = 0 return $n -> bone(); } else { if ($n -> is_inf("+")) { # bnok(+inf,k), 0 < k < +inf return $n -> binf("+"); } else { # bnok(-inf,k), k > 0 my $sign = $k -> is_even() ? "+" : "-"; return $n -> binf($sign); } } } elsif ($k -> is_inf()) { # bnok(n,+/-inf), -inf <= n <= inf return $n -> bnan(); } # At this point, both n and k are real numbers. my $sign = 1; if ($n >= 0) { if ($k < 0 \|\| $k > $n) { return $n -> bzero(); } } else { if ($k >= 0) { # n < 0 and k >= 0: bnok(n,k) = (-1)^k * bnok(-n+k-1,k) $sign = (-1) ** $k; $n -> bneg() -> badd($k) -> bdec(); } elsif ($k <= $n) { # n < 0 and k <= n: bnok(n,k) = (-1)^(n-k) * bnok(-k-1,n-k) $sign = (-1) ($n - $k); my $x0 = $n -> copy(); $n -> bone() -> badd($k) -> bneg(); $k = $k -> copy(); $k -> bneg() -> badd($x0); } else { # n < 0 and n < k < 0: return $n -> bzero(); } } $n->{value} = $LIB->_nok($n->{value}, $k->{value}); $n -> bneg() if $sign == -1; $n->round(@r); } sub buparrow { my $a = shift; my $y = $a -> uparrow(@_); $a -> {value} = $y -> {value}; return $a; } sub uparrow { # Knuth's up-arrow notation buparrow(a, n, b) # # The following is a simple, recursive implementation of the up-arrow # notation, just to show the idea. Such implementations cause "Deep # recursion on subroutine ..." warnings, so we use a faster, non-recursive # algorithm below with @_ as a stack. # # sub buparrow { # my ($a, $n, $b) = @_; # return $a $b if $n == 1; # return $a * $b if $n == 0; # return 1 if $b == 0; # return buparrow($a, $n - 1, buparrow($a, $n, $b - 1)); # } my ($a, $b, $n) = @_; my $class = ref $a; croak("a must be non-negative") if $a < 0; croak("n must be non-negative") if $n < 0; croak("b must be non-negative") if $b < 0; while (@_ >= 3) { # return $a $b if $n == 1; if ($_[-2] == 1) { my ($a, $n, $b) = splice @_, -3; push @_, $a $b; next; } # return $a * $b if $n == 0; if ($_[-2] == 0) { my ($a, $n, $b) = splice @_, -3; push @_, $a * $b; next; } # return 1 if $b == 0; if ($_[-1] == 0) { splice @_, -3; push @_, $class -> bone(); next; } # return buparrow($a, $n - 1, buparrow($a, $n, $b - 1)); my ($a, $n, $b) = splice @_, -3; push @_, ($a, $n - 1, $a, $n, $b - 1); } pop @_; } sub backermann { my $m = shift; my $y = $m -> ackermann(@_); $m -> {value} = $y -> {value}; return $m; } sub ackermann { # Ackermann's function ackermann(m, n) # # The following is a simple, recursive implementation of the ackermann # function, just to show the idea. Such implementations cause "Deep # recursion on subroutine ..." warnings, so we use a faster, non-recursive # algorithm below with @_ as a stack. # # sub ackermann { # my ($m, $n) = @_; # return $n + 1 if $m == 0; # return ackermann($m - 1, 1) if $m > 0 && $n == 0; # return ackermann($m - 1, ackermann($m, $n - 1) if $m > 0 && $n > 0; # } my ($m, $n) = @_; my $class = ref $m; croak("m must be non-negative") if $m < 0; croak("n must be non-negative") if $n < 0; my $two = $class -> new("2"); my $three = $class -> new("3"); my $thirteen = $class -> new("13"); $n = pop; $n = $class -> new($n) unless ref($n); while (@_) { my $m = pop; if ($m > $three) { push @_, (--$m) x $n; while (--$m >= $three) { push @_, $m; } $n = $thirteen; } elsif ($m == $three) { $n = $class -> bone() -> blsft($n + $three) -> bsub($three); } elsif ($m == $two) { $n -> bmul($two) -> badd($three); } elsif ($m >= 0) { $n -> badd($m) -> binc(); } else { die "negative m!"; } } $n; } sub bsin { # Calculate sinus(x) to N digits. Unless upgrading is in effect, returns the # result truncated to an integer. my ($class, $x, @r) = ref($_[0]) ? (undef, @_) : objectify(1, @_); return $x if $x->modify('bsin'); return $x->bnan() if $x->{sign} !~ /^[+-]\z/; # -inf +inf or NaN => NaN return $upgrade->new($x)->bsin(@r) if defined $upgrade; require Math::BigFloat; # calculate the result and truncate it to integer my $t = Math::BigFloat->new($x)->bsin(@r)->as_int(); $x->bone() if $t->is_one(); $x->bzero() if $t->is_zero(); $x->round(@r); } sub bcos { # Calculate cosinus(x) to N digits. Unless upgrading is in effect, returns the # result truncated to an integer. my ($class, $x, @r) = ref($_[0]) ? (undef, @_) : objectify(1, @_); return $x if $x->modify('bcos'); return $x->bnan() if $x->{sign} !~ /^[+-]\z/; # -inf +inf or NaN => NaN return $upgrade->new($x)->bcos(@r) if defined $upgrade; require Math::BigFloat; # calculate the result and truncate it to integer my $t = Math::BigFloat->new($x)->bcos(@r)->as_int(); $x->bone() if $t->is_one(); $x->bzero() if $t->is_zero(); $x->round(@r); } sub batan { # Calculate arcus tangens of x to N digits. Unless upgrading is in effect, returns the # result truncated to an integer. my ($class, $x, @r) = ref($_[0]) ? (undef, @_) : objectify(1, @_); return $x if $x->modify('batan'); return $x->bnan() if $x->{sign} !~ /^[+-]\z/; # -inf +inf or NaN => NaN return $upgrade->new($x)->batan(@r) if defined $upgrade; # calculate the result and truncate it to integer my $tmp = Math::BigFloat->new($x)->batan(@r); $x->{value} = $LIB->_new($tmp->as_int()->bstr()); $x->round(@r); } sub batan2 { # calculate arcus tangens of ($y/$x) # set up parameters my ($class, $y, $x, @r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $y, $x, @r) = objectify(2, @_); } return $y if $y->modify('batan2'); return $y->bnan() if ($y->{sign} eq $nan) \|\| ($x->{sign} eq $nan); # Y X # != 0 -inf result is +- pi if ($x->is_inf() \|\| $y->is_inf()) { # upgrade to Math::BigFloat etc. return $upgrade->new($y)->batan2($upgrade->new($x), @r) if defined $upgrade; if ($y->is_inf()) { if ($x->{sign} eq '-inf') { # calculate 3 pi/4 => 2.3.. => 2 $y->bone(substr($y->{sign}, 0, 1)); $y->bmul($class->new(2)); } elsif ($x->{sign} eq '+inf') { # calculate pi/4 => 0.7 => 0 $y->bzero(); } else { # calculate pi/2 => 1.5 => 1 $y->bone(substr($y->{sign}, 0, 1)); } } else { if ($x->{sign} eq '+inf') { # calculate pi/4 => 0.7 => 0 $y->bzero(); } else { # PI => 3.1415.. => 3 $y->bone(substr($y->{sign}, 0, 1)); $y->bmul($class->new(3)); } } return $y; } return $upgrade->new($y)->batan2($upgrade->new($x), @r) if defined $upgrade; require Math::BigFloat; my $r = Math::BigFloat->new($y) ->batan2(Math::BigFloat->new($x), @r) ->as_int(); $x->{value} = $r->{value}; $x->{sign} = $r->{sign}; $x; } sub bsqrt { # calculate square root of $x my ($class, $x, @r) = ref($_[0]) ? (undef, @_) : objectify(1, @_); return $x if $x->modify('bsqrt'); return $x->bnan() if $x->{sign} !~ /^\+/; # -x or -inf or NaN => NaN return $x if $x->{sign} eq '+inf'; # sqrt(+inf) == inf return $upgrade->bsqrt($x, @r) if defined $upgrade; $x->{value} = $LIB->_sqrt($x->{value}); $x->round(@r); } sub broot { # calculate $y'th root of $x # set up parameters my ($class, $x, $y, @r) = (ref($_[0]), @_); $y = $class->new(2) unless defined $y; # objectify is costly, so avoid it if ((!ref($x)) \|\| (ref($x) ne ref($y))) { ($class, $x, $y, @r) = objectify(2, $class \|\| $class, @_); } return $x if $x->modify('broot'); # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0 return $x->bnan() if $x->{sign} !~ /^\+/ \|\| $y->is_zero() \|\| $y->{sign} !~ /^\+$/; return $x->round(@r) if $x->is_zero() \|\| $x->is_one() \|\| $x->is_inf() \|\| $y->is_one(); return $upgrade->new($x)->broot($upgrade->new($y), @r) if defined $upgrade; $x->{value} = $LIB->_root($x->{value}, $y->{value}); $x->round(@r); } sub bfac { # (BINT or num_str, BINT or num_str) return BINT # compute factorial number from $x, modify $x in place my ($class, $x, @r) = ref($_[0]) ? (undef, @_) : objectify(1, @_); return $x if $x->modify('bfac') \|\| $x->{sign} eq '+inf'; # inf => inf return $x->bnan() if $x->{sign} ne '+'; # NaN, <0 etc => NaN $x->{value} = $LIB->_fac($x->{value}); $x->round(@r); } sub bdfac { # compute double factorial, modify $x in place my ($class, $x, @r) = ref($_[0]) ? (undef, @_) : objectify(1, @_); return $x if $x->modify('bdfac') \|\| $x->{sign} eq '+inf'; # inf => inf return $x->bnan() if $x->{sign} ne '+'; # NaN, <0 etc => NaN croak("bdfac() requires a newer version of the $LIB library.") unless $LIB->can('_dfac'); $x->{value} = $LIB->_dfac($x->{value}); $x->round(@r); } sub bfib { # compute Fibonacci number(s) my ($class, $x, @r) = objectify(1, @_); croak("bfib() requires a newer version of the $LIB library.") unless $LIB->can('_fib'); return $x if $x->modify('bfib'); # List context. if (wantarray) { return () if $x -> is_nan(); croak("bfib() can't return an infinitely long list of numbers") if $x -> is_inf(); # Use the backend library to compute the first $x Fibonacci numbers. my @values = $LIB->_fib($x->{value}); # Make objects out of them. The last element in the array is the # invocand. for (my $i = 0 ; $i < $#values ; ++ $i) { my $fib = $class -> bzero(); $fib -> {value} = $values[$i]; $values[$i] = $fib; } $x -> {value} = $values[-1]; $values[-1] = $x; # If negative, insert sign as appropriate. if ($x -> is_neg()) { for (my $i = 2 ; $i <= $#values ; $i += 2) { $values[$i]{sign} = '-'; } } @values = map { $_ -> round(@r) } @values; return @values; } # Scalar context. else { return $x if $x->modify('bdfac') \|\| $x -> is_inf('+'); return $x->bnan() if $x -> is_nan() \|\| $x -> is_inf('-'); $x->{sign} = $x -> is_neg() && $x -> is_even() ? '-' : '+'; $x->{value} = $LIB->_fib($x->{value}); return $x->round(@r); } } sub blucas { # compute Lucas number(s) my ($class, $x, @r) = objectify(1, @_); croak("blucas() requires a newer version of the $LIB library.") unless $LIB->can('_lucas'); return $x if $x->modify('blucas'); # List context. if (wantarray) { return () if $x -> is_nan(); croak("blucas() can't return an infinitely long list of numbers") if $x -> is_inf(); # Use the backend library to compute the first $x Lucas numbers. my @values = $LIB->_lucas($x->{value}); # Make objects out of them. The last element in the array is the # invocand. for (my $i = 0 ; $i < $#values ; ++ $i) { my $lucas = $class -> bzero(); $lucas -> {value} = $values[$i]; $values[$i] = $lucas; } $x -> {value} = $values[-1]; $values[-1] = $x; # If negative, insert sign as appropriate. if ($x -> is_neg()) { for (my $i = 2 ; $i <= $#values ; $i += 2) { $values[$i]{sign} = '-'; } } @values = map { $_ -> round(@r) } @values; return @values; } # Scalar context. else { return $x if $x -> is_inf('+'); return $x->bnan() if $x -> is_nan() \|\| $x -> is_inf('-'); $x->{sign} = $x -> is_neg() && $x -> is_even() ? '-' : '+'; $x->{value} = $LIB->_lucas($x->{value}); return $x->round(@r); } } sub blsft { # (BINT or num_str, BINT or num_str) return BINT # compute x << y, base n, y >= 0 my ($class, $x, $y, $b, @r); # Objectify the base only when it is defined, since an undefined base, as # in $x->blsft(3) or $x->blog(3, undef) means use the default base 2. if (!ref($_[0]) && $_[0] =~ /^[A-Za-z]\|::/) { # E.g., Math::BigInt->blog(256, 5, 2) ($class, $x, $y, $b, @r) = defined $_[3] ? objectify(3, @_) : objectify(2, @_); } else { # E.g., Math::BigInt::blog(256, 5, 2) or $x->blog(5, 2) ($class, $x, $y, $b, @r) = defined $_[2] ? objectify(3, @_) : objectify(2, @_); } return $x if $x -> modify('blsft'); return $x -> bnan() if ($x -> {sign} !~ /^[+-]$/ \|\| $y -> {sign} !~ /^[+-]$/); return $x -> round(@r) if $y -> is_zero(); $b = defined($b) ? $b -> numify() : 2; # While some of the libraries support an arbitrarily large base, not all of # them do, so rather than returning an incorrect result in those cases, # disallow bases that don't work with all libraries. my $uintmax = ~0; croak("Base is too large.") if $b > $uintmax; return $x -> bnan() if $b <= 0 \|\| $y -> {sign} eq '-'; $x -> {value} = $LIB -> _lsft($x -> {value}, $y -> {value}, $b); $x -> round(@r); } sub brsft { # (BINT or num_str, BINT or num_str) return BINT # compute x >> y, base n, y >= 0 # set up parameters my ($class, $x, $y, $b, @r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, $b, @r) = objectify(2, @_); } return $x if $x -> modify('brsft'); return $x -> bnan() if ($x -> {sign} !~ /^[+-]$/ \|\| $y -> {sign} !~ /^[+-]$/); return $x -> round(@r) if $y -> is_zero(); return $x -> bzero(@r) if $x -> is_zero(); # 0 => 0 $b = 2 if !defined $b; return $x -> bnan() if $b <= 0 \|\| $y -> {sign} eq '-'; # this only works for negative numbers when shifting in base 2 if (($x -> {sign} eq '-') && ($b == 2)) { return $x -> round(@r) if $x -> is_one('-'); # -1 => -1 if (!$y -> is_one()) { # although this is O(NN) in calc (as_bin!) it is O(N) in Pari et # al but perhaps there is a better emulation for two's complement # shift... # if $y != 1, we must simulate it by doing: # convert to bin, flip all bits, shift, and be done $x -> binc(); # -3 => -2 my $bin = $x -> as_bin(); $bin =~ s/^-0b//; # strip '-0b' prefix $bin =~ tr/10/01/; # flip bits # now shift if ($y >= CORE::length($bin)) { $bin = '0'; # shifting to far right creates -1 # 0, because later increment makes # that 1, attached '-' makes it '-1' # because -1 >> x == -1 ! } else { $bin =~ s/.{$y}$//; # cut off at the right side $bin = '1' . $bin; # extend left side by one dummy '1' $bin =~ tr/10/01/; # flip bits back } my $res = $class -> new('0b' . $bin); # add prefix and convert back $res -> binc(); # remember to increment $x -> {value} = $res -> {value}; # take over value return $x -> round(@r); # we are done now, magic, isn't? } # x < 0, n == 2, y == 1 $x -> bdec(); # n == 2, but $y == 1: this fixes it } $x -> {value} = $LIB -> _rsft($x -> {value}, $y -> {value}, $b); $x -> round(@r); } ############################################################################### # Bitwise methods ############################################################################### sub band { #(BINT or num_str, BINT or num_str) return BINT # compute x & y # set up parameters my ($class, $x, $y, @r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, @r) = objectify(2, @_); } return $x if $x->modify('band'); $r[3] = $y; # no push! return $x->bnan() if ($x->{sign} !~ /^[+-]$/ \|\| $y->{sign} !~ /^[+-]$/); if ($x->{sign} eq '+' && $y->{sign} eq '+') { $x->{value} = $LIB->_and($x->{value}, $y->{value}); } else { ($x->{value}, $x->{sign}) = $LIB->_sand($x->{value}, $x->{sign}, $y->{value}, $y->{sign}); } return $x->round(@r); } sub bior { #(BINT or num_str, BINT or num_str) return BINT # compute x \| y # set up parameters my ($class, $x, $y, @r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, @r) = objectify(2, @_); } return $x if $x->modify('bior'); $r[3] = $y; # no push! return $x->bnan() if ($x->{sign} !~ /^[+-]$/ \|\| $y->{sign} !~ /^[+-]$/); if ($x->{sign} eq '+' && $y->{sign} eq '+') { $x->{value} = $LIB->_or($x->{value}, $y->{value}); } else { ($x->{value}, $x->{sign}) = $LIB->_sor($x->{value}, $x->{sign}, $y->{value}, $y->{sign}); } return $x->round(@r); } sub bxor { #(BINT or num_str, BINT or num_str) return BINT # compute x ^ y # set up parameters my ($class, $x, $y, @r) = (ref($_[0]), @_); # objectify is costly, so avoid it if ((!ref($_[0])) \|\| (ref($_[0]) ne ref($_[1]))) { ($class, $x, $y, @r) = objectify(2, @_); } return $x if $x->modify('bxor'); $r[3] = $y; # no push! return $x->bnan() if ($x->{sign} !~ /^[+-]$/ \|\| $y->{sign} !~ /^[+-]$/); if ($x->{sign} eq '+' && $y->{sign} eq '+') { $x->{value} = $LIB->_xor($x->{value}, $y->{value}); } else { ($x->{value}, $x->{sign}) = $LIB->_sxor($x->{value}, $x->{sign}, $y->{value}, $y->{sign}); } return $x->round(@r); } sub bnot { # (num_str or BINT) return BINT # represent ~x as twos-complement number # we don't need $class, so undef instead of ref($_[0]) make it slightly faster my ($class, $x) = ref($_[0]) ? (undef, @_) : objectify(1, @_); return $x if $x->modify('bnot'); $x->binc()->bneg(); # binc already does round } ############################################################################### # Rounding methods ############################################################################### sub round { # Round $self according to given parameters, or given second argument's # parameters or global defaults # for speed reasons, _find_round_parameters is embedded here: my ($self, $a, $p, $r, @args) = @_; # $a accuracy, if given by caller # $p precision, if given by caller # $r round_mode, if given by caller # @args all 'other' arguments (0 for unary, 1 for binary ops) my $class = ref($self); # find out class of argument(s) no strict 'refs'; # now pick $a or $p, but only if we have got "arguments" if (!defined $a) { foreach ($self, @args) { # take the defined one, or if both defined, the one that is smaller $a = $_->{_a} if (defined $_->{_a}) && (!defined $a \|\| $_->{_a} < $a); } } if (!defined $p) { # even if $a is defined, take $p, to signal error for both defined foreach ($self, @args) { # take the defined one, or if both defined, the one that is bigger # -2 > -3, and 3 > 2 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p \|\| $_->{_p} > $p); } } # if still none defined, use globals unless (defined $a \|\| defined $p) { $a = ${"$class\::accuracy"}; $p = ${"$class\::precision"}; } # A == 0 is useless, so undef it to signal no rounding $a = undef if defined $a && $a == 0; # no rounding today? return $self unless defined $a \|\| defined $p; # early out # set A and set P is an fatal error return $self->bnan() if defined $a && defined $p; $r = ${"$class\::round_mode"} unless defined $r; if ($r !~ /^(even\|odd\|[+-]inf\|zero\|trunc\|common)$/) { croak("Unknown round mode '$r'"); } # now round, by calling either bround or bfround: if (defined $a) { $self->bround(int($a), $r) if !defined $self->{_a} \|\| $self->{_a} >= $a; } else { # both can't be undefined due to early out $self->bfround(int($p), $r) if !defined $self->{_p} \|\| $self->{_p} <= $p; } # bround() or bfround() already called bnorm() if nec. $self; } sub bround { # accuracy: +$n preserve $n digits from left, # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF) # no-op for $n == 0 # and overwrite the rest with 0's, return normalized number # do not return $x->bnorm(), but $x my $x = shift; $x = __PACKAGE__->new($x) unless ref $x; my ($scale, $mode) = $x->_scale_a(@_); return $x if !defined $scale \|\| $x->modify('bround'); # no-op if ($x->is_zero() \|\| $scale == 0) { $x->{_a} = $scale if !defined $x->{_a} \|\| $x->{_a} > $scale; # 3 > 2 return $x; } return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN # we have fewer digits than we want to scale to my $len = $x->length(); # convert $scale to a scalar in case it is an object (put's a limit on the # number length, but this would already limited by memory constraints), makes # it faster $scale = $scale->numify() if ref ($scale); # scale < 0, but > -len (not >=!) if (($scale < 0 && $scale < -$len-1) \|\| ($scale >= $len)) { $x->{_a} = $scale if !defined $x->{_a} \|\| $x->{_a} > $scale; # 3 > 2 return $x; } # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or \|-6\| => 6 my ($pad, $digit_round, $digit_after); $pad = $len - $scale; $pad = abs($scale-1) if $scale < 0; # do not use digit(), it is very costly for binary => decimal # getting the entire string is also costly, but we need to do it only once my $xs = $LIB->_str($x->{value}); my $pl = -$pad-1; # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3 $digit_round = '0'; $digit_round = substr($xs, $pl, 1) if $pad <= $len; $pl++; $pl ++ if $pad >= $len; $digit_after = '0'; $digit_after = substr($xs, $pl, 1) if $pad > 0; # in case of 01234 we round down, for 6789 up, and only in case 5 we look # closer at the remaining digits of the original $x, remember decision my $round_up = 1; # default round up $round_up -- if ($mode eq 'trunc') \|\| # trunc by round down ($digit_after =~ /[01234]/) \|\| # round down anyway, # 6789 => round up ($digit_after eq '5') && # not 5000...0000 ($x->_scan_for_nonzero($pad, $xs, $len) == 0) && ( ($mode eq 'even') && ($digit_round =~ /[24680]/) \|\| ($mode eq 'odd') && ($digit_round =~ /[13579]/) \|\| ($mode eq '+inf') && ($x->{sign} eq '-') \|\| ($mode eq '-inf') && ($x->{sign} eq '+') \|\| ($mode eq 'zero') # round down if zero, sign adjusted below ); my $put_back = 0; # not yet modified if (($pad > 0) && ($pad <= $len)) { substr($xs, -$pad, $pad) = '0' x $pad; # replace with '00...' $put_back = 1; # need to put back } elsif ($pad > $len) { $x->bzero(); # round to '0' } if ($round_up) { # what gave test above? $put_back = 1; # need to put back $pad = $len, $xs = '0' x $pad if $scale < 0; # tlr: whack 0.51=>1.0 # we modify directly the string variant instead of creating a number and # adding it, since that is faster (we already have the string) my $c = 0; $pad ++; # for $pad == $len case while ($pad <= $len) { $c = substr($xs, -$pad, 1) + 1; $c = '0' if $c eq '10'; substr($xs, -$pad, 1) = $c; $pad++; last if $c != 0; # no overflow => early out } $xs = '1'.$xs if $c == 0; } $x->{value} = $LIB->_new($xs) if $put_back == 1; # put back, if needed $x->{_a} = $scale if $scale >= 0; if ($scale < 0) { $x->{_a} = $len+$scale; $x->{_a} = 0 if $scale < -$len; } $x; } sub bfround { # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.' # $n == 0 \|\| $n == 1 => round to integer my $x = shift; my $class = ref($x) \|\| $x; $x = $class->new($x) unless ref $x; my ($scale, $mode) = $x->_scale_p(@_); return $x if !defined $scale \|\| $x->modify('bfround'); # no-op # no-op for Math::BigInt objects if $n <= 0 $x->bround($x->length()-$scale, $mode) if $scale > 0; delete $x->{_a}; # delete to save memory $x->{_p} = $scale; # store new _p $x; } sub fround { # Exists to make life easier for switch between MBF and MBI (should we # autoload fxxx() like MBF does for bxxx()?) my $x = shift; $x = __PACKAGE__->new($x) unless ref $x; $x->bround(@_); } sub bfloor { # round towards minus infinity; no-op since it's already integer my ($class, $x, @r) = ref($_[0]) ? (undef, @_) : objectify(1, @_); $x->round(@r); } sub bceil { # round towards plus infinity; no-op since it's already int my ($class, $x, @r) = ref($_[0]) ? (undef, @_) : objectify(1, @_); $x->round(@r); } sub bint { # round towards zero; no-op since it's already integer my ($class, $x, @r) = ref($_[0]) ? (undef, @_) : objectify(1, @_); $x->round(@r); } ############################################################################### # Other mathematical methods ############################################################################### sub bgcd { # (BINT or num_str, BINT or num_str) return BINT # does not modify arguments, but returns new object # GCD -- Euclid's algorithm, variant C (Knuth Vol 3, pg 341 ff) my ($class, @args) = objectify(0, @_); my $x = shift @args; $x = ref($x) && $x -> isa($class) ? $x -> copy() : $class -> new($x); return $class->bnan() if $x->{sign} !~ /^[+-]$/; # x NaN? while (@args) { my $y = shift @args; $y = $class->new($y) unless ref($y) && $y -> isa($class); return $class->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN? $x->{value} = $LIB->_gcd($x->{value}, $y->{value}); last if $LIB->_is_one($x->{value}); } return $x -> babs(); } sub blcm { # (BINT or num_str, BINT or num_str) return BINT # does not modify arguments, but returns new object # Least Common Multiple my ($class, @args) = objectify(0, @_); my $x = shift @args; $x = ref($x) && $x -> isa($class) ? $x -> copy() : $class -> new($x); return $class->bnan() if $x->{sign} !~ /^[+-]$/; # x NaN? while (@args) { my $y = shift @args; $y = $class -> new($y) unless ref($y) && $y -> isa($class); return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y not integer $x -> {value} = $LIB->_lcm($x -> {value}, $y -> {value}); } return $x -> babs(); } ############################################################################### # Object property methods ############################################################################### sub sign { # return the sign of the number: +/-/-inf/+inf/NaN my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); $x->{sign}; } sub digit { # return the nth decimal digit, negative values count backward, 0 is right my ($class, $x, $n) = ref($_[0]) ? (undef, @_) : objectify(1, @_); $n = $n->numify() if ref($n); $LIB->_digit($x->{value}, $n \|\| 0); } sub bdigitsum { # like digitsum(), but assigns the result to the invocand my $x = shift; return $x if $x -> is_nan(); return $x -> bnan() if $x -> is_inf(); $x -> {value} = $LIB -> _digitsum($x -> {value}); $x -> {sign} = '+'; return $x; } sub digitsum { # compute sum of decimal digits and return it my $x = shift; my $class = ref $x; return $class -> bnan() if $x -> is_nan(); return $class -> bnan() if $x -> is_inf(); my $y = $class -> bzero(); $y -> {value} = $LIB -> _digitsum($x -> {value}); return $y; } sub length { my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); my $e = $LIB->_len($x->{value}); wantarray ? ($e, 0) : $e; } sub exponent { # return a copy of the exponent (here always 0, NaN or 1 for $m == 0) my ($class, $x) = ref($_[0]) ? (ref($_[0]), $_[0]) : objectify(1, @_); if ($x->{sign} !~ /^[+-]$/) { my $s = $x->{sign}; $s =~ s/^[+-]//; # NaN, -inf, +inf => NaN or inf return $class->new($s); } return $class->bzero() if $x->is_zero(); # 12300 => 2 trailing zeros => exponent is 2 $class->new($LIB->_zeros($x->{value})); } sub mantissa { # return the mantissa (compatible to Math::BigFloat, e.g. reduced) my ($class, $x) = ref($_[0]) ? (ref($_[0]), $_[0]) : objectify(1, @_); if ($x->{sign} !~ /^[+-]$/) { # for NaN, +inf, -inf: keep the sign return $class->new($x->{sign}); } my $m = $x->copy(); delete $m->{_p}; delete $m->{_a}; # that's a bit inefficient: my $zeros = $LIB->_zeros($m->{value}); $m->brsft($zeros, 10) if $zeros != 0; $m; } sub parts { # return a copy of both the exponent and the mantissa my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); ($x->mantissa(), $x->exponent()); } sub sparts { my $self = shift; my $class = ref $self; croak("sparts() is an instance method, not a class method") unless $class; # Not-a-number. if ($self -> is_nan()) { my $mant = $self -> copy(); # mantissa return $mant unless wantarray; # scalar context my $expo = $class -> bnan(); # exponent return ($mant, $expo); # list context } # Infinity. if ($self -> is_inf()) { my $mant = $self -> copy(); # mantissa return $mant unless wantarray; # scalar context my $expo = $class -> binf('+'); # exponent return ($mant, $expo); # list context } # Finite number. my $mant = $self -> copy(); my $nzeros = $LIB -> _zeros($mant -> {value}); $mant -> brsft($nzeros, 10) if $nzeros != 0; return $mant unless wantarray; my $expo = $class -> new($nzeros); return ($mant, $expo); } sub nparts { my $self = shift; my $class = ref $self; croak("nparts() is an instance method, not a class method") unless $class; # Not-a-number. if ($self -> is_nan()) { my $mant = $self -> copy(); # mantissa return $mant unless wantarray; # scalar context my $expo = $class -> bnan(); # exponent return ($mant, $expo); # list context } # Infinity. if ($self -> is_inf()) { my $mant = $self -> copy(); # mantissa return $mant unless wantarray; # scalar context my $expo = $class -> binf('+'); # exponent return ($mant, $expo); # list context } # Finite number. my ($mant, $expo) = $self -> sparts(); if ($mant -> bcmp(0)) { my ($ndigtot, $ndigfrac) = $mant -> length(); my $expo10adj = $ndigtot - $ndigfrac - 1; if ($expo10adj != 0) { return $upgrade -> new($self) -> nparts() if $upgrade; $mant -> bnan(); return $mant unless wantarray; $expo -> badd($expo10adj); return ($mant, $expo); } } return $mant unless wantarray; return ($mant, $expo); } sub eparts { my $self = shift; my $class = ref $self; croak("eparts() is an instance method, not a class method") unless $class; # Not-a-number and Infinity. return $self -> sparts() if $self -> is_nan() \|\| $self -> is_inf(); # Finite number. my ($mant, $expo) = $self -> sparts(); if ($mant -> bcmp(0)) { my $ndigmant = $mant -> length(); $expo -> badd($ndigmant); # $c is the number of digits that will be in the integer part of the # final mantissa. my $c = $expo -> copy() -> bdec() -> bmod(3) -> binc(); $expo -> bsub($c); if ($ndigmant > $c) { return $upgrade -> new($self) -> eparts() if $upgrade; $mant -> bnan(); return $mant unless wantarray; return ($mant, $expo); } $mant -> blsft($c - $ndigmant, 10); } return $mant unless wantarray; return ($mant, $expo); } sub dparts { my $self = shift; my $class = ref $self; croak("dparts() is an instance method, not a class method") unless $class; my $int = $self -> copy(); return $int unless wantarray; my $frc = $class -> bzero(); return ($int, $frc); } ############################################################################### # String conversion methods ############################################################################### sub bstr { my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); if ($x->{sign} ne '+' && $x->{sign} ne '-') { return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN return 'inf'; # +inf } my $str = $LIB->_str($x->{value}); return $x->{sign} eq '-' ? "-$str" : $str; } # Scientific notation with significand/mantissa as an integer, e.g., "12345" is # written as "1.2345e+4". sub bsstr { my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_); if ($x->{sign} ne '+' && $x->{sign} ne '-') { return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN return 'inf'; # +inf } my ($m, $e) = $x -> parts(); my $str = $LIB->_str($m->{value}) . 'e+' . $LIB->_str($e->{value}); return $x->{sign} eq '-' ? "-$str" : $str; } # Normalized notation, e.g., "12345" is written as "12345e+0". sub bnstr { my $x = shift; if ($x->{sign} ne '+' && $x->{sign} ne '-') { return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN return 'inf'; # +inf } return $x -> bstr() if $x -> is_nan() \|\| $x -> is_inf(); my ($mant, $expo) = $x -> parts(); # The "fraction posision" is the position (offset) for the decimal point # relative to the end of the digit string. my $fracpos = $mant -> length() - 1; if ($fracpos == 0) { my $str = $LIB->_str($mant->{value}) . "e+" . $LIB->_str($expo->{value}); return $x->{sign} eq '-' ? "-$str" : $str; } $expo += $fracpos; my $mantstr = $LIB->_str($mant -> {value}); substr($mantstr, -$fracpos, 0) = '.'; my $str = $mantstr . 'e+' . $LIB->_str($expo -> {value}); return $x->{sign} eq '-' ? "-$str" : $str; } # Engineering notation, e.g., "12345" is written as "12.345e+3". sub bestr { my $x = shift; if ($x->{sign} ne '+' && $x->{sign} ne '-') { return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN return 'inf'; # +inf } my ($mant, $expo) = $x -> parts(); my $sign = $mant -> sign(); $mant -> babs(); my $mantstr = $LIB->_str($mant -> {value}); my $mantlen = CORE::length($mantstr); my $dotidx = 1; $expo += $mantlen - 1; my $c = $expo -> copy() -> bmod(3); $expo -= $c; $dotidx += $c; if ($mantlen < $dotidx) { $mantstr .= "0" x ($dotidx - $mantlen); } elsif ($mantlen > $dotidx) { substr($mantstr, $dotidx, 0) = "."; } my $str = $mantstr . 'e+' . $LIB->_str($expo -> {value}); return $sign eq "-" ? "-$str" : $str; } # Decimal notation, e.g., "12345". sub bdstr { my $x = shift; if ($x->{sign} ne '+' && $x->{sign} ne '-') { return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN return 'inf'; # +inf } my $str = $LIB->_str($x->{value}); return $x->{sign} eq '-' ? "-$str" : $str; } sub to_hex { # return as hex string, with prefixed 0x my $x = shift; $x = __PACKAGE__->new($x) if !ref($x); return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc my $hex = $LIB->_to_hex($x->{value}); return $x->{sign} eq '-' ? "-$hex" : $hex; } sub to_oct { # return as octal string, with prefixed 0 my $x = shift; $x = __PACKAGE__->new($x) if !ref($x); return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc my $oct = $LIB->_to_oct($x->{value}); return $x->{sign} eq '-' ? "-$oct" : $oct; } sub to_bin { # return as binary string, with prefixed 0b my $x = shift; $x = __PACKAGE__->new($x) if !ref($x); return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc my $bin = $LIB->_to_bin($x->{value}); return $x->{sign} eq '-' ? "-$bin" : $bin; } sub to_bytes { # return a byte string my $x = shift; $x = __PACKAGE__->new($x) if !ref($x); croak("to_bytes() requires a finite, non-negative integer") if $x -> is_neg() \|\| ! $x -> is_int(); croak("to_bytes() requires a newer version of the $LIB library.") unless $LIB->can('_to_bytes'); return $LIB->_to_bytes($x->{value}); } sub to_base { # return a base anything string my $x = shift; $x = __PACKAGE__->new($x) if !ref($x); croak("the value to convert must be a finite, non-negative integer") if $x -> is_neg() \|\| !$x -> is_int(); my $base = shift; $base = __PACKAGE__->new($base) unless ref($base); croak("the base must be a finite integer >= 2") if $base < 2 \|\| ! $base -> is_int(); # If no collating sequence is given, pass some of the conversions to # methods optimized for those cases. if (! @_) { return $x -> to_bin() if $base == 2; return $x -> to_oct() if $base == 8; return uc $x -> to_hex() if $base == 16; return $x -> bstr() if $base == 10; } croak("to_base() requires a newer version of the $LIB library.") unless $LIB->can('_to_base'); return $LIB->_to_base($x->{value}, $base -> {value}, @_ ? shift() : ()); } sub as_hex { # return as hex string, with prefixed 0x my $x = shift; $x = __PACKAGE__->new($x) if !ref($x); return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc my $hex = $LIB->_as_hex($x->{value}); return $x->{sign} eq '-' ? "-$hex" : $hex; } sub as_oct { # return as octal string, with prefixed 0 my $x = shift; $x = __PACKAGE__->new($x) if !ref($x); return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc my $oct = $LIB->_as_oct($x->{value}); return $x->{sign} eq '-' ? "-$oct" : $oct; } sub as_bin { # return as binary string, with prefixed 0b my $x = shift; $x = __PACKAGE__->new($x) if !ref($x); return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc my $bin = $LIB->_as_bin($x->{value}); return $x->{sign} eq '-' ? "-$bin" : $bin; } as_bytes = \&to_bytes; ############################################################################### # Other conversion methods ############################################################################### sub numify { # Make a Perl scalar number from a Math::BigInt object. my $x = shift; $x = __PACKAGE__->new($x) unless ref $x; if ($x -> is_nan()) { require Math::Complex; my $inf = Math::Complex::Inf(); return $inf - $inf; } if ($x -> is_inf()) { require Math::Complex; my $inf = Math::Complex::Inf(); return $x -> is_negative() ? -$inf : $inf; } my $num = 0 + $LIB->_num($x->{value}); return $x->{sign} eq '-' ? -$num : $num; } ############################################################################### # Private methods and functions. ############################################################################### sub objectify { # Convert strings and "foreign objects" to the objects we want. # The first argument, $count, is the number of following arguments that # objectify() looks at and converts to objects. The first is a classname. # If the given count is 0, all arguments will be used. # After the count is read, objectify obtains the name of the class to which # the following arguments are converted. If the second argument is a # reference, use the reference type as the class name. Otherwise, if it is # a string that looks like a class name, use that. Otherwise, use $class. # Caller: Gives us: # # $x->badd(1); => ref x, scalar y # Class->badd(1, 2); => classname x (scalar), scalar x, scalar y # Class->badd(Class->(1), 2); => classname x (scalar), ref x, scalar y # Math::BigInt::badd(1, 2); => scalar x, scalar y # A shortcut for the common case $x->unary_op(), in which case the argument # list is (0, $x) or (1, $x). return (ref($_[1]), $_[1]) if @_ == 2 && ($_[0] \|\| 0) == 1 && ref($_[1]); # Check the context. unless (wantarray) { croak(__PACKAGE__ . "::objectify() needs list context"); } # Get the number of arguments to objectify. my $count = shift; # Initialize the output array. my @a = @_; # If the first argument is a reference, use that reference type as our # class name. Otherwise, if the first argument looks like a class name, # then use that as our class name. Otherwise, use the default class name. my $class; if (ref($a[0])) { # reference? $class = ref($a[0]); } elsif ($a[0] =~ /^[A-Z].::/) { # string with class name? $class = shift @a; } else { $class = __PACKAGE__; # default class name } $count \|\|= @a; unshift @a, $class; no strict 'refs'; # What we upgrade to, if anything. my $up = ${"$a[0]::upgrade"}; # Disable downgrading, because Math::BigFloat -> foo('1.0', '2.0') needs # floats. my $down; if (defined ${"$a[0]::downgrade"}) { $down = ${"$a[0]::downgrade"}; ${"$a[0]::downgrade"} = undef; } for my $i (1 .. $count) { my $ref = ref $a[$i]; # Perl scalars are fed to the appropriate constructor. unless ($ref) { $a[$i] = $a[0] -> new($a[$i]); next; } # If it is an object of the right class, all is fine. next if $ref -> isa($a[0]); # Upgrading is OK, so skip further tests if the argument is upgraded. if (defined $up && $ref -> isa($up)) { next; } # See if we can call one of the as_xxx() methods. We don't know whether # the as_xxx() method returns an object or a scalar, so re-check # afterwards. my $recheck = 0; if ($a[0] -> isa('Math::BigInt')) { if ($a[$i] -> can('as_int')) { $a[$i] = $a[$i] -> as_int(); $recheck = 1; } elsif ($a[$i] -> can('as_number')) { $a[$i] = $a[$i] -> as_number(); $recheck = 1; } } elsif ($a[0] -> isa('Math::BigFloat')) { if ($a[$i] -> can('as_float')) { $a[$i] = $a[$i] -> as_float(); $recheck = $1; } } # If we called one of the as_xxx() methods, recheck. if ($recheck) { $ref = ref($a[$i]); # Perl scalars are fed to the appropriate constructor. unless ($ref) { $a[$i] = $a[0] -> new($a[$i]); next; } # If it is an object of the right class, all is fine. next if $ref -> isa($a[0]); } # Last resort. $a[$i] = $a[0] -> new($a[$i]); } # Reset the downgrading. ${"$a[0]::downgrade"} = $down; return @a; } sub import { my $class = shift; $IMPORT++; # remember we did import() my @a; # unrecognized arguments my $warn_or_die = 0; # 0 - no warn, 1 - warn, 2 - die for (my $i = 0; $i <= $#_ ; $i++) { if ($_[$i] eq ':constant') { # this causes overlord er load to step in overload::constant integer => sub { $class->new(shift) }, binary => sub { $class->new(shift) }; } elsif ($_[$i] eq 'upgrade') { # this causes upgrading $upgrade = $_[$i+1]; # or undef to disable $i++; } elsif ($_[$i] =~ /^(lib\|try\|only)\z/) { # this causes a different low lib to take care... $LIB = $_[$i+1] \|\| ''; # try => 0 (no warn) # lib => 1 (warn on fallback) # only => 2 (die on fallback) $warn_or_die = 1 if $_[$i] eq 'lib'; $warn_or_die = 2 if $_[$i] eq 'only'; $i++; } else { push @a, $_[$i]; } } # any non :constant stuff is handled by our parent, Exporter if (@a > 0) { $class->SUPER::import(@a); # need it for subclasses $class->export_to_level(1, $class, @a); # need it for MBF } # try to load core math lib my @c = split /\s,\s/, $LIB; foreach (@c) { tr/a-zA-Z0-9://cd; # limit to sane characters } push @c, \'Calc' # if all fail, try these if $warn_or_die < 2; # but not for "only" $LIB = ''; # signal error foreach my $l (@c) { # fallback libraries are "marked" as \'string', extract string if nec. my $lib = $l; $lib = $$l if ref($l); next unless defined($lib) && CORE::length($lib); $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i; $lib =~ s/\.pm$//; my @parts = split /::/, $lib; # Math::BigInt => Math BigInt $parts[-1] .= '.pm'; # BigInt => BigInt.pm require File::Spec; my $file = File::Spec->catfile(@parts); eval { require $file; }; if ($@ eq '') { $lib->import(); $LIB = $lib; if ($warn_or_die > 0 && ref($l)) { my $msg = "Math::BigInt: couldn't load specified" . " math lib(s), fallback to $lib"; carp($msg) if $warn_or_die == 1; croak($msg) if $warn_or_die == 2; } last; # found a usable one, break } } if ($LIB eq '') { if ($warn_or_die == 2) { croak("Couldn't load specified math lib(s)" . " and fallback disallowed"); } else { croak("Couldn't load any math lib(s), not even fallback to Calc.pm"); } } # notify callbacks foreach my $class (keys %CALLBACKS) { &{$CALLBACKS{$class}}($LIB); } # import done } sub _register_callback { my ($class, $callback) = @_; if (ref($callback) ne 'CODE') { croak("$callback is not a coderef"); } $CALLBACKS{$class} = $callback; } sub _split_dec_string { my $str = shift; if ($str =~ s/ ^ # leading whitespace ( \s ) # optional sign ( [+-]? ) # significand ( \d+ (?: _ \d+ )* (?: \. (?: \d+ (?: _ \d+ )* )? )? \| \. \d+ (?: _ \d+ )* ) # optional exponent (?: [Ee] ( [+-]? ) ( \d+ (?: _ \d+ )* ) )? # trailing stuff ( \D .? )? \z //x) { my $leading = $1; my $significand_sgn = $2 \|\| '+'; my $significand_abs = $3; my $exponent_sgn = $4 \|\| '+'; my $exponent_abs = $5 \|\| '0'; my $trailing = $6; # Remove underscores and leading zeros. $significand_abs =~ tr/_//d; $exponent_abs =~ tr/_//d; $significand_abs =~ s/^0+(.)/$1/; $exponent_abs =~ s/^0+(.)/$1/; # If the significand contains a dot, remove it and adjust the exponent # accordingly. E.g., "1234.56789e+3" -> "123456789e-2" my $idx = index $significand_abs, '.'; if ($idx > -1) { $significand_abs =~ s/0+\z//; substr($significand_abs, $idx, 1) = ''; my $exponent = $exponent_sgn . $exponent_abs; $exponent .= $idx - CORE::length($significand_abs); $exponent_abs = abs $exponent; $exponent_sgn = $exponent < 0 ? '-' : '+'; } return($leading, $significand_sgn, $significand_abs, $exponent_sgn, $exponent_abs, $trailing); } return undef; } sub _split { # input: num_str; output: undef for invalid or # (\$mantissa_sign, \$mantissa_value, \$mantissa_fraction, # \$exp_sign, \$exp_value) # Internal, take apart a string and return the pieces. # Strip leading/trailing whitespace, leading zeros, underscore and reject # invalid input. my $x = shift; # strip white space at front, also extraneous leading zeros $x =~ s/^\s([-]?)0([0-9])/$1$2/g; # will not strip ' .2' $x =~ s/^\s+//; # but this will $x =~ s/\s+$//g; # strip white space at end # shortcut, if nothing to split, return early if ($x =~ /^[+-]?[0-9]+\z/) { $x =~ s/^([+-])0([0-9])/$2/; my $sign = $1 \|\| '+'; return (\$sign, \$x, \'', \'', \0); } # invalid starting char? return if $x !~ /^[+-]?(\.?[0-9]\|0b[0-1]\|0x[0-9a-fA-F])/; return Math::BigInt->from_hex($x) if $x =~ /^[+-]?0x/; # hex string return Math::BigInt->from_bin($x) if $x =~ /^[+-]?0b/; # binary string # strip underscores between digits $x =~ s/([0-9])_([0-9])/$1$2/g; $x =~ s/([0-9])_([0-9])/$1$2/g; # do twice for 1_2_3 # some possible inputs: # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 # 0e999 my ($m, $e, $last) = split /[Ee]/, $x; return if defined $last; # last defined => 1e2E3 or others $e = '0' if !defined $e \|\| $e eq ""; # sign, value for exponent, mantint, mantfrac my ($es, $ev, $mis, $miv, $mfv); # valid exponent? if ($e =~ /^([+-]?)0([0-9]+)$/) # strip leading zeros { $es = $1; $ev = $2; # valid mantissa? return if $m eq '.' \|\| $m eq ''; my ($mi, $mf, $lastf) = split /\./, $m; return if defined $lastf; # lastf defined => 1.2.3 or others $mi = '0' if !defined $mi; $mi .= '0' if $mi =~ /^[\-\+]?$/; $mf = '0' if !defined $mf \|\| $mf eq ''; if ($mi =~ /^([+-]?)0([0-9]+)$/) # strip leading zeros { $mis = $1 \|\| '+'; $miv = $2; return unless ($mf =~ /^([0-9]?)0$/); # strip trailing zeros $mfv = $1; # handle the 0e999 case here $ev = 0 if $miv eq '0' && $mfv eq ''; return (\$mis, \$miv, \$mfv, \$es, \$ev); } } return; # NaN, not a number } sub _trailing_zeros { # return the amount of trailing zeros in $x (as scalar) my $x = shift; $x = __PACKAGE__->new($x) unless ref $x; return 0 if $x->{sign} !~ /^[+-]$/; # NaN, inf, -inf etc $LIB->_zeros($x->{value}); # must handle odd values, 0 etc } sub _scan_for_nonzero { # internal, used by bround() to scan for non-zeros after a '5' my ($x, $pad, $xs, $len) = @_; return 0 if $len == 1; # "5" is trailed by invisible zeros my $follow = $pad - 1; return 0 if $follow > $len \|\| $follow < 1; # use the string form to check whether only '0's follow or not substr ($xs, -$follow) =~ /[^0]/ ? 1 : 0; } sub _find_round_parameters { # After any operation or when calling round(), the result is rounded by # regarding the A & P from arguments, local parameters, or globals. # !!!!!!! If you change this, remember to change round(), too! !!!!!!!!!! # This procedure finds the round parameters, but it is for speed reasons # duplicated in round. Otherwise, it is tested by the testsuite and used # by bdiv(). # returns ($self) or ($self, $a, $p, $r) - sets $self to NaN of both A and P # were requested/defined (locally or globally or both) my ($self, $a, $p, $r, @args) = @_; # $a accuracy, if given by caller # $p precision, if given by caller # $r round_mode, if given by caller # @args all 'other' arguments (0 for unary, 1 for binary ops) my $class = ref($self); # find out class of argument(s) no strict 'refs'; # convert to normal scalar for speed and correctness in inner parts $a = $a->can('numify') ? $a->numify() : "$a" if defined $a && ref($a); $p = $p->can('numify') ? $p->numify() : "$p" if defined $p && ref($p); # now pick $a or $p, but only if we have got "arguments" if (!defined $a) { foreach ($self, @args) { # take the defined one, or if both defined, the one that is smaller $a = $_->{_a} if (defined $_->{_a}) && (!defined $a \|\| $_->{_a} < $a); } } if (!defined $p) { # even if $a is defined, take $p, to signal error for both defined foreach ($self, @args) { # take the defined one, or if both defined, the one that is bigger # -2 > -3, and 3 > 2 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p \|\| $_->{_p} > $p); } } # if still none defined, use globals (#2) $a = ${"$class\::accuracy"} unless defined $a; $p = ${"$class\::precision"} unless defined $p; # A == 0 is useless, so undef it to signal no rounding $a = undef if defined $a && $a == 0; # no rounding today? return ($self) unless defined $a \|\| defined $p; # early out # set A and set P is an fatal error return ($self->bnan()) if defined $a && defined $p; # error $r = ${"$class\::round_mode"} unless defined $r; if ($r !~ /^(even\|odd\|[+-]inf\|zero\|trunc\|common)$/) { croak("Unknown round mode '$r'"); } $a = int($a) if defined $a; $p = int($p) if defined $p; ($self, $a, $p, $r); } ############################################################################### # this method returns 0 if the object can be modified, or 1 if not. # We use a fast constant sub() here, to avoid costly calls. Subclasses # may override it with special code (f.i. Math::BigInt::Constant does so) sub modify () { 0; } 1; __END__ =pod =head1 NAME Math::BigInt - Arbitrary size integer/float math package =head1 SYNOPSIS use Math::BigInt; # or make it faster with huge numbers: install (optional) # Math::BigInt::GMP and always use (it falls back to # pure Perl if the GMP library is not installed): # (See also the L<MATH LIBRARY> section!) # warns if Math::BigInt::GMP cannot be found use Math::BigInt lib => 'GMP'; # to suppress the warning use this: # use Math::BigInt try => 'GMP'; # dies if GMP cannot be loaded: # use Math::BigInt only => 'GMP'; my $str = '1234567890'; my @values = (64, 74, 18); my $n = 1; my $sign = '-'; # Configuration methods (may be used as class methods and instance methods) Math::BigInt->accuracy(); # get class accuracy Math::BigInt->accuracy($n); # set class accuracy Math::BigInt->precision(); # get class precision Math::BigInt->precision($n); # set class precision Math::BigInt->round_mode(); # get class rounding mode Math::BigInt->round_mode($m); # set global round mode, must be one of # 'even', 'odd', '+inf', '-inf', 'zero', # 'trunc', or 'common' Math::BigInt->config(); # return hash with configuration # Constructor methods (when the class methods below are used as instance # methods, the value is assigned the invocand) $x = Math::BigInt->new($str); # defaults to 0 $x = Math::BigInt->new('0x123'); # from hexadecimal $x = Math::BigInt->new('0b101'); # from binary $x = Math::BigInt->from_hex('cafe'); # from hexadecimal $x = Math::BigInt->from_oct('377'); # from octal $x = Math::BigInt->from_bin('1101'); # from binary $x = Math::BigInt->from_base('why', 36); # from any base $x = Math::BigInt->bzero(); # create a +0 $x = Math::BigInt->bone(); # create a +1 $x = Math::BigInt->bone('-'); # create a -1 $x = Math::BigInt->binf(); # create a +inf $x = Math::BigInt->binf('-'); # create a -inf $x = Math::BigInt->bnan(); # create a Not-A-Number $x = Math::BigInt->bpi(); # returns pi $y = $x->copy(); # make a copy (unlike $y = $x) $y = $x->as_int(); # return as a Math::BigInt # Boolean methods (these don't modify the invocand) $x->is_zero(); # if $x is 0 $x->is_one(); # if $x is +1 $x->is_one("+"); # ditto $x->is_one("-"); # if $x is -1 $x->is_inf(); # if $x is +inf or -inf $x->is_inf("+"); # if $x is +inf $x->is_inf("-"); # if $x is -inf $x->is_nan(); # if $x is NaN $x->is_positive(); # if $x > 0 $x->is_pos(); # ditto $x->is_negative(); # if $x < 0 $x->is_neg(); # ditto $x->is_odd(); # if $x is odd $x->is_even(); # if $x is even $x->is_int(); # if $x is an integer # Comparison methods $x->bcmp($y); # compare numbers (undef, < 0, == 0, > 0) $x->bacmp($y); # compare absolutely (undef, < 0, == 0, > 0) $x->beq($y); # true if and only if $x == $y $x->bne($y); # true if and only if $x != $y $x->blt($y); # true if and only if $x < $y $x->ble($y); # true if and only if $x <= $y $x->bgt($y); # true if and only if $x > $y $x->bge($y); # true if and only if $x >= $y # Arithmetic methods $x->bneg(); # negation $x->babs(); # absolute value $x->bsgn(); # sign function (-1, 0, 1, or NaN) $x->bnorm(); # normalize (no-op) $x->binc(); # increment $x by 1 $x->bdec(); # decrement $x by 1 $x->badd($y); # addition (add $y to $x) $x->bsub($y); # subtraction (subtract $y from $x) $x->bmul($y); # multiplication (multiply $x by $y) $x->bmuladd($y,$z); # $x = $x * $y + $z $x->bdiv($y); # division (floored), set $x to quotient # return (quo,rem) or quo if scalar $x->btdiv($y); # division (truncated), set $x to quotient # return (quo,rem) or quo if scalar $x->bmod($y); # modulus (x % y) $x->btmod($y); # modulus (truncated) $x->bmodinv($mod); # modular multiplicative inverse $x->bmodpow($y,$mod); # modular exponentiation (($x $y) % $mod) $x->bpow($y); # power of arguments (x y) $x->blog(); # logarithm of $x to base e (Euler's number) $x->blog($base); # logarithm of $x to base $base (e.g., base 2) $x->bexp(); # calculate e ** $x where e is Euler's number $x->bnok($y); # x over y (binomial coefficient n over k) $x->buparrow($n, $y); # Knuth's up-arrow notation $x->backermann($y); # the Ackermann function $x->bsin(); # sine $x->bcos(); # cosine $x->batan(); # inverse tangent $x->batan2($y); # two-argument inverse tangent $x->bsqrt(); # calculate square root $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root) $x->bfac(); # factorial of $x (1234..$x) $x->blsft($n); # left shift $n places in base 2 $x->blsft($n,$b); # left shift $n places in base $b # returns (quo,rem) or quo (scalar context) $x->brsft($n); # right shift $n places in base 2 $x->brsft($n,$b); # right shift $n places in base $b # returns (quo,rem) or quo (scalar context) # Bitwise methods $x->band($y); # bitwise and $x->bior($y); # bitwise inclusive or $x->bxor($y); # bitwise exclusive or $x->bnot(); # bitwise not (two's complement) # Rounding methods $x->round($A,$P,$mode); # round to accuracy or precision using # rounding mode $mode $x->bround($n); # accuracy: preserve $n digits $x->bfround($n); # $n > 0: round to $nth digit left of dec. point # $n < 0: round to $nth digit right of dec. point $x->bfloor(); # round towards minus infinity $x->bceil(); # round towards plus infinity $x->bint(); # round towards zero # Other mathematical methods $x->bgcd($y); # greatest common divisor $x->blcm($y); # least common multiple # Object property methods (do not modify the invocand) $x->sign(); # the sign, either +, - or NaN $x->digit($n); # the nth digit, counting from the right $x->digit(-$n); # the nth digit, counting from the left $x->length(); # return number of digits in number ($xl,$f) = $x->length(); # length of number and length of fraction # part, latter is always 0 digits long # for Math::BigInt objects $x->mantissa(); # return (signed) mantissa as a Math::BigInt $x->exponent(); # return exponent as a Math::BigInt $x->parts(); # return (mantissa,exponent) as a Math::BigInt $x->sparts(); # mantissa and exponent (as integers) $x->nparts(); # mantissa and exponent (normalised) $x->eparts(); # mantissa and exponent (engineering notation) $x->dparts(); # integer and fraction part # Conversion methods (do not modify the invocand) $x->bstr(); # decimal notation, possibly zero padded $x->bsstr(); # string in scientific notation with integers $x->bnstr(); # string in normalized notation $x->bestr(); # string in engineering notation $x->bdstr(); # string in decimal notation $x->to_hex(); # as signed hexadecimal string $x->to_bin(); # as signed binary string $x->to_oct(); # as signed octal string $x->to_bytes(); # as byte string $x->to_base($b); # as string in any base $x->as_hex(); # as signed hexadecimal string with prefixed 0x $x->as_bin(); # as signed binary string with prefixed 0b $x->as_oct(); # as signed octal string with prefixed 0 # Other conversion methods $x->numify(); # return as scalar (might overflow or underflow) =head1 DESCRIPTION Math::BigInt provides support for arbitrary precision integers. Overloading is also provided for Perl operators. =head2 Input Input values to these routines may be any scalar number or string that looks like a number and represents an integer. =over =item * Leading and trailing whitespace is ignored. =item * Leading and trailing zeros are ignored. =item * If the string has a "0x" prefix, it is interpreted as a hexadecimal number. =item * If the string has a "0b" prefix, it is interpreted as a binary number. =item * One underline is allowed between any two digits. =item * If the string can not be interpreted, NaN is returned. =back Octal numbers are typically prefixed by "0", but since leading zeros are stripped, these methods can not automatically recognize octal numbers, so use the constructor from_oct() to interpret octal strings. Some examples of valid string input Input string Resulting value 123 123 1.23e2 123 12300e-2 123 0xcafe 51966 0b1101 13 67_538_754 67538754 -4_5_6.7_8_9e+0_1_0 -4567890000000 Input given as scalar numbers might lose precision. Quote your input to ensure that no digits are lost: $x = Math::BigInt->new( 56789012345678901234 ); # bad $x = Math::BigInt->new('56789012345678901234'); # good Currently, Math::BigInt->new() defaults to 0, while Math::BigInt->new('') results in 'NaN'. This might change in the future, so use always the following explicit forms to get a zero or NaN: $zero = Math::BigInt->bzero(); $nan = Math::BigInt->bnan(); =head2 Output Output values are usually Math::BigInt objects. Boolean operators C<is_zero()>, C<is_one()>, C<is_inf()>, etc. return true or false. Comparison operators C<bcmp()> and C<bacmp()>) return -1, 0, 1, or undef. =head1 METHODS =head2 Configuration methods Each of the methods below (except config(), accuracy() and precision()) accepts three additional parameters. These arguments C<$A>, C<$P> and C<$R> are C<accuracy>, C<precision> and C<round_mode>. Please see the section about L</ACCURACY and PRECISION> for more information. Setting a class variable effects all object instance that are created afterwards. =over =item accuracy() Math::BigInt->accuracy(5); # set class accuracy $x->accuracy(5); # set instance accuracy $A = Math::BigInt->accuracy(); # get class accuracy $A = $x->accuracy(); # get instance accuracy Set or get the accuracy, i.e., the number of significant digits. The accuracy must be an integer. If the accuracy is set to C<undef>, no rounding is done. Alternatively, one can round the results explicitly using one of L</round()>, L</bround()> or L</bfround()> or by passing the desired accuracy to the method as an additional parameter: my $x = Math::BigInt->new(30000); my $y = Math::BigInt->new(7); print scalar $x->copy()->bdiv($y, 2); # prints 4300 print scalar $x->copy()->bdiv($y)->bround(2); # prints 4300 Please see the section about L</ACCURACY and PRECISION> for further details. $y = Math::BigInt->new(1234567); # $y is not rounded Math::BigInt->accuracy(4); # set class accuracy to 4 $x = Math::BigInt->new(1234567); # $x is rounded automatically print "$x $y"; # prints "1235000 1234567" print $x->accuracy(); # prints "4" print $y->accuracy(); # also prints "4", since # class accuracy is 4 Math::BigInt->accuracy(5); # set class accuracy to 5 print $x->accuracy(); # prints "4", since instance # accuracy is 4 print $y->accuracy(); # prints "5", since no instance # accuracy, and class accuracy is 5 Note: Each class has it's own globals separated from Math::BigInt, but it is possible to subclass Math::BigInt and make the globals of the subclass aliases to the ones from Math::BigInt. =item precision() Math::BigInt->precision(-2); # set class precision $x->precision(-2); # set instance precision $P = Math::BigInt->precision(); # get class precision $P = $x->precision(); # get instance precision Set or get the precision, i.e., the place to round relative to the decimal point. The precision must be a integer. Setting the precision to $P means that each number is rounded up or down, depending on the rounding mode, to the nearest multiple of 10*$P. If the precision is set to C<undef>, no rounding is done. You might want to use L</accuracy()> instead. With L</accuracy()> you set the number of digits each result should have, with L</precision()> you set the place where to round. Please see the section about L</ACCURACY and PRECISION> for further details. $y = Math::BigInt->new(1234567); # $y is not rounded Math::BigInt->precision(4); # set class precision to 4 $x = Math::BigInt->new(1234567); # $x is rounded automatically print $x; # prints "1230000" Note: Each class has its own globals separated from Math::BigInt, but it is possible to subclass Math::BigInt and make the globals of the subclass aliases to the ones from Math::BigInt. =item div_scale() Set/get the fallback accuracy. This is the accuracy used when neither accuracy nor precision is set explicitly. It is used when a computation might otherwise attempt to return an infinite number of digits. =item round_mode() Set/get the rounding mode. =item upgrade() Set/get the class for upgrading. When a computation might result in a non-integer, the operands are upgraded to this class. This is used for instance by L<bignum>. The default is C<undef>, thus the following operation creates a Math::BigInt, not a Math::BigFloat: my $i = Math::BigInt->new(123); my $f = Math::BigFloat->new('123.1'); print $i + $f, "\n"; # prints 246 =item downgrade() Set/get the class for downgrading. The default is C<undef>. Downgrading is not done by Math::BigInt. =item modify() $x->modify('bpowd'); This method returns 0 if the object can be modified with the given operation, or 1 if not. This is used for instance by L<Math::BigInt::Constant>. =item config() Math::BigInt->config("trap_nan" => 1); # set $accu = Math::BigInt->config("accuracy"); # get Set or get class variables. Read-only parameters are marked as RO. Read-write parameters are marked as RW. The following parameters are supported. Parameter RO/RW Description Example ============================================================ lib RO Name of the math backend library Math::BigInt::Calc lib_version RO Version of the math backend library 0.30 class RO The class of config you just called Math::BigRat version RO version number of the class you used 0.10 upgrade RW To which class numbers are upgraded undef downgrade RW To which class numbers are downgraded undef precision RW Global precision undef accuracy RW Global accuracy undef round_mode RW Global round mode even div_scale RW Fallback accuracy for division etc. 40 trap_nan RW Trap NaNs undef trap_inf RW Trap +inf/-inf undef =back =head2 Constructor methods =over =item new() $x = Math::BigInt->new($str,$A,$P,$R); Creates a new Math::BigInt object from a scalar or another Math::BigInt object. The input is accepted as decimal, hexadecimal (with leading '0x') or binary (with leading '0b'). See L</Input> for more info on accepted input formats. =item from_hex() $x = Math::BigInt->from_hex("0xcafe"); # input is hexadecimal Interpret input as a hexadecimal string. A "0x" or "x" prefix is optional. A single underscore character may be placed right after the prefix, if present, or between any two digits. If the input is invalid, a NaN is returned. =item from_oct() $x = Math::BigInt->from_oct("0775"); # input is octal Interpret the input as an octal string and return the corresponding value. A "0" (zero) prefix is optional. A single underscore character may be placed right after the prefix, if present, or between any two digits. If the input is invalid, a NaN is returned. =item from_bin() $x = Math::BigInt->from_bin("0b10011"); # input is binary Interpret the input as a binary string. A "0b" or "b" prefix is optional. A single underscore character may be placed right after the prefix, if present, or between any two digits. If the input is invalid, a NaN is returned. =item from_bytes() $x = Math::BigInt->from_bytes("\xf3\x6b"); # $x = 62315 Interpret the input as a byte string, assuming big endian byte order. The output is always a non-negative, finite integer. In some special cases, from_bytes() matches the conversion done by unpack(): $b = "\x4e"; # one char byte string $x = Math::BigInt->from_bytes($b); # = 78 $y = unpack "C", $b; # ditto, but scalar $b = "\xf3\x6b"; # two char byte string $x = Math::BigInt->from_bytes($b); # = 62315 $y = unpack "S>", $b; # ditto, but scalar $b = "\x2d\xe0\x49\xad"; # four char byte string $x = Math::BigInt->from_bytes($b); # = 769673645 $y = unpack "L>", $b; # ditto, but scalar $b = "\x2d\xe0\x49\xad\x2d\xe0\x49\xad"; # eight char byte string $x = Math::BigInt->from_bytes($b); # = 3305723134637787565 $y = unpack "Q>", $b; # ditto, but scalar =item from_base() Given a string, a base, and an optional collation sequence, interpret the string as a number in the given base. The collation sequence describes the value of each character in the string. If a collation sequence is not given, a default collation sequence is used. If the base is less than or equal to 36, the collation sequence is the string consisting of the 36 characters "0" to "9" and "A" to "Z". In this case, the letter case in the input is ignored. If the base is greater than 36, and smaller than or equal to 62, the collation sequence is the string consisting of the 62 characters "0" to "9", "A" to "Z", and "a" to "z". A base larger than 62 requires the collation sequence to be specified explicitly. These examples show standard binary, octal, and hexadecimal conversion. All cases return 250. $x = Math::BigInt->from_base("11111010", 2); $x = Math::BigInt->from_base("372", 8); $x = Math::BigInt->from_base("fa", 16); When the base is less than or equal to 36, and no collation sequence is given, the letter case is ignored, so both of these also return 250: $x = Math::BigInt->from_base("6Y", 16); $x = Math::BigInt->from_base("6y", 16); When the base greater than 36, and no collation sequence is given, the default collation sequence contains both uppercase and lowercase letters, so the letter case in the input is not ignored: $x = Math::BigInt->from_base("6S", 37); # $x is 250 $x = Math::BigInt->from_base("6s", 37); # $x is 276 $x = Math::BigInt->from_base("121", 3); # $x is 16 $x = Math::BigInt->from_base("XYZ", 36); # $x is 44027 $x = Math::BigInt->from_base("Why", 42); # $x is 58314 The collation sequence can be any set of unique characters. These two cases are equivalent $x = Math::BigInt->from_base("100", 2, "01"); # $x is 4 $x = Math::BigInt->from_base("\|--", 2, "-\|"); # $x is 4 =item bzero() $x = Math::BigInt->bzero(); $x->bzero(); Returns a new Math::BigInt object representing zero. If used as an instance method, assigns the value to the invocand. =item bone() $x = Math::BigInt->bone(); # +1 $x = Math::BigInt->bone("+"); # +1 $x = Math::BigInt->bone("-"); # -1 $x->bone(); # +1 $x->bone("+"); # +1 $x->bone('-'); # -1 Creates a new Math::BigInt object representing one. The optional argument is either '-' or '+', indicating whether you want plus one or minus one. If used as an instance method, assigns the value to the invocand. =item binf() $x = Math::BigInt->binf($sign); Creates a new Math::BigInt object representing infinity. The optional argument is either '-' or '+', indicating whether you want infinity or minus infinity. If used as an instance method, assigns the value to the invocand. $x->binf(); $x->binf('-'); =item bnan() $x = Math::BigInt->bnan(); Creates a new Math::BigInt object representing NaN (Not A Number). If used as an instance method, assigns the value to the invocand. $x->bnan(); =item bpi() $x = Math::BigInt->bpi(100); # 3 $x->bpi(100); # 3 Creates a new Math::BigInt object representing PI. If used as an instance method, assigns the value to the invocand. With Math::BigInt this always returns 3. If upgrading is in effect, returns PI, rounded to N digits with the current rounding mode: use Math::BigFloat; use Math::BigInt upgrade => "Math::BigFloat"; print Math::BigInt->bpi(3), "\n"; # 3.14 print Math::BigInt->bpi(100), "\n"; # 3.1415.... =item copy() $x->copy(); # make a true copy of $x (unlike $y = $x) =item as_int() =item as_number() These methods are called when Math::BigInt encounters an object it doesn't know how to handle. For instance, assume $x is a Math::BigInt, or subclass thereof, and $y is defined, but not a Math::BigInt, or subclass thereof. If you do $x -> badd($y); $y needs to be converted into an object that $x can deal with. This is done by first checking if $y is something that $x might be upgraded to. If that is the case, no further attempts are made. The next is to see if $y supports the method C<as_int()>. If it does, C<as_int()> is called, but if it doesn't, the next thing is to see if $y supports the method C<as_number()>. If it does, C<as_number()> is called. The method C<as_int()> (and C<as_number()>) is expected to return either an object that has the same class as $x, a subclass thereof, or a string that C<ref($x)-E<gt>new()> can parse to create an object. C<as_number()> is an alias to C<as_int()>. C<as_number> was introduced in v1.22, while C<as_int()> was introduced in v1.68. In Math::BigInt, C<as_int()> has the same effect as C<copy()>. =back =head2 Boolean methods None of these methods modify the invocand object. =over =item is_zero() $x->is_zero(); # true if $x is 0 Returns true if the invocand is zero and false otherwise. =item is_one( [ SIGN ]) $x->is_one(); # true if $x is +1 $x->is_one("+"); # ditto $x->is_one("-"); # true if $x is -1 Returns true if the invocand is one and false otherwise. =item is_finite() $x->is_finite(); # true if $x is not +inf, -inf or NaN Returns true if the invocand is a finite number, i.e., it is neither +inf, -inf, nor NaN. =item is_inf( [ SIGN ] ) $x->is_inf(); # true if $x is +inf $x->is_inf("+"); # ditto $x->is_inf("-"); # true if $x is -inf Returns true if the invocand is infinite and false otherwise. =item is_nan() $x->is_nan(); # true if $x is NaN =item is_positive() =item is_pos() $x->is_positive(); # true if > 0 $x->is_pos(); # ditto Returns true if the invocand is positive and false otherwise. A C<NaN> is neither positive nor negative. =item is_negative() =item is_neg() $x->is_negative(); # true if < 0 $x->is_neg(); # ditto Returns true if the invocand is negative and false otherwise. A C<NaN> is neither positive nor negative. =item is_non_positive() $x->is_non_positive(); # true if <= 0 Returns true if the invocand is negative or zero. =item is_non_negative() $x->is_non_negative(); # true if >= 0 Returns true if the invocand is positive or zero. =item is_odd() $x->is_odd(); # true if odd, false for even Returns true if the invocand is odd and false otherwise. C<NaN>, C<+inf>, and C<-inf> are neither odd nor even. =item is_even() $x->is_even(); # true if $x is even Returns true if the invocand is even and false otherwise. C<NaN>, C<+inf>, C<-inf> are not integers and are neither odd nor even. =item is_int() $x->is_int(); # true if $x is an integer Returns true if the invocand is an integer and false otherwise. C<NaN>, C<+inf>, C<-inf> are not integers. =back =head2 Comparison methods None of these methods modify the invocand object. Note that a C<NaN> is neither less than, greater than, or equal to anything else, even a C<NaN>. =over =item bcmp() $x->bcmp($y); Returns -1, 0, 1 depending on whether $x is less than, equal to, or grater than $y. Returns undef if any operand is a NaN. =item bacmp() $x->bacmp($y); Returns -1, 0, 1 depending on whether the absolute value of $x is less than, equal to, or grater than the absolute value of $y. Returns undef if any operand is a NaN. =item beq() $x -> beq($y); Returns true if and only if $x is equal to $y, and false otherwise. =item bne() $x -> bne($y); Returns true if and only if $x is not equal to $y, and false otherwise. =item blt() $x -> blt($y); Returns true if and only if $x is equal to $y, and false otherwise. =item ble() $x -> ble($y); Returns true if and only if $x is less than or equal to $y, and false otherwise. =item bgt() $x -> bgt($y); Returns true if and only if $x is greater than $y, and false otherwise. =item bge() $x -> bge($y); Returns true if and only if $x is greater than or equal to $y, and false otherwise. =back =head2 Arithmetic methods These methods modify the invocand object and returns it. =over =item bneg() $x->bneg(); Negate the number, e.g. change the sign between '+' and '-', or between '+inf' and '-inf', respectively. Does nothing for NaN or zero. =item babs() $x->babs(); Set the number to its absolute value, e.g. change the sign from '-' to '+' and from '-inf' to '+inf', respectively. Does nothing for NaN or positive numbers. =item bsgn() $x->bsgn(); Signum function. Set the number to -1, 0, or 1, depending on whether the number is negative, zero, or positive, respectively. Does not modify NaNs. =item bnorm() $x->bnorm(); # normalize (no-op) Normalize the number. This is a no-op and is provided only for backwards compatibility. =item binc() $x->binc(); # increment x by 1 =item bdec() $x->bdec(); # decrement x by 1 =item badd() $x->badd($y); # addition (add $y to $x) =item bsub() $x->bsub($y); # subtraction (subtract $y from $x) =item bmul() $x->bmul($y); # multiplication (multiply $x by $y) =item bmuladd() $x->bmuladd($y,$z); Multiply $x by $y, and then add $z to the result, This method was added in v1.87 of Math::BigInt (June 2007). =item bdiv() $x->bdiv($y); # divide, set $x to quotient Divides $x by $y by doing floored division (F-division), where the quotient is the floored (rounded towards negative infinity) quotient of the two operands. In list context, returns the quotient and the remainder. The remainder is either zero or has the same sign as the second operand. In scalar context, only the quotient is returned. The quotient is always the greatest integer less than or equal to the real-valued quotient of the two operands, and the remainder (when it is non-zero) always has the same sign as the second operand; so, for example, 1 / 4 => ( 0, 1) 1 / -4 => (-1, -3) -3 / 4 => (-1, 1) -3 / -4 => ( 0, -3) -11 / 2 => (-5, 1) 11 / -2 => (-5, -1) The behavior of the overloaded operator % agrees with the behavior of Perl's built-in % operator (as documented in the perlop manpage), and the equation $x == ($x / $y) $y + ($x % $y) holds true for any finite $x and finite, non-zero $y. Perl's "use integer" might change the behaviour of % and / for scalars. This is because under 'use integer' Perl does what the underlying C library thinks is right, and this varies. However, "use integer" does not change the way things are done with Math::BigInt objects. =item btdiv() $x->btdiv($y); # divide, set $x to quotient Divides $x by $y by doing truncated division (T-division), where quotient is the truncated (rouneded towards zero) quotient of the two operands. In list context, returns the quotient and the remainder. The remainder is either zero or has the same sign as the first operand. In scalar context, only the quotient is returned. =item bmod() $x->bmod($y); # modulus (x % y) Returns $x modulo $y, i.e., the remainder after floored division (F-division). This method is like Perl's % operator. See L</bdiv()>. =item btmod() $x->btmod($y); # modulus Returns the remainer after truncated division (T-division). See L</btdiv()>. =item bmodinv() $x->bmodinv($mod); # modular multiplicative inverse Returns the multiplicative inverse of C<$x> modulo C<$mod>. If $y = $x -> copy() -> bmodinv($mod) then C<$y> is the number closest to zero, and with the same sign as C<$mod>, satisfying ($x * $y) % $mod = 1 % $mod If C<$x> and C<$y> are non-zero, they must be relative primes, i.e., C<bgcd($y, $mod)==1>. 'C<NaN>' is returned when no modular multiplicative inverse exists. =item bmodpow() $num->bmodpow($exp,$mod); # modular exponentiation # ($num$exp % $mod) Returns the value of C<$num> taken to the power C<$exp> in the modulus C<$mod> using binary exponentiation. C<bmodpow> is far superior to writing $num $exp % $mod because it is much faster - it reduces internal variables into the modulus whenever possible, so it operates on smaller numbers. C<bmodpow> also supports negative exponents. bmodpow($num, -1, $mod) is exactly equivalent to bmodinv($num, $mod) =item bpow() $x->bpow($y); # power of arguments (x y) C<bpow()> (and the rounding functions) now modifies the first argument and returns it, unlike the old code which left it alone and only returned the result. This is to be consistent with C<badd()> etc. The first three modifies $x, the last one won't: print bpow($x,$i),"\n"; # modify $x print $x->bpow($i),"\n"; # ditto print $x = $i,"\n"; # the same print $x $i,"\n"; # leave $x alone The form C<$x = $y> is faster than C<$x = $x $y;>, though. =item blog() $x->blog($base, $accuracy); # logarithm of x to the base $base If C<$base> is not defined, Euler's number (e) is used: print $x->blog(undef, 100); # log(x) to 100 digits =item bexp() $x->bexp($accuracy); # calculate e X Calculates the expression C<e ** $x> where C<e> is Euler's number. This method was added in v1.82 of Math::BigInt (April 2007). See also L</blog()>. =item bnok() $x->bnok($y); # x over y (binomial coefficient n over k) Calculates the binomial coefficient n over k, also called the "choose" function, which is ( n ) n! \| \| = -------- ( k ) k!(n-k)! when n and k are non-negative. This method implements the full Kronenburg extension (Kronenburg, M.J. "The Binomial Coefficient for Negative Arguments." 18 May 2011. http://arxiv.org/abs/1105.3689/) illustrated by the following pseudo-code: if n >= 0 and k >= 0: return binomial(n, k) if k >= 0: return (-1)^kbinomial(-n+k-1, k) if k <= n: return (-1)^(n-k)binomial(-k-1, n-k) else return 0 The behaviour is identical to the behaviour of the Maple and Mathematica function for negative integers n, k. =item buparrow() =item uparrow() $a -> buparrow($n, $b); # modifies $a $x = $a -> uparrow($n, $b); # does not modify $a This method implements Knuth's up-arrow notation, where $n is a non-negative integer representing the number of up-arrows. $n = 0 gives multiplication, $n = 1 gives exponentiation, $n = 2 gives tetration, $n = 3 gives hexation etc. The following illustrates the relation between the first values of $n. See L<https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation>. =item backermann() =item ackermann() $m -> backermann($n); # modifies $a $x = $m -> ackermann($n); # does not modify $a This method implements the Ackermann function: / n + 1 if m = 0 A(m, n) = \| A(m-1, 1) if m > 0 and n = 0 \ A(m-1, A(m, n-1)) if m > 0 and n > 0 Its value grows rapidly, even for small inputs. For example, A(4, 2) is an integer of 19729 decimal digits. See https://en.wikipedia.org/wiki/Ackermann_function =item bsin() my $x = Math::BigInt->new(1); print $x->bsin(100), "\n"; Calculate the sine of $x, modifying $x in place. In Math::BigInt, unless upgrading is in effect, the result is truncated to an integer. This method was added in v1.87 of Math::BigInt (June 2007). =item bcos() my $x = Math::BigInt->new(1); print $x->bcos(100), "\n"; Calculate the cosine of $x, modifying $x in place. In Math::BigInt, unless upgrading is in effect, the result is truncated to an integer. This method was added in v1.87 of Math::BigInt (June 2007). =item batan() my $x = Math::BigFloat->new(0.5); print $x->batan(100), "\n"; Calculate the arcus tangens of $x, modifying $x in place. In Math::BigInt, unless upgrading is in effect, the result is truncated to an integer. This method was added in v1.87 of Math::BigInt (June 2007). =item batan2() my $x = Math::BigInt->new(1); my $y = Math::BigInt->new(1); print $y->batan2($x), "\n"; Calculate the arcus tangens of C<$y> divided by C<$x>, modifying $y in place. In Math::BigInt, unless upgrading is in effect, the result is truncated to an integer. This method was added in v1.87 of Math::BigInt (June 2007). =item bsqrt() $x->bsqrt(); # calculate square root C<bsqrt()> returns the square root truncated to an integer. If you want a better approximation of the square root, then use: $x = Math::BigFloat->new(12); Math::BigFloat->precision(0); Math::BigFloat->round_mode('even'); print $x->copy->bsqrt(),"\n"; # 4 Math::BigFloat->precision(2); print $x->bsqrt(),"\n"; # 3.46 print $x->bsqrt(3),"\n"; # 3.464 =item broot() $x->broot($N); Calculates the N'th root of C<$x>. =item bfac() $x->bfac(); # factorial of $x (1234..$x) Returns the factorial of C<$x>, i.e., the product of all positive integers up to and including C<$x>. =item bdfac() $x->bdfac(); # double factorial of $x (1234..$x) Returns the double factorial of C<$x>. If C<$x> is an even integer, returns the product of all positive, even integers up to and including C<$x>, i.e., 246...$x. If C<$x> is an odd integer, returns the product of all positive, odd integers, i.e., 135...$x. =item bfib() $F = $n->bfib(); # a single Fibonacci number @F = $n->bfib(); # a list of Fibonacci numbers In scalar context, returns a single Fibonacci number. In list context, returns a list of Fibonacci numbers. The invocand is the last element in the output. The Fibonacci sequence is defined by F(0) = 0 F(1) = 1 F(n) = F(n-1) + F(n-2) In list context, F(0) and F(n) is the first and last number in the output, respectively. For example, if $n is 12, then C<< @F = $n->bfib() >> returns the following values, F(0) to F(12): 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 The sequence can also be extended to negative index n using the re-arranged recurrence relation F(n-2) = F(n) - F(n-1) giving the bidirectional sequence n -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 F(n) 13 -8 5 -3 2 -1 1 0 1 1 2 3 5 8 13 If $n is -12, the following values, F(0) to F(12), are returned: 0, 1, -1, 2, -3, 5, -8, 13, -21, 34, -55, 89, -144 =item blucas() $F = $n->blucas(); # a single Lucas number @F = $n->blucas(); # a list of Lucas numbers In scalar context, returns a single Lucas number. In list context, returns a list of Lucas numbers. The invocand is the last element in the output. The Lucas sequence is defined by L(0) = 2 L(1) = 1 L(n) = L(n-1) + L(n-2) In list context, L(0) and L(n) is the first and last number in the output, respectively. For example, if $n is 12, then C<< @L = $n->blucas() >> returns the following values, L(0) to L(12): 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322 The sequence can also be extended to negative index n using the re-arranged recurrence relation L(n-2) = L(n) - L(n-1) giving the bidirectional sequence n -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 L(n) 29 -18 11 -7 4 -3 1 2 1 3 4 7 11 18 29 If $n is -12, the following values, L(0) to L(-12), are returned: 2, 1, -3, 4, -7, 11, -18, 29, -47, 76, -123, 199, -322 =item brsft() $x->brsft($n); # right shift $n places in base 2 $x->brsft($n, $b); # right shift $n places in base $b The latter is equivalent to $x -> bdiv($b -> copy() -> bpow($n)) =item blsft() $x->blsft($n); # left shift $n places in base 2 $x->blsft($n, $b); # left shift $n places in base $b The latter is equivalent to $x -> bmul($b -> copy() -> bpow($n)) =back =head2 Bitwise methods =over =item band() $x->band($y); # bitwise and =item bior() $x->bior($y); # bitwise inclusive or =item bxor() $x->bxor($y); # bitwise exclusive or =item bnot() $x->bnot(); # bitwise not (two's complement) Two's complement (bitwise not). This is equivalent to, but faster than, $x->binc()->bneg(); =back =head2 Rounding methods =over =item round() $x->round($A,$P,$round_mode); Round $x to accuracy C<$A> or precision C<$P> using the round mode C<$round_mode>. =item bround() $x->bround($N); # accuracy: preserve $N digits Rounds $x to an accuracy of $N digits. =item bfround() $x->bfround($N); Rounds to a multiple of 10*$N. Examples: Input N Result 123456.123456 3 123500 123456.123456 2 123450 123456.123456 -2 123456.12 123456.123456 -3 123456.123 =item bfloor() $x->bfloor(); Round $x towards minus infinity, i.e., set $x to the largest integer less than or equal to $x. =item bceil() $x->bceil(); Round $x towards plus infinity, i.e., set $x to the smallest integer greater than or equal to $x). =item bint() $x->bint(); Round $x towards zero. =back =head2 Other mathematical methods =over =item bgcd() $x -> bgcd($y); # GCD of $x and $y $x -> bgcd($y, $z, ...); # GCD of $x, $y, $z, ... Returns the greatest common divisor (GCD). =item blcm() $x -> blcm($y); # LCM of $x and $y $x -> blcm($y, $z, ...); # LCM of $x, $y, $z, ... Returns the least common multiple (LCM). =back =head2 Object property methods =over =item sign() $x->sign(); Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN. If you want $x to have a certain sign, use one of the following methods: $x->babs(); # '+' $x->babs()->bneg(); # '-' $x->bnan(); # 'NaN' $x->binf(); # '+inf' $x->binf('-'); # '-inf' =item digit() $x->digit($n); # return the nth digit, counting from right If C<$n> is negative, returns the digit counting from left. =item digitsum() $x->digitsum(); Computes the sum of the base 10 digits and returns it. =item bdigitsum() $x->bdigitsum(); Computes the sum of the base 10 digits and assigns the result to the invocand. =item length() $x->length(); ($xl, $fl) = $x->length(); Returns the number of digits in the decimal representation of the number. In list context, returns the length of the integer and fraction part. For Math::BigInt objects, the length of the fraction part is always 0. The following probably doesn't do what you expect: $c = Math::BigInt->new(123); print $c->length(),"\n"; # prints 30 It prints both the number of digits in the number and in the fraction part since print calls C<length()> in list context. Use something like: print scalar $c->length(),"\n"; # prints 3 =item mantissa() $x->mantissa(); Return the signed mantissa of $x as a Math::BigInt. =item exponent() $x->exponent(); Return the exponent of $x as a Math::BigInt. =item parts() $x->parts(); Returns the significand (mantissa) and the exponent as integers. In Math::BigFloat, both are returned as Math::BigInt objects. =item sparts() Returns the significand (mantissa) and the exponent as integers. In scalar context, only the significand is returned. The significand is the integer with the smallest absolute value. The output of C<sparts()> corresponds to the output from C<bsstr()>. In Math::BigInt, this method is identical to C<parts()>. =item nparts() Returns the significand (mantissa) and exponent corresponding to normalized notation. In scalar context, only the significand is returned. For finite non-zero numbers, the significand's absolute value is greater than or equal to 1 and less than 10. The output of C<nparts()> corresponds to the output from C<bnstr()>. In Math::BigInt, if the significand can not be represented as an integer, upgrading is performed or NaN is returned. =item eparts() Returns the significand (mantissa) and exponent corresponding to engineering notation. In scalar context, only the significand is returned. For finite non-zero numbers, the significand's absolute value is greater than or equal to 1 and less than 1000, and the exponent is a multiple of 3. The output of C<eparts()> corresponds to the output from C<bestr()>. In Math::BigInt, if the significand can not be represented as an integer, upgrading is performed or NaN is returned. =item dparts() Returns the integer part and the fraction part. If the fraction part can not be represented as an integer, upgrading is performed or NaN is returned. The output of C<dparts()> corresponds to the output from C<bdstr()>. =back =head2 String conversion methods =over =item bstr() Returns a string representing the number using decimal notation. In Math::BigFloat, the output is zero padded according to the current accuracy or precision, if any of those are defined. =item bsstr() Returns a string representing the number using scientific notation where both the significand (mantissa) and the exponent are integers. The output corresponds to the output from C<sparts()>. 123 is returned as "123e+0" 1230 is returned as "123e+1" 12300 is returned as "123e+2" 12000 is returned as "12e+3" 10000 is returned as "1e+4" =item bnstr() Returns a string representing the number using normalized notation, the most common variant of scientific notation. For finite non-zero numbers, the absolute value of the significand is greater than or equal to 1 and less than 10. The output corresponds to the output from C<nparts()>. 123 is returned as "1.23e+2" 1230 is returned as "1.23e+3" 12300 is returned as "1.23e+4" 12000 is returned as "1.2e+4" 10000 is returned as "1e+4" =item bestr() Returns a string representing the number using engineering notation. For finite non-zero numbers, the absolute value of the significand is greater than or equal to 1 and less than 1000, and the exponent is a multiple of 3. The output corresponds to the output from C<eparts()>. 123 is returned as "123e+0" 1230 is returned as "1.23e+3" 12300 is returned as "12.3e+3" 12000 is returned as "12e+3" 10000 is returned as "10e+3" =item bdstr() Returns a string representing the number using decimal notation. The output corresponds to the output from C<dparts()>. 123 is returned as "123" 1230 is returned as "1230" 12300 is returned as "12300" 12000 is returned as "12000" 10000 is returned as "10000" =item to_hex() $x->to_hex(); Returns a hexadecimal string representation of the number. See also from_hex(). =item to_bin() $x->to_bin(); Returns a binary string representation of the number. See also from_bin(). =item to_oct() $x->to_oct(); Returns an octal string representation of the number. See also from_oct(). =item to_bytes() $x = Math::BigInt->new("1667327589"); $s = $x->to_bytes(); # $s = "cafe" Returns a byte string representation of the number using big endian byte order. The invocand must be a non-negative, finite integer. See also from_bytes(). =item to_base() $x = Math::BigInt->new("250"); $x->to_base(2); # returns "11111010" $x->to_base(8); # returns "372" $x->to_base(16); # returns "fa" Returns a string representation of the number in the given base. If a collation sequence is given, the collation sequence determines which characters are used in the output. Here are some more examples $x = Math::BigInt->new("16")->to_base(3); # returns "121" $x = Math::BigInt->new("44027")->to_base(36); # returns "XYZ" $x = Math::BigInt->new("58314")->to_base(42); # returns "Why" $x = Math::BigInt->new("4")->to_base(2, "-\|"); # returns "\|--" See from_base() for information and examples. =item as_hex() $x->as_hex(); As, C<to_hex()>, but with a "0x" prefix. =item as_bin() $x->as_bin(); As, C<to_bin()>, but with a "0b" prefix. =item as_oct() $x->as_oct(); As, C<to_oct()>, but with a "0" prefix. =item as_bytes() This is just an alias for C<to_bytes()>. =back =head2 Other conversion methods =over =item numify() print $x->numify(); Returns a Perl scalar from $x. It is used automatically whenever a scalar is needed, for instance in array index operations. =back =head1 ACCURACY and PRECISION Math::BigInt and Math::BigFloat have full support for accuracy and precision based rounding, both automatically after every operation, as well as manually. This section describes the accuracy/precision handling in Math::BigInt and Math::BigFloat as it used to be and as it is now, complete with an explanation of all terms and abbreviations. Not yet implemented things (but with correct description) are marked with '!', things that need to be answered are marked with '?'. In the next paragraph follows a short description of terms used here (because these may differ from terms used by others people or documentation). During the rest of this document, the shortcuts A (for accuracy), P (for precision), F (fallback) and R (rounding mode) are be used. =head2 Precision P Precision is a fixed number of digits before (positive) or after (negative) the decimal point. For example, 123.45 has a precision of -2. 0 means an integer like 123 (or 120). A precision of 2 means at least two digits to the left of the decimal point are zero, so 123 with P = 1 becomes 120. Note that numbers with zeros before the decimal point may have different precisions, because 1200 can have P = 0, 1 or 2 (depending on what the initial value was). It could also have p < 0, when the digits after the decimal point are zero. The string output (of floating point numbers) is padded with zeros: Initial value P A Result String ------------------------------------------------------------ 1234.01 -3 1000 1000 1234 -2 1200 1200 1234.5 -1 1230 1230 1234.001 1 1234 1234.0 1234.01 0 1234 1234 1234.01 2 1234.01 1234.01 1234.01 5 1234.01 1234.01000 For Math::BigInt objects, no padding occurs. =head2 Accuracy A Number of significant digits. Leading zeros are not counted. A number may have an accuracy greater than the non-zero digits when there are zeros in it or trailing zeros. For example, 123.456 has A of 6, 10203 has 5, 123.0506 has 7, 123.45000 has 8 and 0.000123 has 3. The string output (of floating point numbers) is padded with zeros: Initial value P A Result String ------------------------------------------------------------ 1234.01 3 1230 1230 1234.01 6 1234.01 1234.01 1234.1 8 1234.1 1234.1000 For Math::BigInt objects, no padding occurs. =head2 Fallback F When both A and P are undefined, this is used as a fallback accuracy when dividing numbers. =head2 Rounding mode R When rounding a number, different 'styles' or 'kinds' of rounding are possible. (Note that random rounding, as in Math::Round, is not implemented.) =head3 Directed rounding These round modes always round in the same direction. =over =item 'trunc' Round towards zero. Remove all digits following the rounding place, i.e., replace them with zeros. Thus, 987.65 rounded to tens (P=1) becomes 980, and rounded to the fourth significant digit becomes 987.6 (A=4). 123.456 rounded to the second place after the decimal point (P=-2) becomes 123.46. This corresponds to the IEEE 754 rounding mode 'roundTowardZero'. =back =head3 Rounding to nearest These rounding modes round to the nearest digit. They differ in how they determine which way to round in the ambiguous case when there is a tie. =over =item 'even' Round towards the nearest even digit, e.g., when rounding to nearest integer, -5.5 becomes -6, 4.5 becomes 4, but 4.501 becomes 5. This corresponds to the IEEE 754 rounding mode 'roundTiesToEven'. =item 'odd' Round towards the nearest odd digit, e.g., when rounding to nearest integer, 4.5 becomes 5, -5.5 becomes -5, but 5.501 becomes 6. This corresponds to the IEEE 754 rounding mode 'roundTiesToOdd'. =item '+inf' Round towards plus infinity, i.e., always round up. E.g., when rounding to the nearest integer, 4.5 becomes 5, -5.5 becomes -5, and 4.501 also becomes 5. This corresponds to the IEEE 754 rounding mode 'roundTiesToPositive'. =item '-inf' Round towards minus infinity, i.e., always round down. E.g., when rounding to the nearest integer, 4.5 becomes 4, -5.5 becomes -6, but 4.501 becomes 5. This corresponds to the IEEE 754 rounding mode 'roundTiesToNegative'. =item 'zero' Round towards zero, i.e., round positive numbers down and negative numbers up. E.g., when rounding to the nearest integer, 4.5 becomes 4, -5.5 becomes -5, but 4.501 becomes 5. This corresponds to the IEEE 754 rounding mode 'roundTiesToZero'. =item 'common' Round away from zero, i.e., round to the number with the largest absolute value. E.g., when rounding to the nearest integer, -1.5 becomes -2, 1.5 becomes 2 and 1.49 becomes 1. This corresponds to the IEEE 754 rounding mode 'roundTiesToAway'. =back The handling of A & P in MBI/MBF (the old core code shipped with Perl versions <= 5.7.2) is like this: =over =item Precision bfround($p) is able to round to $p number of digits after the decimal point * otherwise P is unused =item Accuracy (significant digits) * bround($a) rounds to $a significant digits * only bdiv() and bsqrt() take A as (optional) parameter + other operations simply create the same number (bneg etc), or more (bmul) of digits + rounding/truncating is only done when explicitly calling one of bround or bfround, and never for Math::BigInt (not implemented) * bsqrt() simply hands its accuracy argument over to bdiv. * the documentation and the comment in the code indicate two different ways on how bdiv() determines the maximum number of digits it should calculate, and the actual code does yet another thing POD: max($Math::BigFloat::div_scale,length(dividend)+length(divisor)) Comment: result has at most max(scale, length(dividend), length(divisor)) digits Actual code: scale = max(scale, length(dividend)-1,length(divisor)-1); scale += length(divisor) - length(dividend); So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3). Actually, the 'difference' added to the scale is cal- culated from the number of "significant digits" in dividend and divisor, which is derived by looking at the length of the man- tissa. Which is wrong, since it includes the + sign (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange assumption that 124 has 3 significant digits, while 120/7 will get you '17', not '17.1' since 120 is thought to have 2 signif- icant digits. The rounding after the division then uses the remainder and $y to determine whether it must round up or down. ? I have no idea which is the right way. That's why I used a slightly more ? simple scheme and tweaked the few failing testcases to match it. =back This is how it works now: =over =item Setting/Accessing * You can set the A global via Math::BigInt->accuracy() or Math::BigFloat->accuracy() or whatever class you are using. * You can also set P globally by using Math::SomeClass->precision() likewise. * Globals are classwide, and not inherited by subclasses. * to undefine A, use Math::SomeCLass->accuracy(undef); * to undefine P, use Math::SomeClass->precision(undef); * Setting Math::SomeClass->accuracy() clears automatically Math::SomeClass->precision(), and vice versa. * To be valid, A must be > 0, P can have any value. * If P is negative, this means round to the P'th place to the right of the decimal point; positive values mean to the left of the decimal point. P of 0 means round to integer. * to find out the current global A, use Math::SomeClass->accuracy() * to find out the current global P, use Math::SomeClass->precision() * use $x->accuracy() respective $x->precision() for the local setting of $x. * Please note that $x->accuracy() respective $x->precision() return eventually defined global A or P, when $x's A or P is not set. =item Creating numbers * When you create a number, you can give the desired A or P via: $x = Math::BigInt->new($number,$A,$P); * Only one of A or P can be defined, otherwise the result is NaN * If no A or P is give ($x = Math::BigInt->new($number) form), then the globals (if set) will be used. Thus changing the global defaults later on will not change the A or P of previously created numbers (i.e., A and P of $x will be what was in effect when $x was created) * If given undef for A and P, NO rounding will occur, and the globals will NOT be used. This is used by subclasses to create numbers without suffering rounding in the parent. Thus a subclass is able to have its own globals enforced upon creation of a number by using $x = Math::BigInt->new($number,undef,undef): use Math::BigInt::SomeSubclass; use Math::BigInt; Math::BigInt->accuracy(2); Math::BigInt::SomeSubClass->accuracy(3); $x = Math::BigInt::SomeSubClass->new(1234); $x is now 1230, and not 1200. A subclass might choose to implement this otherwise, e.g. falling back to the parent's A and P. =item Usage * If A or P are enabled/defined, they are used to round the result of each operation according to the rules below * Negative P is ignored in Math::BigInt, since Math::BigInt objects never have digits after the decimal point * Math::BigFloat uses Math::BigInt internally, but setting A or P inside Math::BigInt as globals does not tamper with the parts of a Math::BigFloat. A flag is used to mark all Math::BigFloat numbers as 'never round'. =item Precedence * It only makes sense that a number has only one of A or P at a time. If you set either A or P on one object, or globally, the other one will be automatically cleared. * If two objects are involved in an operation, and one of them has A in effect, and the other P, this results in an error (NaN). * A takes precedence over P (Hint: A comes before P). If neither of them is defined, nothing is used, i.e. the result will have as many digits as it can (with an exception for bdiv/bsqrt) and will not be rounded. * There is another setting for bdiv() (and thus for bsqrt()). If neither of A or P is defined, bdiv() will use a fallback (F) of $div_scale digits. If either the dividend's or the divisor's mantissa has more digits than the value of F, the higher value will be used instead of F. This is to limit the digits (A) of the result (just consider what would happen with unlimited A and P in the case of 1/3 :-) * bdiv will calculate (at least) 4 more digits than required (determined by A, P or F), and, if F is not used, round the result (this will still fail in the case of a result like 0.12345000000001 with A or P of 5, but this can not be helped - or can it?) * Thus you can have the math done by on Math::Big* class in two modi: + never round (this is the default): This is done by setting A and P to undef. No math operation will round the result, with bdiv() and bsqrt() as exceptions to guard against overflows. You must explicitly call bround(), bfround() or round() (the latter with parameters). Note: Once you have rounded a number, the settings will 'stick' on it and 'infect' all other numbers engaged in math operations with it, since local settings have the highest precedence. So, to get SaferRound[tm], use a copy() before rounding like this: $x = Math::BigFloat->new(12.34); $y = Math::BigFloat->new(98.76); $z = $x * $y; # 1218.6984 print $x->copy()->bround(3); # 12.3 (but A is now 3!) $z = $x * $y; # still 1218.6984, without # copy would have been 1210! + round after each op: After each single operation (except for testing like is_zero()), the method round() is called and the result is rounded appropriately. By setting proper values for A and P, you can have all-the-same-A or all-the-same-P modes. For example, Math::Currency might set A to undef, and P to -2, globally. ?Maybe an extra option that forbids local A & P settings would be in order, ?so that intermediate rounding does not 'poison' further math? =item Overriding globals * you will be able to give A, P and R as an argument to all the calculation routines; the second parameter is A, the third one is P, and the fourth is R (shift right by one for binary operations like badd). P is used only if the first parameter (A) is undefined. These three parameters override the globals in the order detailed as follows, i.e. the first defined value wins: (local: per object, global: global default, parameter: argument to sub) + parameter A + parameter P + local A (if defined on both of the operands: smaller one is taken) + local P (if defined on both of the operands: bigger one is taken) + global A + global P + global F * bsqrt() will hand its arguments to bdiv(), as it used to, only now for two arguments (A and P) instead of one =item Local settings * You can set A or P locally by using $x->accuracy() or $x->precision() and thus force different A and P for different objects/numbers. * Setting A or P this way immediately rounds $x to the new value. * $x->accuracy() clears $x->precision(), and vice versa. =item Rounding * the rounding routines will use the respective global or local settings. bround() is for accuracy rounding, while bfround() is for precision * the two rounding functions take as the second parameter one of the following rounding modes (R): 'even', 'odd', '+inf', '-inf', 'zero', 'trunc', 'common' * you can set/get the global R by using Math::SomeClass->round_mode() or by setting $Math::SomeClass::round_mode * after each operation, $result->round() is called, and the result may eventually be rounded (that is, if A or P were set either locally, globally or as parameter to the operation) * to manually round a number, call $x->round($A,$P,$round_mode); this will round the number by using the appropriate rounding function and then normalize it. * rounding modifies the local settings of the number: $x = Math::BigFloat->new(123.456); $x->accuracy(5); $x->bround(4); Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy() will be 4 from now on. =item Default values * R: 'even' * F: 40 * A: undef * P: undef =item Remarks * The defaults are set up so that the new code gives the same results as the old code (except in a few cases on bdiv): + Both A and P are undefined and thus will not be used for rounding after each operation. + round() is thus a no-op, unless given extra parameters A and P =back =head1 Infinity and Not a Number While Math::BigInt has extensive handling of inf and NaN, certain quirks remain. =over =item oct()/hex() These perl routines currently (as of Perl v.5.8.6) cannot handle passed inf. te@linux:~> perl -wle 'print 2 3333' Inf te@linux:~> perl -wle 'print 2 3333 == 2 3333' 1 te@linux:~> perl -wle 'print oct(2 3333)' 0 te@linux:~> perl -wle 'print hex(2 3333)' Illegal hexadecimal digit 'I' ignored at -e line 1. 0 The same problems occur if you pass them Math::BigInt->binf() objects. Since overloading these routines is not possible, this cannot be fixed from Math::BigInt. =back =head1 INTERNALS You should neither care about nor depend on the internal representation; it might change without notice. Use B<ONLY> method calls like C<< $x->sign(); >> instead relying on the internal representation. =head2 MATH LIBRARY Math with the numbers is done (by default) by a module called C<Math::BigInt::Calc>. This is equivalent to saying: use Math::BigInt try => 'Calc'; You can change this backend library by using: use Math::BigInt try => 'GMP'; B<Note>: General purpose packages should not be explicit about the library to use; let the script author decide which is best. If your script works with huge numbers and Calc is too slow for them, you can also for the loading of one of these libraries and if none of them can be used, the code dies: use Math::BigInt only => 'GMP,Pari'; The following would first try to find Math::BigInt::Foo, then Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc: use Math::BigInt try => 'Foo,Math::BigInt::Bar'; The library that is loaded last is used. Note that this can be overwritten at any time by loading a different library, and numbers constructed with different libraries cannot be used in math operations together. =head3 What library to use? B<Note>: General purpose packages should not be explicit about the library to use; let the script author decide which is best. L<Math::BigInt::GMP> and L<Math::BigInt::Pari> are in cases involving big numbers much faster than Calc, however it is slower when dealing with very small numbers (less than about 20 digits) and when converting very large numbers to decimal (for instance for printing, rounding, calculating their length in decimal etc). So please select carefully what library you want to use. Different low-level libraries use different formats to store the numbers. However, you should B<NOT> depend on the number having a specific format internally. See the respective math library module documentation for further details. =head2 SIGN The sign is either '+', '-', 'NaN', '+inf' or '-inf'. A sign of 'NaN' is used to represent the result when input arguments are not numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively minus infinity. You get '+inf' when dividing a positive number by 0, and '-inf' when dividing any negative number by 0. =head1 EXAMPLES use Math::BigInt; sub bigint { Math::BigInt->new(shift); } $x = Math::BigInt->bstr("1234") # string "1234" $x = "$x"; # same as bstr() $x = Math::BigInt->bneg("1234"); # Math::BigInt "-1234" $x = Math::BigInt->babs("-12345"); # Math::BigInt "12345" $x = Math::BigInt->bnorm("-0.00"); # Math::BigInt "0" $x = bigint(1) + bigint(2); # Math::BigInt "3" $x = bigint(1) + "2"; # ditto (auto-Math::BigIntify of "2") $x = bigint(1); # Math::BigInt "1" $x = $x + 5 / 2; # Math::BigInt "3" $x = $x 3; # Math::BigInt "27" $x = 2; # Math::BigInt "54" $x = Math::BigInt->new(0); # Math::BigInt "0" $x--; # Math::BigInt "-1" $x = Math::BigInt->badd(4,5) # Math::BigInt "9" print $x->bsstr(); # 9e+0 Examples for rounding: use Math::BigFloat; use Test::More; $x = Math::BigFloat->new(123.4567); $y = Math::BigFloat->new(123.456789); Math::BigFloat->accuracy(4); # no more A than 4 is ($x->copy()->bround(),123.4); # even rounding print $x->copy()->bround(),"\n"; # 123.4 Math::BigFloat->round_mode('odd'); # round to odd print $x->copy()->bround(),"\n"; # 123.5 Math::BigFloat->accuracy(5); # no more A than 5 Math::BigFloat->round_mode('odd'); # round to odd print $x->copy()->bround(),"\n"; # 123.46 $y = $x->copy()->bround(4),"\n"; # A = 4: 123.4 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4 Math::BigFloat->accuracy(undef); # A not important now Math::BigFloat->precision(2); # P important print $x->copy()->bnorm(),"\n"; # 123.46 print $x->copy()->bround(),"\n"; # 123.46 Examples for converting: my $x = Math::BigInt->new('0b1'.'01' x 123); print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n"; =head1 Autocreating constants After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal and binary constants in the given scope are converted to C<Math::BigInt>. This conversion happens at compile time. In particular, perl -MMath::BigInt=:constant -e 'print 2100,"\n"' prints the integer value of C<2100>. Note that without conversion of constants the expression 2100 is calculated using Perl scalars. Please note that strings and floating point constants are not affected, so that use Math::BigInt qw/:constant/; $x = 1234567890123456789012345678901234567890 + 123456789123456789; $y = '1234567890123456789012345678901234567890' + '123456789123456789'; does not give you what you expect. You need an explicit Math::BigInt->new() around one of the operands. You should also quote large constants to protect loss of precision: use Math::BigInt; $x = Math::BigInt->new('1234567889123456789123456789123456789'); Without the quotes Perl would convert the large number to a floating point constant at compile time and then hand the result to Math::BigInt, which results in an truncated result or a NaN. This also applies to integers that look like floating point constants: use Math::BigInt ':constant'; print ref(123e2),"\n"; print ref(123.2e2),"\n"; prints nothing but newlines. Use either L<bignum> or L<Math::BigFloat> to get this to work. =head1 PERFORMANCE Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x must be made in the second case. For long numbers, the copy can eat up to 20% of the work (in the case of addition/subtraction, less for multiplication/division). If $y is very small compared to $x, the form $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes more time then the actual addition. With a technique called copy-on-write, the cost of copying with overload could be minimized or even completely avoided. A test implementation of COW did show performance gains for overloaded math, but introduced a performance loss due to a constant overhead for all other operations. So Math::BigInt does currently not COW. The rewritten version of this module (vs. v0.01) is slower on certain operations, like C<new()>, C<bstr()> and C<numify()>. The reason are that it does now more work and handles much more cases. The time spent in these operations is usually gained in the other math operations so that code on the average should get (much) faster. If they don't, please contact the author. Some operations may be slower for small numbers, but are significantly faster for big numbers. Other operations are now constant (O(1), like C<bneg()>, C<babs()> etc), instead of O(N) and thus nearly always take much less time. These optimizations were done on purpose. If you find the Calc module to slow, try to install any of the replacement modules and see if they help you. =head2 Alternative math libraries You can use an alternative library to drive Math::BigInt. See the section L</MATH LIBRARY> for more information. For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>. =head1 SUBCLASSING =head2 Subclassing Math::BigInt The basic design of Math::BigInt allows simple subclasses with very little work, as long as a few simple rules are followed: =over =item The public API must remain consistent, i.e. if a sub-class is overloading addition, the sub-class must use the same name, in this case badd(). The reason for this is that Math::BigInt is optimized to call the object methods directly. =item * The private object hash keys like C<< $x->{sign} >> may not be changed, but additional keys can be added, like C<< $x->{_custom} >>. =item * Accessor functions are available for all existing object hash keys and should be used instead of directly accessing the internal hash keys. The reason for this is that Math::BigInt itself has a pluggable interface which permits it to support different storage methods. =back More complex sub-classes may have to replicate more of the logic internal of Math::BigInt if they need to change more basic behaviors. A subclass that needs to merely change the output only needs to overload C<bstr()>. All other object methods and overloaded functions can be directly inherited from the parent class. At the very minimum, any subclass needs to provide its own C<new()> and can store additional hash keys in the object. There are also some package globals that must be defined, e.g.: # Globals $accuracy = undef; $precision = -2; # round to 2 decimal places $round_mode = 'even'; $div_scale = 40; Additionally, you might want to provide the following two globals to allow auto-upgrading and auto-downgrading to work correctly: $upgrade = undef; $downgrade = undef; This allows Math::BigInt to correctly retrieve package globals from the subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or t/Math/BigFloat/SubClass.pm completely functional subclass examples. Don't forget to use overload; in your subclass to automatically inherit the overloading from the parent. If you like, you can change part of the overloading, look at Math::String for an example. =head1 UPGRADING When used like this: use Math::BigInt upgrade => 'Foo::Bar'; certain operations 'upgrade' their calculation and thus the result to the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat: use Math::BigInt upgrade => 'Math::BigFloat'; As a shortcut, you can use the module L<bignum>: use bignum; Also good for one-liners: perl -Mbignum -le 'print 2 ** 255' This makes it possible to mix arguments of different classes (as in 2.5 + 2) as well es preserve accuracy (as in sqrt(3)). Beware: This feature is not fully implemented yet. =head2 Auto-upgrade The following methods upgrade themselves unconditionally; that is if upgrade is in effect, they always hands up their work: div bsqrt blog bexp bpi bsin bcos batan batan2 All other methods upgrade themselves only when one (or all) of their arguments are of the class mentioned in $upgrade. =head1 EXPORTS C<Math::BigInt> exports nothing by default, but can export the following methods: bgcd blcm =head1 CAVEATS Some things might not work as you expect them. Below is documented what is known to be troublesome: =over =item Comparing numbers as strings Both C<bstr()> and C<bsstr()> as well as stringify via overload drop the leading '+'. This is to be consistent with Perl and to make C<cmp> (especially with overloading) to work as you expect. It also solves problems with C<Test.pm> and L<Test::More>, which stringify arguments before comparing them. Mark Biggar said, when asked about to drop the '+' altogether, or make only C<cmp> work: I agree (with the first alternative), don't add the '+' on positive numbers. It's not as important anymore with the new internal form for numbers. It made doing things like abs and neg easier, but those have to be done differently now anyway. So, the following examples now works as expected: use Test::More tests => 1; use Math::BigInt; my $x = Math::BigInt -> new(33); my $y = Math::BigInt -> new(33); is($x,33, 'multiplication'); print "$x eq 9" if $x eq $y; print "$x eq 9" if $x eq '9'; print "$x eq 9" if $x eq 33; Additionally, the following still works: print "$x == 9" if $x == $y; print "$x == 9" if $x == 9; print "$x == 9" if $x == 33; There is now a C<bsstr()> method to get the string in scientific notation aka C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr() for comparison, but Perl represents some numbers as 100 and others as 1e+308. If in doubt, convert both arguments to Math::BigInt before comparing them as strings: use Test::More tests => 3; use Math::BigInt; $x = Math::BigInt->new('1e56'); $y = 1e56; is($x,$y); # fails is($x->bsstr(),$y); # okay $y = Math::BigInt->new($y); is($x,$y); # okay Alternatively, simply use C<< <=> >> for comparisons, this always gets it right. There is not yet a way to get a number automatically represented as a string that matches exactly the way Perl represents it. See also the section about L<Infinity and Not a Number> for problems in comparing NaNs. =item int() C<int()> returns (at least for Perl v5.7.1 and up) another Math::BigInt, not a Perl scalar: $x = Math::BigInt->new(123); $y = int($x); # 123 as a Math::BigInt $x = Math::BigFloat->new(123.45); $y = int($x); # 123 as a Math::BigFloat If you want a real Perl scalar, use C<numify()>: $y = $x->numify(); # 123 as a scalar This is seldom necessary, though, because this is done automatically, like when you access an array: $z = $array[$x]; # does work automatically =item Modifying and = Beware of: $x = Math::BigFloat->new(5); $y = $x; This makes a second reference to the B<same> object and stores it in $y. Thus anything that modifies $x (except overloaded operators) also modifies $y, and vice versa. Or in other words, C<=> is only safe if you modify your Math::BigInt objects only via overloaded math. As soon as you use a method call it breaks: $x->bmul(2); print "$x, $y\n"; # prints '10, 10' If you want a true copy of $x, use: $y = $x->copy(); You can also chain the calls like this, this first makes a copy and then multiply it by 2: $y = $x->copy()->bmul(2); See also the documentation for overload.pm regarding C<=>. =item Overloading -$x The following: $x = -$x; is slower than $x->bneg(); since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant needs to preserve $x since it does not know that it later gets overwritten. This makes a copy of $x and takes O(N), but $x->bneg() is O(1). =item Mixing different object types With overloaded operators, it is the first (dominating) operand that determines which method is called. Here are some examples showing what actually gets called in various cases. use Math::BigInt; use Math::BigFloat; $mbf = Math::BigFloat->new(5); $mbi2 = Math::BigInt->new(5); $mbi = Math::BigInt->new(2); # what actually gets called: $float = $mbf + $mbi; # $mbf->badd($mbi) $float = $mbf / $mbi; # $mbf->bdiv($mbi) $integer = $mbi + $mbf; # $mbi->badd($mbf) $integer = $mbi2 / $mbi; # $mbi2->bdiv($mbi) $integer = $mbi2 / $mbf; # $mbi2->bdiv($mbf) For instance, Math::BigInt->bdiv() always returns a Math::BigInt, regardless of whether the second operant is a Math::BigFloat. To get a Math::BigFloat you either need to call the operation manually, make sure each operand already is a Math::BigFloat, or cast to that type via Math::BigFloat->new(): $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5 Beware of casting the entire expression, as this would cast the result, at which point it is too late: $float = Math::BigFloat->new($mbi2 / $mbi); # = 2 Beware also of the order of more complicated expressions like: $integer = ($mbi2 + $mbi) / $mbf; # int / float => int $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto If in doubt, break the expression into simpler terms, or cast all operands to the desired resulting type. Scalar values are a bit different, since: $float = 2 + $mbf; $float = $mbf + 2; both result in the proper type due to the way the overloaded math works. This section also applies to other overloaded math packages, like Math::String. One solution to you problem might be autoupgrading\|upgrading. See the pragmas L<bignum>, L<bigint> and L<bigrat> for an easy way to do this. =back =head1 BUGS Please report any bugs or feature requests to C<bug-math-bigint at rt.cpan.org>, or through the web interface at L<https://rt.cpan.org/Ticket/Create.html?Queue=Math-BigInt> (requires login). We will be notified, and then you'll automatically be notified of progress on your bug as I make changes. =head1 SUPPORT You can find documentation for this module with the perldoc command. perldoc Math::BigInt You can also look for information at: =over 4 =item RT: CPAN's request tracker L<https://rt.cpan.org/Public/Dist/Display.html?Name=Math-BigInt> =item * AnnoCPAN: Annotated CPAN documentation L<http://annocpan.org/dist/Math-BigInt> =item * CPAN Ratings L<https://cpanratings.perl.org/dist/Math-BigInt> =item * MetaCPAN L<https://metacpan.org/release/Math-BigInt> =item * CPAN Testers Matrix L<http://matrix.cpantesters.org/?dist=Math-BigInt> =item * The Bignum mailing list =over 4 =item * Post to mailing list C<bignum at lists.scsys.co.uk> =item * View mailing list L<http://lists.scsys.co.uk/pipermail/bignum/> =item * Subscribe/Unsubscribe L<http://lists.scsys.co.uk/cgi-bin/mailman/listinfo/bignum> =back =back =head1 LICENSE This program is free software; you may redistribute it and/or modify it under the same terms as Perl itself. =head1 SEE ALSO L<Math::BigFloat> and L<Math::BigRat> as well as the backends L<Math::BigInt::FastCalc>, L<Math::BigInt::GMP>, and L<Math::BigInt::Pari>. The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest because they solve the autoupgrading/downgrading issue, at least partly. =head1 AUTHORS =over 4 =item * Mark Biggar, overloaded interface by Ilya Zakharevich, 1996-2001. =item * Completely rewritten by Tels L<http://bloodgate.com>, 2001-2008. =item * Florian Ragwitz E<lt>flora@cpan.orgE<gt>, 2010. =item * Peter John Acklam E<lt>pjacklam@online.noE<gt>, 2011-. =back Many people contributed in one or more ways to the final beast, see the file CREDITS for an (incomplete) list. If you miss your name, please drop me a mail. Thank you! =cut	212.67 KB	2019-10-20 12:42

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