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📁 ..	-	2025-05-25 12:29
📄 Calc.pm package Math::BigInt::Calc; use 5.006001; use strict; use warnings; use Carp qw< carp croak >; use Math::BigInt::Lib; our $VERSION = '1.999818'; our @ISA = ('Math::BigInt::Lib'); # Package to store unsigned big integers in decimal and do math with them # Internally the numbers are stored in an array with at least 1 element, no # leading zero parts (except the first) and in base 1eX where X is determined # automatically at loading time to be the maximum possible value # todo: # - fully remove funky $# stuff in div() (maybe - that code scares me...) # USE_MUL: due to problems on certain os (os390, posix-bc) "* 1e-5" is used # instead of "/ 1e5" at some places, (marked with USE_MUL). Other platforms # BS2000, some Crays need USE_DIV instead. # The BEGIN block is used to determine which of the two variants gives the # correct result. # Beware of things like: # $i = $i * $y + $car; $car = int($i / $BASE); $i = $i % $BASE; # This works on x86, but fails on ARM (SA1100, iPAQ) due to who knows what # reasons. So, use this instead (slower, but correct): # $i = $i * $y + $car; $car = int($i / $BASE); $i -= $BASE * $car; ############################################################################## # global constants, flags and accessory # constants for easier life my ($BASE, $BASE_LEN, $RBASE, $MAX_VAL); my ($AND_BITS, $XOR_BITS, $OR_BITS); my ($AND_MASK, $XOR_MASK, $OR_MASK); sub _base_len { # Set/get the BASE_LEN and assorted other, related values. # Used only by the testsuite, the set variant is used only by the BEGIN # block below: my ($class, $b, $int) = @_; if (defined $b) { no warnings "redefine"; if ($] >= 5.008 && $int && $b > 7) { $BASE_LEN = $b; _mul = \&_mul_use_div_64; _div = \&_div_use_div_64; $BASE = int("1e" . $BASE_LEN); $MAX_VAL = $BASE-1; return $BASE_LEN unless wantarray; return ($BASE_LEN, $BASE, $AND_BITS, $XOR_BITS, $OR_BITS, $BASE_LEN, $MAX_VAL); } # find whether we can use mul or div in mul()/div() $BASE_LEN = $b + 1; my $caught = 0; while (--$BASE_LEN > 5) { $BASE = int("1e" . $BASE_LEN); $RBASE = abs('1e-' . $BASE_LEN); # see USE_MUL $caught = 0; $caught += 1 if (int($BASE * $RBASE) != 1); # should be 1 $caught += 2 if (int($BASE / $BASE) != 1); # should be 1 last if $caught != 3; } $BASE = int("1e" . $BASE_LEN); $RBASE = abs('1e-' . $BASE_LEN); # see USE_MUL $MAX_VAL = $BASE-1; # ($caught & 1) != 0 => cannot use MUL # ($caught & 2) != 0 => cannot use DIV if ($caught == 2) # 2 { # must USE_MUL since we cannot use DIV _mul = \&_mul_use_mul; _div = \&_div_use_mul; } else # 0 or 1 { # can USE_DIV instead _mul = \&_mul_use_div; _div = \&_div_use_div; } } return $BASE_LEN unless wantarray; return ($BASE_LEN, $BASE, $AND_BITS, $XOR_BITS, $OR_BITS, $BASE_LEN, $MAX_VAL); } sub _new { # Given a string representing an integer, returns a reference to an array # of integers, where each integer represents a chunk of the original input # integer. my ($class, $str) = @_; #unless ($str =~ /^([1-9]\d\|0)\z/) { # croak("Invalid input string '$str'"); #} my $input_len = length($str) - 1; # Shortcut for small numbers. return bless [ $str ], $class if $input_len < $BASE_LEN; my $format = "a" . (($input_len % $BASE_LEN) + 1); $format .= $] < 5.008 ? "a$BASE_LEN" x int($input_len / $BASE_LEN) : "(a$BASE_LEN)"; my $self = [ reverse(map { 0 + $_ } unpack($format, $str)) ]; return bless $self, $class; } BEGIN { # from Daniel Pfeiffer: determine largest group of digits that is precisely # multipliable with itself plus carry # Test now changed to expect the proper pattern, not a result off by 1 or 2 my ($e, $num) = 3; # lowest value we will use is 3+1-1 = 3 do { $num = '9' x ++$e; $num = $num + 1; } while $num =~ /9{$e}0{$e}/; # must be a certain pattern $e--; # last test failed, so retract one step # the limits below brush the problems with the test above under the rug: # the test should be able to find the proper $e automatically $e = 5 if $^O =~ /^uts/; # UTS get's some special treatment $e = 5 if $^O =~ /^unicos/; # unicos is also problematic (6 seems to work # there, but we play safe) my $int = 0; if ($e > 7) { use integer; my $e1 = 7; $num = 7; do { $num = ('9' x ++$e1) + 0; $num = $num + 1; } while ("$num" =~ /9{$e1}0{$e1}/); # must be a certain pattern $e1--; # last test failed, so retract one step if ($e1 > 7) { $int = 1; $e = $e1; } } __PACKAGE__ -> _base_len($e, $int); # set and store use integer; # find out how many bits _and, _or and _xor can take (old default = 16) # I don't think anybody has yet 128 bit scalars, so let's play safe. local $^W = 0; # don't warn about 'nonportable number' $AND_BITS = 15; $XOR_BITS = 15; $OR_BITS = 15; # find max bits, we will not go higher than numberofbits that fit into $BASE # to make _and etc simpler (and faster for smaller, slower for large numbers) my $max = 16; while (2 $max < $BASE) { $max++; } { no integer; $max = 16 if $] < 5.006; # older Perls might not take >16 too well } my ($x, $y, $z); do { $AND_BITS++; $x = CORE::oct('0b' . '1' x $AND_BITS); $y = $x & $x; $z = (2 $AND_BITS) - 1; } while ($AND_BITS < $max && $x == $z && $y == $x); $AND_BITS --; # retreat one step do { $XOR_BITS++; $x = CORE::oct('0b' . '1' x $XOR_BITS); $y = $x ^ 0; $z = (2 $XOR_BITS) - 1; } while ($XOR_BITS < $max && $x == $z && $y == $x); $XOR_BITS --; # retreat one step do { $OR_BITS++; $x = CORE::oct('0b' . '1' x $OR_BITS); $y = $x \| $x; $z = (2 $OR_BITS) - 1; } while ($OR_BITS < $max && $x == $z && $y == $x); $OR_BITS--; # retreat one step $AND_MASK = __PACKAGE__->_new(( 2 $AND_BITS )); $XOR_MASK = __PACKAGE__->_new(( 2 $XOR_BITS )); $OR_MASK = __PACKAGE__->_new(( 2 ** $OR_BITS )); # We can compute the approximate length no faster than the real length: _alen = \&_len; } ############################################################################### sub _zero { # create a zero my $class = shift; return bless [ 0 ], $class; } sub _one { # create a one my $class = shift; return bless [ 1 ], $class; } sub _two { # create a two my $class = shift; return bless [ 2 ], $class; } sub _ten { # create a 10 my $class = shift; bless [ 10 ], $class; } sub _1ex { # create a 1Ex my $class = shift; my $rem = $_[0] % $BASE_LEN; # remainder my $parts = $_[0] / $BASE_LEN; # parts # 000000, 000000, 100 bless [ (0) x $parts, '1' . ('0' x $rem) ], $class; } sub _copy { # make a true copy my $class = shift; return bless [ @{ $_[0] } ], $class; } # catch and throw away sub import { } ############################################################################## # convert back to string and number sub _str { # Convert number from internal base 1eN format to string format. Internal # format is always normalized, i.e., no leading zeros. my $ary = $_[1]; my $idx = $#$ary; # index of last element if ($idx < 0) { # should not happen croak("$_[1] has no elements"); } # Handle first one differently, since it should not have any leading zeros. my $ret = int($ary->[$idx]); if ($idx > 0) { # Interestingly, the pre-padd method uses more time. # The old grep variant takes longer (14 vs. 10 sec). my $z = '0' x ($BASE_LEN - 1); while (--$idx >= 0) { $ret .= substr($z . $ary->[$idx], -$BASE_LEN); } } $ret; } sub _num { # Make a Perl scalar number (int/float) from a BigInt object. my $x = $_[1]; return $x->[0] if @$x == 1; # below $BASE # Start with the most significant element and work towards the least # significant element. Avoid multiplying "inf" (which happens if the number # overflows) with "0" (if there are zero elements in $x) since this gives # "nan" which propagates to the output. my $num = 0; for (my $i = $#$x ; $i >= 0 ; --$i) { $num = $BASE; $num += $x -> [$i]; } return $num; } ############################################################################## # actual math code sub _add { # (ref to int_num_array, ref to int_num_array) # # Routine to add two base 1eX numbers stolen from Knuth Vol 2 Algorithm A # pg 231. There are separate routines to add and sub as per Knuth pg 233. # This routine modifies array x, but not y. my ($c, $x, $y) = @_; # $x + 0 => $x return $x if @$y == 1 && $y->[0] == 0; # 0 + $y => $y->copy if (@$x == 1 && $x->[0] == 0) { @$x = @$y; return $x; } # For each in Y, add Y to X and carry. If after that, something is left in # X, foreach in X add carry to X and then return X, carry. Trades one # "$j++" for having to shift arrays. my $i; my $car = 0; my $j = 0; for $i (@$y) { $x->[$j] -= $BASE if $car = (($x->[$j] += $i + $car) >= $BASE) ? 1 : 0; $j++; } while ($car != 0) { $x->[$j] -= $BASE if $car = (($x->[$j] += $car) >= $BASE) ? 1 : 0; $j++; } $x; } sub _inc { # (ref to int_num_array, ref to int_num_array) # Add 1 to $x, modify $x in place my ($c, $x) = @_; for my $i (@$x) { return $x if ($i += 1) < $BASE; # early out $i = 0; # overflow, next } push @$x, 1 if $x->[-1] == 0; # last overflowed, so extend $x; } sub _dec { # (ref to int_num_array, ref to int_num_array) # Sub 1 from $x, modify $x in place my ($c, $x) = @_; my $MAX = $BASE - 1; # since MAX_VAL based on BASE for my $i (@$x) { last if ($i -= 1) >= 0; # early out $i = $MAX; # underflow, next } pop @$x if $x->[-1] == 0 && @$x > 1; # last underflowed (but leave 0) $x; } sub _sub { # (ref to int_num_array, ref to int_num_array, swap) # # Subtract base 1eX numbers -- stolen from Knuth Vol 2 pg 232, $x > $y # subtract Y from X by modifying x in place my ($c, $sx, $sy, $s) = @_; my $car = 0; my $i; my $j = 0; if (!$s) { for $i (@$sx) { last unless defined $sy->[$j] \|\| $car; $i += $BASE if $car = (($i -= ($sy->[$j] \|\| 0) + $car) < 0); $j++; } # might leave leading zeros, so fix that return __strip_zeros($sx); } for $i (@$sx) { # We can't do an early out if $x < $y, since we need to copy the high # chunks from $y. Found by Bob Mathews. #last unless defined $sy->[$j] \|\| $car; $sy->[$j] += $BASE if $car = ($sy->[$j] = $i - ($sy->[$j] \|\| 0) - $car) < 0; $j++; } # might leave leading zeros, so fix that __strip_zeros($sy); } sub _mul_use_mul { # (ref to int_num_array, ref to int_num_array) # multiply two numbers in internal representation # modifies first arg, second need not be different from first my ($c, $xv, $yv) = @_; if (@$yv == 1) { # shortcut for two very short numbers (improved by Nathan Zook) works # also if xv and yv are the same reference, and handles also $x == 0 if (@$xv == 1) { if (($xv->[0] = $yv->[0]) >= $BASE) { my $rem = $xv->[0] % $BASE; $xv->[1] = ($xv->[0] - $rem) $RBASE; $xv->[0] = $rem; } return $xv; } # $x * 0 => 0 if ($yv->[0] == 0) { @$xv = (0); return $xv; } # multiply a large number a by a single element one, so speed up my $y = $yv->[0]; my $car = 0; my $rem; foreach my $i (@$xv) { $i = $i * $y + $car; $rem = $i % $BASE; $car = ($i - $rem) * $RBASE; $i = $rem; } push @$xv, $car if $car != 0; return $xv; } # shortcut for result $x == 0 => result = 0 return $xv if @$xv == 1 && $xv->[0] == 0; # since multiplying $x with $x fails, make copy in this case $yv = $c->_copy($xv) if $xv == $yv; # same references? my @prod = (); my ($prod, $rem, $car, $cty, $xi, $yi); for $xi (@$xv) { $car = 0; $cty = 0; # looping through this if $xi == 0 is silly - so optimize it away! $xi = (shift(@prod) \|\| 0), next if $xi == 0; for $yi (@$yv) { $prod = $xi * $yi + ($prod[$cty] \|\| 0) + $car; $rem = $prod % $BASE; $car = int(($prod - $rem) * $RBASE); $prod[$cty++] = $rem; } $prod[$cty] += $car if $car; # need really to check for 0? $xi = shift(@prod) \|\| 0; # \|\| 0 makes v5.005_3 happy } push @$xv, @prod; $xv; } sub _mul_use_div_64 { # (ref to int_num_array, ref to int_num_array) # multiply two numbers in internal representation # modifies first arg, second need not be different from first # works for 64 bit integer with "use integer" my ($c, $xv, $yv) = @_; use integer; if (@$yv == 1) { # shortcut for two very short numbers (improved by Nathan Zook) works # also if xv and yv are the same reference, and handles also $x == 0 if (@$xv == 1) { if (($xv->[0] = $yv->[0]) >= $BASE) { $xv->[0] = $xv->[0] - ($xv->[1] = $xv->[0] / $BASE) $BASE; } return $xv; } # $x * 0 => 0 if ($yv->[0] == 0) { @$xv = (0); return $xv; } # multiply a large number a by a single element one, so speed up my $y = $yv->[0]; my $car = 0; foreach my $i (@$xv) { #$i = $i * $y + $car; $car = $i / $BASE; $i -= $car * $BASE; $i = $i * $y + $car; $i -= ($car = $i / $BASE) * $BASE; } push @$xv, $car if $car != 0; return $xv; } # shortcut for result $x == 0 => result = 0 return $xv if @$xv == 1 && $xv->[0] == 0; # since multiplying $x with $x fails, make copy in this case $yv = $c->_copy($xv) if $xv == $yv; # same references? my @prod = (); my ($prod, $car, $cty, $xi, $yi); for $xi (@$xv) { $car = 0; $cty = 0; # looping through this if $xi == 0 is silly - so optimize it away! $xi = (shift(@prod) \|\| 0), next if $xi == 0; for $yi (@$yv) { $prod = $xi * $yi + ($prod[$cty] \|\| 0) + $car; $prod[$cty++] = $prod - ($car = $prod / $BASE) * $BASE; } $prod[$cty] += $car if $car; # need really to check for 0? $xi = shift(@prod) \|\| 0; # \|\| 0 makes v5.005_3 happy } push @$xv, @prod; $xv; } sub _mul_use_div { # (ref to int_num_array, ref to int_num_array) # multiply two numbers in internal representation # modifies first arg, second need not be different from first my ($c, $xv, $yv) = @_; if (@$yv == 1) { # shortcut for two very short numbers (improved by Nathan Zook) works # also if xv and yv are the same reference, and handles also $x == 0 if (@$xv == 1) { if (($xv->[0] = $yv->[0]) >= $BASE) { my $rem = $xv->[0] % $BASE; $xv->[1] = ($xv->[0] - $rem) / $BASE; $xv->[0] = $rem; } return $xv; } # $x 0 => 0 if ($yv->[0] == 0) { @$xv = (0); return $xv; } # multiply a large number a by a single element one, so speed up my $y = $yv->[0]; my $car = 0; my $rem; foreach my $i (@$xv) { $i = $i * $y + $car; $rem = $i % $BASE; $car = ($i - $rem) / $BASE; $i = $rem; } push @$xv, $car if $car != 0; return $xv; } # shortcut for result $x == 0 => result = 0 return $xv if @$xv == 1 && $xv->[0] == 0; # since multiplying $x with $x fails, make copy in this case $yv = $c->_copy($xv) if $xv == $yv; # same references? my @prod = (); my ($prod, $rem, $car, $cty, $xi, $yi); for $xi (@$xv) { $car = 0; $cty = 0; # looping through this if $xi == 0 is silly - so optimize it away! $xi = (shift(@prod) \|\| 0), next if $xi == 0; for $yi (@$yv) { $prod = $xi * $yi + ($prod[$cty] \|\| 0) + $car; $rem = $prod % $BASE; $car = ($prod - $rem) / $BASE; $prod[$cty++] = $rem; } $prod[$cty] += $car if $car; # need really to check for 0? $xi = shift(@prod) \|\| 0; # \|\| 0 makes v5.005_3 happy } push @$xv, @prod; $xv; } sub _div_use_mul { # ref to array, ref to array, modify first array and return remainder if # in list context my ($c, $x, $yorg) = @_; # the general div algorithm here is about O(NN) and thus quite slow, so # we first check for some special cases and use shortcuts to handle them. # if both numbers have only one element: if (@$x == 1 && @$yorg == 1) { # shortcut, $yorg and $x are two small numbers my $rem = [ $x->[0] % $yorg->[0] ]; bless $rem, $c; $x->[0] = ($x->[0] - $rem->[0]) / $yorg->[0]; return ($x, $rem) if wantarray; return $x; } # if x has more than one, but y has only one element: if (@$yorg == 1) { my $rem; $rem = $c->_mod($c->_copy($x), $yorg) if wantarray; # shortcut, $y is < $BASE my $j = @$x; my $r = 0; my $y = $yorg->[0]; my $b; while ($j-- > 0) { $b = $r $BASE + $x->[$j]; $r = $b % $y; $x->[$j] = ($b - $r) / $y; } pop(@$x) if @$x > 1 && $x->[-1] == 0; # remove any trailing zero return ($x, $rem) if wantarray; return $x; } # now x and y have more than one element # check whether y has more elements than x, if so, the result is 0 if (@$yorg > @$x) { my $rem; $rem = $c->_copy($x) if wantarray; # make copy @$x = 0; # set to 0 return ($x, $rem) if wantarray; # including remainder? return $x; # only x, which is [0] now } # check whether the numbers have the same number of elements, in that case # the result will fit into one element and can be computed efficiently if (@$yorg == @$x) { my $cmp = 0; for (my $j = $#$x ; $j >= 0 ; --$j) { last if $cmp = $x->[$j] - $yorg->[$j]; } if ($cmp == 0) { # x = y @$x = 1; return $x, $c->_zero() if wantarray; return $x; } if ($cmp < 0) { # x < y if (wantarray) { my $rem = $c->_copy($x); @$x = 0; return $x, $rem; } @$x = 0; return $x; } } # all other cases: my $y = $c->_copy($yorg); # always make copy to preserve my $tmp = $y->[-1] + 1; my $rem = $BASE % $tmp; my $dd = ($BASE - $rem) / $tmp; if ($dd != 1) { my $car = 0; for my $xi (@$x) { $xi = $xi * $dd + $car; $xi -= ($car = int($xi * $RBASE)) * $BASE; # see USE_MUL } push(@$x, $car); $car = 0; for my $yi (@$y) { $yi = $yi * $dd + $car; $yi -= ($car = int($yi * $RBASE)) * $BASE; # see USE_MUL } } else { push(@$x, 0); } # @q will accumulate the final result, $q contains the current computed # part of the final result my @q = (); my ($v2, $v1) = @$y[-2, -1]; $v2 = 0 unless $v2; while ($#$x > $#$y) { my ($u2, $u1, $u0) = @$x[-3 .. -1]; $u2 = 0 unless $u2; #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" # if $v1 == 0; my $tmp = $u0 * $BASE + $u1; my $rem = $tmp % $v1; my $q = $u0 == $v1 ? $MAX_VAL : (($tmp - $rem) / $v1); --$q while $v2 * $q > ($u0 * $BASE + $u1 - $q * $v1) * $BASE + $u2; if ($q) { my $prd; my ($car, $bar) = (0, 0); for (my $yi = 0, my $xi = $#$x - $#$y - 1; $yi <= $#$y; ++$yi, ++$xi) { $prd = $q * $y->[$yi] + $car; $prd -= ($car = int($prd * $RBASE)) * $BASE; # see USE_MUL $x->[$xi] += $BASE if $bar = (($x->[$xi] -= $prd + $bar) < 0); } if ($x->[-1] < $car + $bar) { $car = 0; --$q; for (my $yi = 0, my $xi = $#$x - $#$y - 1; $yi <= $#$y; ++$yi, ++$xi) { $x->[$xi] -= $BASE if $car = (($x->[$xi] += $y->[$yi] + $car) >= $BASE); } } } pop(@$x); unshift(@q, $q); } if (wantarray) { my $d = bless [], $c; if ($dd != 1) { my $car = 0; my ($prd, $rem); for my $xi (reverse @$x) { $prd = $car * $BASE + $xi; $rem = $prd % $dd; $tmp = ($prd - $rem) / $dd; $car = $rem; unshift @$d, $tmp; } } else { @$d = @$x; } @$x = @q; __strip_zeros($x); __strip_zeros($d); return ($x, $d); } @$x = @q; __strip_zeros($x); $x; } sub _div_use_div_64 { # ref to array, ref to array, modify first array and return remainder if # in list context # This version works on integers use integer; my ($c, $x, $yorg) = @_; # the general div algorithm here is about O(NN) and thus quite slow, so # we first check for some special cases and use shortcuts to handle them. # if both numbers have only one element: if (@$x == 1 && @$yorg == 1) { # shortcut, $yorg and $x are two small numbers if (wantarray) { my $rem = [ $x->[0] % $yorg->[0] ]; bless $rem, $c; $x->[0] = $x->[0] / $yorg->[0]; return ($x, $rem); } else { $x->[0] = $x->[0] / $yorg->[0]; return $x; } } # if x has more than one, but y has only one element: if (@$yorg == 1) { my $rem; $rem = $c->_mod($c->_copy($x), $yorg) if wantarray; # shortcut, $y is < $BASE my $j = @$x; my $r = 0; my $y = $yorg->[0]; my $b; while ($j-- > 0) { $b = $r $BASE + $x->[$j]; $r = $b % $y; $x->[$j] = $b / $y; } pop(@$x) if @$x > 1 && $x->[-1] == 0; # remove any trailing zero return ($x, $rem) if wantarray; return $x; } # now x and y have more than one element # check whether y has more elements than x, if so, the result is 0 if (@$yorg > @$x) { my $rem; $rem = $c->_copy($x) if wantarray; # make copy @$x = 0; # set to 0 return ($x, $rem) if wantarray; # including remainder? return $x; # only x, which is [0] now } # check whether the numbers have the same number of elements, in that case # the result will fit into one element and can be computed efficiently if (@$yorg == @$x) { my $cmp = 0; for (my $j = $#$x ; $j >= 0 ; --$j) { last if $cmp = $x->[$j] - $yorg->[$j]; } if ($cmp == 0) { # x = y @$x = 1; return $x, $c->_zero() if wantarray; return $x; } if ($cmp < 0) { # x < y if (wantarray) { my $rem = $c->_copy($x); @$x = 0; return $x, $rem; } @$x = 0; return $x; } } # all other cases: my $y = $c->_copy($yorg); # always make copy to preserve my $tmp; my $dd = $BASE / ($y->[-1] + 1); if ($dd != 1) { my $car = 0; for my $xi (@$x) { $xi = $xi * $dd + $car; $xi -= ($car = $xi / $BASE) * $BASE; } push(@$x, $car); $car = 0; for my $yi (@$y) { $yi = $yi * $dd + $car; $yi -= ($car = $yi / $BASE) * $BASE; } } else { push(@$x, 0); } # @q will accumulate the final result, $q contains the current computed # part of the final result my @q = (); my ($v2, $v1) = @$y[-2, -1]; $v2 = 0 unless $v2; while ($#$x > $#$y) { my ($u2, $u1, $u0) = @$x[-3 .. -1]; $u2 = 0 unless $u2; #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" # if $v1 == 0; my $tmp = $u0 * $BASE + $u1; my $rem = $tmp % $v1; my $q = $u0 == $v1 ? $MAX_VAL : (($tmp - $rem) / $v1); --$q while $v2 * $q > ($u0 * $BASE + $u1 - $q * $v1) * $BASE + $u2; if ($q) { my $prd; my ($car, $bar) = (0, 0); for (my $yi = 0, my $xi = $#$x - $#$y - 1; $yi <= $#$y; ++$yi, ++$xi) { $prd = $q * $y->[$yi] + $car; $prd -= ($car = int($prd / $BASE)) * $BASE; $x->[$xi] += $BASE if $bar = (($x->[$xi] -= $prd + $bar) < 0); } if ($x->[-1] < $car + $bar) { $car = 0; --$q; for (my $yi = 0, my $xi = $#$x - $#$y - 1; $yi <= $#$y; ++$yi, ++$xi) { $x->[$xi] -= $BASE if $car = (($x->[$xi] += $y->[$yi] + $car) >= $BASE); } } } pop(@$x); unshift(@q, $q); } if (wantarray) { my $d = bless [], $c; if ($dd != 1) { my $car = 0; my $prd; for my $xi (reverse @$x) { $prd = $car * $BASE + $xi; $car = $prd - ($tmp = $prd / $dd) * $dd; unshift @$d, $tmp; } } else { @$d = @$x; } @$x = @q; __strip_zeros($x); __strip_zeros($d); return ($x, $d); } @$x = @q; __strip_zeros($x); $x; } sub _div_use_div { # ref to array, ref to array, modify first array and return remainder if # in list context my ($c, $x, $yorg) = @_; # the general div algorithm here is about O(NN) and thus quite slow, so # we first check for some special cases and use shortcuts to handle them. # if both numbers have only one element: if (@$x == 1 && @$yorg == 1) { # shortcut, $yorg and $x are two small numbers my $rem = [ $x->[0] % $yorg->[0] ]; bless $rem, $c; $x->[0] = ($x->[0] - $rem->[0]) / $yorg->[0]; return ($x, $rem) if wantarray; return $x; } # if x has more than one, but y has only one element: if (@$yorg == 1) { my $rem; $rem = $c->_mod($c->_copy($x), $yorg) if wantarray; # shortcut, $y is < $BASE my $j = @$x; my $r = 0; my $y = $yorg->[0]; my $b; while ($j-- > 0) { $b = $r $BASE + $x->[$j]; $r = $b % $y; $x->[$j] = ($b - $r) / $y; } pop(@$x) if @$x > 1 && $x->[-1] == 0; # remove any trailing zero return ($x, $rem) if wantarray; return $x; } # now x and y have more than one element # check whether y has more elements than x, if so, the result is 0 if (@$yorg > @$x) { my $rem; $rem = $c->_copy($x) if wantarray; # make copy @$x = 0; # set to 0 return ($x, $rem) if wantarray; # including remainder? return $x; # only x, which is [0] now } # check whether the numbers have the same number of elements, in that case # the result will fit into one element and can be computed efficiently if (@$yorg == @$x) { my $cmp = 0; for (my $j = $#$x ; $j >= 0 ; --$j) { last if $cmp = $x->[$j] - $yorg->[$j]; } if ($cmp == 0) { # x = y @$x = 1; return $x, $c->_zero() if wantarray; return $x; } if ($cmp < 0) { # x < y if (wantarray) { my $rem = $c->_copy($x); @$x = 0; return $x, $rem; } @$x = 0; return $x; } } # all other cases: my $y = $c->_copy($yorg); # always make copy to preserve my $tmp = $y->[-1] + 1; my $rem = $BASE % $tmp; my $dd = ($BASE - $rem) / $tmp; if ($dd != 1) { my $car = 0; for my $xi (@$x) { $xi = $xi * $dd + $car; $rem = $xi % $BASE; $car = ($xi - $rem) / $BASE; $xi = $rem; } push(@$x, $car); $car = 0; for my $yi (@$y) { $yi = $yi * $dd + $car; $rem = $yi % $BASE; $car = ($yi - $rem) / $BASE; $yi = $rem; } } else { push(@$x, 0); } # @q will accumulate the final result, $q contains the current computed # part of the final result my @q = (); my ($v2, $v1) = @$y[-2, -1]; $v2 = 0 unless $v2; while ($#$x > $#$y) { my ($u2, $u1, $u0) = @$x[-3 .. -1]; $u2 = 0 unless $u2; #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" # if $v1 == 0; my $tmp = $u0 * $BASE + $u1; my $rem = $tmp % $v1; my $q = $u0 == $v1 ? $MAX_VAL : (($tmp - $rem) / $v1); --$q while $v2 * $q > ($u0 * $BASE + $u1 - $q * $v1) * $BASE + $u2; if ($q) { my $prd; my ($car, $bar) = (0, 0); for (my $yi = 0, my $xi = $#$x - $#$y - 1; $yi <= $#$y; ++$yi, ++$xi) { $prd = $q * $y->[$yi] + $car; $rem = $prd % $BASE; $car = ($prd - $rem) / $BASE; $prd -= $car * $BASE; $x->[$xi] += $BASE if $bar = (($x->[$xi] -= $prd + $bar) < 0); } if ($x->[-1] < $car + $bar) { $car = 0; --$q; for (my $yi = 0, my $xi = $#$x - $#$y - 1; $yi <= $#$y; ++$yi, ++$xi) { $x->[$xi] -= $BASE if $car = (($x->[$xi] += $y->[$yi] + $car) >= $BASE); } } } pop(@$x); unshift(@q, $q); } if (wantarray) { my $d = bless [], $c; if ($dd != 1) { my $car = 0; my ($prd, $rem); for my $xi (reverse @$x) { $prd = $car * $BASE + $xi; $rem = $prd % $dd; $tmp = ($prd - $rem) / $dd; $car = $rem; unshift @$d, $tmp; } } else { @$d = @$x; } @$x = @q; __strip_zeros($x); __strip_zeros($d); return ($x, $d); } @$x = @q; __strip_zeros($x); $x; } ############################################################################## # testing sub _acmp { # Internal absolute post-normalized compare (ignore signs) # ref to array, ref to array, return <0, 0, >0 # Arrays must have at least one entry; this is not checked for. my ($c, $cx, $cy) = @_; # shortcut for short numbers return (($cx->[0] <=> $cy->[0]) <=> 0) if @$cx == 1 && @$cy == 1; # fast comp based on number of array elements (aka pseudo-length) my $lxy = (@$cx - @$cy) # or length of first element if same number of elements (aka difference 0) \|\| # need int() here because sometimes the last element is '00018' vs '18' (length(int($cx->[-1])) - length(int($cy->[-1]))); return -1 if $lxy < 0; # already differs, ret return 1 if $lxy > 0; # ditto # manual way (abort if unequal, good for early ne) my $a; my $j = @$cx; while (--$j >= 0) { last if $a = $cx->[$j] - $cy->[$j]; } $a <=> 0; } sub _len { # compute number of digits in base 10 # int() because add/sub sometimes leaves strings (like '00005') instead of # '5' in this place, thus causing length() to report wrong length my $cx = $_[1]; (@$cx - 1) * $BASE_LEN + length(int($cx->[-1])); } sub _digit { # Return the nth digit. Zero is rightmost, so _digit(123, 0) gives 3. # Negative values count from the left, so _digit(123, -1) gives 1. my ($c, $x, $n) = @_; my $len = _len('', $x); $n += $len if $n < 0; # -1 last, -2 second-to-last # Math::BigInt::Calc returns 0 if N is out of range, but this is not done # by the other backend libraries. return "0" if $n < 0 \|\| $n >= $len; # return 0 for digits out of range my $elem = int($n / $BASE_LEN); # index of array element my $digit = $n % $BASE_LEN; # index of digit within the element substr("0" x $BASE_LEN . "$x->[$elem]", -1 - $digit, 1); } sub _zeros { # Return number of trailing zeros in decimal. # Check each array element for having 0 at end as long as elem == 0 # Upon finding a elem != 0, stop. my $x = $_[1]; return 0 if @$x == 1 && $x->[0] == 0; my $zeros = 0; foreach my $elem (@$x) { if ($elem != 0) { $elem =~ /[^0](0)\z/; $zeros += length($1); # count trailing zeros last; # early out } $zeros += $BASE_LEN; } $zeros; } ############################################################################## # _is_ routines sub _is_zero { # return true if arg is zero @{$_[1]} == 1 && $_[1]->[0] == 0 ? 1 : 0; } sub _is_even { # return true if arg is even $_[1]->[0] & 1 ? 0 : 1; } sub _is_odd { # return true if arg is odd $_[1]->[0] & 1 ? 1 : 0; } sub _is_one { # return true if arg is one @{$_[1]} == 1 && $_[1]->[0] == 1 ? 1 : 0; } sub _is_two { # return true if arg is two @{$_[1]} == 1 && $_[1]->[0] == 2 ? 1 : 0; } sub _is_ten { # return true if arg is ten @{$_[1]} == 1 && $_[1]->[0] == 10 ? 1 : 0; } sub __strip_zeros { # Internal normalization function that strips leading zeros from the array. # Args: ref to array my $x = shift; push @$x, 0 if @$x == 0; # div might return empty results, so fix it return $x if @$x == 1; # early out #print "strip: cnt $cnt i $i\n"; # '0', '3', '4', '0', '0', # 0 1 2 3 4 # cnt = 5, i = 4 # i = 4 # i = 3 # => fcnt = cnt - i (5-2 => 3, cnt => 5-1 = 4, throw away from 4th pos) # >= 1: skip first part (this can be zero) my $i = $#$x; while ($i > 0) { last if $x->[$i] != 0; $i--; } $i++; splice(@$x, $i) if $i < @$x; $x; } ############################################################################### # check routine to test internal state for corruptions sub _check { # used by the test suite my ($class, $x) = @_; my $msg = $class -> SUPER::_check($x); return $msg if $msg; my $n; eval { $n = @$x }; return "Not an array reference" unless $@ eq ''; return "Reference to an empty array" unless $n > 0; # The following fails with Math::BigInt::FastCalc because a # Math::BigInt::FastCalc "object" is an unblessed array ref. # #return 0 unless ref($x) eq $class; for (my $i = 0 ; $i <= $#$x ; ++ $i) { my $e = $x -> [$i]; return "Element at index $i is undefined" unless defined $e; return "Element at index $i is a '" . ref($e) . "', which is not a scalar" unless ref($e) eq ""; # It would be better to use the regex /^([1-9]\d\|0)\z/, but that fails # in Math::BigInt::FastCalc, because it sometimes creates array # elements like "000000". return "Element at index $i is '$e', which does not look like an" . " normal integer" unless $e =~ /^\d+\z/; return "Element at index $i is '$e', which is not smaller than" . " the base '$BASE'" if $e >= $BASE; return "Element at index $i (last element) is zero" if $#$x > 0 && $i == $#$x && $e == 0; } return 0; } ############################################################################### sub _mod { # if possible, use mod shortcut my ($c, $x, $yo) = @_; # slow way since $y too big if (@$yo > 1) { my ($xo, $rem) = $c->_div($x, $yo); @$x = @$rem; return $x; } my $y = $yo->[0]; # if both are single element arrays if (@$x == 1) { $x->[0] %= $y; return $x; } # if @$x has more than one element, but @$y is a single element my $b = $BASE % $y; if ($b == 0) { # when BASE % Y == 0 then (B BASE) % Y == 0 # (B * BASE) % $y + A % Y => A % Y # so need to consider only last element: O(1) $x->[0] %= $y; } elsif ($b == 1) { # else need to go through all elements in @$x: O(N), but loop is a bit # simplified my $r = 0; foreach (@$x) { $r = ($r + $_) % $y; # not much faster, but heh... #$r += $_ % $y; $r %= $y; } $r = 0 if $r == $y; $x->[0] = $r; } else { # else need to go through all elements in @$x: O(N) my $r = 0; my $bm = 1; foreach (@$x) { $r = ($_ * $bm + $r) % $y; $bm = ($bm * $b) % $y; #$r += ($_ % $y) * $bm; #$bm = $b; #$bm %= $y; #$r %= $y; } $r = 0 if $r == $y; $x->[0] = $r; } @$x = $x->[0]; # keep one element of @$x return $x; } ############################################################################## # shifts sub _rsft { my ($c, $x, $y, $n) = @_; if ($n != 10) { $n = $c->_new($n); return scalar $c->_div($x, $c->_pow($n, $y)); } # shortcut (faster) for shifting by 10) # multiples of $BASE_LEN my $dst = 0; # destination my $src = $c->_num($y); # as normal int my $xlen = (@$x - 1) $BASE_LEN + length(int($x->[-1])); if ($src >= $xlen or ($src == $xlen and !defined $x->[1])) { # 12345 67890 shifted right by more than 10 digits => 0 splice(@$x, 1); # leave only one element $x->[0] = 0; # set to zero return $x; } my $rem = $src % $BASE_LEN; # remainder to shift $src = int($src / $BASE_LEN); # source if ($rem == 0) { splice(@$x, 0, $src); # even faster, 38.4 => 39.3 } else { my $len = @$x - $src; # elems to go my $vd; my $z = '0' x $BASE_LEN; $x->[ @$x ] = 0; # avoid \|\| 0 test inside loop while ($dst < $len) { $vd = $z . $x->[$src]; $vd = substr($vd, -$BASE_LEN, $BASE_LEN - $rem); $src++; $vd = substr($z . $x->[$src], -$rem, $rem) . $vd; $vd = substr($vd, -$BASE_LEN, $BASE_LEN) if length($vd) > $BASE_LEN; $x->[$dst] = int($vd); $dst++; } splice(@$x, $dst) if $dst > 0; # kill left-over array elems pop(@$x) if $x->[-1] == 0 && @$x > 1; # kill last element if 0 } # else rem == 0 $x; } sub _lsft { my ($c, $x, $n, $b) = @_; return $x if $c->_is_zero($x) \|\| $c->_is_zero($n); # For backwards compatibility, allow the base $b to be a scalar. $b = $c->_new($b) unless ref $b; # If the base is a power of 10, use shifting, since the internal # representation is in base 10eX. my $bstr = $c->_str($b); if ($bstr =~ /^1(0+)\z/) { # Adjust $n so that we're shifting in base 10. Do this by multiplying # $n by the base 10 logarithm of $b: $b $n = 10 (log10($b) * $n). my $log10b = length($1); $n = $c->_mul($c->_new($log10b), $n); $n = $c->_num($n); # shift-len as normal int # $q is the number of places to shift the elements within the array, # and $r is the number of places to shift the values within the # elements. my $r = $n % $BASE_LEN; my $q = ($n - $r) / $BASE_LEN; # If we must shift the values within the elements ... if ($r) { my $i = @$x; # index $x->[$i] = 0; # initialize most significant element my $z = '0' x $BASE_LEN; my $vd; while ($i >= 0) { $vd = $x->[$i]; $vd = $z . $vd; $vd = substr($vd, $r - $BASE_LEN, $BASE_LEN - $r); $vd .= $i > 0 ? substr($z . $x->[$i - 1], -$BASE_LEN, $r) : '0' x $r; $vd = substr($vd, -$BASE_LEN, $BASE_LEN) if length($vd) > $BASE_LEN; $x->[$i] = int($vd); # e.g., "0...048" -> 48 etc. $i--; } pop(@$x) if $x->[-1] == 0; # if most significant element is zero } # If we must shift the elements within the array ... if ($q) { unshift @$x, (0) x $q; } } else { $x = $c->_mul($x, $c->_pow($b, $n)); } return $x; } sub _pow { # power of $x to $y # ref to array, ref to array, return ref to array my ($c, $cx, $cy) = @_; if (@$cy == 1 && $cy->[0] == 0) { splice(@$cx, 1); $cx->[0] = 1; # y == 0 => x => 1 return $cx; } if ((@$cx == 1 && $cx->[0] == 1) \|\| # x == 1 (@$cy == 1 && $cy->[0] == 1)) # or y == 1 { return $cx; } if (@$cx == 1 && $cx->[0] == 0) { splice (@$cx, 1); $cx->[0] = 0; # 0 ** y => 0 (if not y <= 0) return $cx; } my $pow2 = $c->_one(); my $y_bin = $c->_as_bin($cy); $y_bin =~ s/^0b//; my $len = length($y_bin); while (--$len > 0) { $c->_mul($pow2, $cx) if substr($y_bin, $len, 1) eq '1'; # is odd? $c->_mul($cx, $cx); } $c->_mul($cx, $pow2); $cx; } sub _nok { # Return binomial coefficient (n over k). # Given refs to arrays, return ref to array. # First input argument is modified. my ($c, $n, $k) = @_; # If k > n/2, or, equivalently, 2k > n, compute nok(n, k) as # nok(n, n-k), to minimize the number if iterations in the loop. { my $twok = $c->_mul($c->_two(), $c->_copy($k)); # 2 k if ($c->_acmp($twok, $n) > 0) { # if 2k > n $k = $c->_sub($c->_copy($n), $k); # k = n - k } } # Example: # # / 7 \ 7! 1234 * 567 5 * 6 * 7 6 7 # \| \| = --------- = --------------- = --------- = 5 * - * - # \ 3 / (7-3)! 3! 1234 123 1 * 2 * 3 2 3 if ($c->_is_zero($k)) { @$n = 1; } else { # Make a copy of the original n, since we'll be modifying n in-place. my $n_orig = $c->_copy($n); # n = 5, f = 6, d = 2 (cf. example above) $c->_sub($n, $k); $c->_inc($n); my $f = $c->_copy($n); $c->_inc($f); my $d = $c->_two(); # while f <= n (the original n, that is) ... while ($c->_acmp($f, $n_orig) <= 0) { # n = (n * f / d) == 5 * 6 / 2 (cf. example above) $c->_mul($n, $f); $c->_div($n, $d); # f = 7, d = 3 (cf. example above) $c->_inc($f); $c->_inc($d); } } return $n; } my @factorials = ( 1, 1, 2, 23, 234, 2345, 23456, 234567, ); sub _fac { # factorial of $x # ref to array, return ref to array my ($c, $cx) = @_; if ((@$cx == 1) && ($cx->[0] <= 7)) { $cx->[0] = $factorials[$cx->[0]]; # 0 => 1, 1 => 1, 2 => 2 etc. return $cx; } if ((@$cx == 1) && # we do this only if $x >= 12 and $x <= 7000 ($cx->[0] >= 12 && $cx->[0] < 7000)) { # Calculate (k-j) (k-j+1) ... k .. (k+j-1) * (k + j) # See http://blogten.blogspot.com/2007/01/calculating-n.html # The above series can be expressed as factors: # k * k - (j - i) * 2 # We cache kk, and calculate (j j) as the sum of the first j odd integers # This will not work when N exceeds the storage of a Perl scalar, however, # in this case the algorithm would be way too slow to terminate, anyway. # As soon as the last element of $cx is 0, we split it up and remember # how many zeors we got so far. The reason is that n! will accumulate # zeros at the end rather fast. my $zero_elements = 0; # If n is even, set n = n -1 my $k = $c->_num($cx); my $even = 1; if (($k & 1) == 0) { $even = $k; $k --; } # set k to the center point $k = ($k + 1) / 2; # print "k $k even: $even\n"; # now calculate k * k my $k2 = $k * $k; my $odd = 1; my $sum = 1; my $i = $k - 1; # keep reference to x my $new_x = $c->_new($k * $even); @$cx = @$new_x; if ($cx->[0] == 0) { $zero_elements ++; shift @$cx; } # print STDERR "x = ", $c->_str($cx), "\n"; my $BASE2 = int(sqrt($BASE))-1; my $j = 1; while ($j <= $i) { my $m = ($k2 - $sum); $odd += 2; $sum += $odd; $j++; while ($j <= $i && ($m < $BASE2) && (($k2 - $sum) < $BASE2)) { $m = ($k2 - $sum); $odd += 2; $sum += $odd; $j++; # print STDERR "\n k2 $k2 m $m sum $sum odd $odd\n"; sleep(1); } if ($m < $BASE) { $c->_mul($cx, [$m]); } else { $c->_mul($cx, $c->_new($m)); } if ($cx->[0] == 0) { $zero_elements ++; shift @$cx; } # print STDERR "Calculate $k2 - $sum = $m (x = ", $c->_str($cx), ")\n"; } # multiply in the zeros again unshift @$cx, (0) x $zero_elements; return $cx; } # go forward until $base is exceeded limit is either $x steps (steps == 100 # means a result always too high) or $base. my $steps = 100; $steps = $cx->[0] if @$cx == 1; my $r = 2; my $cf = 3; my $step = 2; my $last = $r; while ($r $cf < $BASE && $step < $steps) { $last = $r; $r = $cf++; $step++; } if ((@$cx == 1) && $step == $cx->[0]) { # completely done, so keep reference to $x and return $cx->[0] = $r; return $cx; } # now we must do the left over steps my $n; # steps still to do if (@$cx == 1) { $n = $cx->[0]; } else { $n = $c->_copy($cx); } # Set $cx to the last result below $BASE (but keep ref to $x) $cx->[0] = $last; splice (@$cx, 1); # As soon as the last element of $cx is 0, we split it up and remember # how many zeors we got so far. The reason is that n! will accumulate # zeros at the end rather fast. my $zero_elements = 0; # do left-over steps fit into a scalar? if (ref $n eq 'ARRAY') { # No, so use slower inc() & cmp() # ($n is at least $BASE here) my $base_2 = int(sqrt($BASE)) - 1; #print STDERR "base_2: $base_2\n"; while ($step < $base_2) { if ($cx->[0] == 0) { $zero_elements ++; shift @$cx; } my $b = $step ($step + 1); $step += 2; $c->_mul($cx, [$b]); } $step = [$step]; while ($c->_acmp($step, $n) <= 0) { if ($cx->[0] == 0) { $zero_elements ++; shift @$cx; } $c->_mul($cx, $step); $c->_inc($step); } } else { # Yes, so we can speed it up slightly # print "# left over steps $n\n"; my $base_4 = int(sqrt(sqrt($BASE))) - 2; #print STDERR "base_4: $base_4\n"; my $n4 = $n - 4; while ($step < $n4 && $step < $base_4) { if ($cx->[0] == 0) { $zero_elements ++; shift @$cx; } my $b = $step * ($step + 1); $step += 2; $b = $step ($step + 1); $step += 2; $c->_mul($cx, [$b]); } my $base_2 = int(sqrt($BASE)) - 1; my $n2 = $n - 2; #print STDERR "base_2: $base_2\n"; while ($step < $n2 && $step < $base_2) { if ($cx->[0] == 0) { $zero_elements ++; shift @$cx; } my $b = $step * ($step + 1); $step += 2; $c->_mul($cx, [$b]); } # do what's left over while ($step <= $n) { $c->_mul($cx, [$step]); $step++; if ($cx->[0] == 0) { $zero_elements ++; shift @$cx; } } } # multiply in the zeros again unshift @$cx, (0) x $zero_elements; $cx; # return result } sub _log_int { # calculate integer log of $x to base $base # ref to array, ref to array - return ref to array my ($c, $x, $base) = @_; # X == 0 => NaN return if @$x == 1 && $x->[0] == 0; # BASE 0 or 1 => NaN return if @$base == 1 && $base->[0] < 2; # X == 1 => 0 (is exact) if (@$x == 1 && $x->[0] == 1) { @$x = 0; return $x, 1; } my $cmp = $c->_acmp($x, $base); # X == BASE => 1 (is exact) if ($cmp == 0) { @$x = 1; return $x, 1; } # 1 < X < BASE => 0 (is truncated) if ($cmp < 0) { @$x = 0; return $x, 0; } my $x_org = $c->_copy($x); # preserve x # Compute a guess for the result based on: # $guess = int ( length_in_base_10(X) / ( log(base) / log(10) ) ) my $len = $c->_len($x_org); my $log = log($base->[-1]) / log(10); # for each additional element in $base, we add $BASE_LEN to the result, # based on the observation that log($BASE, 10) is BASE_LEN and # log(xy) == log(x) + log(y): $log += (@$base - 1) $BASE_LEN; # calculate now a guess based on the values obtained above: my $res = int($len / $log); @$x = $res; my $trial = $c->_pow($c->_copy($base), $x); my $acmp = $c->_acmp($trial, $x_org); # Did we get the exact result? return $x, 1 if $acmp == 0; # Too small? while ($acmp < 0) { $c->_mul($trial, $base); $c->_inc($x); $acmp = $c->_acmp($trial, $x_org); } # Too big? while ($acmp > 0) { $c->_div($trial, $base); $c->_dec($x); $acmp = $c->_acmp($trial, $x_org); } return $x, 1 if $acmp == 0; # result is exact return $x, 0; # result is too small } # for debugging: use constant DEBUG => 0; my $steps = 0; sub steps { $steps }; sub _sqrt { # square-root of $x in place # Compute a guess of the result (by rule of thumb), then improve it via # Newton's method. my ($c, $x) = @_; if (@$x == 1) { # fits into one Perl scalar, so result can be computed directly $x->[0] = int(sqrt($x->[0])); return $x; } my $y = $c->_copy($x); # hopefully _len/2 is < $BASE, the -1 is to always undershot the guess # since our guess will "grow" my $l = int(($c->_len($x)-1) / 2); my $lastelem = $x->[-1]; # for guess my $elems = @$x - 1; # not enough digits, but could have more? if ((length($lastelem) <= 3) && ($elems > 1)) { # right-align with zero pad my $len = length($lastelem) & 1; print "$lastelem => " if DEBUG; $lastelem .= substr($x->[-2] . '0' x $BASE_LEN, 0, $BASE_LEN); # former odd => make odd again, or former even to even again $lastelem = $lastelem / 10 if (length($lastelem) & 1) != $len; print "$lastelem\n" if DEBUG; } # construct $x (instead of $c->_lsft($x, $l, 10) my $r = $l % $BASE_LEN; # 10000 00000 00000 00000 ($BASE_LEN=5) $l = int($l / $BASE_LEN); print "l = $l " if DEBUG; splice @$x, $l; # keep ref($x), but modify it # we make the first part of the guess not '1000...0' but int(sqrt($lastelem)) # that gives us: # 14400 00000 => sqrt(14400) => guess first digits to be 120 # 144000 000000 => sqrt(144000) => guess 379 print "$lastelem (elems $elems) => " if DEBUG; $lastelem = $lastelem / 10 if ($elems & 1 == 1); # odd or even? my $g = sqrt($lastelem); $g =~ s/\.//; # 2.345 => 2345 $r -= 1 if $elems & 1 == 0; # 70 => 7 # padd with zeros if result is too short $x->[$l--] = int(substr($g . '0' x $r, 0, $r+1)); print "now ", $x->[-1] if DEBUG; print " would have been ", int('1' . '0' x $r), "\n" if DEBUG; # If @$x > 1, we could compute the second elem of the guess, too, to create # an even better guess. Not implemented yet. Does it improve performance? $x->[$l--] = 0 while ($l >= 0); # all other digits of guess are zero print "start x= ", $c->_str($x), "\n" if DEBUG; my $two = $c->_two(); my $last = $c->_zero(); my $lastlast = $c->_zero(); $steps = 0 if DEBUG; while ($c->_acmp($last, $x) != 0 && $c->_acmp($lastlast, $x) != 0) { $steps++ if DEBUG; $lastlast = $c->_copy($last); $last = $c->_copy($x); $c->_add($x, $c->_div($c->_copy($y), $x)); $c->_div($x, $two ); print " x= ", $c->_str($x), "\n" if DEBUG; } print "\nsteps in sqrt: $steps, " if DEBUG; $c->_dec($x) if $c->_acmp($y, $c->_mul($c->_copy($x), $x)) < 0; # overshot? print " final ", $x->[-1], "\n" if DEBUG; $x; } sub _root { # Take n'th root of $x in place. my ($c, $x, $n) = @_; # Small numbers. if (@$x == 1 && @$n == 1) { # Result can be computed directly. Adjust initial result for numerical # errors, e.g., int(1000(1/3)) is 2, not 3. my $y = int($x->[0] (1 / $n->[0])); my $yp1 = $y + 1; $y = $yp1 if $yp1 ** $n->[0] == $x->[0]; $x->[0] = $y; return $x; } # If x <= n, the result is always (truncated to) 1. if ((@$x > 1 \|\| $x -> [0] > 0) && # if x is non-zero ... $c -> _acmp($x, $n) <= 0) # ... and x <= n { my $one = $x -> _one(); @$x = @$one; return $x; } # If $n is a power of two, take sqrt($x) repeatedly, e.g., root($x, 4) = # sqrt(sqrt($x)), root($x, 8) = sqrt(sqrt(sqrt($x))). my $b = $c -> _as_bin($n); if ($b =~ /0b1(0+)$/) { my $count = length($1); # 0b100 => len('00') => 2 my $cnt = $count; # counter for loop unshift @$x, 0; # add one element, together with one # more below in the loop this makes 2 while ($cnt-- > 0) { # 'Inflate' $x by adding one element, basically computing # $x * $BASE * $BASE. This gives us more $BASE_LEN digits for # result since len(sqrt($X)) approx == len($x) / 2. unshift @$x, 0; # Calculate sqrt($x), $x is now one element to big, again. In the # next round we make that two, again. $c -> _sqrt($x); } # $x is now one element too big, so truncate result by removing it. shift @$x; return $x; } my $DEBUG = 0; # Now the general case. This works by finding an initial guess. If this # guess is incorrect, a relatively small delta is chosen. This delta is # used to find a lower and upper limit for the correct value. The delta is # doubled in each iteration. When a lower and upper limit is found, # bisection is applied to narrow down the region until we have the correct # value. # Split x into mantissa and exponent in base 10, so that # # x = xm * 10^xe, where 0 < xm < 1 and xe is an integer my $x_str = $c -> _str($x); my $xm = "." . $x_str; my $xe = length($x_str); # From this we compute the base 10 logarithm of x # # log_10(x) = log_10(xm) + log_10(xe^10) # = log(xm)/log(10) + xe # # and then the base 10 logarithm of y, where y = x^(1/n) # # log_10(y) = log_10(x)/n my $log10x = log($xm) / log(10) + $xe; my $log10y = $log10x / $c -> _num($n); # And from this we compute ym and ye, the mantissa and exponent (in # base 10) of y, where 1 < ym <= 10 and ye is an integer. my $ye = int $log10y; my $ym = 10 ** ($log10y - $ye); # Finally, we scale the mantissa and exponent to incraese the integer # part of ym, before building the string representing our guess of y. if ($DEBUG) { print "\n"; print "xm = $xm\n"; print "xe = $xe\n"; print "log10x = $log10x\n"; print "log10y = $log10y\n"; print "ym = $ym\n"; print "ye = $ye\n"; print "\n"; } my $d = $ye < 15 ? $ye : 15; $ym = 10 * $d; $ye -= $d; my $y_str = sprintf('%.0f', $ym) . "0" x $ye; my $y = $c -> _new($y_str); if ($DEBUG) { print "ym = $ym\n"; print "ye = $ye\n"; print "\n"; print "y_str = $y_str (initial guess)\n"; print "\n"; } # See if our guess y is correct. my $trial = $c -> _pow($c -> _copy($y), $n); my $acmp = $c -> _acmp($trial, $x); if ($acmp == 0) { @$x = @$y; return $x; } # Find a lower and upper limit for the correct value of y. Start off with a # delta value that is approximately the size of the accuracy of the guess. my $lower; my $upper; my $delta = $c -> _new("1" . ("0" x $ye)); my $two = $c -> _two(); if ($acmp < 0) { $lower = $y; while ($acmp < 0) { $upper = $c -> _add($c -> _copy($lower), $delta); if ($DEBUG) { print "lower = $lower\n"; print "upper = $upper\n"; print "delta = $delta\n"; print "\n"; } $acmp = $c -> _acmp($c -> _pow($c -> _copy($upper), $n), $x); if ($acmp == 0) { @$x = @$upper; return $x; } $delta = $c -> _mul($delta, $two); } } elsif ($acmp > 0) { $upper = $y; while ($acmp > 0) { if ($c -> _acmp($upper, $delta) <= 0) { $lower = $c -> _zero(); last; } $lower = $c -> _sub($c -> _copy($upper), $delta); if ($DEBUG) { print "lower = $lower\n"; print "upper = $upper\n"; print "delta = $delta\n"; print "\n"; } $acmp = $c -> _acmp($c -> _pow($c -> _copy($lower), $n), $x); if ($acmp == 0) { @$x = @$lower; return $x; } $delta = $c -> _mul($delta, $two); } } # Use bisection to narrow down the interval. my $one = $c -> _one(); { $delta = $c -> _sub($c -> _copy($upper), $lower); if ($c -> _acmp($delta, $one) <= 0) { @$x = @$lower; return $x; } if ($DEBUG) { print "lower = $lower\n"; print "upper = $upper\n"; print "delta = $delta\n"; print "\n"; } $delta = $c -> _div($delta, $two); my $middle = $c -> _add($c -> _copy($lower), $delta); $acmp = $c -> _acmp($c -> _pow($c -> _copy($middle), $n), $x); if ($acmp < 0) { $lower = $middle; } elsif ($acmp > 0) { $upper = $middle; } else { @$x = @$middle; return $x; } redo; } $x; } ############################################################################## # binary stuff sub _and { my ($c, $x, $y) = @_; # the shortcut makes equal, large numbers _really_ fast, and makes only a # very small performance drop for small numbers (e.g. something with less # than 32 bit) Since we optimize for large numbers, this is enabled. return $x if $c->_acmp($x, $y) == 0; # shortcut my $m = $c->_one(); my ($xr, $yr); my $mask = $AND_MASK; my $x1 = $c->_copy($x); my $y1 = $c->_copy($y); my $z = $c->_zero(); use integer; until ($c->_is_zero($x1) \|\| $c->_is_zero($y1)) { ($x1, $xr) = $c->_div($x1, $mask); ($y1, $yr) = $c->_div($y1, $mask); $c->_add($z, $c->_mul([ 0 + $xr->[0] & 0 + $yr->[0] ], $m)); $c->_mul($m, $mask); } @$x = @$z; return $x; } sub _xor { my ($c, $x, $y) = @_; return $c->_zero() if $c->_acmp($x, $y) == 0; # shortcut (see -and) my $m = $c->_one(); my ($xr, $yr); my $mask = $XOR_MASK; my $x1 = $c->_copy($x); my $y1 = $c->_copy($y); # make copy my $z = $c->_zero(); use integer; until ($c->_is_zero($x1) \|\| $c->_is_zero($y1)) { ($x1, $xr) = $c->_div($x1, $mask); ($y1, $yr) = $c->_div($y1, $mask); # make ints() from $xr, $yr (see _and()) #$b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b = $BASE; } #$b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ $b; $b = $BASE; } #$c->_add($x, $c->_mul($c->_new($xrr ^ $yrr)), $m) ); $c->_add($z, $c->_mul([ 0 + $xr->[0] ^ 0 + $yr->[0] ], $m)); $c->_mul($m, $mask); } # the loop stops when the shorter of the two numbers is exhausted # the remainder of the longer one will survive bit-by-bit, so we simple # multiply-add it in $c->_add($z, $c->_mul($x1, $m) ) if !$c->_is_zero($x1); $c->_add($z, $c->_mul($y1, $m) ) if !$c->_is_zero($y1); @$x = @$z; return $x; } sub _or { my ($c, $x, $y) = @_; return $x if $c->_acmp($x, $y) == 0; # shortcut (see _and) my $m = $c->_one(); my ($xr, $yr); my $mask = $OR_MASK; my $x1 = $c->_copy($x); my $y1 = $c->_copy($y); # make copy my $z = $c->_zero(); use integer; until ($c->_is_zero($x1) \|\| $c->_is_zero($y1)) { ($x1, $xr) = $c->_div($x1, $mask); ($y1, $yr) = $c->_div($y1, $mask); # make ints() from $xr, $yr (see _and()) # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ $b; $b = $BASE; } # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ $b; $b = $BASE; } # $c->_add($x, $c->_mul(_new( $c, ($xrr \| $yrr) ), $m) ); $c->_add($z, $c->_mul([ 0 + $xr->[0] \| 0 + $yr->[0] ], $m)); $c->_mul($m, $mask); } # the loop stops when the shorter of the two numbers is exhausted # the remainder of the longer one will survive bit-by-bit, so we simple # multiply-add it in $c->_add($z, $c->_mul($x1, $m) ) if !$c->_is_zero($x1); $c->_add($z, $c->_mul($y1, $m) ) if !$c->_is_zero($y1); @$x = @$z; return $x; } sub _as_hex { # convert a decimal number to hex (ref to array, return ref to string) my ($c, $x) = @_; # fits into one element (handle also 0x0 case) return sprintf("0x%x", $x->[0]) if @$x == 1; my $x1 = $c->_copy($x); my $es = ''; my ($xr, $h, $x10000); if ($] >= 5.006) { $x10000 = [ 0x10000 ]; $h = 'h4'; } else { $x10000 = [ 0x1000 ]; $h = 'h3'; } while (@$x1 != 1 \|\| $x1->[0] != 0) # _is_zero() { ($x1, $xr) = $c->_div($x1, $x10000); $es .= unpack($h, pack('V', $xr->[0])); } $es = reverse $es; $es =~ s/^[0]+//; # strip leading zeros '0x' . $es; # return result prepended with 0x } sub _as_bin { # convert a decimal number to bin (ref to array, return ref to string) my ($c, $x) = @_; # fits into one element (and Perl recent enough), handle also 0b0 case # handle zero case for older Perls if ($] <= 5.005 && @$x == 1 && $x->[0] == 0) { my $t = '0b0'; return $t; } if (@$x == 1 && $] >= 5.006) { my $t = sprintf("0b%b", $x->[0]); return $t; } my $x1 = $c->_copy($x); my $es = ''; my ($xr, $b, $x10000); if ($] >= 5.006) { $x10000 = [ 0x10000 ]; $b = 'b16'; } else { $x10000 = [ 0x1000 ]; $b = 'b12'; } while (!(@$x1 == 1 && $x1->[0] == 0)) # _is_zero() { ($x1, $xr) = $c->_div($x1, $x10000); $es .= unpack($b, pack('v', $xr->[0])); } $es = reverse $es; $es =~ s/^[0]+//; # strip leading zeros '0b' . $es; # return result prepended with 0b } sub _as_oct { # convert a decimal number to octal (ref to array, return ref to string) my ($c, $x) = @_; # fits into one element (handle also 0 case) return sprintf("0%o", $x->[0]) if @$x == 1; my $x1 = $c->_copy($x); my $es = ''; my $xr; my $x1000 = [ 0100000 ]; while (@$x1 != 1 \|\| $x1->[0] != 0) # _is_zero() { ($x1, $xr) = $c->_div($x1, $x1000); $es .= reverse sprintf("%05o", $xr->[0]); } $es = reverse $es; $es =~ s/^0+//; # strip leading zeros '0' . $es; # return result prepended with 0 } sub _from_oct { # convert a octal number to decimal (string, return ref to array) my ($c, $os) = @_; # for older Perls, play safe my $m = [ 0100000 ]; my $d = 5; # 5 digits at a time my $mul = $c->_one(); my $x = $c->_zero(); my $len = int((length($os) - 1) / $d); # $d digit parts, w/o the '0' my $val; my $i = -$d; while ($len >= 0) { $val = substr($os, $i, $d); # get oct digits $val = CORE::oct($val); $i -= $d; $len --; my $adder = [ $val ]; $c->_add($x, $c->_mul($adder, $mul)) if $val != 0; $c->_mul($mul, $m) if $len >= 0; # skip last mul } $x; } sub _from_hex { # convert a hex number to decimal (string, return ref to array) my ($c, $hs) = @_; my $m = $c->_new(0x10000000); # 28 bit at a time (<32 bit!) my $d = 7; # 7 digits at a time my $mul = $c->_one(); my $x = $c->_zero(); my $len = int((length($hs) - 2) / $d); # $d digit parts, w/o the '0x' my $val; my $i = -$d; while ($len >= 0) { $val = substr($hs, $i, $d); # get hex digits $val =~ s/^0x// if $len == 0; # for last part only because $val = CORE::hex($val); # hex does not like wrong chars $i -= $d; $len --; my $adder = [ $val ]; # if the resulting number was to big to fit into one element, create a # two-element version (bug found by Mark Lakata - Thanx!) if (CORE::length($val) > $BASE_LEN) { $adder = $c->_new($val); } $c->_add($x, $c->_mul($adder, $mul)) if $val != 0; $c->_mul($mul, $m) if $len >= 0; # skip last mul } $x; } sub _from_bin { # convert a hex number to decimal (string, return ref to array) my ($c, $bs) = @_; # instead of converting X (8) bit at a time, it is faster to "convert" the # number to hex, and then call _from_hex. my $hs = $bs; $hs =~ s/^[+-]?0b//; # remove sign and 0b my $l = length($hs); # bits $hs = '0' x (8 - ($l % 8)) . $hs if ($l % 8) != 0; # padd left side w/ 0 my $h = '0x' . unpack('H', pack ('B', $hs)); # repack as hex $c->_from_hex($h); } ############################################################################## # special modulus functions sub _modinv { # modular multiplicative inverse my ($c, $x, $y) = @_; # modulo zero if ($c->_is_zero($y)) { return undef, undef; } # modulo one if ($c->_is_one($y)) { return $c->_zero(), '+'; } my $u = $c->_zero(); my $v = $c->_one(); my $a = $c->_copy($y); my $b = $c->_copy($x); # Euclid's Algorithm for bgcd(), only that we calc bgcd() ($a) and the result # ($u) at the same time. See comments in BigInt for why this works. my $q; my $sign = 1; { ($a, $q, $b) = ($b, $c->_div($a, $b)); # step 1 last if $c->_is_zero($b); my $t = $c->_add( # step 2: $c->_mul($c->_copy($v), $q), # t = v q $u); # + u $u = $v; # u = v $v = $t; # v = t $sign = -$sign; redo; } # if the gcd is not 1, then return NaN return (undef, undef) unless $c->_is_one($a); ($v, $sign == 1 ? '+' : '-'); } sub _modpow { # modulus of power ($x $y) % $z my ($c, $num, $exp, $mod) = @_; # a^b (mod 1) = 0 for all a and b if ($c->_is_one($mod)) { @$num = 0; return $num; } # 0^a (mod m) = 0 if m != 0, a != 0 # 0^0 (mod m) = 1 if m != 0 if ($c->_is_zero($num)) { if ($c->_is_zero($exp)) { @$num = 1; } else { @$num = 0; } return $num; } # $num = $c->_mod($num, $mod); # this does not make it faster my $acc = $c->_copy($num); my $t = $c->_one(); my $expbin = $c->_as_bin($exp); $expbin =~ s/^0b//; my $len = length($expbin); while (--$len >= 0) { if (substr($expbin, $len, 1) eq '1') { # is_odd $t = $c->_mul($t, $acc); $t = $c->_mod($t, $mod); } $acc = $c->_mul($acc, $acc); $acc = $c->_mod($acc, $mod); } @$num = @$t; $num; } sub _gcd { # Greatest common divisor. my ($c, $x, $y) = @_; # gcd(0, 0) = 0 # gcd(0, a) = a, if a != 0 if (@$x == 1 && $x->[0] == 0) { if (@$y == 1 && $y->[0] == 0) { @$x = 0; } else { @$x = @$y; } return $x; } # Until $y is zero ... until (@$y == 1 && $y->[0] == 0) { # Compute remainder. $c->_mod($x, $y); # Swap $x and $y. my $tmp = $c->_copy($x); @$x = @$y; $y = $tmp; # no deref here; that would modify input $y } return $x; } 1; =pod =head1 NAME Math::BigInt::Calc - Pure Perl module to support Math::BigInt =head1 SYNOPSIS # to use it with Math::BigInt use Math::BigInt lib => 'Calc'; # to use it with Math::BigFloat use Math::BigFloat lib => 'Calc'; # to use it with Math::BigRat use Math::BigRat lib => 'Calc'; =head1 DESCRIPTION Math::BigInt::Calc inherits from Math::BigInt::Lib. In this library, the numbers are represented in base B = 10N, where N is the largest possible value that does not cause overflow in the intermediate computations. The base B elements are stored in an array, with the least significant element stored in array element zero. There are no leading zero elements, except a single zero element when the number is zero. For instance, if B = 10000, the number 1234567890 is represented internally as [7890, 3456, 12]. =head1 SEE ALSO L<Math::BigInt::Lib> for a description of the API. Alternative libraries L<Math::BigInt::FastCalc>, L<Math::BigInt::GMP>, and L<Math::BigInt::Pari>. Some of the modules that use these libraries L<Math::BigInt>, L<Math::BigFloat>, and L<Math::BigRat>. =cut	71.83 KB	2019-10-20 12:42
📄 Lib.pm package Math::BigInt::Lib; use 5.006001; use strict; use warnings; our $VERSION = '1.999818'; use Carp; use overload # overload key: with_assign '+' => sub { my $class = ref $_[0]; my $x = $class -> _copy($_[0]); my $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]); return $class -> _add($x, $y); }, '-' => sub { my $class = ref $_[0]; my ($x, $y); if ($_[2]) { # if swapped $y = $_[0]; $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } else { $x = $class -> _copy($_[0]); $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } return $class -> _sub($x, $y); }, '' => sub { my $class = ref $_[0]; my $x = $class -> _copy($_[0]); my $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]); return $class -> _mul($x, $y); }, '/' => sub { my $class = ref $_[0]; my ($x, $y); if ($_[2]) { # if swapped $y = $_[0]; $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } else { $x = $class -> _copy($_[0]); $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } return $class -> _div($x, $y); }, '%' => sub { my $class = ref $_[0]; my ($x, $y); if ($_[2]) { # if swapped $y = $_[0]; $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } else { $x = $class -> _copy($_[0]); $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } return $class -> _mod($x, $y); }, '' => sub { my $class = ref $_[0]; my ($x, $y); if ($_[2]) { # if swapped $y = $_[0]; $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } else { $x = $class -> _copy($_[0]); $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } return $class -> _pow($x, $y); }, '<<' => sub { my $class = ref $_[0]; my ($x, $y); if ($_[2]) { # if swapped $y = $class -> _num($_[0]); $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } else { $x = $_[0]; $y = ref($_[1]) ? $class -> _num($_[1]) : $_[1]; } return $class -> _blsft($x, $y); }, '>>' => sub { my $class = ref $_[0]; my ($x, $y); if ($_[2]) { # if swapped $y = $_[0]; $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } else { $x = $class -> _copy($_[0]); $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } return $class -> _brsft($x, $y); }, # overload key: num_comparison '<' => sub { my $class = ref $_[0]; my ($x, $y); if ($_[2]) { # if swapped $y = $_[0]; $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } else { $x = $class -> _copy($_[0]); $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } return $class -> _acmp($x, $y) < 0; }, '<=' => sub { my $class = ref $_[0]; my ($x, $y); if ($_[2]) { # if swapped $y = $_[0]; $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } else { $x = $class -> _copy($_[0]); $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } return $class -> _acmp($x, $y) <= 0; }, '>' => sub { my $class = ref $_[0]; my ($x, $y); if ($_[2]) { # if swapped $y = $_[0]; $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } else { $x = $class -> _copy($_[0]); $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } return $class -> _acmp($x, $y) > 0; }, '>=' => sub { my $class = ref $_[0]; my ($x, $y); if ($_[2]) { # if swapped $y = $_[0]; $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } else { $x = $class -> _copy($_[0]); $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } return $class -> _acmp($x, $y) >= 0; }, '==' => sub { my $class = ref $_[0]; my $x = $class -> _copy($_[0]); my $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]); return $class -> _acmp($x, $y) == 0; }, '!=' => sub { my $class = ref $_[0]; my $x = $class -> _copy($_[0]); my $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]); return $class -> _acmp($x, $y) != 0; }, # overload key: 3way_comparison '<=>' => sub { my $class = ref $_[0]; my ($x, $y); if ($_[2]) { # if swapped $y = $_[0]; $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } else { $x = $class -> _copy($_[0]); $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } return $class -> _acmp($x, $y); }, # overload key: binary '&' => sub { my $class = ref $_[0]; my ($x, $y); if ($_[2]) { # if swapped $y = $_[0]; $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } else { $x = $class -> _copy($_[0]); $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } return $class -> _and($x, $y); }, '\|' => sub { my $class = ref $_[0]; my ($x, $y); if ($_[2]) { # if swapped $y = $_[0]; $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } else { $x = $class -> _copy($_[0]); $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } return $class -> _or($x, $y); }, '^' => sub { my $class = ref $_[0]; my ($x, $y); if ($_[2]) { # if swapped $y = $_[0]; $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } else { $x = $class -> _copy($_[0]); $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]); } return $class -> _xor($x, $y); }, # overload key: func 'abs' => sub { $_[0] }, 'sqrt' => sub { my $class = ref $_[0]; return $class -> _sqrt($class -> _copy($_[0])); }, 'int' => sub { $_[0] }, # overload key: conversion 'bool' => sub { ref($_[0]) -> _is_zero($_[0]) ? '' : 1; }, '""' => sub { ref($_[0]) -> _str($_[0]); }, '0+' => sub { ref($_[0]) -> _num($_[0]); }, '=' => sub { ref($_[0]) -> _copy($_[0]); }, ; sub _new { croak "@{[(caller 0)[3]]} method not implemented"; } sub _zero { my $class = shift; return $class -> _new("0"); } sub _one { my $class = shift; return $class -> _new("1"); } sub _two { my $class = shift; return $class -> _new("2"); } sub _ten { my $class = shift; return $class -> _new("10"); } sub _1ex { my ($class, $exp) = @_; $exp = $class -> _num($exp) if ref($exp); return $class -> _new("1" . ("0" x $exp)); } sub _copy { my ($class, $x) = @_; return $class -> _new($class -> _str($x)); } # catch and throw away sub import { } ############################################################################## # convert back to string and number sub _str { # Convert number from internal base 1eN format to string format. Internal # format is always normalized, i.e., no leading zeros. croak "@{[(caller 0)[3]]} method not implemented"; } sub _num { my ($class, $x) = @_; 0 + $class -> _str($x); } ############################################################################## # actual math code sub _add { croak "@{[(caller 0)[3]]} method not implemented"; } sub _sub { croak "@{[(caller 0)[3]]} method not implemented"; } sub _mul { my ($class, $x, $y) = @_; my $sum = $class -> _zero(); my $i = $class -> _zero(); while ($class -> _acmp($i, $y) < 0) { $sum = $class -> _add($sum, $x); $i = $class -> _inc($i); } return $sum; } sub _div { my ($class, $x, $y) = @_; croak "@{[(caller 0)[3]]} requires non-zero divisor" if $class -> _is_zero($y); my $r = $class -> _copy($x); my $q = $class -> _zero(); while ($class -> _acmp($r, $y) >= 0) { $q = $class -> _inc($q); $r = $class -> _sub($r, $y); } return $q, $r if wantarray; return $q; } sub _inc { my ($class, $x) = @_; $class -> _add($x, $class -> _one()); } sub _dec { my ($class, $x) = @_; $class -> _sub($x, $class -> _one()); } ############################################################################## # testing sub _acmp { # Compare two (absolute) values. Return -1, 0, or 1. my ($class, $x, $y) = @_; my $xstr = $class -> _str($x); my $ystr = $class -> _str($y); length($xstr) <=> length($ystr) \|\| $xstr cmp $ystr; } sub _len { my ($class, $x) = @_; CORE::length($class -> _str($x)); } sub _alen { my ($class, $x) = @_; $class -> _len($x); } sub _digit { my ($class, $x, $n) = @_; substr($class ->_str($x), -($n+1), 1); } sub _digitsum { my ($class, $x) = @_; my $len = $class -> _len($x); my $sum = $class -> _zero(); for (my $i = 0 ; $i < $len ; ++$i) { my $digit = $class -> _digit($x, $i); $digit = $class -> _new($digit); $sum = $class -> _add($sum, $digit); } return $sum; } sub _zeros { my ($class, $x) = @_; my $str = $class -> _str($x); $str =~ /[^0](0)\z/ ? CORE::length($1) : 0; } ############################################################################## # _is_* routines sub _is_zero { # return true if arg is zero my ($class, $x) = @_; $class -> _str($x) == 0; } sub _is_even { # return true if arg is even my ($class, $x) = @_; substr($class -> _str($x), -1, 1) % 2 == 0; } sub _is_odd { # return true if arg is odd my ($class, $x) = @_; substr($class -> _str($x), -1, 1) % 2 != 0; } sub _is_one { # return true if arg is one my ($class, $x) = @_; $class -> _str($x) == 1; } sub _is_two { # return true if arg is two my ($class, $x) = @_; $class -> _str($x) == 2; } sub _is_ten { # return true if arg is ten my ($class, $x) = @_; $class -> _str($x) == 10; } ############################################################################### # check routine to test internal state for corruptions sub _check { # used by the test suite my ($class, $x) = @_; return "Input is undefined" unless defined $x; return "$x is not a reference" unless ref($x); return 0; } ############################################################################### sub _mod { # modulus my ($class, $x, $y) = @_; croak "@{[(caller 0)[3]]} requires non-zero second operand" if $class -> _is_zero($y); if ($class -> can('_div')) { $x = $class -> _copy($x); my ($q, $r) = $class -> _div($x, $y); return $r; } else { my $r = $class -> _copy($x); while ($class -> _acmp($r, $y) >= 0) { $r = $class -> _sub($r, $y); } return $r; } } ############################################################################## # shifts sub _rsft { my ($class, $x, $n, $b) = @_; $b = $class -> _new($b) unless ref $b; return scalar $class -> _div($x, $class -> _pow($class -> _copy($b), $n)); } sub _lsft { my ($class, $x, $n, $b) = @_; $b = $class -> _new($b) unless ref $b; return $class -> _mul($x, $class -> _pow($class -> _copy($b), $n)); } sub _pow { # power of $x to $y my ($class, $x, $y) = @_; if ($class -> _is_zero($y)) { return $class -> _one(); # y == 0 => x => 1 } if (($class -> _is_one($x)) \|\| # x == 1 ($class -> _is_one($y))) # or y == 1 { return $x; } if ($class -> _is_zero($x)) { return $class -> _zero(); # 0 ** y => 0 (if not y <= 0) } my $pow2 = $class -> _one(); my $y_bin = $class -> _as_bin($y); $y_bin =~ s/^0b//; my $len = length($y_bin); while (--$len > 0) { $pow2 = $class -> _mul($pow2, $x) if substr($y_bin, $len, 1) eq '1'; $x = $class -> _mul($x, $x); } $x = $class -> _mul($x, $pow2); return $x; } sub _nok { # Return binomial coefficient (n over k). my ($class, $n, $k) = @_; # If k > n/2, or, equivalently, 2k > n, compute nok(n, k) as # nok(n, n-k), to minimize the number if iterations in the loop. { my $twok = $class -> _mul($class -> _two(), $class -> _copy($k)); if ($class -> _acmp($twok, $n) > 0) { $k = $class -> _sub($class -> _copy($n), $k); } } # Example: # # / 7 \ 7! 1234 * 567 5 * 6 * 7 # \| \| = --------- = --------------- = --------- = ((5 * 6) / 2 * 7) / 3 # \ 3 / (7-3)! 3! 1234 123 1 * 2 * 3 # # Equivalently, _nok(11, 5) is computed as # # (((((((7 * 8) / 2) * 9) / 3) * 10) / 4) * 11) / 5 if ($class -> _is_zero($k)) { return $class -> _one(); } # Make a copy of the original n, in case the subclass modifies n in-place. my $n_orig = $class -> _copy($n); # n = 5, f = 6, d = 2 (cf. example above) $n = $class -> _sub($n, $k); $n = $class -> _inc($n); my $f = $class -> _copy($n); $f = $class -> _inc($f); my $d = $class -> _two(); # while f <= n (the original n, that is) ... while ($class -> _acmp($f, $n_orig) <= 0) { $n = $class -> _mul($n, $f); $n = $class -> _div($n, $d); $f = $class -> _inc($f); $d = $class -> _inc($d); } return $n; } sub _fac { # factorial my ($class, $x) = @_; my $two = $class -> _two(); if ($class -> _acmp($x, $two) < 0) { return $class -> _one(); } my $i = $class -> _copy($x); while ($class -> _acmp($i, $two) > 0) { $i = $class -> _dec($i); $x = $class -> _mul($x, $i); } return $x; } sub _dfac { # double factorial my ($class, $x) = @_; my $two = $class -> _two(); if ($class -> _acmp($x, $two) < 0) { return $class -> _one(); } my $i = $class -> _copy($x); while ($class -> _acmp($i, $two) > 0) { $i = $class -> _sub($i, $two); $x = $class -> _mul($x, $i); } return $x; } sub _log_int { # calculate integer log of $x to base $base # calculate integer log of $x to base $base # ref to array, ref to array - return ref to array my ($class, $x, $base) = @_; # X == 0 => NaN return if $class -> _is_zero($x); $base = $class -> _new(2) unless defined($base); $base = $class -> _new($base) unless ref($base); # BASE 0 or 1 => NaN return if $class -> _is_zero($base) \|\| $class -> _is_one($base); # X == 1 => 0 (is exact) if ($class -> _is_one($x)) { return $class -> _zero(), 1; } my $cmp = $class -> _acmp($x, $base); # X == BASE => 1 (is exact) if ($cmp == 0) { return $class -> _one(), 1; } # 1 < X < BASE => 0 (is truncated) if ($cmp < 0) { return $class -> _zero(), 0; } my $y; # log(x) / log(b) = log(xm * 10^xe) / log(bm * 10^be) # = (log(xm) + xe(log(10))) / (log(bm) + belog(10)) { my $x_str = $class -> _str($x); my $b_str = $class -> _str($base); my $xm = "." . $x_str; my $bm = "." . $b_str; my $xe = length($x_str); my $be = length($b_str); my $log10 = log(10); my $guess = int((log($xm) + $xe * $log10) / (log($bm) + $be * $log10)); $y = $class -> _new($guess); } my $trial = $class -> _pow($class -> _copy($base), $y); my $acmp = $class -> _acmp($trial, $x); # Did we get the exact result? return $y, 1 if $acmp == 0; # Too small? while ($acmp < 0) { $trial = $class -> _mul($trial, $base); $y = $class -> _inc($y); $acmp = $class -> _acmp($trial, $x); } # Too big? while ($acmp > 0) { $trial = $class -> _div($trial, $base); $y = $class -> _dec($y); $acmp = $class -> _acmp($trial, $x); } return $y, 1 if $acmp == 0; # result is exact return $y, 0; # result is too small } sub _sqrt { # square-root of $y in place my ($class, $y) = @_; return $y if $class -> _is_zero($y); my $y_str = $class -> _str($y); my $y_len = length($y_str); # Compute the guess $x. my $xm; my $xe; if ($y_len % 2 == 0) { $xm = sqrt("." . $y_str); $xe = $y_len / 2; $xm = sprintf "%.0f", int($xm * 1e15); $xe -= 15; } else { $xm = sqrt(".0" . $y_str); $xe = ($y_len + 1) / 2; $xm = sprintf "%.0f", int($xm * 1e16); $xe -= 16; } my $x; if ($xe < 0) { $x = substr $xm, 0, length($xm) + $xe; } else { $x = $xm . ("0" x $xe); } $x = $class -> _new($x); # Newton's method for computing square root of y # # x(i+1) = x(i) - f(x(i)) / f'(x(i)) # = x(i) - (x(i)^2 - y) / (2 * x(i)) # use if x(i)^2 > y # = y(i) + (y - x(i)^2) / (2 * x(i)) # use if x(i)^2 < y # Determine if x, our guess, is too small, correct, or too large. my $xsq = $class -> _mul($class -> _copy($x), $x); # x(i)^2 my $acmp = $class -> _acmp($xsq, $y); # x(i)^2 <=> y # Only assign a value to this variable if we will be using it. my $two; $two = $class -> _two() if $acmp != 0; # If x is too small, do one iteration of Newton's method. Since the # function f(x) = x^2 - y is concave and monotonically increasing, the next # guess for x will either be correct or too large. if ($acmp < 0) { # x(i+1) = x(i) + (y - x(i)^2) / (2 * x(i)) my $numer = $class -> _sub($class -> _copy($y), $xsq); # y - x(i)^2 my $denom = $class -> _mul($class -> _copy($two), $x); # 2 * x(i) my $delta = $class -> _div($numer, $denom); unless ($class -> _is_zero($delta)) { $x = $class -> _add($x, $delta); $xsq = $class -> _mul($class -> _copy($x), $x); # x(i)^2 $acmp = $class -> _acmp($xsq, $y); # x(i)^2 <=> y } } # If our guess for x is too large, apply Newton's method repeatedly until # we either have got the correct value, or the delta is zero. while ($acmp > 0) { # x(i+1) = x(i) - (x(i)^2 - y) / (2 * x(i)) my $numer = $class -> _sub($xsq, $y); # x(i)^2 - y my $denom = $class -> _mul($class -> _copy($two), $x); # 2 * x(i) my $delta = $class -> _div($numer, $denom); last if $class -> _is_zero($delta); $x = $class -> _sub($x, $delta); $xsq = $class -> _mul($class -> _copy($x), $x); # x(i)^2 $acmp = $class -> _acmp($xsq, $y); # x(i)^2 <=> y } # When the delta is zero, our value for x might still be too large. We # require that the outout is either exact or too small (i.e., rounded down # to the nearest integer), so do a final check. while ($acmp > 0) { $x = $class -> _dec($x); $xsq = $class -> _mul($class -> _copy($x), $x); # x(i)^2 $acmp = $class -> _acmp($xsq, $y); # x(i)^2 <=> y } return $x; } sub _root { my ($class, $y, $n) = @_; return $y if $class -> _is_zero($y) \|\| $class -> _is_one($y) \|\| $class -> _is_one($n); # If y <= n, the result is always (truncated to) 1. return $class -> _one() if $class -> _acmp($y, $n) <= 0; # Compute the initial guess x of y^(1/n). When n is large, Newton's method # converges slowly if the "guess" (initial value) is poor, so we need a # good guess. It the guess is too small, the next guess will be too large, # and from then on all guesses are too large. my $DEBUG = 0; # Split y into mantissa and exponent in base 10, so that # # y = xm * 10^xe, where 0 < xm < 1 and xe is an integer my $y_str = $class -> _str($y); my $ym = "." . $y_str; my $ye = length($y_str); # From this compute the approximate base 10 logarithm of y # # log_10(y) = log_10(ym) + log_10(ye^10) # = log(ym)/log(10) + ye my $log10y = log($ym) / log(10) + $ye; # And from this compute the approximate base 10 logarithm of x, where # x = y^(1/n) # # log_10(x) = log_10(y)/n my $log10x = $log10y / $class -> _num($n); # From this compute xm and xe, the mantissa and exponent (in base 10) of x, # where 1 < xm <= 10 and xe is an integer. my $xe = int $log10x; my $xm = 10 ** ($log10x - $xe); # Scale the mantissa and exponent to increase the integer part of ym, which # gives us better accuracy. if ($DEBUG) { print "\n"; print "y_str = $y_str\n"; print "ym = $ym\n"; print "ye = $ye\n"; print "log10y = $log10y\n"; print "log10x = $log10x\n"; print "xm = $xm\n"; print "xe = $xe\n"; } my $d = $xe < 15 ? $xe : 15; $xm = 10 * $d; $xe -= $d; if ($DEBUG) { print "\n"; print "xm = $xm\n"; print "xe = $xe\n"; } # If the mantissa is not an integer, round up to nearest integer, and then # convert the number to a string. It is important to always round up due to # how Newton's method behaves in this case. If the initial guess is too # small, the next guess will be too large, after which every succeeding # guess converges the correct value from above. Now, if the initial guess # is too small and n is large, the next guess will be much too large and # require a large number of iterations to get close to the solution. # Because of this, we are likely to find the solution faster if we make # sure the initial guess is not too small. my $xm_int = int($xm); my $x_str = sprintf '%.0f', $xm > $xm_int ? $xm_int + 1 : $xm_int; $x_str .= "0" x $xe; my $x = $class -> _new($x_str); if ($DEBUG) { print "xm = $xm\n"; print "xe = $xe\n"; print "\n"; print "x_str = $x_str (initial guess)\n"; print "\n"; } # Use Newton's method for computing n'th root of y. # # x(i+1) = x(i) - f(x(i)) / f'(x(i)) # = x(i) - (x(i)^n - y) / (n * x(i)^(n-1)) # use if x(i)^n > y # = x(i) + (y - x(i)^n) / (n * x(i)^(n-1)) # use if x(i)^n < y # Determine if x, our guess, is too small, correct, or too large. Rather # than computing x(i)^n and x(i)^(n-1) directly, compute x(i)^(n-1) and # then the same value multiplied by x. my $nm1 = $class -> _dec($class -> _copy($n)); # n-1 my $xpownm1 = $class -> _pow($class -> _copy($x), $nm1); # x(i)^(n-1) my $xpown = $class -> _mul($class -> _copy($xpownm1), $x); # x(i)^n my $acmp = $class -> _acmp($xpown, $y); # x(i)^n <=> y if ($DEBUG) { print "\n"; print "x = ", $class -> _str($x), "\n"; print "x^n = ", $class -> _str($xpown), "\n"; print "y = ", $class -> _str($y), "\n"; print "acmp = $acmp\n"; } # If x is too small, do one iteration of Newton's method. Since the # function f(x) = x^n - y is concave and monotonically increasing, the next # guess for x will either be correct or too large. if ($acmp < 0) { # x(i+1) = x(i) + (y - x(i)^n) / (n * x(i)^(n-1)) my $numer = $class -> _sub($class -> _copy($y), $xpown); # y - x(i)^n my $denom = $class -> _mul($class -> _copy($n), $xpownm1); # n * x(i)^(n-1) my $delta = $class -> _div($numer, $denom); if ($DEBUG) { print "\n"; print "numer = ", $class -> _str($numer), "\n"; print "denom = ", $class -> _str($denom), "\n"; print "delta = ", $class -> _str($delta), "\n"; } unless ($class -> _is_zero($delta)) { $x = $class -> _add($x, $delta); $xpownm1 = $class -> _pow($class -> _copy($x), $nm1); # x(i)^(n-1) $xpown = $class -> _mul($class -> _copy($xpownm1), $x); # x(i)^n $acmp = $class -> _acmp($xpown, $y); # x(i)^n <=> y if ($DEBUG) { print "\n"; print "x = ", $class -> _str($x), "\n"; print "x^n = ", $class -> _str($xpown), "\n"; print "y = ", $class -> _str($y), "\n"; print "acmp = $acmp\n"; } } } # If our guess for x is too large, apply Newton's method repeatedly until # we either have got the correct value, or the delta is zero. while ($acmp > 0) { # x(i+1) = x(i) - (x(i)^n - y) / (n * x(i)^(n-1)) my $numer = $class -> _sub($class -> _copy($xpown), $y); # x(i)^n - y my $denom = $class -> _mul($class -> _copy($n), $xpownm1); # n * x(i)^(n-1) if ($DEBUG) { print "numer = ", $class -> _str($numer), "\n"; print "denom = ", $class -> _str($denom), "\n"; } my $delta = $class -> _div($numer, $denom); if ($DEBUG) { print "delta = ", $class -> _str($delta), "\n"; } last if $class -> _is_zero($delta); $x = $class -> _sub($x, $delta); $xpownm1 = $class -> _pow($class -> _copy($x), $nm1); # x(i)^(n-1) $xpown = $class -> _mul($class -> _copy($xpownm1), $x); # x(i)^n $acmp = $class -> _acmp($xpown, $y); # x(i)^n <=> y if ($DEBUG) { print "\n"; print "x = ", $class -> _str($x), "\n"; print "x^n = ", $class -> _str($xpown), "\n"; print "y = ", $class -> _str($y), "\n"; print "acmp = $acmp\n"; } } # When the delta is zero, our value for x might still be too large. We # require that the outout is either exact or too small (i.e., rounded down # to the nearest integer), so do a final check. while ($acmp > 0) { $x = $class -> _dec($x); $xpown = $class -> _pow($class -> _copy($x), $n); # x(i)^n $acmp = $class -> _acmp($xpown, $y); # x(i)^n <=> y } return $x; } ############################################################################## # binary stuff sub _and { my ($class, $x, $y) = @_; return $x if $class -> _acmp($x, $y) == 0; my $m = $class -> _one(); my $mask = $class -> _new("32768"); my ($xr, $yr); # remainders after division my $xc = $class -> _copy($x); my $yc = $class -> _copy($y); my $z = $class -> _zero(); until ($class -> _is_zero($xc) \|\| $class -> _is_zero($yc)) { ($xc, $xr) = $class -> _div($xc, $mask); ($yc, $yr) = $class -> _div($yc, $mask); my $bits = $class -> _new($class -> _num($xr) & $class -> _num($yr)); $z = $class -> _add($z, $class -> _mul($bits, $m)); $m = $class -> _mul($m, $mask); } return $z; } sub _xor { my ($class, $x, $y) = @_; return $class -> _zero() if $class -> _acmp($x, $y) == 0; my $m = $class -> _one(); my $mask = $class -> _new("32768"); my ($xr, $yr); # remainders after division my $xc = $class -> _copy($x); my $yc = $class -> _copy($y); my $z = $class -> _zero(); until ($class -> _is_zero($xc) \|\| $class -> _is_zero($yc)) { ($xc, $xr) = $class -> _div($xc, $mask); ($yc, $yr) = $class -> _div($yc, $mask); my $bits = $class -> _new($class -> _num($xr) ^ $class -> _num($yr)); $z = $class -> _add($z, $class -> _mul($bits, $m)); $m = $class -> _mul($m, $mask); } # The loop above stops when the smallest of the two numbers is exhausted. # The remainder of the longer one will survive bit-by-bit, so we simple # multiply-add it in. $z = $class -> _add($z, $class -> _mul($xc, $m)) unless $class -> _is_zero($xc); $z = $class -> _add($z, $class -> _mul($yc, $m)) unless $class -> _is_zero($yc); return $z; } sub _or { my ($class, $x, $y) = @_; return $x if $class -> _acmp($x, $y) == 0; # shortcut (see _and) my $m = $class -> _one(); my $mask = $class -> _new("32768"); my ($xr, $yr); # remainders after division my $xc = $class -> _copy($x); my $yc = $class -> _copy($y); my $z = $class -> _zero(); until ($class -> _is_zero($xc) \|\| $class -> _is_zero($yc)) { ($xc, $xr) = $class -> _div($xc, $mask); ($yc, $yr) = $class -> _div($yc, $mask); my $bits = $class -> _new($class -> _num($xr) \| $class -> _num($yr)); $z = $class -> _add($z, $class -> _mul($bits, $m)); $m = $class -> _mul($m, $mask); } # The loop above stops when the smallest of the two numbers is exhausted. # The remainder of the longer one will survive bit-by-bit, so we simple # multiply-add it in. $z = $class -> _add($z, $class -> _mul($xc, $m)) unless $class -> _is_zero($xc); $z = $class -> _add($z, $class -> _mul($yc, $m)) unless $class -> _is_zero($yc); return $z; } sub _sand { my ($class, $x, $sx, $y, $sy) = @_; return ($class -> _zero(), '+') if $class -> _is_zero($x) \|\| $class -> _is_zero($y); my $sign = $sx eq '-' && $sy eq '-' ? '-' : '+'; my ($bx, $by); if ($sx eq '-') { # if x is negative # two's complement: inc (dec unsigned value) and flip all "bits" in $bx $bx = $class -> _copy($x); $bx = $class -> _dec($bx); $bx = $class -> _as_hex($bx); $bx =~ s/^-?0x//; $bx =~ tr<0123456789abcdef> <\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>; } else { # if x is positive $bx = $class -> _as_hex($x); # get binary representation $bx =~ s/^-?0x//; $bx =~ tr<fedcba9876543210> <\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>; } if ($sy eq '-') { # if y is negative # two's complement: inc (dec unsigned value) and flip all "bits" in $by $by = $class -> _copy($y); $by = $class -> _dec($by); $by = $class -> _as_hex($by); $by =~ s/^-?0x//; $by =~ tr<0123456789abcdef> <\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>; } else { $by = $class -> _as_hex($y); # get binary representation $by =~ s/^-?0x//; $by =~ tr<fedcba9876543210> <\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>; } # now we have bit-strings from X and Y, reverse them for padding $bx = reverse $bx; $by = reverse $by; # padd the shorter string my $xx = "\x00"; $xx = "\x0f" if $sx eq '-'; my $yy = "\x00"; $yy = "\x0f" if $sy eq '-'; my $diff = CORE::length($bx) - CORE::length($by); if ($diff > 0) { # if $yy eq "\x00", we can cut $bx, otherwise we need to padd $by $by .= $yy x $diff; } elsif ($diff < 0) { # if $xx eq "\x00", we can cut $by, otherwise we need to padd $bx $bx .= $xx x abs($diff); } # and the strings together my $r = $bx & $by; # and reverse the result again $bx = reverse $r; # One of $bx or $by was negative, so need to flip bits in the result. In both # cases (one or two of them negative, or both positive) we need to get the # characters back. if ($sign eq '-') { $bx =~ tr<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00> <0123456789abcdef>; } else { $bx =~ tr<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00> <fedcba9876543210>; } # leading zeros will be stripped by _from_hex() $bx = '0x' . $bx; $bx = $class -> _from_hex($bx); $bx = $class -> _inc($bx) if $sign eq '-'; # avoid negative zero $sign = '+' if $class -> _is_zero($bx); return $bx, $sign; } sub _sxor { my ($class, $x, $sx, $y, $sy) = @_; return ($class -> _zero(), '+') if $class -> _is_zero($x) && $class -> _is_zero($y); my $sign = $sx ne $sy ? '-' : '+'; my ($bx, $by); if ($sx eq '-') { # if x is negative # two's complement: inc (dec unsigned value) and flip all "bits" in $bx $bx = $class -> _copy($x); $bx = $class -> _dec($bx); $bx = $class -> _as_hex($bx); $bx =~ s/^-?0x//; $bx =~ tr<0123456789abcdef> <\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>; } else { # if x is positive $bx = $class -> _as_hex($x); # get binary representation $bx =~ s/^-?0x//; $bx =~ tr<fedcba9876543210> <\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>; } if ($sy eq '-') { # if y is negative # two's complement: inc (dec unsigned value) and flip all "bits" in $by $by = $class -> _copy($y); $by = $class -> _dec($by); $by = $class -> _as_hex($by); $by =~ s/^-?0x//; $by =~ tr<0123456789abcdef> <\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>; } else { $by = $class -> _as_hex($y); # get binary representation $by =~ s/^-?0x//; $by =~ tr<fedcba9876543210> <\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>; } # now we have bit-strings from X and Y, reverse them for padding $bx = reverse $bx; $by = reverse $by; # padd the shorter string my $xx = "\x00"; $xx = "\x0f" if $sx eq '-'; my $yy = "\x00"; $yy = "\x0f" if $sy eq '-'; my $diff = CORE::length($bx) - CORE::length($by); if ($diff > 0) { # if $yy eq "\x00", we can cut $bx, otherwise we need to padd $by $by .= $yy x $diff; } elsif ($diff < 0) { # if $xx eq "\x00", we can cut $by, otherwise we need to padd $bx $bx .= $xx x abs($diff); } # xor the strings together my $r = $bx ^ $by; # and reverse the result again $bx = reverse $r; # One of $bx or $by was negative, so need to flip bits in the result. In both # cases (one or two of them negative, or both positive) we need to get the # characters back. if ($sign eq '-') { $bx =~ tr<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00> <0123456789abcdef>; } else { $bx =~ tr<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00> <fedcba9876543210>; } # leading zeros will be stripped by _from_hex() $bx = '0x' . $bx; $bx = $class -> _from_hex($bx); $bx = $class -> _inc($bx) if $sign eq '-'; # avoid negative zero $sign = '+' if $class -> _is_zero($bx); return $bx, $sign; } sub _sor { my ($class, $x, $sx, $y, $sy) = @_; return ($class -> _zero(), '+') if $class -> _is_zero($x) && $class -> _is_zero($y); my $sign = $sx eq '-' \|\| $sy eq '-' ? '-' : '+'; my ($bx, $by); if ($sx eq '-') { # if x is negative # two's complement: inc (dec unsigned value) and flip all "bits" in $bx $bx = $class -> _copy($x); $bx = $class -> _dec($bx); $bx = $class -> _as_hex($bx); $bx =~ s/^-?0x//; $bx =~ tr<0123456789abcdef> <\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>; } else { # if x is positive $bx = $class -> _as_hex($x); # get binary representation $bx =~ s/^-?0x//; $bx =~ tr<fedcba9876543210> <\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>; } if ($sy eq '-') { # if y is negative # two's complement: inc (dec unsigned value) and flip all "bits" in $by $by = $class -> _copy($y); $by = $class -> _dec($by); $by = $class -> _as_hex($by); $by =~ s/^-?0x//; $by =~ tr<0123456789abcdef> <\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>; } else { $by = $class -> _as_hex($y); # get binary representation $by =~ s/^-?0x//; $by =~ tr<fedcba9876543210> <\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>; } # now we have bit-strings from X and Y, reverse them for padding $bx = reverse $bx; $by = reverse $by; # padd the shorter string my $xx = "\x00"; $xx = "\x0f" if $sx eq '-'; my $yy = "\x00"; $yy = "\x0f" if $sy eq '-'; my $diff = CORE::length($bx) - CORE::length($by); if ($diff > 0) { # if $yy eq "\x00", we can cut $bx, otherwise we need to padd $by $by .= $yy x $diff; } elsif ($diff < 0) { # if $xx eq "\x00", we can cut $by, otherwise we need to padd $bx $bx .= $xx x abs($diff); } # or the strings together my $r = $bx \| $by; # and reverse the result again $bx = reverse $r; # One of $bx or $by was negative, so need to flip bits in the result. In both # cases (one or two of them negative, or both positive) we need to get the # characters back. if ($sign eq '-') { $bx =~ tr<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00> <0123456789abcdef>; } else { $bx =~ tr<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00> <fedcba9876543210>; } # leading zeros will be stripped by _from_hex() $bx = '0x' . $bx; $bx = $class -> _from_hex($bx); $bx = $class -> _inc($bx) if $sign eq '-'; # avoid negative zero $sign = '+' if $class -> _is_zero($bx); return $bx, $sign; } sub _to_bin { # convert the number to a string of binary digits without prefix my ($class, $x) = @_; my $str = ''; my $tmp = $class -> _copy($x); my $chunk = $class -> _new("16777216"); # 2^24 = 24 binary digits my $rem; until ($class -> _acmp($tmp, $chunk) < 0) { ($tmp, $rem) = $class -> _div($tmp, $chunk); $str = sprintf("%024b", $class -> _num($rem)) . $str; } unless ($class -> _is_zero($tmp)) { $str = sprintf("%b", $class -> _num($tmp)) . $str; } return length($str) ? $str : '0'; } sub _to_oct { # convert the number to a string of octal digits without prefix my ($class, $x) = @_; my $str = ''; my $tmp = $class -> _copy($x); my $chunk = $class -> _new("16777216"); # 2^24 = 8 octal digits my $rem; until ($class -> _acmp($tmp, $chunk) < 0) { ($tmp, $rem) = $class -> _div($tmp, $chunk); $str = sprintf("%08o", $class -> _num($rem)) . $str; } unless ($class -> _is_zero($tmp)) { $str = sprintf("%o", $class -> _num($tmp)) . $str; } return length($str) ? $str : '0'; } sub _to_hex { # convert the number to a string of hexadecimal digits without prefix my ($class, $x) = @_; my $str = ''; my $tmp = $class -> _copy($x); my $chunk = $class -> _new("16777216"); # 2^24 = 6 hexadecimal digits my $rem; until ($class -> _acmp($tmp, $chunk) < 0) { ($tmp, $rem) = $class -> _div($tmp, $chunk); $str = sprintf("%06x", $class -> _num($rem)) . $str; } unless ($class -> _is_zero($tmp)) { $str = sprintf("%x", $class -> _num($tmp)) . $str; } return length($str) ? $str : '0'; } sub _as_bin { # convert the number to a string of binary digits with prefix my ($class, $x) = @_; return '0b' . $class -> _to_bin($x); } sub _as_oct { # convert the number to a string of octal digits with prefix my ($class, $x) = @_; return '0' . $class -> _to_oct($x); # yes, 0 becomes "00" } sub _as_hex { # convert the number to a string of hexadecimal digits with prefix my ($class, $x) = @_; return '0x' . $class -> _to_hex($x); } sub _to_bytes { # convert the number to a string of bytes my ($class, $x) = @_; my $str = ''; my $tmp = $class -> _copy($x); my $chunk = $class -> _new("65536"); my $rem; until ($class -> _is_zero($tmp)) { ($tmp, $rem) = $class -> _div($tmp, $chunk); $str = pack('n', $class -> _num($rem)) . $str; } $str =~ s/^\0+//; return length($str) ? $str : "\x00"; } _as_bytes = \&_to_bytes; sub _to_base { # convert the number to a string of digits in various bases my $class = shift; my $x = shift; my $base = shift; $base = $class -> _new($base) unless ref($base); my $collseq; if (@_) { $collseq = shift(); } else { if ($class -> _acmp($base, $class -> _new("94")) <= 0) { $collseq = '0123456789' # 48 .. 57 . 'ABCDEFGHIJKLMNOPQRSTUVWXYZ' # 65 .. 90 . 'abcdefghijklmnopqrstuvwxyz' # 97 .. 122 . '!"#$%&\'()+,-./' # 33 .. 47 . ':;<=>?@' # 58 .. 64 . '[\\]^_`' # 91 .. 96 . '{\|}~'; # 123 .. 126 } else { croak "When base > 94, a collation sequence must be given"; } } my @collseq = split '', $collseq; my $str = ''; my $tmp = $class -> _copy($x); my $rem; until ($class -> _is_zero($tmp)) { ($tmp, $rem) = $class -> _div($tmp, $base); my $num = $class -> _num($rem); croak "no character to represent '$num' in collation sequence", " (collation sequence is too short)" if $num > $#collseq; my $chr = $collseq[$num]; $str = $chr . $str; } return "0" unless length $str; return $str; } sub _from_hex { # Convert a string of hexadecimal digits to a number. my ($class, $hex) = @_; $hex =~ s/^0[xX]//; # Find the largest number of hexadecimal digits that we can safely use with # 32 bit integers. There are 4 bits pr hexadecimal digit, and we use only # 31 bits to play safe. This gives us int(31 / 4) = 7. my $len = length $hex; my $rem = 1 + ($len - 1) % 7; # Do the first chunk. my $ret = $class -> _new(int hex substr $hex, 0, $rem); return $ret if $rem == $len; # Do the remaining chunks, if any. my $shift = $class -> _new(1 << (4 * 7)); for (my $offset = $rem ; $offset < $len ; $offset += 7) { my $part = int hex substr $hex, $offset, 7; $ret = $class -> _mul($ret, $shift); $ret = $class -> _add($ret, $class -> _new($part)); } return $ret; } sub _from_oct { # Convert a string of octal digits to a number. my ($class, $oct) = @_; # Find the largest number of octal digits that we can safely use with 32 # bit integers. There are 3 bits pr octal digit, and we use only 31 bits to # play safe. This gives us int(31 / 3) = 10. my $len = length $oct; my $rem = 1 + ($len - 1) % 10; # Do the first chunk. my $ret = $class -> _new(int oct substr $oct, 0, $rem); return $ret if $rem == $len; # Do the remaining chunks, if any. my $shift = $class -> _new(1 << (3 * 10)); for (my $offset = $rem ; $offset < $len ; $offset += 10) { my $part = int oct substr $oct, $offset, 10; $ret = $class -> _mul($ret, $shift); $ret = $class -> _add($ret, $class -> _new($part)); } return $ret; } sub _from_bin { # Convert a string of binary digits to a number. my ($class, $bin) = @_; $bin =~ s/^0[bB]//; # The largest number of binary digits that we can safely use with 32 bit # integers is 31. We use only 31 bits to play safe. my $len = length $bin; my $rem = 1 + ($len - 1) % 31; # Do the first chunk. my $ret = $class -> _new(int oct '0b' . substr $bin, 0, $rem); return $ret if $rem == $len; # Do the remaining chunks, if any. my $shift = $class -> _new(1 << 31); for (my $offset = $rem ; $offset < $len ; $offset += 31) { my $part = int oct '0b' . substr $bin, $offset, 31; $ret = $class -> _mul($ret, $shift); $ret = $class -> _add($ret, $class -> _new($part)); } return $ret; } sub _from_bytes { # convert string of bytes to a number my ($class, $str) = @_; my $x = $class -> _zero(); my $base = $class -> _new("256"); my $n = length($str); for (my $i = 0 ; $i < $n ; ++$i) { $x = $class -> _mul($x, $base); my $byteval = $class -> _new(unpack 'C', substr($str, $i, 1)); $x = $class -> _add($x, $byteval); } return $x; } sub _from_base { # convert a string to a decimal number my $class = shift; my $str = shift; my $base = shift; $base = $class -> _new($base) unless ref($base); my $n = length($str); my $x = $class -> _zero(); my $collseq; if (@_) { $collseq = shift(); } else { if ($class -> _acmp($base, $class -> _new("36")) <= 0) { $str = uc $str; $collseq = '0123456789' . 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'; } elsif ($class -> _acmp($base, $class -> _new("94")) <= 0) { $collseq = '0123456789' # 48 .. 57 . 'ABCDEFGHIJKLMNOPQRSTUVWXYZ' # 65 .. 90 . 'abcdefghijklmnopqrstuvwxyz' # 97 .. 122 . '!"#$%&\'()+,-./' # 33 .. 47 . ':;<=>?@' # 58 .. 64 . '[\\]^_`' # 91 .. 96 . '{\|}~'; # 123 .. 126 } else { croak "When base > 94, a collation sequence must be given"; } $collseq = substr $collseq, 0, $class -> _num($base); } # Create a mapping from each character in the collation sequence to the # corresponding integer. Check for duplicates in the collation sequence. my @collseq = split '', $collseq; my %collseq; for my $num (0 .. $#collseq) { my $chr = $collseq[$num]; die "duplicate character '$chr' in collation sequence" if exists $collseq{$chr}; $collseq{$chr} = $num; } for (my $i = 0 ; $i < $n ; ++$i) { my $chr = substr($str, $i, 1); die "input character '$chr' does not exist in collation sequence" unless exists $collseq{$chr}; $x = $class -> _mul($x, $base); my $num = $class -> _new($collseq{$chr}); $x = $class -> _add($x, $num); } return $x; } ############################################################################## # special modulus functions sub _modinv { # modular multiplicative inverse my ($class, $x, $y) = @_; # modulo zero if ($class -> _is_zero($y)) { return (undef, undef); } # modulo one if ($class -> _is_one($y)) { return ($class -> _zero(), '+'); } my $u = $class -> _zero(); my $v = $class -> _one(); my $a = $class -> _copy($y); my $b = $class -> _copy($x); # Euclid's Algorithm for bgcd(). my $q; my $sign = 1; { ($a, $q, $b) = ($b, $class -> _div($a, $b)); last if $class -> _is_zero($b); my $vq = $class -> _mul($class -> _copy($v), $q); my $t = $class -> _add($vq, $u); $u = $v; $v = $t; $sign = -$sign; redo; } # if the gcd is not 1, there exists no modular multiplicative inverse return (undef, undef) unless $class -> _is_one($a); ($v, $sign == 1 ? '+' : '-'); } sub _modpow { # modulus of power ($x * $y) % $z my ($class, $num, $exp, $mod) = @_; # a^b (mod 1) = 0 for all a and b if ($class -> _is_one($mod)) { return $class -> _zero(); } # 0^a (mod m) = 0 if m != 0, a != 0 # 0^0 (mod m) = 1 if m != 0 if ($class -> _is_zero($num)) { return $class -> _is_zero($exp) ? $class -> _one() : $class -> _zero(); } # $num = $class -> _mod($num, $mod); # this does not make it faster my $acc = $class -> _copy($num); my $t = $class -> _one(); my $expbin = $class -> _as_bin($exp); $expbin =~ s/^0b//; my $len = length($expbin); while (--$len >= 0) { if (substr($expbin, $len, 1) eq '1') { $t = $class -> _mul($t, $acc); $t = $class -> _mod($t, $mod); } $acc = $class -> _mul($acc, $acc); $acc = $class -> _mod($acc, $mod); } return $t; } sub _gcd { # Greatest common divisor. my ($class, $x, $y) = @_; # gcd(0, 0) = 0 # gcd(0, a) = a, if a != 0 if ($class -> _acmp($x, $y) == 0) { return $class -> _copy($x); } if ($class -> _is_zero($x)) { if ($class -> _is_zero($y)) { return $class -> _zero(); } else { return $class -> _copy($y); } } else { if ($class -> _is_zero($y)) { return $class -> _copy($x); } else { # Until $y is zero ... $x = $class -> _copy($x); until ($class -> _is_zero($y)) { # Compute remainder. $x = $class -> _mod($x, $y); # Swap $x and $y. my $tmp = $x; $x = $class -> _copy($y); $y = $tmp; } return $x; } } } sub _lcm { # Least common multiple. my ($class, $x, $y) = @_; # lcm(0, x) = 0 for all x return $class -> _zero() if ($class -> _is_zero($x) \|\| $class -> _is_zero($y)); my $gcd = $class -> _gcd($class -> _copy($x), $y); $x = $class -> _div($x, $gcd); $x = $class -> _mul($x, $y); return $x; } sub _lucas { my ($class, $n) = @_; $n = $class -> _num($n) if ref $n; # In list context, use lucas(n) = lucas(n-1) + lucas(n-2) if (wantarray) { my @y; push @y, $class -> _two(); return @y if $n == 0; push @y, $class -> _one(); return @y if $n == 1; for (my $i = 2 ; $i <= $n ; ++ $i) { $y[$i] = $class -> _add($class -> _copy($y[$i - 1]), $y[$i - 2]); } return @y; } require Scalar::Util; # In scalar context use that lucas(n) = fib(n-1) + fib(n+1). # # Remember that _fib() behaves differently in scalar context and list # context, so we must add scalar() to get the desired behaviour. return $class -> _two() if $n == 0; return $class -> _add(scalar $class -> _fib($n - 1), scalar $class -> _fib($n + 1)); } sub _fib { my ($class, $n) = @_; $n = $class -> _num($n) if ref $n; # In list context, use fib(n) = fib(n-1) + fib(n-2) if (wantarray) { my @y; push @y, $class -> _zero(); return @y if $n == 0; push @y, $class -> _one(); return @y if $n == 1; for (my $i = 2 ; $i <= $n ; ++ $i) { $y[$i] = $class -> _add($class -> _copy($y[$i - 1]), $y[$i - 2]); } return @y; } # In scalar context use a fast algorithm that is much faster than the # recursive algorith used in list context. my $cache = {}; my $two = $class -> _two(); my $fib; $fib = sub { my $n = shift; return $class -> _zero() if $n <= 0; return $class -> _one() if $n <= 2; return $cache -> {$n} if exists $cache -> {$n}; my $k = int($n / 2); my $a = $fib -> ($k + 1); my $b = $fib -> ($k); my $y; if ($n % 2 == 1) { # aa + bb $y = $class -> _add($class -> _mul($class -> _copy($a), $a), $class -> _mul($class -> _copy($b), $b)); } else { # (2a - b)b $y = $class -> _mul($class -> _sub($class -> _mul( $class -> _copy($two), $a), $b), $b); } $cache -> {$n} = $y; return $y; }; return $fib -> ($n); } ############################################################################## ############################################################################## 1; __END__ =pod =head1 NAME Math::BigInt::Lib - virtual parent class for Math::BigInt libraries =head1 SYNOPSIS # In the backend library for Math::BigInt et al. package Math::BigInt::MyBackend; use Math::BigInt::lib; our @ISA = qw< Math::BigInt::lib >; sub _new { ... } sub _str { ... } sub _add { ... } str _sub { ... } ... # In your main program. use Math::BigInt lib => 'MyBackend'; =head1 DESCRIPTION This module provides support for big integer calculations. It is not intended to be used directly, but rather as a parent class for backend libraries used by Math::BigInt, Math::BigFloat, Math::BigRat, and related modules. Other backend libraries include Math::BigInt::Calc, Math::BigInt::FastCalc, Math::BigInt::GMP, and Math::BigInt::Pari. In order to allow for multiple big integer libraries, Math::BigInt was rewritten to use a plug-in library for core math routines. Any module which conforms to the API can be used by Math::BigInt by using this in your program: use Math::BigInt lib => 'libname'; 'libname' is either the long name, like 'Math::BigInt::Pari', or only the short version, like 'Pari'. =head2 General Notes A library only needs to deal with unsigned big integers. Testing of input parameter validity is done by the caller, so there is no need to worry about underflow (e.g., in C<_sub()> and C<_dec()>) or about division by zero (e.g., in C<_div()> and C<_mod()>)) or similar cases. Some libraries use methods that don't modify their argument, and some libraries don't even use objects, but rather unblessed references. Because of this, liberary methods are always called as class methods, not instance methods: $x = Class -> method($x, $y); # like this $x = $x -> method($y); # not like this ... $x -> method($y); # ... or like this And with boolean methods $bool = Class -> method($x, $y); # like this $bool = $x -> method($y); # not like this Return values are always objects, strings, Perl scalars, or true/false for comparison routines. =head3 API version =over 4 =item CLASS-E<gt>api_version() This method is no longer used and can be omitted. Methods that are not implemented by a subclass will be inherited from this class. =back =head3 Constructors The following methods are mandatory: _new(), _str(), _add(), and _sub(). However, computations will be very slow without _mul() and _div(). =over 4 =item CLASS-E<gt>_new(STR) Convert a string representing an unsigned decimal number to an object representing the same number. The input is normalized, i.e., it matches C<^(0\|[1-9]\d)$>. =item CLASS-E<gt>_zero() Return an object representing the number zero. =item CLASS-E<gt>_one() Return an object representing the number one. =item CLASS-E<gt>_two() Return an object representing the number two. =item CLASS-E<gt>_ten() Return an object representing the number ten. =item CLASS-E<gt>_from_bin(STR) Return an object given a string representing a binary number. The input has a '0b' prefix and matches the regular expression C<^0[bB](0\|1[01])$>. =item CLASS-E<gt>_from_oct(STR) Return an object given a string representing an octal number. The input has a '0' prefix and matches the regular expression C<^0[1-7]$>. =item CLASS-E<gt>_from_hex(STR) Return an object given a string representing a hexadecimal number. The input has a '0x' prefix and matches the regular expression C<^0x(0\|[1-9a-fA-F][\da-fA-F])$>. =item CLASS-E<gt>_from_bytes(STR) Returns an object given a byte string representing the number. The byte string is in big endian byte order, so the two-byte input string "\x01\x00" should give an output value representing the number 256. =item CLASS-E<gt>_from_base(STR, BASE, COLLSEQ) Returns an object given a string STR, a base BASE, and a collation sequence COLLSEQ. Each character in STR represents a numerical value identical to the character's position in COLLSEQ. All characters in STR must be present in COLLSEQ. If BASE is less than or equal to 94, and a collation sequence is not specified, the following default collation sequence is used. It contains of all the 94 printable ASCII characters except space/blank: 0123456789 # ASCII 48 to 57 ABCDEFGHIJKLMNOPQRSTUVWXYZ # ASCII 65 to 90 abcdefghijklmnopqrstuvwxyz # ASCII 97 to 122 !"#$%&'()+,-./ # ASCII 33 to 47 :;<=>?@ # ASCII 58 to 64 [\]^_` # ASCII 91 to 96 {\|}~ # ASCII 123 to 126 If the default collation sequence is used, and the BASE is less than or equal to 36, the letter case in STR is ignored. For instance, with base 3 and collation sequence "-/\|", the character "-" represents 0, "/" represents 1, and "\|" represents 2. So if STR is "/\|-", the output is 1 3*2 + 2 3*1 + 0 3*0 = 15. The following examples show standard binary, octal, decimal, and hexadecimal conversion. All examples return 250. $x = $class -> _from_base("11111010", 2) $x = $class -> _from_base("372", 8) $x = $class -> _from_base("250", 10) $x = $class -> _from_base("FA", 16) Some more examples, all returning 250: $x = $class -> _from_base("100021", 3) $x = $class -> _from_base("3322", 4) $x = $class -> _from_base("2000", 5) $x = $class -> _from_base("caaa", 5, "abcde") $x = $class -> _from_base("42", 62) $x = $class -> _from_base("2!", 94) =back =head3 Mathematical functions =over 4 =item CLASS-E<gt>_add(OBJ1, OBJ2) Returns the result of adding OBJ2 to OBJ1. =item CLASS-E<gt>_mul(OBJ1, OBJ2) Returns the result of multiplying OBJ2 and OBJ1. =item CLASS-E<gt>_div(OBJ1, OBJ2) In scalar context, returns the quotient after dividing OBJ1 by OBJ2 and truncating the result to an integer. In list context, return the quotient and the remainder. =item CLASS-E<gt>_sub(OBJ1, OBJ2, FLAG) =item CLASS-E<gt>_sub(OBJ1, OBJ2) Returns the result of subtracting OBJ2 by OBJ1. If C<flag> is false or omitted, OBJ1 might be modified. If C<flag> is true, OBJ2 might be modified. =item CLASS-E<gt>_dec(OBJ) Returns the result after decrementing OBJ by one. =item CLASS-E<gt>_inc(OBJ) Returns the result after incrementing OBJ by one. =item CLASS-E<gt>_mod(OBJ1, OBJ2) Returns OBJ1 modulo OBJ2, i.e., the remainder after dividing OBJ1 by OBJ2. =item CLASS-E<gt>_sqrt(OBJ) Returns the square root of OBJ, truncated to an integer. =item CLASS-E<gt>_root(OBJ, N) Returns the Nth root of OBJ, truncated to an integer. =item CLASS-E<gt>_fac(OBJ) Returns the factorial of OBJ, i.e., the product of all positive integers up to and including OBJ. =item CLASS-E<gt>_dfac(OBJ) Returns the double factorial of OBJ. If OBJ is an even integer, returns the product of all positive, even integers up to and including OBJ, i.e., 246...OBJ. If OBJ is an odd integer, returns the product of all positive, odd integers, i.e., 135...OBJ. =item CLASS-E<gt>_pow(OBJ1, OBJ2) Returns OBJ1 raised to the power of OBJ2. By convention, 00 = 1. =item CLASS-E<gt>_modinv(OBJ1, OBJ2) Returns the modular multiplicative inverse, i.e., return OBJ3 so that (OBJ3 OBJ1) % OBJ2 = 1 % OBJ2 The result is returned as two arguments. If the modular multiplicative inverse does not exist, both arguments are undefined. Otherwise, the arguments are a number (object) and its sign ("+" or "-"). The output value, with its sign, must either be a positive value in the range 1,2,...,OBJ2-1 or the same value subtracted OBJ2. For instance, if the input arguments are objects representing the numbers 7 and 5, the method must either return an object representing the number 3 and a "+" sign, since (37) % 5 = 1 % 5, or an object representing the number 2 and a "-" sign, since (-27) % 5 = 1 % 5. =item CLASS-E<gt>_modpow(OBJ1, OBJ2, OBJ3) Returns the modular exponentiation, i.e., (OBJ1 OBJ2) % OBJ3. =item CLASS-E<gt>_rsft(OBJ, N, B) Returns the result after shifting OBJ N digits to thee right in base B. This is equivalent to performing integer division by BN and discarding the remainder, except that it might be much faster. For instance, if the object $obj represents the hexadecimal number 0xabcde, then C<_rsft($obj, 2, 16)> returns an object representing the number 0xabc. The "remainer", 0xde, is discarded and not returned. =item CLASS-E<gt>_lsft(OBJ, N, B) Returns the result after shifting OBJ N digits to the left in base B. This is equivalent to multiplying by B*N, except that it might be much faster. =item CLASS-E<gt>_log_int(OBJ, B) Returns the logarithm of OBJ to base BASE truncted to an integer. This method has two output arguments, the OBJECT and a STATUS. The STATUS is Perl scalar; it is 1 if OBJ is the exact result, 0 if the result was truncted to give OBJ, and undef if it is unknown whether OBJ is the exact result. =item CLASS-E<gt>_gcd(OBJ1, OBJ2) Returns the greatest common divisor of OBJ1 and OBJ2. =item CLASS-E<gt>_lcm(OBJ1, OBJ2) Return the least common multiple of OBJ1 and OBJ2. =item CLASS-E<gt>_fib(OBJ) In scalar context, returns the nth Fibonacci number: _fib(0) returns 0, _fib(1) returns 1, _fib(2) returns 1, _fib(3) returns 2 etc. In list context, returns the Fibonacci numbers from F(0) to F(n): 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... =item CLASS-E<gt>_lucas(OBJ) In scalar context, returns the nth Lucas number: _lucas(0) returns 2, _lucas(1) returns 1, _lucas(2) returns 3, etc. In list context, returns the Lucas numbers from L(0) to L(n): 2, 1, 3, 4, 7, 11, 18, 29,47, 76, ... =back =head3 Bitwise operators =over 4 =item CLASS-E<gt>_and(OBJ1, OBJ2) Returns bitwise and. =item CLASS-E<gt>_or(OBJ1, OBJ2) Returns bitwise or. =item CLASS-E<gt>_xor(OBJ1, OBJ2) Returns bitwise exclusive or. =item CLASS-E<gt>_sand(OBJ1, OBJ2, SIGN1, SIGN2) Returns bitwise signed and. =item CLASS-E<gt>_sor(OBJ1, OBJ2, SIGN1, SIGN2) Returns bitwise signed or. =item CLASS-E<gt>_sxor(OBJ1, OBJ2, SIGN1, SIGN2) Returns bitwise signed exclusive or. =back =head3 Boolean operators =over 4 =item CLASS-E<gt>_is_zero(OBJ) Returns a true value if OBJ is zero, and false value otherwise. =item CLASS-E<gt>_is_one(OBJ) Returns a true value if OBJ is one, and false value otherwise. =item CLASS-E<gt>_is_two(OBJ) Returns a true value if OBJ is two, and false value otherwise. =item CLASS-E<gt>_is_ten(OBJ) Returns a true value if OBJ is ten, and false value otherwise. =item CLASS-E<gt>_is_even(OBJ) Return a true value if OBJ is an even integer, and a false value otherwise. =item CLASS-E<gt>_is_odd(OBJ) Return a true value if OBJ is an even integer, and a false value otherwise. =item CLASS-E<gt>_acmp(OBJ1, OBJ2) Compare OBJ1 and OBJ2 and return -1, 0, or 1, if OBJ1 is numerically less than, equal to, or larger than OBJ2, respectively. =back =head3 String conversion =over 4 =item CLASS-E<gt>_str(OBJ) Returns a string representing OBJ in decimal notation. The returned string should have no leading zeros, i.e., it should match C<^(0\|[1-9]\d)$>. =item CLASS-E<gt>_to_bin(OBJ) Returns the binary string representation of OBJ. =item CLASS-E<gt>_to_oct(OBJ) Returns the octal string representation of the number. =item CLASS-E<gt>_to_hex(OBJ) Returns the hexadecimal string representation of the number. =item CLASS-E<gt>_to_bytes(OBJ) Returns a byte string representation of OBJ. The byte string is in big endian byte order, so if OBJ represents the number 256, the output should be the two-byte string "\x01\x00". =item CLASS-E<gt>_to_base(OBJ, BASE, COLLSEQ) Returns a string representation of OBJ in base BASE with collation sequence COLLSEQ. $val = $class -> _new("210"); $str = $class -> _to_base($val, 10, "xyz") # $str is "zyx" $val = $class -> _new("32"); $str = $class -> _to_base($val, 2, "-\|") # $str is "\|-----" See _from_base() for more information. =item CLASS-E<gt>_as_bin(OBJ) Like C<_to_bin()> but with a '0b' prefix. =item CLASS-E<gt>_as_oct(OBJ) Like C<_to_oct()> but with a '0' prefix. =item CLASS-E<gt>_as_hex(OBJ) Like C<_to_hex()> but with a '0x' prefix. =item CLASS-E<gt>_as_bytes(OBJ) This is an alias to C<_to_bytes()>. =back =head3 Numeric conversion =over 4 =item CLASS-E<gt>_num(OBJ) Returns a Perl scalar number representing the number OBJ as close as possible. Since Perl scalars have limited precision, the returned value might not be exactly the same as OBJ. =back =head3 Miscellaneous =over 4 =item CLASS-E<gt>_copy(OBJ) Returns a true copy OBJ. =item CLASS-E<gt>_len(OBJ) Returns the number of the decimal digits in OBJ. The output is a Perl scalar. =item CLASS-E<gt>_zeros(OBJ) Returns the number of trailing decimal zeros. The output is a Perl scalar. The number zero has no trailing decimal zeros. =item CLASS-E<gt>_digit(OBJ, N) Returns the Nth digit in OBJ as a Perl scalar. N is a Perl scalar, where zero refers to the rightmost (least significant) digit, and negative values count from the left (most significant digit). If $obj represents the number 123, then CLASS->_digit($obj, 0) # returns 3 CLASS->_digit($obj, 1) # returns 2 CLASS->_digit($obj, 2) # returns 1 CLASS->_digit($obj, -1) # returns 1 =item CLASS-E<gt>_digitsum(OBJ) Returns the sum of the base 10 digits. =item CLASS-E<gt>_check(OBJ) Returns true if the object is invalid and false otherwise. Preferably, the true value is a string describing the problem with the object. This is a check routine to test the internal state of the object for corruption. =item CLASS-E<gt>_set(OBJ) xxx =back =head2 API version 2 The following methods are required for an API version of 2 or greater. =head3 Constructors =over 4 =item CLASS-E<gt>_1ex(N) Return an object representing the number 10*N where N E<gt>= 0 is a Perl scalar. =back =head3 Mathematical functions =over 4 =item CLASS-E<gt>_nok(OBJ1, OBJ2) Return the binomial coefficient OBJ1 over OBJ1. =back =head3 Miscellaneous =over 4 =item CLASS-E<gt>_alen(OBJ) Return the approximate number of decimal digits of the object. The output is a Perl scalar. =back =head1 WRAP YOUR OWN If you want to port your own favourite C library for big numbers to the Math::BigInt interface, you can take any of the already existing modules as a rough guideline. You should really wrap up the latest Math::BigInt and Math::BigFloat testsuites with your module, and replace in them any of the following: use Math::BigInt; by this: use Math::BigInt lib => 'yourlib'; This way you ensure that your library really works 100% within Math::BigInt. =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::Calc 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 AUTHOR Peter John Acklam, E<lt>pjacklam@online.noE<gt> Code and documentation based on the Math::BigInt::Calc module by Tels E<lt>nospam-abuse@bloodgate.comE<gt> =head1 SEE ALSO L<Math::BigInt>, L<Math::BigInt::Calc>, L<Math::BigInt::GMP>, L<Math::BigInt::FastCalc> and L<Math::BigInt::Pari>. =cut	69.2 KB	2019-10-20 12:42

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