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#line 1 "Math/BigInt/Calc.pm"
package Math::BigInt::Calc;
use 5.006002;
use strict;
# use warnings; # do not use warnings for older Perls
our $VERSION = '1.998';
# 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 * $car;
##############################################################################
# global constants, flags and accessory
# announce that we are compatible with MBI v1.83 and up
sub api_version () { 2; }
# constants for easier life
my ($BASE,$BASE_LEN,$RBASE,$MAX_VAL);
my ($AND_BITS,$XOR_BITS,$OR_BITS);
my ($AND_MASK,$XOR_MASK,$OR_MASK);
sub _base_len
{
# Set/get the BASE_LEN and assorted other, connected values.
# Used only by the testsuite, the set variant is used only by the BEGIN
# block below:
shift;
my ($b, $int) = @_;
if (defined $b)
{
# avoid redefinitions
undef &_mul;
undef &_div;
if ($] >= 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
{
# (ref to string) return ref to num_array
# Convert a number from string format (without sign) to internal base
# 1ex format. Assumes normalized value as input.
my $il = length($_[1])-1;
# < BASE_LEN due len-1 above
return [ int($_[1]) ] if $il < $BASE_LEN; # shortcut for short numbers
# this leaves '00000' instead of int 0 and will be corrected after any op
[ reverse(unpack("a" . ($il % $BASE_LEN+1)
. ("a$BASE_LEN" x ($il / $BASE_LEN)), $_[1])) ];
}
BEGIN
{
# from Daniel Pfeiffer: determine largest group of digits that is precisely
# multipliable with itself plus carry
# Test now changed to expect the proper pattern, not a result off by 1 or 2
my ($e, $num) = 3; # lowest value we will use is 3+1-1 = 3
do
{
$num = ('9' x ++$e) + 0;
$num *= $num + 1.0;
} while ("$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
[ 0 ];
}
sub _one
{
# create a one
[ 1 ];
}
sub _two
{
# create a two (used internally for shifting)
[ 2 ];
}
sub _ten
{
# create a 10 (used internally for shifting)
[ 10 ];
}
sub _1ex
{
# create a 1Ex
my $rem = $_[1] % $BASE_LEN; # remainder
my $parts = $_[1] / $BASE_LEN; # parts
# 000000, 000000, 100
[ (0) x $parts, '1' . ('0' x $rem) ];
}
sub _copy
{
# make a true copy
[ @{$_[1]} ];
}
# catch and throw away
sub import { }
##############################################################################
# convert back to string and number
sub _str
{
# (ref to BINT) return num_str
# Convert number from internal base 100000 format to string format.
# internal format is always normalized (no leading zeros, "-0" => "+0")
my $ar = $_[1];
my $l = scalar @$ar; # number of parts
if ($l < 1) # should not happen
{
require Carp;
Carp::croak("$_[1] has no elements");
}
my $ret = "";
# handle first one different to strip leading zeros from it (there are no
# leading zero parts in internal representation)
$l --; $ret .= int($ar->[$l]); $l--;
# 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 ($l >= 0)
{
$ret .= substr($z.$ar->[$l],-$BASE_LEN); # fastest way I could think of
$l--;
}
$ret;
}
sub _num
{
# Make a Perl scalar number (int/float) from a BigInt object.
my $x = $_[1];
return 0 + $x->[0] if scalar @$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) = @_;
return $x if (@$y == 1) && $y->[0] == 0; # $x + 0 => $x
if ((@$x == 1) && $x->[0] == 0) # 0 + $y => $y->copy
{
# twice as slow as $x = [ @$y ], but nec. to retain $x as ref :(
@$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) == 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 is < than $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)
{
$xv->[0] = $xv->[0] - ($xv->[1] = int($xv->[0] * $RBASE)) * $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; $car = int($i * $RBASE); $i -= $car * $BASE;
}
push @$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
$yv = [@$xv] if $xv == $yv; # same references?
my @prod = (); my ($prod,$car,$cty,$xi,$yi);
for $xi (@$xv)
{
$car = 0; $cty = 0;
# slow variant
# for $yi (@$yv)
# {
# $prod = $xi * $yi + ($prod[$cty] || 0) + $car;
# $prod[$cty++] =
# $prod - ($car = int($prod * RBASE)) * $BASE; # see USE_MUL
# }
# $prod[$cty] += $car if $car; # need really to check for 0?
# $xi = shift @prod;
# faster variant
# 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;
## this is actually a tad slower
## $prod = $prod[$cty]; $prod += ($car + $xi * $yi); # no ||0 here
$prod[$cty++] =
$prod - ($car = int($prod * $RBASE)) * $BASE; # see USE_MUL
}
$prod[$cty] += $car if $car; # need really to check for 0?
$xi = shift @prod || 0; # || 0 makes v5.005_3 happy
}
push @$xv, @prod;
# can't have leading zeros
# __strip_zeros($xv);
$xv;
}
sub _mul_use_div_64
{
# (ref to int_num_array, ref to int_num_array)
# multiply two numbers in internal representation
# modifies first arg, second need not be different from first
# works for 64 bit integer with "use integer"
my ($c,$xv,$yv) = @_;
use integer;
if (@$yv == 1)
{
# shortcut for two small numbers, also handles $x == 0
if (@$xv == 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->[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
$yv = [@$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 small numbers, also handles $x == 0
if (@$xv == 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->[0] *= $yv->[0]) >= $BASE)
{
$xv->[0] =
$xv->[0] - ($xv->[1] = int($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; $car = int($i / $BASE); $i -= $car * $BASE;
# This (together with use integer;) does not work on 32-bit Perls
#$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
$yv = [@$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 = int($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;
# can't have leading zeros
# __strip_zeros($xv);
$xv;
}
sub _div_use_mul
{
# ref to array, ref to array, modify first array and return remainder if
# in list context
# see comments in _div_use_div() for more explanations
my ($c,$x,$yorg) = @_;
# the general div algorithm here is about O(N*N) and thus quite slow, so
# we first check for some special cases and use shortcuts to handle them.
# This works, because we store the numbers in a chunked format where each
# element contains 5..7 digits (depending on system).
# if both numbers have only one element:
if (@$x == 1 && @$yorg == 1)
{
# shortcut, $yorg and $x are two small numbers
if (wantarray)
{
my $r = [ $x->[0] % $yorg->[0] ];
$x->[0] = int($x->[0] / $yorg->[0]);
return ($x,$r);
}
else
{
$x->[0] = int($x->[0] / $yorg->[0]);
return $x;
}
}
# if x has more than one, but y has only one element:
if (@$yorg == 1)
{
my $rem;
$rem = _mod($c,[ @$x ],$yorg) if wantarray;
# shortcut, $y is < $BASE
my $j = scalar @$x; my $r = 0;
my $y = $yorg->[0]; my $b;
while ($j-- > 0)
{
$b = $r * $BASE + $x->[$j];
$x->[$j] = int($b/$y);
$r = $b % $y;
}
pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero
return ($x,$rem) if wantarray;
return $x;
}
# now x and y have more than one element
# check whether y has more elements than x, if yet, the result will be 0
if (@$yorg > @$x)
{
my $rem;
$rem = [@$x] if wantarray; # make copy
splice (@$x,1); # keep ref to original array
$x->[0] = 0; # set to 0
return ($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 $rem;
# if $yorg has more digits than $x (it's leading element is longer than
# the one from $x), the result will also be 0:
if (length(int($yorg->[-1])) > length(int($x->[-1])))
{
$rem = [@$x] if wantarray; # make copy
splice (@$x,1); # keep ref to org array
$x->[0] = 0; # set to 0
return ($x,$rem) if wantarray; # including remainder?
return $x;
}
# now calculate $x / $yorg
if (length(int($yorg->[-1])) == length(int($x->[-1])))
{
# same length, so make full compare
my $a = 0; my $j = scalar @$x - 1;
# manual way (abort if unequal, good for early ne)
while ($j >= 0)
{
last if ($a = $x->[$j] - $yorg->[$j]); $j--;
}
# $a contains the result of the compare between X and Y
# a < 0: x < y, a == 0: x == y, a > 0: x > y
if ($a <= 0)
{
$rem = [ 0 ]; # a = 0 => x == y => rem 0
$rem = [@$x] if $a != 0; # a < 0 => x < y => rem = x
splice(@$x,1); # keep single element
$x->[0] = 0; # if $a < 0
$x->[0] = 1 if $a == 0; # $x == $y
return ($x,$rem) if wantarray;
return $x;
}
# $x >= $y, so proceed normally
}
}
# all other cases:
my $y = [ @$yorg ]; # always make copy to preserve
my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0);
$car = $bar = $prd = 0;
if (($dd = int($BASE/($y->[-1]+1))) != 1)
{
for $xi (@$x)
{
$xi = $xi * $dd + $car;
$xi -= ($car = int($xi * $RBASE)) * $BASE; # see USE_MUL
}
push(@$x, $car); $car = 0;
for $yi (@$y)
{
$yi = $yi * $dd + $car;
$yi -= ($car = int($yi * $RBASE)) * $BASE; # see USE_MUL
}
}
else
{
push(@$x, 0);
}
@q = (); ($v2,$v1) = @$y[-2,-1];
$v2 = 0 unless $v2;
while ($#$x > $#$y)
{
($u2,$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;
$q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$BASE+$u1)/$v1));
--$q while ($v2*$q > ($u0*$BASE+$u1-$q*$v1)*$BASE+$u2);
if ($q)
{
($car, $bar) = (0,0);
for ($yi = 0, $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 ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi)
{
$x->[$xi] -= $BASE
if ($car = (($x->[$xi] += $y->[$yi] + $car) >= $BASE));
}
}
}
pop(@$x);
unshift(@q, $q);
}
if (wantarray)
{
@d = ();
if ($dd != 1)
{
$car = 0;
for $xi (reverse @$x)
{
$prd = $car * $BASE + $xi;
$car = $prd - ($tmp = int($prd / $dd)) * $dd; # see USE_MUL
unshift(@d, $tmp);
}
}
else
{
@d = @$x;
}
@$x = @q;
my $d = \@d;
__strip_zeros($x);
__strip_zeros($d);
return ($x,$d);
}
@$x = @q;
__strip_zeros($x);
$x;
}
sub _div_use_div_64
{
# ref to array, ref to array, modify first array and return remainder if
# in list context
# This version works on 64 bit integers
my ($c,$x,$yorg) = @_;
use integer;
# the general div algorithm here is about O(N*N) and thus quite slow, so
# we first check for some special cases and use shortcuts to handle them.
# This works, because we store the numbers in a chunked format where each
# element contains 5..7 digits (depending on system).
# if both numbers have only one element:
if (@$x == 1 && @$yorg == 1)
{
# shortcut, $yorg and $x are two small numbers
if (wantarray)
{
my $r = [ $x->[0] % $yorg->[0] ];
$x->[0] = int($x->[0] / $yorg->[0]);
return ($x,$r);
}
else
{
$x->[0] = int($x->[0] / $yorg->[0]);
return $x;
}
}
# if x has more than one, but y has only one element:
if (@$yorg == 1)
{
my $rem;
$rem = _mod($c,[ @$x ],$yorg) if wantarray;
# shortcut, $y is < $BASE
my $j = scalar @$x; my $r = 0;
my $y = $yorg->[0]; my $b;
while ($j-- > 0)
{
$b = $r * $BASE + $x->[$j];
$x->[$j] = int($b/$y);
$r = $b % $y;
}
pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero
return ($x,$rem) if wantarray;
return $x;
}
# now x and y have more than one element
# check whether y has more elements than x, if yet, the result will be 0
if (@$yorg > @$x)
{
my $rem;
$rem = [@$x] if wantarray; # make copy
splice (@$x,1); # keep ref to original array
$x->[0] = 0; # set to 0
return ($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 $rem;
# if $yorg has more digits than $x (it's leading element is longer than
# the one from $x), the result will also be 0:
if (length(int($yorg->[-1])) > length(int($x->[-1])))
{
$rem = [@$x] if wantarray; # make copy
splice (@$x,1); # keep ref to org array
$x->[0] = 0; # set to 0
return ($x,$rem) if wantarray; # including remainder?
return $x;
}
# now calculate $x / $yorg
if (length(int($yorg->[-1])) == length(int($x->[-1])))
{
# same length, so make full compare
my $a = 0; my $j = scalar @$x - 1;
# manual way (abort if unequal, good for early ne)
while ($j >= 0)
{
last if ($a = $x->[$j] - $yorg->[$j]); $j--;
}
# $a contains the result of the compare between X and Y
# a < 0: x < y, a == 0: x == y, a > 0: x > y
if ($a <= 0)
{
$rem = [ 0 ]; # a = 0 => x == y => rem 0
$rem = [@$x] if $a != 0; # a < 0 => x < y => rem = x
splice(@$x,1); # keep single element
$x->[0] = 0; # if $a < 0
$x->[0] = 1 if $a == 0; # $x == $y
return ($x,$rem) if wantarray; # including remainder?
return $x;
}
# $x >= $y, so proceed normally
}
}
# all other cases:
my $y = [ @$yorg ]; # always make copy to preserve
my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0);
$car = $bar = $prd = 0;
if (($dd = int($BASE/($y->[-1]+1))) != 1)
{
for $xi (@$x)
{
$xi = $xi * $dd + $car;
$xi -= ($car = int($xi / $BASE)) * $BASE;
}
push(@$x, $car); $car = 0;
for $yi (@$y)
{
$yi = $yi * $dd + $car;
$yi -= ($car = int($yi / $BASE)) * $BASE;
}
}
else
{
push(@$x, 0);
}
# @q will accumulate the final result, $q contains the current computed
# part of the final result
@q = (); ($v2,$v1) = @$y[-2,-1];
$v2 = 0 unless $v2;
while ($#$x > $#$y)
{
($u2,$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;
$q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$BASE+$u1)/$v1));
--$q while ($v2*$q > ($u0*$BASE+$u1-$q*$v1)*$BASE+$u2);
if ($q)
{
($car, $bar) = (0,0);
for ($yi = 0, $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 ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi)
{
$x->[$xi] -= $BASE
if ($car = (($x->[$xi] += $y->[$yi] + $car) >= $BASE));
}
}
}
pop(@$x); unshift(@q, $q);
}
if (wantarray)
{
@d = ();
if ($dd != 1)
{
$car = 0;
for $xi (reverse @$x)
{
$prd = $car * $BASE + $xi;
$car = $prd - ($tmp = int($prd / $dd)) * $dd;
unshift(@d, $tmp);
}
}
else
{
@d = @$x;
}
@$x = @q;
my $d = \@d;
__strip_zeros($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(N*N) and thus quite slow, so
# we first check for some special cases and use shortcuts to handle them.
# This works, because we store the numbers in a chunked format where each
# element contains 5..7 digits (depending on system).
# if both numbers have only one element:
if (@$x == 1 && @$yorg == 1)
{
# shortcut, $yorg and $x are two small numbers
if (wantarray)
{
my $r = [ $x->[0] % $yorg->[0] ];
$x->[0] = int($x->[0] / $yorg->[0]);
return ($x,$r);
}
else
{
$x->[0] = int($x->[0] / $yorg->[0]);
return $x;
}
}
# if x has more than one, but y has only one element:
if (@$yorg == 1)
{
my $rem;
$rem = _mod($c,[ @$x ],$yorg) if wantarray;
# shortcut, $y is < $BASE
my $j = scalar @$x; my $r = 0;
my $y = $yorg->[0]; my $b;
while ($j-- > 0)
{
$b = $r * $BASE + $x->[$j];
$x->[$j] = int($b/$y);
$r = $b % $y;
}
pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero
return ($x,$rem) if wantarray;
return $x;
}
# now x and y have more than one element
# check whether y has more elements than x, if yet, the result will be 0
if (@$yorg > @$x)
{
my $rem;
$rem = [@$x] if wantarray; # make copy
splice (@$x,1); # keep ref to original array
$x->[0] = 0; # set to 0
return ($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 $rem;
# if $yorg has more digits than $x (it's leading element is longer than
# the one from $x), the result will also be 0:
if (length(int($yorg->[-1])) > length(int($x->[-1])))
{
$rem = [@$x] if wantarray; # make copy
splice (@$x,1); # keep ref to org array
$x->[0] = 0; # set to 0
return ($x,$rem) if wantarray; # including remainder?
return $x;
}
# now calculate $x / $yorg
if (length(int($yorg->[-1])) == length(int($x->[-1])))
{
# same length, so make full compare
my $a = 0; my $j = scalar @$x - 1;
# manual way (abort if unequal, good for early ne)
while ($j >= 0)
{
last if ($a = $x->[$j] - $yorg->[$j]); $j--;
}
# $a contains the result of the compare between X and Y
# a < 0: x < y, a == 0: x == y, a > 0: x > y
if ($a <= 0)
{
$rem = [ 0 ]; # a = 0 => x == y => rem 0
$rem = [@$x] if $a != 0; # a < 0 => x < y => rem = x
splice(@$x,1); # keep single element
$x->[0] = 0; # if $a < 0
$x->[0] = 1 if $a == 0; # $x == $y
return ($x,$rem) if wantarray; # including remainder?
return $x;
}
# $x >= $y, so proceed normally
}
}
# all other cases:
my $y = [ @$yorg ]; # always make copy to preserve
my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0);
$car = $bar = $prd = 0;
if (($dd = int($BASE/($y->[-1]+1))) != 1)
{
for $xi (@$x)
{
$xi = $xi * $dd + $car;
$xi -= ($car = int($xi / $BASE)) * $BASE;
}
push(@$x, $car); $car = 0;
for $yi (@$y)
{
$yi = $yi * $dd + $car;
$yi -= ($car = int($yi / $BASE)) * $BASE;
}
}
else
{
push(@$x, 0);
}
# @q will accumulate the final result, $q contains the current computed
# part of the final result
@q = (); ($v2,$v1) = @$y[-2,-1];
$v2 = 0 unless $v2;
while ($#$x > $#$y)
{
($u2,$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;
$q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$BASE+$u1)/$v1));
--$q while ($v2*$q > ($u0*$BASE+$u1-$q*$v1)*$BASE+$u2);
if ($q)
{
($car, $bar) = (0,0);
for ($yi = 0, $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 ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi)
{
$x->[$xi] -= $BASE
if ($car = (($x->[$xi] += $y->[$yi] + $car) >= $BASE));
}
}
}
pop(@$x); unshift(@q, $q);
}
if (wantarray)
{
@d = ();
if ($dd != 1)
{
$car = 0;
for $xi (reverse @$x)
{
$prd = $car * $BASE + $xi;
$car = $prd - ($tmp = int($prd / $dd)) * $dd;
unshift(@d, $tmp);
}
}
else
{
@d = @$x;
}
@$x = @q;
my $d = \@d;
__strip_zeros($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] <=> $cy->[0]) <=> 0)
if scalar @$cx == scalar @$cy && scalar @$cx == 1;
# fast comp based on number of array elements (aka pseudo-length)
my $lxy = (scalar @$cx - scalar @$cy)
# or length of first element if same number of elements (aka difference 0)
||
# need int() here because sometimes the last element is '00018' vs '18'
(length(int($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 $j = scalar @$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
return "0" if $n < 0 || $n >= $len; # return 0 for digits out of range
my $elem = int($n / $BASE_LEN); # which array element
my $digit = $n % $BASE_LEN; # which digit in this element
substr("$x->[$elem]", -$digit-1, 1);
}
sub _zeros
{
# return amount of trailing zeros in decimal
# check each array elem in _m for having 0 at end as long as elem == 0
# Upon finding a elem != 0, stop
my $x = $_[1];
return 0 if scalar @$x == 1 && $x->[0] == 0;
my $zeros = 0; my $elem;
foreach my $e (@$x)
{
if ($e != 0)
{
$elem = "$e"; # preserve x
$elem =~ s/.*?(0*$)/$1/; # strip anything not zero
$zeros *= $BASE_LEN; # elems * 5
$zeros += length($elem); # count trailing zeros
last; # early out
}
$zeros ++; # real else branch: 50% slower!
}
$zeros;
}
##############################################################################
# _is_* routines
sub _is_zero
{
# return true if arg is zero
(((scalar @{$_[1]} == 1) && ($_[1]->[0] == 0))) <=> 0;
}
sub _is_even
{
# return true if arg is even
(!($_[1]->[0] & 1)) <=> 0;
}
sub _is_odd
{
# return true if arg is odd
(($_[1]->[0] & 1)) <=> 0;
}
sub _is_one
{
# return true if arg is one
(scalar @{$_[1]} == 1) && ($_[1]->[0] == 1) <=> 0;
}
sub _is_two
{
# return true if arg is two
(scalar @{$_[1]} == 1) && ($_[1]->[0] == 2) <=> 0;
}
sub _is_ten
{
# return true if arg is ten
(scalar @{$_[1]} == 1) && ($_[1]->[0] == 10) <=> 0;
}
sub __strip_zeros
{
# internal normalization function that strips leading zeros from the array
# args: ref to array
my $s = shift;
my $cnt = scalar @$s; # get count of parts
my $i = $cnt-1;
push @$s,0 if $i < 0; # div might return empty results, so fix it
return $s if @$s == 1; # early out
#print "strip: cnt $cnt i $i\n";
# '0', '3', '4', '0', '0',
# 0 1 2 3 4
# cnt = 5, i = 4
# i = 4
# i = 3
# => fcnt = cnt - i (5-2 => 3, cnt => 5-1 = 4, throw away from 4th pos)
# >= 1: skip first part (this can be zero)
while ($i > 0) { last if $s->[$i] != 0; $i--; }
$i++; splice @$s,$i if ($i < $cnt); # $i cant be 0
$s;
}
###############################################################################
# check routine to test internal state for corruptions
sub _check
{
# used by the test suite
my $x = $_[1];
return "$x is not a reference" if !ref($x);
# are all parts are valid?
my $i = 0; my $j = scalar @$x; my ($e,$try);
while ($i < $j)
{
$e = $x->[$i]; $e = 'undef' unless defined $e;
$try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e)";
last if $e !~ /^[+]?[0-9]+$/;
$try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (stringify)";
last if "$e" !~ /^[+]?[0-9]+$/;
$try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (cat-stringify)";
last if '' . "$e" !~ /^[+]?[0-9]+$/;
$try = ' < 0 || >= $BASE; '."($x, $e)";
last if $e <0 || $e >= $BASE;
# this test is disabled, since new/bnorm and certain ops (like early out
# in add/sub) are allowed/expected to leave '00000' in some elements
#$try = '=~ /^00+/; '."($x, $e)";
#last if $e =~ /^00+/;
$i++;
}
return "Illegal part '$e' at pos $i (tested: $try)" if $i < $j;
0;
}
###############################################################################
sub _mod
{
# if possible, use mod shortcut
my ($c,$x,$yo) = @_;
# slow way since $y too big
if (scalar @$yo > 1)
{
my ($xo,$rem) = _div($c,$x,$yo);
@$x = @$rem;
return $x;
}
my $y = $yo->[0];
# if both are single element arrays
if (scalar @$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)
{
$n = _new($c,$n); return _div($c,$x, _pow($c,$n,$y));
}
# shortcut (faster) for shifting by 10)
# multiples of $BASE_LEN
my $dst = 0; # destination
my $src = _num($c,$y); # as normal int
my $xlen = (@$x-1)*$BASE_LEN+length(int($x->[-1])); # len of x in digits
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 $len = scalar @$x - $src; # elems to go
my $vd; my $z = '0'x $BASE_LEN;
$x->[scalar @$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,$x,$y,$n) = @_;
if ($n != 10)
{
$n = _new($c,$n); return _mul($c,$x, _pow($c,$n,$y));
}
# shortcut (faster) for shifting by 10) since we are in base 10eX
# multiples of $BASE_LEN:
my $src = scalar @$x; # source
my $len = _num($c,$y); # shift-len as normal int
my $rem = $len % $BASE_LEN; # remainder to shift
my $dst = $src + int($len/$BASE_LEN); # destination
my $vd; # further speedup
$x->[$src] = 0; # avoid first ||0 for speed
my $z = '0' x $BASE_LEN;
while ($src >= 0)
{
$vd = $x->[$src]; $vd = $z.$vd;
$vd = substr($vd,-$BASE_LEN+$rem,$BASE_LEN-$rem);
$vd .= $src > 0 ? substr($z.$x->[$src-1],-$BASE_LEN,$rem) : '0' x $rem;
$vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN;
$x->[$dst] = int($vd);
$dst--; $src--;
}
# set lowest parts to 0
while ($dst >= 0) { $x->[$dst--] = 0; }
# fix spurious last zero element
splice @$x,-1 if $x->[-1] == 0;
$x;
}
sub _pow
{
# power of $x to $y
# ref to array, ref to array, return ref to array
my ($c,$cx,$cy) = @_;
if (scalar @$cy == 1 && $cy->[0] == 0)
{
splice (@$cx,1); $cx->[0] = 1; # y == 0 => x => 1
return $cx;
}
if ((scalar @$cx == 1 && $cx->[0] == 1) || # x == 1
(scalar @$cy == 1 && $cy->[0] == 1)) # or y == 1
{
return $cx;
}
if (scalar @$cx == 1 && $cx->[0] == 0)
{
splice (@$cx,1); $cx->[0] = 0; # 0 ** y => 0 (if not y <= 0)
return $cx;
}
my $pow2 = _one();
my $y_bin = _as_bin($c,$cy); $y_bin =~ s/^0b//;
my $len = length($y_bin);
while (--$len > 0)
{
_mul($c,$pow2,$cx) if substr($y_bin,$len,1) eq '1'; # is odd?
_mul($c,$cx,$cx);
}
_mul($c,$cx,$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, 2*k > n, compute nok(n, k) as
# nok(n, n-k), to minimize the number if iterations in the loop.
{
my $twok = _mul($c, _two($c), _copy($c, $k)); # 2 * k
if (_acmp($c, $twok, $n) > 0) { # if 2*k > n
$k = _sub($c, _copy($c, $n), $k); # k = n - k
}
}
# Example:
#
# / 7 \ 7! 1*2*3*4 * 5*6*7 5 * 6 * 7 6 7
# | | = --------- = --------------- = --------- = 5 * - * -
# \ 3 / (7-3)! 3! 1*2*3*4 * 1*2*3 1 * 2 * 3 2 3
if (_is_zero($c, $k)) {
@$n = 1;
}
else {
# Make a copy of the original n, since we'll be modifying n in-place.
my $n_orig = _copy($c, $n);
# n = 5, f = 6, d = 2 (cf. example above)
_sub($c, $n, $k);
_inc($c, $n);
my $f = _copy($c, $n);
_inc($c, $f);
my $d = _two($c);
# while f <= n (the original n, that is) ...
while (_acmp($c, $f, $n_orig) <= 0) {
# n = (n * f / d) == 5 * 6 / 2 (cf. example above)
_mul($c, $n, $f);
_div($c, $n, $d);
# f = 7, d = 3 (cf. example above)
_inc($c, $f);
_inc($c, $d);
}
}
return $n;
}
my @factorials = (
1,
1,
2,
2*3,
2*3*4,
2*3*4*5,
2*3*4*5*6,
2*3*4*5*6*7,
);
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 k*k, 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 to 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 $k = _num($c,$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 $new_x = _new($c, $k * $even);
@$cx = @$new_x;
if ($cx->[0] == 0)
{
$zero_elements ++; shift @$cx;
}
# print STDERR "x = ", _str($c,$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 < $BASE)
{
_mul($c,$cx,[$m]);
}
else
{
_mul($c,$cx,$c->_new($m));
}
if ($cx->[0] == 0)
{
$zero_elements ++; shift @$cx;
}
# print STDERR "Calculate $k2 - $sum = $m (x = ", _str($c,$cx),")\n";
}
# multiply in the zeros again
unshift @$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 $n; # steps still to do
if (scalar @$cx == 1)
{
$n = $cx->[0];
}
else
{
$n = _copy($c,$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); $step += 2;
_mul($c,$cx,[$b]);
}
$step = [$step];
while (_acmp($c,$step,$n) <= 0)
{
if ($cx->[0] == 0)
{
$zero_elements ++; shift @$cx;
}
_mul($c,$cx,$step); _inc($c,$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); $step += 2;
_mul($c,$cx,[$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); $step += 2;
_mul($c,$cx,[$b]);
}
# do what's left over
while ($step <= $n)
{
_mul($c,$cx,[$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 (scalar @$x == 1 && $x->[0] == 0);
# BASE 0 or 1 => NaN
return if (scalar @$base == 1 && $base->[0] < 2);
my $cmp = _acmp($c,$x,$base); # X == BASE => 1
if ($cmp == 0)
{
splice (@$x,1); $x->[0] = 1;
return ($x,1)
}
# X < BASE
if ($cmp < 0)
{
splice (@$x,1); $x->[0] = 0;
return ($x,undef);
}
my $x_org = _copy($c,$x); # preserve x
splice(@$x,1); $x->[0] = 1; # keep ref to $x
# Compute a guess for the result based on:
# $guess = int ( length_in_base_10(X) / ( log(base) / log(10) ) )
my $len = _len($c,$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(x*y) == log(x) + log(y):
$log += ((scalar @$base)-1) * $BASE_LEN;
# calculate now a guess based on the values obtained above:
my $res = int($len / $log);
$x->[0] = $res;
my $trial = _pow ($c, _copy($c, $base), $x);
my $a = _acmp($c,$trial,$x_org);
# print STDERR "# trial ", _str($c,$x)," was: $a (0 = exact, -1 too small, +1 too big)\n";
# found an exact result?
return ($x,1) if $a == 0;
if ($a > 0)
{
# or too big
_div($c,$trial,$base); _dec($c, $x);
while (($a = _acmp($c,$trial,$x_org)) > 0)
{
# print STDERR "# big _log_int at ", _str($c,$x), "\n";
_div($c,$trial,$base); _dec($c, $x);
}
# result is now exact (a == 0), or too small (a < 0)
return ($x, $a == 0 ? 1 : 0);
}
# else: result was to small
_mul($c,$trial,$base);
# did we now get the right result?
$a = _acmp($c,$trial,$x_org);
if ($a == 0) # yes, exactly
{
_inc($c, $x);
return ($x,1);
}
return ($x,0) if $a > 0;
# Result still too small (we should come here only if the estimate above
# was very off base):
# Now let the normal trial run obtain the real result
# Simple loop that increments $x by 2 in each step, possible overstepping
# the real result
my $base_mul = _mul($c, _copy($c,$base), $base); # $base * $base
while (($a = _acmp($c,$trial,$x_org)) < 0)
{
# print STDERR "# small _log_int at ", _str($c,$x), "\n";
_mul($c,$trial,$base_mul); _add($c, $x, [2]);
}
my $exact = 1;
if ($a > 0)
{
# overstepped the result
_dec($c, $x);
_div($c,$trial,$base);
$a = _acmp($c,$trial,$x_org);
if ($a > 0)
{
_dec($c, $x);
}
$exact = 0 if $a != 0; # a = -1 => not exact result, a = 0 => exact
}
($x,$exact); # return result
}
# 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,$x) = @_;
if (scalar @$x == 1)
{
# fits into one Perl scalar, so result can be computed directly
$x->[0] = int(sqrt($x->[0]));
return $x;
}
my $y = _copy($c,$x);
# hopefully _len/2 is < $BASE, the -1 is to always undershot the guess
# since our guess will "grow"
my $l = int((_len($c,$x)-1) / 2);
my $lastelem = $x->[-1]; # for guess
my $elems = scalar @$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 $x (instead of _lsft($c,$x,$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 ($l >= 0); # all other digits of guess are zero
print "start x= ",_str($c,$x),"\n" if DEBUG;
my $two = _two();
my $last = _zero();
my $lastlast = _zero();
$steps = 0 if DEBUG;
while (_acmp($c,$last,$x) != 0 && _acmp($c,$lastlast,$x) != 0)
{
$steps++ if DEBUG;
$lastlast = _copy($c,$last);
$last = _copy($c,$x);
_add($c,$x, _div($c,_copy($c,$y),$x));
_div($c,$x, $two );
print " x= ",_str($c,$x),"\n" if DEBUG;
}
print "\nsteps in sqrt: $steps, " if DEBUG;
_dec($c,$x) if _acmp($c,$y,_mul($c,_copy($c,$x),$x)) < 0; # overshot?
print " final ",$x->[-1],"\n" if DEBUG;
$x;
}
sub _root
{
# take n'th root of $x in place (n >= 3)
my ($c,$x,$n) = @_;
if (scalar @$x == 1)
{
if (scalar @$n > 1)
{
# result will always be smaller than 2 so trunc to 1 at once
$x->[0] = 1;
}
else
{
# fits into one Perl scalar, so result can be computed directly
# cannot use int() here, because it rounds wrongly (try
# (81 ** 3) ** (1/3) to see what I mean)
#$x->[0] = int( $x->[0] ** (1 / $n->[0]) );
# round to 8 digits, then truncate result to integer
$x->[0] = int ( sprintf ("%.8f", $x->[0] ** (1 / $n->[0]) ) );
}
return $x;
}
# we know now that X is more than one element long
# if $n is a power of two, we can repeatedly take sqrt($X) and find the
# proper result, because sqrt(sqrt($x)) == root($x,4)
my $b = _as_bin($c,$n);
if ($b =~ /0b1(0+)$/)
{
my $count = CORE::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), $x is now one element to big, again. In the next
# round we make that two, again.
_sqrt($c,$x);
}
# $x is now one element to big, so truncate result by removing it
splice (@$x,0,1);
}
else
{
# trial computation by starting with 2,4,8,16 etc until we overstep
my $step;
my $trial = _two();
# while still to do more than X steps
do
{
$step = _two();
while (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) < 0)
{
_mul ($c, $step, [2]);
_add ($c, $trial, $step);
}
# hit exactly?
if (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) == 0)
{
@$x = @$trial; # make copy while preserving ref to $x
return $x;
}
# overstepped, so go back on step
_sub($c, $trial, $step);
} while (scalar @$step > 1 || $step->[0] > 128);
# reset step to 2
$step = _two();
# add two, because $trial cannot be exactly the result (otherwise we would
# already have found it)
_add($c, $trial, $step);
# and now add more and more (2,4,6,8,10 etc)
while (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) < 0)
{
_add ($c, $trial, $step);
}
# hit not exactly? (overstepped)
if (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) > 0)
{
_dec($c,$trial);
}
# hit not exactly? (overstepped)
# 80 too small, 81 slightly too big, 82 too big
if (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) > 0)
{
_dec ($c, $trial);
}
@$x = @$trial; # make copy while preserving ref to $x
return $x;
}
$x;
}
##############################################################################
# binary stuff
sub _and
{
my ($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 $x if _acmp($c,$x,$y) == 0; # shortcut
my $m = _one(); my ($xr,$yr);
my $mask = $AND_MASK;
my $x1 = $x;
my $y1 = _copy($c,$y); # make copy
$x = _zero();
my ($b,$xrr,$yrr);
use integer;
while (!_is_zero($c,$x1) && !_is_zero($c,$y1))
{
($x1, $xr) = _div($c,$x1,$mask);
($y1, $yr) = _div($c,$y1,$mask);
# make ints() from $xr, $yr
# this is when the AND_BITS are greater than $BASE and is slower for
# small (<256 bits) numbers, but faster for large numbers. Disabled
# due to KISS principle
# $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; }
# $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; }
# _add($c,$x, _mul($c, _new( $c, ($xrr & $yrr) ), $m) );
# 0+ due to '&' doesn't work in strings
_add($c,$x, _mul($c, [ 0+$xr->[0] & 0+$yr->[0] ], $m) );
_mul($c,$m,$mask);
}
$x;
}
sub _xor
{
my ($c,$x,$y) = @_;
return _zero() if _acmp($c,$x,$y) == 0; # shortcut (see -and)
my $m = _one(); my ($xr,$yr);
my $mask = $XOR_MASK;
my $x1 = $x;
my $y1 = _copy($c,$y); # make copy
$x = _zero();
my ($b,$xrr,$yrr);
use integer;
while (!_is_zero($c,$x1) && !_is_zero($c,$y1))
{
($x1, $xr) = _div($c,$x1,$mask);
($y1, $yr) = _div($c,$y1,$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 *= $BASE; }
#_add($c,$x, _mul($c, _new( $c, ($xrr ^ $yrr) ), $m) );
# 0+ due to '^' doesn't work in strings
_add($c,$x, _mul($c, [ 0+$xr->[0] ^ 0+$yr->[0] ], $m) );
_mul($c,$m,$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
_add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1);
_add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1);
$x;
}
sub _or
{
my ($c,$x,$y) = @_;
return $x if _acmp($c,$x,$y) == 0; # shortcut (see _and)
my $m = _one(); my ($xr,$yr);
my $mask = $OR_MASK;
my $x1 = $x;
my $y1 = _copy($c,$y); # make copy
$x = _zero();
my ($b,$xrr,$yrr);
use integer;
while (!_is_zero($c,$x1) && !_is_zero($c,$y1))
{
($x1, $xr) = _div($c,$x1,$mask);
($y1, $yr) = _div($c,$y1,$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 *= $BASE; }
# _add($c,$x, _mul($c, _new( $c, ($xrr | $yrr) ), $m) );
# 0+ due to '|' doesn't work in strings
_add($c,$x, _mul($c, [ 0+$xr->[0] | 0+$yr->[0] ], $m) );
_mul($c,$m,$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
_add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1);
_add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1);
$x;
}
sub _as_hex
{
# convert a decimal number to hex (ref to array, return ref to string)
my ($c,$x) = @_;
# fits into one element (handle also 0x0 case)
return sprintf("0x%x",$x->[0]) if @$x == 1;
my $x1 = _copy($c,$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, $xr) = _div($c,$x1,$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 $x1 = _copy($c,$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, $xr) = _div($c,$x1,$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 $x1 = _copy($c,$x);
my $es = '';
my $xr;
my $x1000 = [ 0100000 ];
while (@$x1 != 1 || $x1->[0] != 0) # _is_zero()
{
($x1, $xr) = _div($c,$x1,$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 = _one();
my $x = _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 = [ $val ];
_add ($c, $x, _mul ($c, $adder, $mul ) ) if $val != 0;
_mul ($c, $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 $m = _new($c, 0x10000000); # 28 bit at a time (<32 bit!)
my $d = 7; # 7 digits at a time
if ($] <= 5.006)
{
# for older Perls, play safe
$m = [ 0x10000 ]; # 16 bit at a time (<32 bit!)
$d = 4; # 4 digits at a time
}
my $mul = _one();
my $x = _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)
{
$adder = _new($c,$val);
}
_add ($c, $x, _mul ($c, $adder, $mul ) ) if $val != 0;
_mul ($c, $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,$y) = @_;
# modulo zero
if (_is_zero($c, $y)) {
return (undef, undef);
}
# modulo one
if (_is_one($c, $y)) {
return (_zero($c), '+');
}
my $u = _zero($c);
my $v = _one($c);
my $a = _copy($c,$y);
my $b = _copy($c,$x);
# Euclid's Algorithm for bgcd(), only that we calc bgcd() ($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, _div($c, $a, $b)); # step 1
last if _is_zero($c, $b);
my $t = _add($c, # step 2:
_mul($c, _copy($c, $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 _is_one($c, $a);
($v, $sign == 1 ? '+' : '-');
}
sub _modpow
{
# modulus of power ($x ** $y) % $z
my ($c,$num,$exp,$mod) = @_;
# a^b (mod 1) = 0 for all a and b
if (_is_one($c,$mod))
{
@$num = 0;
return $num;
}
# 0^a (mod m) = 0 if m != 0, a != 0
# 0^0 (mod m) = 1 if m != 0
if (_is_zero($c, $num)) {
if (_is_zero($c, $exp)) {
@$num = 1;
} else {
@$num = 0;
}
return $num;
}
# $num = _mod($c,$num,$mod); # this does not make it faster
my $acc = _copy($c,$num); my $t = _one();
my $expbin = _as_bin($c,$exp); $expbin =~ s/^0b//;
my $len = length($expbin);
while (--$len >= 0)
{
if ( substr($expbin,$len,1) eq '1') # is_odd
{
_mul($c,$t,$acc);
$t = _mod($c,$t,$mod);
}
_mul($c,$acc,$acc);
$acc = _mod($c,$acc,$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 && $y->[0] == 0) {
# Compute remainder.
_mod($c, $x, $y);
# Swap $x and $y.
my $tmp = [ @$x ];
@$x = @$y;
$y = $tmp; # no deref here; that would modify input $y
}
return $x;
}
##############################################################################
##############################################################################
1;
__END__
#line 2971