| Server IP : 66.29.132.122 / Your IP : 3.147.63.2 Web Server : LiteSpeed System : Linux business142.web-hosting.com 4.18.0-553.lve.el8.x86_64 #1 SMP Mon May 27 15:27:34 UTC 2024 x86_64 User : admazpex ( 531) PHP Version : 7.2.34 Disable Function : NONE MySQL : OFF | cURL : ON | WGET : ON | Perl : ON | Python : ON | Sudo : OFF | Pkexec : OFF Directory : /usr/share/perl5/vendor_perl/Math/BigInt/ |
Upload File : |
package Math::BigInt::Calc;
use 5.006001;
use strict;
use warnings;
use Carp;
use Math::BigInt::Lib;
our $VERSION = '1.999811';
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 * $car;
##############################################################################
# global constants, flags and accessory
# announce that we are compatible with MBI v1.83 and up
sub api_version () { 2; }
# constants for easier life
my ($BASE, $BASE_LEN, $RBASE, $MAX_VAL);
my ($AND_BITS, $XOR_BITS, $OR_BITS);
my ($AND_MASK, $XOR_MASK, $OR_MASK);
sub _base_len {
# Set/get the BASE_LEN and assorted other, related values.
# Used only by the testsuite, the set variant is used only by the BEGIN
# block below:
my ($class, $b, $int) = @_;
if (defined $b) {
# avoid redefinitions
undef &_mul;
undef &_div;
if ($] >= 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, $MAX_VAL);
}
sub _new {
# Given a string representing an integer, returns a reference to an array
# of integers, where each integer represents a chunk of the original input
# integer.
my ($class, $str) = @_;
#unless ($str =~ /^([1-9]\d*|0)\z/) {
# require Carp;
# Carp::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 ($idx < 0) { # should not happen
require Carp;
Carp::croak("$_[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) {
$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 = $c->_copy($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 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 = $c->_copy($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 = 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 $rem = [ $x->[0] % $yorg->[0] ];
bless $rem, $c;
$x->[0] = int($x->[0] / $yorg->[0]);
return ($x, $rem);
} 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 = $c->_mod($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];
$x->[$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 = $c->_copy($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) {
# 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]))) {
my $rem = $c->_copy($x) if wantarray; # make copy
@$x = 0; # set to 0
return ($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 $j = @$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) {
# a = 0 => x == y => rem 0
# a < 0 => x < y => rem = x
my $rem = $a == 0 ? $c->_zero() : $c->_copy($x);
@$x = 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 = $c->_copy($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) {
my $d = bless [], $c;
if ($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;
__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 $rem = [ $x->[0] % $yorg->[0] ];
bless $rem, $c;
$x->[0] = int($x->[0] / $yorg->[0]);
return ($x, $rem);
} 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 = $c->_mod($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];
$x->[$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 = $c->_copy($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 $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 = $c->_copy($x) if wantarray; # make copy
@$x = 0; # set to 0
return ($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 $j = @$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 = $c->_zero(); # a = 0 => x == y => rem 0
$rem = $c->_copy($x) if $a != 0; # a < 0 => x < y => rem = x
@$x = 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 = $c->_copy($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) {
my $d = bless [], $c;
if ($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;
__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 $rem = [ $x->[0] % $yorg->[0] ];
bless $rem, $c;
$x->[0] = int($x->[0] / $yorg->[0]);
return ($x, $rem);
} 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 = $c->_mod($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];
$x->[$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 = $c->_copy($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 $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 = $c->_copy($x) if wantarray; # make copy
@$x = 0; # set to 0
return ($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 $j = @$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 = $c->_zero(); # a = 0 => x == y => rem 0
$rem = $c->_copy($x) if $a != 0; # a < 0 => x < y => rem = x
@$x = 0;
$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 = $c->_copy($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) {
my $d = bless [], $c;
if ($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;
__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);
# Handle the special case when the base is a power of 10. Don't check
# whether log($b)/log(10) is an integer, because log(1000)/log(10) is not
# exactly 3.
my $log10 = sprintf "%.0f", log($b) / log(10);
if ($b == 10 ** $log10) {
$b = 10;
$n = $c->_mul($n, $c->_new($log10));
# shortcut (faster) for shifting by 10) since we are in base 10eX
# multiples of $BASE_LEN:
my $src = @$x; # source
my $len = $c->_num($n); # 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;
return $x;
} else {
$b = $c->_new($b);
#print $c->_str($b);
return $c->_mul($x, $c->_pow($b, $n));
}
}
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, 2*k > 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 2*k > n
$k = $c->_sub($c->_copy($n), $k); # k = n - k
}
}
# Example:
#
# / 7 \ 7! 1*2*3*4 * 5*6*7 5 * 6 * 7 6 7
# | | = --------- = --------------- = --------- = 5 * - * -
# \ 3 / (7-3)! 3! 1*2*3*4 * 1*2*3 1 * 2 * 3 2 3
if ($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,
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 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(x*y) == 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;
my $zero = $c -> _zero();
while ($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 = 10**N, 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