2474 lines
69 KiB
Perl
2474 lines
69 KiB
Perl
package Math::BigInt::Lib;
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use 5.006001;
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use strict;
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use warnings;
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our $VERSION = '1.999818';
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use Carp;
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use overload
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# overload key: with_assign
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'+' => sub {
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my $class = ref $_[0];
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my $x = $class -> _copy($_[0]);
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my $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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return $class -> _add($x, $y);
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},
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'-' => sub {
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my $class = ref $_[0];
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my ($x, $y);
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if ($_[2]) { # if swapped
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$y = $_[0];
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$x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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} else {
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$x = $class -> _copy($_[0]);
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$y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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}
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return $class -> _sub($x, $y);
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},
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'*' => sub {
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my $class = ref $_[0];
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my $x = $class -> _copy($_[0]);
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my $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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return $class -> _mul($x, $y);
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},
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'/' => sub {
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my $class = ref $_[0];
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my ($x, $y);
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if ($_[2]) { # if swapped
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$y = $_[0];
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$x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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} else {
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$x = $class -> _copy($_[0]);
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$y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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}
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return $class -> _div($x, $y);
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},
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'%' => sub {
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my $class = ref $_[0];
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my ($x, $y);
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if ($_[2]) { # if swapped
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$y = $_[0];
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$x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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} else {
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$x = $class -> _copy($_[0]);
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$y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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}
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return $class -> _mod($x, $y);
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},
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'**' => sub {
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my $class = ref $_[0];
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my ($x, $y);
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if ($_[2]) { # if swapped
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$y = $_[0];
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$x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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} else {
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$x = $class -> _copy($_[0]);
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$y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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}
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return $class -> _pow($x, $y);
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},
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'<<' => sub {
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my $class = ref $_[0];
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my ($x, $y);
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if ($_[2]) { # if swapped
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$y = $class -> _num($_[0]);
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$x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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} else {
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$x = $_[0];
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$y = ref($_[1]) ? $class -> _num($_[1]) : $_[1];
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}
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return $class -> _blsft($x, $y);
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},
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'>>' => sub {
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my $class = ref $_[0];
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my ($x, $y);
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if ($_[2]) { # if swapped
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$y = $_[0];
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$x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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} else {
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$x = $class -> _copy($_[0]);
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$y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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}
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return $class -> _brsft($x, $y);
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},
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# overload key: num_comparison
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'<' => sub {
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my $class = ref $_[0];
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my ($x, $y);
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if ($_[2]) { # if swapped
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$y = $_[0];
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$x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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} else {
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$x = $class -> _copy($_[0]);
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$y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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}
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return $class -> _acmp($x, $y) < 0;
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},
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'<=' => sub {
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my $class = ref $_[0];
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my ($x, $y);
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if ($_[2]) { # if swapped
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$y = $_[0];
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$x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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} else {
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$x = $class -> _copy($_[0]);
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$y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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}
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return $class -> _acmp($x, $y) <= 0;
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},
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'>' => sub {
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my $class = ref $_[0];
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my ($x, $y);
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if ($_[2]) { # if swapped
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$y = $_[0];
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$x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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} else {
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$x = $class -> _copy($_[0]);
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$y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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}
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return $class -> _acmp($x, $y) > 0;
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},
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'>=' => sub {
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my $class = ref $_[0];
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my ($x, $y);
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if ($_[2]) { # if swapped
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$y = $_[0];
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$x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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} else {
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$x = $class -> _copy($_[0]);
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$y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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}
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return $class -> _acmp($x, $y) >= 0;
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},
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'==' => sub {
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my $class = ref $_[0];
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my $x = $class -> _copy($_[0]);
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my $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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return $class -> _acmp($x, $y) == 0;
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},
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'!=' => sub {
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my $class = ref $_[0];
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my $x = $class -> _copy($_[0]);
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my $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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return $class -> _acmp($x, $y) != 0;
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},
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# overload key: 3way_comparison
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'<=>' => sub {
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my $class = ref $_[0];
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my ($x, $y);
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if ($_[2]) { # if swapped
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$y = $_[0];
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$x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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} else {
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$x = $class -> _copy($_[0]);
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$y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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}
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return $class -> _acmp($x, $y);
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},
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# overload key: binary
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'&' => sub {
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my $class = ref $_[0];
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my ($x, $y);
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if ($_[2]) { # if swapped
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$y = $_[0];
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$x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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} else {
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$x = $class -> _copy($_[0]);
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$y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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}
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return $class -> _and($x, $y);
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},
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'|' => sub {
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my $class = ref $_[0];
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my ($x, $y);
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if ($_[2]) { # if swapped
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$y = $_[0];
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$x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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} else {
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$x = $class -> _copy($_[0]);
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$y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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}
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return $class -> _or($x, $y);
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},
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'^' => sub {
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my $class = ref $_[0];
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my ($x, $y);
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if ($_[2]) { # if swapped
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$y = $_[0];
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$x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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} else {
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$x = $class -> _copy($_[0]);
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$y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
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}
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return $class -> _xor($x, $y);
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},
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# overload key: func
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'abs' => sub { $_[0] },
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'sqrt' => sub {
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my $class = ref $_[0];
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return $class -> _sqrt($class -> _copy($_[0]));
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},
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'int' => sub { $_[0] },
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# overload key: conversion
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'bool' => sub { ref($_[0]) -> _is_zero($_[0]) ? '' : 1; },
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'""' => sub { ref($_[0]) -> _str($_[0]); },
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'0+' => sub { ref($_[0]) -> _num($_[0]); },
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'=' => sub { ref($_[0]) -> _copy($_[0]); },
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;
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sub _new {
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croak "@{[(caller 0)[3]]} method not implemented";
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}
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sub _zero {
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my $class = shift;
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return $class -> _new("0");
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}
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sub _one {
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my $class = shift;
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return $class -> _new("1");
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}
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sub _two {
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my $class = shift;
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return $class -> _new("2");
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}
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sub _ten {
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my $class = shift;
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return $class -> _new("10");
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}
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sub _1ex {
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my ($class, $exp) = @_;
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$exp = $class -> _num($exp) if ref($exp);
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return $class -> _new("1" . ("0" x $exp));
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}
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sub _copy {
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my ($class, $x) = @_;
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return $class -> _new($class -> _str($x));
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}
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# catch and throw away
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sub import { }
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##############################################################################
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# convert back to string and number
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sub _str {
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# Convert number from internal base 1eN format to string format. Internal
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# format is always normalized, i.e., no leading zeros.
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croak "@{[(caller 0)[3]]} method not implemented";
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}
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sub _num {
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my ($class, $x) = @_;
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0 + $class -> _str($x);
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}
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##############################################################################
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# actual math code
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sub _add {
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croak "@{[(caller 0)[3]]} method not implemented";
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}
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sub _sub {
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croak "@{[(caller 0)[3]]} method not implemented";
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}
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sub _mul {
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my ($class, $x, $y) = @_;
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my $sum = $class -> _zero();
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my $i = $class -> _zero();
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while ($class -> _acmp($i, $y) < 0) {
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$sum = $class -> _add($sum, $x);
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$i = $class -> _inc($i);
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}
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return $sum;
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}
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sub _div {
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my ($class, $x, $y) = @_;
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croak "@{[(caller 0)[3]]} requires non-zero divisor"
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if $class -> _is_zero($y);
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my $r = $class -> _copy($x);
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my $q = $class -> _zero();
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while ($class -> _acmp($r, $y) >= 0) {
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$q = $class -> _inc($q);
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$r = $class -> _sub($r, $y);
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}
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return $q, $r if wantarray;
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return $q;
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}
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sub _inc {
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my ($class, $x) = @_;
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$class -> _add($x, $class -> _one());
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}
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sub _dec {
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my ($class, $x) = @_;
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$class -> _sub($x, $class -> _one());
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}
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##############################################################################
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# testing
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sub _acmp {
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# Compare two (absolute) values. Return -1, 0, or 1.
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my ($class, $x, $y) = @_;
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my $xstr = $class -> _str($x);
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my $ystr = $class -> _str($y);
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length($xstr) <=> length($ystr) || $xstr cmp $ystr;
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}
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sub _len {
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my ($class, $x) = @_;
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CORE::length($class -> _str($x));
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}
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sub _alen {
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my ($class, $x) = @_;
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$class -> _len($x);
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}
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sub _digit {
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my ($class, $x, $n) = @_;
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substr($class ->_str($x), -($n+1), 1);
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}
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sub _digitsum {
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my ($class, $x) = @_;
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my $len = $class -> _len($x);
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my $sum = $class -> _zero();
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for (my $i = 0 ; $i < $len ; ++$i) {
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my $digit = $class -> _digit($x, $i);
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$digit = $class -> _new($digit);
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$sum = $class -> _add($sum, $digit);
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}
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return $sum;
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}
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sub _zeros {
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my ($class, $x) = @_;
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my $str = $class -> _str($x);
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$str =~ /[^0](0*)\z/ ? CORE::length($1) : 0;
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}
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##############################################################################
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# _is_* routines
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sub _is_zero {
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# return true if arg is zero
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my ($class, $x) = @_;
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$class -> _str($x) == 0;
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}
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sub _is_even {
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# return true if arg is even
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my ($class, $x) = @_;
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substr($class -> _str($x), -1, 1) % 2 == 0;
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}
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sub _is_odd {
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# return true if arg is odd
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my ($class, $x) = @_;
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substr($class -> _str($x), -1, 1) % 2 != 0;
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}
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sub _is_one {
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# return true if arg is one
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my ($class, $x) = @_;
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$class -> _str($x) == 1;
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}
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sub _is_two {
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# return true if arg is two
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my ($class, $x) = @_;
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$class -> _str($x) == 2;
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}
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sub _is_ten {
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# return true if arg is ten
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my ($class, $x) = @_;
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$class -> _str($x) == 10;
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}
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###############################################################################
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# check routine to test internal state for corruptions
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sub _check {
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# used by the test suite
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my ($class, $x) = @_;
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return "Input is undefined" unless defined $x;
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return "$x is not a reference" unless ref($x);
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return 0;
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}
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###############################################################################
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sub _mod {
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# modulus
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my ($class, $x, $y) = @_;
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croak "@{[(caller 0)[3]]} requires non-zero second operand"
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if $class -> _is_zero($y);
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if ($class -> can('_div')) {
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$x = $class -> _copy($x);
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my ($q, $r) = $class -> _div($x, $y);
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return $r;
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} else {
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my $r = $class -> _copy($x);
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while ($class -> _acmp($r, $y) >= 0) {
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$r = $class -> _sub($r, $y);
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}
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return $r;
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}
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}
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##############################################################################
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# shifts
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sub _rsft {
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my ($class, $x, $n, $b) = @_;
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$b = $class -> _new($b) unless ref $b;
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return scalar $class -> _div($x, $class -> _pow($class -> _copy($b), $n));
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}
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sub _lsft {
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my ($class, $x, $n, $b) = @_;
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$b = $class -> _new($b) unless ref $b;
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return $class -> _mul($x, $class -> _pow($class -> _copy($b), $n));
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}
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sub _pow {
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# power of $x to $y
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my ($class, $x, $y) = @_;
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if ($class -> _is_zero($y)) {
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return $class -> _one(); # y == 0 => x => 1
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}
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if (($class -> _is_one($x)) || # x == 1
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($class -> _is_one($y))) # or y == 1
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{
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return $x;
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}
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if ($class -> _is_zero($x)) {
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return $class -> _zero(); # 0 ** y => 0 (if not y <= 0)
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}
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my $pow2 = $class -> _one();
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my $y_bin = $class -> _as_bin($y);
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$y_bin =~ s/^0b//;
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my $len = length($y_bin);
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while (--$len > 0) {
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$pow2 = $class -> _mul($pow2, $x) if substr($y_bin, $len, 1) eq '1';
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$x = $class -> _mul($x, $x);
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}
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$x = $class -> _mul($x, $pow2);
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return $x;
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}
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sub _nok {
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# Return binomial coefficient (n over k).
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my ($class, $n, $k) = @_;
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# If k > n/2, or, equivalently, 2*k > n, compute nok(n, k) as
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# nok(n, n-k), to minimize the number if iterations in the loop.
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{
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my $twok = $class -> _mul($class -> _two(), $class -> _copy($k));
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if ($class -> _acmp($twok, $n) > 0) {
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$k = $class -> _sub($class -> _copy($n), $k);
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}
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}
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# Example:
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#
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# / 7 \ 7! 1*2*3*4 * 5*6*7 5 * 6 * 7
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# | | = --------- = --------------- = --------- = ((5 * 6) / 2 * 7) / 3
|
|
# \ 3 / (7-3)! 3! 1*2*3*4 * 1*2*3 1 * 2 * 3
|
|
#
|
|
# Equivalently, _nok(11, 5) is computed as
|
|
#
|
|
# (((((((7 * 8) / 2) * 9) / 3) * 10) / 4) * 11) / 5
|
|
|
|
if ($class -> _is_zero($k)) {
|
|
return $class -> _one();
|
|
}
|
|
|
|
# Make a copy of the original n, in case the subclass modifies n in-place.
|
|
|
|
my $n_orig = $class -> _copy($n);
|
|
|
|
# n = 5, f = 6, d = 2 (cf. example above)
|
|
|
|
$n = $class -> _sub($n, $k);
|
|
$n = $class -> _inc($n);
|
|
|
|
my $f = $class -> _copy($n);
|
|
$f = $class -> _inc($f);
|
|
|
|
my $d = $class -> _two();
|
|
|
|
# while f <= n (the original n, that is) ...
|
|
|
|
while ($class -> _acmp($f, $n_orig) <= 0) {
|
|
$n = $class -> _mul($n, $f);
|
|
$n = $class -> _div($n, $d);
|
|
$f = $class -> _inc($f);
|
|
$d = $class -> _inc($d);
|
|
}
|
|
|
|
return $n;
|
|
}
|
|
|
|
sub _fac {
|
|
# factorial
|
|
my ($class, $x) = @_;
|
|
|
|
my $two = $class -> _two();
|
|
|
|
if ($class -> _acmp($x, $two) < 0) {
|
|
return $class -> _one();
|
|
}
|
|
|
|
my $i = $class -> _copy($x);
|
|
while ($class -> _acmp($i, $two) > 0) {
|
|
$i = $class -> _dec($i);
|
|
$x = $class -> _mul($x, $i);
|
|
}
|
|
|
|
return $x;
|
|
}
|
|
|
|
sub _dfac {
|
|
# double factorial
|
|
my ($class, $x) = @_;
|
|
|
|
my $two = $class -> _two();
|
|
|
|
if ($class -> _acmp($x, $two) < 0) {
|
|
return $class -> _one();
|
|
}
|
|
|
|
my $i = $class -> _copy($x);
|
|
while ($class -> _acmp($i, $two) > 0) {
|
|
$i = $class -> _sub($i, $two);
|
|
$x = $class -> _mul($x, $i);
|
|
}
|
|
|
|
return $x;
|
|
}
|
|
|
|
sub _log_int {
|
|
# calculate integer log of $x to base $base
|
|
# calculate integer log of $x to base $base
|
|
# ref to array, ref to array - return ref to array
|
|
my ($class, $x, $base) = @_;
|
|
|
|
# X == 0 => NaN
|
|
return if $class -> _is_zero($x);
|
|
|
|
$base = $class -> _new(2) unless defined($base);
|
|
$base = $class -> _new($base) unless ref($base);
|
|
|
|
# BASE 0 or 1 => NaN
|
|
return if $class -> _is_zero($base) || $class -> _is_one($base);
|
|
|
|
# X == 1 => 0 (is exact)
|
|
if ($class -> _is_one($x)) {
|
|
return $class -> _zero(), 1;
|
|
}
|
|
|
|
my $cmp = $class -> _acmp($x, $base);
|
|
|
|
# X == BASE => 1 (is exact)
|
|
if ($cmp == 0) {
|
|
return $class -> _one(), 1;
|
|
}
|
|
|
|
# 1 < X < BASE => 0 (is truncated)
|
|
if ($cmp < 0) {
|
|
return $class -> _zero(), 0;
|
|
}
|
|
|
|
my $y;
|
|
|
|
# log(x) / log(b) = log(xm * 10^xe) / log(bm * 10^be)
|
|
# = (log(xm) + xe*(log(10))) / (log(bm) + be*log(10))
|
|
|
|
{
|
|
my $x_str = $class -> _str($x);
|
|
my $b_str = $class -> _str($base);
|
|
my $xm = "." . $x_str;
|
|
my $bm = "." . $b_str;
|
|
my $xe = length($x_str);
|
|
my $be = length($b_str);
|
|
my $log10 = log(10);
|
|
my $guess = int((log($xm) + $xe * $log10) / (log($bm) + $be * $log10));
|
|
$y = $class -> _new($guess);
|
|
}
|
|
|
|
my $trial = $class -> _pow($class -> _copy($base), $y);
|
|
my $acmp = $class -> _acmp($trial, $x);
|
|
|
|
# Did we get the exact result?
|
|
|
|
return $y, 1 if $acmp == 0;
|
|
|
|
# Too small?
|
|
|
|
while ($acmp < 0) {
|
|
$trial = $class -> _mul($trial, $base);
|
|
$y = $class -> _inc($y);
|
|
$acmp = $class -> _acmp($trial, $x);
|
|
}
|
|
|
|
# Too big?
|
|
|
|
while ($acmp > 0) {
|
|
$trial = $class -> _div($trial, $base);
|
|
$y = $class -> _dec($y);
|
|
$acmp = $class -> _acmp($trial, $x);
|
|
}
|
|
|
|
return $y, 1 if $acmp == 0; # result is exact
|
|
return $y, 0; # result is too small
|
|
}
|
|
|
|
sub _sqrt {
|
|
# square-root of $y in place
|
|
my ($class, $y) = @_;
|
|
|
|
return $y if $class -> _is_zero($y);
|
|
|
|
my $y_str = $class -> _str($y);
|
|
my $y_len = length($y_str);
|
|
|
|
# Compute the guess $x.
|
|
|
|
my $xm;
|
|
my $xe;
|
|
if ($y_len % 2 == 0) {
|
|
$xm = sqrt("." . $y_str);
|
|
$xe = $y_len / 2;
|
|
$xm = sprintf "%.0f", int($xm * 1e15);
|
|
$xe -= 15;
|
|
} else {
|
|
$xm = sqrt(".0" . $y_str);
|
|
$xe = ($y_len + 1) / 2;
|
|
$xm = sprintf "%.0f", int($xm * 1e16);
|
|
$xe -= 16;
|
|
}
|
|
|
|
my $x;
|
|
if ($xe < 0) {
|
|
$x = substr $xm, 0, length($xm) + $xe;
|
|
} else {
|
|
$x = $xm . ("0" x $xe);
|
|
}
|
|
|
|
$x = $class -> _new($x);
|
|
|
|
# Newton's method for computing square root of y
|
|
#
|
|
# x(i+1) = x(i) - f(x(i)) / f'(x(i))
|
|
# = x(i) - (x(i)^2 - y) / (2 * x(i)) # use if x(i)^2 > y
|
|
# = y(i) + (y - x(i)^2) / (2 * x(i)) # use if x(i)^2 < y
|
|
|
|
# Determine if x, our guess, is too small, correct, or too large.
|
|
|
|
my $xsq = $class -> _mul($class -> _copy($x), $x); # x(i)^2
|
|
my $acmp = $class -> _acmp($xsq, $y); # x(i)^2 <=> y
|
|
|
|
# Only assign a value to this variable if we will be using it.
|
|
|
|
my $two;
|
|
$two = $class -> _two() if $acmp != 0;
|
|
|
|
# If x is too small, do one iteration of Newton's method. Since the
|
|
# function f(x) = x^2 - y is concave and monotonically increasing, the next
|
|
# guess for x will either be correct or too large.
|
|
|
|
if ($acmp < 0) {
|
|
|
|
# x(i+1) = x(i) + (y - x(i)^2) / (2 * x(i))
|
|
|
|
my $numer = $class -> _sub($class -> _copy($y), $xsq); # y - x(i)^2
|
|
my $denom = $class -> _mul($class -> _copy($two), $x); # 2 * x(i)
|
|
my $delta = $class -> _div($numer, $denom);
|
|
|
|
unless ($class -> _is_zero($delta)) {
|
|
$x = $class -> _add($x, $delta);
|
|
$xsq = $class -> _mul($class -> _copy($x), $x); # x(i)^2
|
|
$acmp = $class -> _acmp($xsq, $y); # x(i)^2 <=> y
|
|
}
|
|
}
|
|
|
|
# If our guess for x is too large, apply Newton's method repeatedly until
|
|
# we either have got the correct value, or the delta is zero.
|
|
|
|
while ($acmp > 0) {
|
|
|
|
# x(i+1) = x(i) - (x(i)^2 - y) / (2 * x(i))
|
|
|
|
my $numer = $class -> _sub($xsq, $y); # x(i)^2 - y
|
|
my $denom = $class -> _mul($class -> _copy($two), $x); # 2 * x(i)
|
|
my $delta = $class -> _div($numer, $denom);
|
|
last if $class -> _is_zero($delta);
|
|
|
|
$x = $class -> _sub($x, $delta);
|
|
$xsq = $class -> _mul($class -> _copy($x), $x); # x(i)^2
|
|
$acmp = $class -> _acmp($xsq, $y); # x(i)^2 <=> y
|
|
}
|
|
|
|
# When the delta is zero, our value for x might still be too large. We
|
|
# require that the outout is either exact or too small (i.e., rounded down
|
|
# to the nearest integer), so do a final check.
|
|
|
|
while ($acmp > 0) {
|
|
$x = $class -> _dec($x);
|
|
$xsq = $class -> _mul($class -> _copy($x), $x); # x(i)^2
|
|
$acmp = $class -> _acmp($xsq, $y); # x(i)^2 <=> y
|
|
}
|
|
|
|
return $x;
|
|
}
|
|
|
|
sub _root {
|
|
my ($class, $y, $n) = @_;
|
|
|
|
return $y if $class -> _is_zero($y) || $class -> _is_one($y) ||
|
|
$class -> _is_one($n);
|
|
|
|
# If y <= n, the result is always (truncated to) 1.
|
|
|
|
return $class -> _one() if $class -> _acmp($y, $n) <= 0;
|
|
|
|
# Compute the initial guess x of y^(1/n). When n is large, Newton's method
|
|
# converges slowly if the "guess" (initial value) is poor, so we need a
|
|
# good guess. It the guess is too small, the next guess will be too large,
|
|
# and from then on all guesses are too large.
|
|
|
|
my $DEBUG = 0;
|
|
|
|
# Split y into mantissa and exponent in base 10, so that
|
|
#
|
|
# y = xm * 10^xe, where 0 < xm < 1 and xe is an integer
|
|
|
|
my $y_str = $class -> _str($y);
|
|
my $ym = "." . $y_str;
|
|
my $ye = length($y_str);
|
|
|
|
# From this compute the approximate base 10 logarithm of y
|
|
#
|
|
# log_10(y) = log_10(ym) + log_10(ye^10)
|
|
# = log(ym)/log(10) + ye
|
|
|
|
my $log10y = log($ym) / log(10) + $ye;
|
|
|
|
# And from this compute the approximate base 10 logarithm of x, where
|
|
# x = y^(1/n)
|
|
#
|
|
# log_10(x) = log_10(y)/n
|
|
|
|
my $log10x = $log10y / $class -> _num($n);
|
|
|
|
# From this compute xm and xe, the mantissa and exponent (in base 10) of x,
|
|
# where 1 < xm <= 10 and xe is an integer.
|
|
|
|
my $xe = int $log10x;
|
|
my $xm = 10 ** ($log10x - $xe);
|
|
|
|
# Scale the mantissa and exponent to increase the integer part of ym, which
|
|
# gives us better accuracy.
|
|
|
|
if ($DEBUG) {
|
|
print "\n";
|
|
print "y_str = $y_str\n";
|
|
print "ym = $ym\n";
|
|
print "ye = $ye\n";
|
|
print "log10y = $log10y\n";
|
|
print "log10x = $log10x\n";
|
|
print "xm = $xm\n";
|
|
print "xe = $xe\n";
|
|
}
|
|
|
|
my $d = $xe < 15 ? $xe : 15;
|
|
$xm *= 10 ** $d;
|
|
$xe -= $d;
|
|
|
|
if ($DEBUG) {
|
|
print "\n";
|
|
print "xm = $xm\n";
|
|
print "xe = $xe\n";
|
|
}
|
|
|
|
# If the mantissa is not an integer, round up to nearest integer, and then
|
|
# convert the number to a string. It is important to always round up due to
|
|
# how Newton's method behaves in this case. If the initial guess is too
|
|
# small, the next guess will be too large, after which every succeeding
|
|
# guess converges the correct value from above. Now, if the initial guess
|
|
# is too small and n is large, the next guess will be much too large and
|
|
# require a large number of iterations to get close to the solution.
|
|
# Because of this, we are likely to find the solution faster if we make
|
|
# sure the initial guess is not too small.
|
|
|
|
my $xm_int = int($xm);
|
|
my $x_str = sprintf '%.0f', $xm > $xm_int ? $xm_int + 1 : $xm_int;
|
|
$x_str .= "0" x $xe;
|
|
|
|
my $x = $class -> _new($x_str);
|
|
|
|
if ($DEBUG) {
|
|
print "xm = $xm\n";
|
|
print "xe = $xe\n";
|
|
print "\n";
|
|
print "x_str = $x_str (initial guess)\n";
|
|
print "\n";
|
|
}
|
|
|
|
# Use Newton's method for computing n'th root of y.
|
|
#
|
|
# x(i+1) = x(i) - f(x(i)) / f'(x(i))
|
|
# = x(i) - (x(i)^n - y) / (n * x(i)^(n-1)) # use if x(i)^n > y
|
|
# = x(i) + (y - x(i)^n) / (n * x(i)^(n-1)) # use if x(i)^n < y
|
|
|
|
# Determine if x, our guess, is too small, correct, or too large. Rather
|
|
# than computing x(i)^n and x(i)^(n-1) directly, compute x(i)^(n-1) and
|
|
# then the same value multiplied by x.
|
|
|
|
my $nm1 = $class -> _dec($class -> _copy($n)); # n-1
|
|
my $xpownm1 = $class -> _pow($class -> _copy($x), $nm1); # x(i)^(n-1)
|
|
my $xpown = $class -> _mul($class -> _copy($xpownm1), $x); # x(i)^n
|
|
my $acmp = $class -> _acmp($xpown, $y); # x(i)^n <=> y
|
|
|
|
if ($DEBUG) {
|
|
print "\n";
|
|
print "x = ", $class -> _str($x), "\n";
|
|
print "x^n = ", $class -> _str($xpown), "\n";
|
|
print "y = ", $class -> _str($y), "\n";
|
|
print "acmp = $acmp\n";
|
|
}
|
|
|
|
# If x is too small, do one iteration of Newton's method. Since the
|
|
# function f(x) = x^n - y is concave and monotonically increasing, the next
|
|
# guess for x will either be correct or too large.
|
|
|
|
if ($acmp < 0) {
|
|
|
|
# x(i+1) = x(i) + (y - x(i)^n) / (n * x(i)^(n-1))
|
|
|
|
my $numer = $class -> _sub($class -> _copy($y), $xpown); # y - x(i)^n
|
|
my $denom = $class -> _mul($class -> _copy($n), $xpownm1); # n * x(i)^(n-1)
|
|
my $delta = $class -> _div($numer, $denom);
|
|
|
|
if ($DEBUG) {
|
|
print "\n";
|
|
print "numer = ", $class -> _str($numer), "\n";
|
|
print "denom = ", $class -> _str($denom), "\n";
|
|
print "delta = ", $class -> _str($delta), "\n";
|
|
}
|
|
|
|
unless ($class -> _is_zero($delta)) {
|
|
$x = $class -> _add($x, $delta);
|
|
$xpownm1 = $class -> _pow($class -> _copy($x), $nm1); # x(i)^(n-1)
|
|
$xpown = $class -> _mul($class -> _copy($xpownm1), $x); # x(i)^n
|
|
$acmp = $class -> _acmp($xpown, $y); # x(i)^n <=> y
|
|
|
|
if ($DEBUG) {
|
|
print "\n";
|
|
print "x = ", $class -> _str($x), "\n";
|
|
print "x^n = ", $class -> _str($xpown), "\n";
|
|
print "y = ", $class -> _str($y), "\n";
|
|
print "acmp = $acmp\n";
|
|
}
|
|
}
|
|
}
|
|
|
|
# If our guess for x is too large, apply Newton's method repeatedly until
|
|
# we either have got the correct value, or the delta is zero.
|
|
|
|
while ($acmp > 0) {
|
|
|
|
# x(i+1) = x(i) - (x(i)^n - y) / (n * x(i)^(n-1))
|
|
|
|
my $numer = $class -> _sub($class -> _copy($xpown), $y); # x(i)^n - y
|
|
my $denom = $class -> _mul($class -> _copy($n), $xpownm1); # n * x(i)^(n-1)
|
|
|
|
if ($DEBUG) {
|
|
print "numer = ", $class -> _str($numer), "\n";
|
|
print "denom = ", $class -> _str($denom), "\n";
|
|
}
|
|
|
|
my $delta = $class -> _div($numer, $denom);
|
|
|
|
if ($DEBUG) {
|
|
print "delta = ", $class -> _str($delta), "\n";
|
|
}
|
|
|
|
last if $class -> _is_zero($delta);
|
|
|
|
$x = $class -> _sub($x, $delta);
|
|
$xpownm1 = $class -> _pow($class -> _copy($x), $nm1); # x(i)^(n-1)
|
|
$xpown = $class -> _mul($class -> _copy($xpownm1), $x); # x(i)^n
|
|
$acmp = $class -> _acmp($xpown, $y); # x(i)^n <=> y
|
|
|
|
if ($DEBUG) {
|
|
print "\n";
|
|
print "x = ", $class -> _str($x), "\n";
|
|
print "x^n = ", $class -> _str($xpown), "\n";
|
|
print "y = ", $class -> _str($y), "\n";
|
|
print "acmp = $acmp\n";
|
|
}
|
|
}
|
|
|
|
# When the delta is zero, our value for x might still be too large. We
|
|
# require that the outout is either exact or too small (i.e., rounded down
|
|
# to the nearest integer), so do a final check.
|
|
|
|
while ($acmp > 0) {
|
|
$x = $class -> _dec($x);
|
|
$xpown = $class -> _pow($class -> _copy($x), $n); # x(i)^n
|
|
$acmp = $class -> _acmp($xpown, $y); # x(i)^n <=> y
|
|
}
|
|
|
|
return $x;
|
|
}
|
|
|
|
##############################################################################
|
|
# binary stuff
|
|
|
|
sub _and {
|
|
my ($class, $x, $y) = @_;
|
|
|
|
return $x if $class -> _acmp($x, $y) == 0;
|
|
|
|
my $m = $class -> _one();
|
|
my $mask = $class -> _new("32768");
|
|
|
|
my ($xr, $yr); # remainders after division
|
|
|
|
my $xc = $class -> _copy($x);
|
|
my $yc = $class -> _copy($y);
|
|
my $z = $class -> _zero();
|
|
|
|
until ($class -> _is_zero($xc) || $class -> _is_zero($yc)) {
|
|
($xc, $xr) = $class -> _div($xc, $mask);
|
|
($yc, $yr) = $class -> _div($yc, $mask);
|
|
my $bits = $class -> _new($class -> _num($xr) & $class -> _num($yr));
|
|
$z = $class -> _add($z, $class -> _mul($bits, $m));
|
|
$m = $class -> _mul($m, $mask);
|
|
}
|
|
|
|
return $z;
|
|
}
|
|
|
|
sub _xor {
|
|
my ($class, $x, $y) = @_;
|
|
|
|
return $class -> _zero() if $class -> _acmp($x, $y) == 0;
|
|
|
|
my $m = $class -> _one();
|
|
my $mask = $class -> _new("32768");
|
|
|
|
my ($xr, $yr); # remainders after division
|
|
|
|
my $xc = $class -> _copy($x);
|
|
my $yc = $class -> _copy($y);
|
|
my $z = $class -> _zero();
|
|
|
|
until ($class -> _is_zero($xc) || $class -> _is_zero($yc)) {
|
|
($xc, $xr) = $class -> _div($xc, $mask);
|
|
($yc, $yr) = $class -> _div($yc, $mask);
|
|
my $bits = $class -> _new($class -> _num($xr) ^ $class -> _num($yr));
|
|
$z = $class -> _add($z, $class -> _mul($bits, $m));
|
|
$m = $class -> _mul($m, $mask);
|
|
}
|
|
|
|
# The loop above stops when the smallest of the two numbers is exhausted.
|
|
# The remainder of the longer one will survive bit-by-bit, so we simple
|
|
# multiply-add it in.
|
|
|
|
$z = $class -> _add($z, $class -> _mul($xc, $m))
|
|
unless $class -> _is_zero($xc);
|
|
$z = $class -> _add($z, $class -> _mul($yc, $m))
|
|
unless $class -> _is_zero($yc);
|
|
|
|
return $z;
|
|
}
|
|
|
|
sub _or {
|
|
my ($class, $x, $y) = @_;
|
|
|
|
return $x if $class -> _acmp($x, $y) == 0; # shortcut (see _and)
|
|
|
|
my $m = $class -> _one();
|
|
my $mask = $class -> _new("32768");
|
|
|
|
my ($xr, $yr); # remainders after division
|
|
|
|
my $xc = $class -> _copy($x);
|
|
my $yc = $class -> _copy($y);
|
|
my $z = $class -> _zero();
|
|
|
|
until ($class -> _is_zero($xc) || $class -> _is_zero($yc)) {
|
|
($xc, $xr) = $class -> _div($xc, $mask);
|
|
($yc, $yr) = $class -> _div($yc, $mask);
|
|
my $bits = $class -> _new($class -> _num($xr) | $class -> _num($yr));
|
|
$z = $class -> _add($z, $class -> _mul($bits, $m));
|
|
$m = $class -> _mul($m, $mask);
|
|
}
|
|
|
|
# The loop above stops when the smallest of the two numbers is exhausted.
|
|
# The remainder of the longer one will survive bit-by-bit, so we simple
|
|
# multiply-add it in.
|
|
|
|
$z = $class -> _add($z, $class -> _mul($xc, $m))
|
|
unless $class -> _is_zero($xc);
|
|
$z = $class -> _add($z, $class -> _mul($yc, $m))
|
|
unless $class -> _is_zero($yc);
|
|
|
|
return $z;
|
|
}
|
|
|
|
sub _sand {
|
|
my ($class, $x, $sx, $y, $sy) = @_;
|
|
|
|
return ($class -> _zero(), '+')
|
|
if $class -> _is_zero($x) || $class -> _is_zero($y);
|
|
|
|
my $sign = $sx eq '-' && $sy eq '-' ? '-' : '+';
|
|
|
|
my ($bx, $by);
|
|
|
|
if ($sx eq '-') { # if x is negative
|
|
# two's complement: inc (dec unsigned value) and flip all "bits" in $bx
|
|
$bx = $class -> _copy($x);
|
|
$bx = $class -> _dec($bx);
|
|
$bx = $class -> _as_hex($bx);
|
|
$bx =~ s/^-?0x//;
|
|
$bx =~ tr<0123456789abcdef>
|
|
<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>;
|
|
} else { # if x is positive
|
|
$bx = $class -> _as_hex($x); # get binary representation
|
|
$bx =~ s/^-?0x//;
|
|
$bx =~ tr<fedcba9876543210>
|
|
<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>;
|
|
}
|
|
|
|
if ($sy eq '-') { # if y is negative
|
|
# two's complement: inc (dec unsigned value) and flip all "bits" in $by
|
|
$by = $class -> _copy($y);
|
|
$by = $class -> _dec($by);
|
|
$by = $class -> _as_hex($by);
|
|
$by =~ s/^-?0x//;
|
|
$by =~ tr<0123456789abcdef>
|
|
<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>;
|
|
} else {
|
|
$by = $class -> _as_hex($y); # get binary representation
|
|
$by =~ s/^-?0x//;
|
|
$by =~ tr<fedcba9876543210>
|
|
<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>;
|
|
}
|
|
|
|
# now we have bit-strings from X and Y, reverse them for padding
|
|
$bx = reverse $bx;
|
|
$by = reverse $by;
|
|
|
|
# padd the shorter string
|
|
my $xx = "\x00"; $xx = "\x0f" if $sx eq '-';
|
|
my $yy = "\x00"; $yy = "\x0f" if $sy eq '-';
|
|
my $diff = CORE::length($bx) - CORE::length($by);
|
|
if ($diff > 0) {
|
|
# if $yy eq "\x00", we can cut $bx, otherwise we need to padd $by
|
|
$by .= $yy x $diff;
|
|
} elsif ($diff < 0) {
|
|
# if $xx eq "\x00", we can cut $by, otherwise we need to padd $bx
|
|
$bx .= $xx x abs($diff);
|
|
}
|
|
|
|
# and the strings together
|
|
my $r = $bx & $by;
|
|
|
|
# and reverse the result again
|
|
$bx = reverse $r;
|
|
|
|
# One of $bx or $by was negative, so need to flip bits in the result. In both
|
|
# cases (one or two of them negative, or both positive) we need to get the
|
|
# characters back.
|
|
if ($sign eq '-') {
|
|
$bx =~ tr<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>
|
|
<0123456789abcdef>;
|
|
} else {
|
|
$bx =~ tr<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>
|
|
<fedcba9876543210>;
|
|
}
|
|
|
|
# leading zeros will be stripped by _from_hex()
|
|
$bx = '0x' . $bx;
|
|
$bx = $class -> _from_hex($bx);
|
|
|
|
$bx = $class -> _inc($bx) if $sign eq '-';
|
|
|
|
# avoid negative zero
|
|
$sign = '+' if $class -> _is_zero($bx);
|
|
|
|
return $bx, $sign;
|
|
}
|
|
|
|
sub _sxor {
|
|
my ($class, $x, $sx, $y, $sy) = @_;
|
|
|
|
return ($class -> _zero(), '+')
|
|
if $class -> _is_zero($x) && $class -> _is_zero($y);
|
|
|
|
my $sign = $sx ne $sy ? '-' : '+';
|
|
|
|
my ($bx, $by);
|
|
|
|
if ($sx eq '-') { # if x is negative
|
|
# two's complement: inc (dec unsigned value) and flip all "bits" in $bx
|
|
$bx = $class -> _copy($x);
|
|
$bx = $class -> _dec($bx);
|
|
$bx = $class -> _as_hex($bx);
|
|
$bx =~ s/^-?0x//;
|
|
$bx =~ tr<0123456789abcdef>
|
|
<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>;
|
|
} else { # if x is positive
|
|
$bx = $class -> _as_hex($x); # get binary representation
|
|
$bx =~ s/^-?0x//;
|
|
$bx =~ tr<fedcba9876543210>
|
|
<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>;
|
|
}
|
|
|
|
if ($sy eq '-') { # if y is negative
|
|
# two's complement: inc (dec unsigned value) and flip all "bits" in $by
|
|
$by = $class -> _copy($y);
|
|
$by = $class -> _dec($by);
|
|
$by = $class -> _as_hex($by);
|
|
$by =~ s/^-?0x//;
|
|
$by =~ tr<0123456789abcdef>
|
|
<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>;
|
|
} else {
|
|
$by = $class -> _as_hex($y); # get binary representation
|
|
$by =~ s/^-?0x//;
|
|
$by =~ tr<fedcba9876543210>
|
|
<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>;
|
|
}
|
|
|
|
# now we have bit-strings from X and Y, reverse them for padding
|
|
$bx = reverse $bx;
|
|
$by = reverse $by;
|
|
|
|
# padd the shorter string
|
|
my $xx = "\x00"; $xx = "\x0f" if $sx eq '-';
|
|
my $yy = "\x00"; $yy = "\x0f" if $sy eq '-';
|
|
my $diff = CORE::length($bx) - CORE::length($by);
|
|
if ($diff > 0) {
|
|
# if $yy eq "\x00", we can cut $bx, otherwise we need to padd $by
|
|
$by .= $yy x $diff;
|
|
} elsif ($diff < 0) {
|
|
# if $xx eq "\x00", we can cut $by, otherwise we need to padd $bx
|
|
$bx .= $xx x abs($diff);
|
|
}
|
|
|
|
# xor the strings together
|
|
my $r = $bx ^ $by;
|
|
|
|
# and reverse the result again
|
|
$bx = reverse $r;
|
|
|
|
# One of $bx or $by was negative, so need to flip bits in the result. In both
|
|
# cases (one or two of them negative, or both positive) we need to get the
|
|
# characters back.
|
|
if ($sign eq '-') {
|
|
$bx =~ tr<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>
|
|
<0123456789abcdef>;
|
|
} else {
|
|
$bx =~ tr<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>
|
|
<fedcba9876543210>;
|
|
}
|
|
|
|
# leading zeros will be stripped by _from_hex()
|
|
$bx = '0x' . $bx;
|
|
$bx = $class -> _from_hex($bx);
|
|
|
|
$bx = $class -> _inc($bx) if $sign eq '-';
|
|
|
|
# avoid negative zero
|
|
$sign = '+' if $class -> _is_zero($bx);
|
|
|
|
return $bx, $sign;
|
|
}
|
|
|
|
sub _sor {
|
|
my ($class, $x, $sx, $y, $sy) = @_;
|
|
|
|
return ($class -> _zero(), '+')
|
|
if $class -> _is_zero($x) && $class -> _is_zero($y);
|
|
|
|
my $sign = $sx eq '-' || $sy eq '-' ? '-' : '+';
|
|
|
|
my ($bx, $by);
|
|
|
|
if ($sx eq '-') { # if x is negative
|
|
# two's complement: inc (dec unsigned value) and flip all "bits" in $bx
|
|
$bx = $class -> _copy($x);
|
|
$bx = $class -> _dec($bx);
|
|
$bx = $class -> _as_hex($bx);
|
|
$bx =~ s/^-?0x//;
|
|
$bx =~ tr<0123456789abcdef>
|
|
<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>;
|
|
} else { # if x is positive
|
|
$bx = $class -> _as_hex($x); # get binary representation
|
|
$bx =~ s/^-?0x//;
|
|
$bx =~ tr<fedcba9876543210>
|
|
<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>;
|
|
}
|
|
|
|
if ($sy eq '-') { # if y is negative
|
|
# two's complement: inc (dec unsigned value) and flip all "bits" in $by
|
|
$by = $class -> _copy($y);
|
|
$by = $class -> _dec($by);
|
|
$by = $class -> _as_hex($by);
|
|
$by =~ s/^-?0x//;
|
|
$by =~ tr<0123456789abcdef>
|
|
<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>;
|
|
} else {
|
|
$by = $class -> _as_hex($y); # get binary representation
|
|
$by =~ s/^-?0x//;
|
|
$by =~ tr<fedcba9876543210>
|
|
<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>;
|
|
}
|
|
|
|
# now we have bit-strings from X and Y, reverse them for padding
|
|
$bx = reverse $bx;
|
|
$by = reverse $by;
|
|
|
|
# padd the shorter string
|
|
my $xx = "\x00"; $xx = "\x0f" if $sx eq '-';
|
|
my $yy = "\x00"; $yy = "\x0f" if $sy eq '-';
|
|
my $diff = CORE::length($bx) - CORE::length($by);
|
|
if ($diff > 0) {
|
|
# if $yy eq "\x00", we can cut $bx, otherwise we need to padd $by
|
|
$by .= $yy x $diff;
|
|
} elsif ($diff < 0) {
|
|
# if $xx eq "\x00", we can cut $by, otherwise we need to padd $bx
|
|
$bx .= $xx x abs($diff);
|
|
}
|
|
|
|
# or the strings together
|
|
my $r = $bx | $by;
|
|
|
|
# and reverse the result again
|
|
$bx = reverse $r;
|
|
|
|
# One of $bx or $by was negative, so need to flip bits in the result. In both
|
|
# cases (one or two of them negative, or both positive) we need to get the
|
|
# characters back.
|
|
if ($sign eq '-') {
|
|
$bx =~ tr<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>
|
|
<0123456789abcdef>;
|
|
} else {
|
|
$bx =~ tr<\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00>
|
|
<fedcba9876543210>;
|
|
}
|
|
|
|
# leading zeros will be stripped by _from_hex()
|
|
$bx = '0x' . $bx;
|
|
$bx = $class -> _from_hex($bx);
|
|
|
|
$bx = $class -> _inc($bx) if $sign eq '-';
|
|
|
|
# avoid negative zero
|
|
$sign = '+' if $class -> _is_zero($bx);
|
|
|
|
return $bx, $sign;
|
|
}
|
|
|
|
sub _to_bin {
|
|
# convert the number to a string of binary digits without prefix
|
|
my ($class, $x) = @_;
|
|
my $str = '';
|
|
my $tmp = $class -> _copy($x);
|
|
my $chunk = $class -> _new("16777216"); # 2^24 = 24 binary digits
|
|
my $rem;
|
|
until ($class -> _acmp($tmp, $chunk) < 0) {
|
|
($tmp, $rem) = $class -> _div($tmp, $chunk);
|
|
$str = sprintf("%024b", $class -> _num($rem)) . $str;
|
|
}
|
|
unless ($class -> _is_zero($tmp)) {
|
|
$str = sprintf("%b", $class -> _num($tmp)) . $str;
|
|
}
|
|
return length($str) ? $str : '0';
|
|
}
|
|
|
|
sub _to_oct {
|
|
# convert the number to a string of octal digits without prefix
|
|
my ($class, $x) = @_;
|
|
my $str = '';
|
|
my $tmp = $class -> _copy($x);
|
|
my $chunk = $class -> _new("16777216"); # 2^24 = 8 octal digits
|
|
my $rem;
|
|
until ($class -> _acmp($tmp, $chunk) < 0) {
|
|
($tmp, $rem) = $class -> _div($tmp, $chunk);
|
|
$str = sprintf("%08o", $class -> _num($rem)) . $str;
|
|
}
|
|
unless ($class -> _is_zero($tmp)) {
|
|
$str = sprintf("%o", $class -> _num($tmp)) . $str;
|
|
}
|
|
return length($str) ? $str : '0';
|
|
}
|
|
|
|
sub _to_hex {
|
|
# convert the number to a string of hexadecimal digits without prefix
|
|
my ($class, $x) = @_;
|
|
my $str = '';
|
|
my $tmp = $class -> _copy($x);
|
|
my $chunk = $class -> _new("16777216"); # 2^24 = 6 hexadecimal digits
|
|
my $rem;
|
|
until ($class -> _acmp($tmp, $chunk) < 0) {
|
|
($tmp, $rem) = $class -> _div($tmp, $chunk);
|
|
$str = sprintf("%06x", $class -> _num($rem)) . $str;
|
|
}
|
|
unless ($class -> _is_zero($tmp)) {
|
|
$str = sprintf("%x", $class -> _num($tmp)) . $str;
|
|
}
|
|
return length($str) ? $str : '0';
|
|
}
|
|
|
|
sub _as_bin {
|
|
# convert the number to a string of binary digits with prefix
|
|
my ($class, $x) = @_;
|
|
return '0b' . $class -> _to_bin($x);
|
|
}
|
|
|
|
sub _as_oct {
|
|
# convert the number to a string of octal digits with prefix
|
|
my ($class, $x) = @_;
|
|
return '0' . $class -> _to_oct($x); # yes, 0 becomes "00"
|
|
}
|
|
|
|
sub _as_hex {
|
|
# convert the number to a string of hexadecimal digits with prefix
|
|
my ($class, $x) = @_;
|
|
return '0x' . $class -> _to_hex($x);
|
|
}
|
|
|
|
sub _to_bytes {
|
|
# convert the number to a string of bytes
|
|
my ($class, $x) = @_;
|
|
my $str = '';
|
|
my $tmp = $class -> _copy($x);
|
|
my $chunk = $class -> _new("65536");
|
|
my $rem;
|
|
until ($class -> _is_zero($tmp)) {
|
|
($tmp, $rem) = $class -> _div($tmp, $chunk);
|
|
$str = pack('n', $class -> _num($rem)) . $str;
|
|
}
|
|
$str =~ s/^\0+//;
|
|
return length($str) ? $str : "\x00";
|
|
}
|
|
|
|
*_as_bytes = \&_to_bytes;
|
|
|
|
sub _to_base {
|
|
# convert the number to a string of digits in various bases
|
|
my $class = shift;
|
|
my $x = shift;
|
|
my $base = shift;
|
|
$base = $class -> _new($base) unless ref($base);
|
|
|
|
my $collseq;
|
|
if (@_) {
|
|
$collseq = shift();
|
|
} else {
|
|
if ($class -> _acmp($base, $class -> _new("94")) <= 0) {
|
|
$collseq = '0123456789' # 48 .. 57
|
|
. 'ABCDEFGHIJKLMNOPQRSTUVWXYZ' # 65 .. 90
|
|
. 'abcdefghijklmnopqrstuvwxyz' # 97 .. 122
|
|
. '!"#$%&\'()*+,-./' # 33 .. 47
|
|
. ':;<=>?@' # 58 .. 64
|
|
. '[\\]^_`' # 91 .. 96
|
|
. '{|}~'; # 123 .. 126
|
|
} else {
|
|
croak "When base > 94, a collation sequence must be given";
|
|
}
|
|
}
|
|
|
|
my @collseq = split '', $collseq;
|
|
|
|
my $str = '';
|
|
my $tmp = $class -> _copy($x);
|
|
my $rem;
|
|
until ($class -> _is_zero($tmp)) {
|
|
($tmp, $rem) = $class -> _div($tmp, $base);
|
|
my $num = $class -> _num($rem);
|
|
croak "no character to represent '$num' in collation sequence",
|
|
" (collation sequence is too short)" if $num > $#collseq;
|
|
my $chr = $collseq[$num];
|
|
$str = $chr . $str;
|
|
}
|
|
return "0" unless length $str;
|
|
return $str;
|
|
}
|
|
|
|
sub _from_hex {
|
|
# Convert a string of hexadecimal digits to a number.
|
|
|
|
my ($class, $hex) = @_;
|
|
$hex =~ s/^0[xX]//;
|
|
|
|
# Find the largest number of hexadecimal digits that we can safely use with
|
|
# 32 bit integers. There are 4 bits pr hexadecimal digit, and we use only
|
|
# 31 bits to play safe. This gives us int(31 / 4) = 7.
|
|
|
|
my $len = length $hex;
|
|
my $rem = 1 + ($len - 1) % 7;
|
|
|
|
# Do the first chunk.
|
|
|
|
my $ret = $class -> _new(int hex substr $hex, 0, $rem);
|
|
return $ret if $rem == $len;
|
|
|
|
# Do the remaining chunks, if any.
|
|
|
|
my $shift = $class -> _new(1 << (4 * 7));
|
|
for (my $offset = $rem ; $offset < $len ; $offset += 7) {
|
|
my $part = int hex substr $hex, $offset, 7;
|
|
$ret = $class -> _mul($ret, $shift);
|
|
$ret = $class -> _add($ret, $class -> _new($part));
|
|
}
|
|
|
|
return $ret;
|
|
}
|
|
|
|
sub _from_oct {
|
|
# Convert a string of octal digits to a number.
|
|
|
|
my ($class, $oct) = @_;
|
|
|
|
# Find the largest number of octal digits that we can safely use with 32
|
|
# bit integers. There are 3 bits pr octal digit, and we use only 31 bits to
|
|
# play safe. This gives us int(31 / 3) = 10.
|
|
|
|
my $len = length $oct;
|
|
my $rem = 1 + ($len - 1) % 10;
|
|
|
|
# Do the first chunk.
|
|
|
|
my $ret = $class -> _new(int oct substr $oct, 0, $rem);
|
|
return $ret if $rem == $len;
|
|
|
|
# Do the remaining chunks, if any.
|
|
|
|
my $shift = $class -> _new(1 << (3 * 10));
|
|
for (my $offset = $rem ; $offset < $len ; $offset += 10) {
|
|
my $part = int oct substr $oct, $offset, 10;
|
|
$ret = $class -> _mul($ret, $shift);
|
|
$ret = $class -> _add($ret, $class -> _new($part));
|
|
}
|
|
|
|
return $ret;
|
|
}
|
|
|
|
sub _from_bin {
|
|
# Convert a string of binary digits to a number.
|
|
|
|
my ($class, $bin) = @_;
|
|
$bin =~ s/^0[bB]//;
|
|
|
|
# The largest number of binary digits that we can safely use with 32 bit
|
|
# integers is 31. We use only 31 bits to play safe.
|
|
|
|
my $len = length $bin;
|
|
my $rem = 1 + ($len - 1) % 31;
|
|
|
|
# Do the first chunk.
|
|
|
|
my $ret = $class -> _new(int oct '0b' . substr $bin, 0, $rem);
|
|
return $ret if $rem == $len;
|
|
|
|
# Do the remaining chunks, if any.
|
|
|
|
my $shift = $class -> _new(1 << 31);
|
|
for (my $offset = $rem ; $offset < $len ; $offset += 31) {
|
|
my $part = int oct '0b' . substr $bin, $offset, 31;
|
|
$ret = $class -> _mul($ret, $shift);
|
|
$ret = $class -> _add($ret, $class -> _new($part));
|
|
}
|
|
|
|
return $ret;
|
|
}
|
|
|
|
sub _from_bytes {
|
|
# convert string of bytes to a number
|
|
my ($class, $str) = @_;
|
|
my $x = $class -> _zero();
|
|
my $base = $class -> _new("256");
|
|
my $n = length($str);
|
|
for (my $i = 0 ; $i < $n ; ++$i) {
|
|
$x = $class -> _mul($x, $base);
|
|
my $byteval = $class -> _new(unpack 'C', substr($str, $i, 1));
|
|
$x = $class -> _add($x, $byteval);
|
|
}
|
|
return $x;
|
|
}
|
|
|
|
sub _from_base {
|
|
# convert a string to a decimal number
|
|
my $class = shift;
|
|
my $str = shift;
|
|
my $base = shift;
|
|
$base = $class -> _new($base) unless ref($base);
|
|
|
|
my $n = length($str);
|
|
my $x = $class -> _zero();
|
|
|
|
my $collseq;
|
|
if (@_) {
|
|
$collseq = shift();
|
|
} else {
|
|
if ($class -> _acmp($base, $class -> _new("36")) <= 0) {
|
|
$str = uc $str;
|
|
$collseq = '0123456789' . 'ABCDEFGHIJKLMNOPQRSTUVWXYZ';
|
|
} elsif ($class -> _acmp($base, $class -> _new("94")) <= 0) {
|
|
$collseq = '0123456789' # 48 .. 57
|
|
. 'ABCDEFGHIJKLMNOPQRSTUVWXYZ' # 65 .. 90
|
|
. 'abcdefghijklmnopqrstuvwxyz' # 97 .. 122
|
|
. '!"#$%&\'()*+,-./' # 33 .. 47
|
|
. ':;<=>?@' # 58 .. 64
|
|
. '[\\]^_`' # 91 .. 96
|
|
. '{|}~'; # 123 .. 126
|
|
} else {
|
|
croak "When base > 94, a collation sequence must be given";
|
|
}
|
|
$collseq = substr $collseq, 0, $class -> _num($base);
|
|
}
|
|
|
|
# Create a mapping from each character in the collation sequence to the
|
|
# corresponding integer. Check for duplicates in the collation sequence.
|
|
|
|
my @collseq = split '', $collseq;
|
|
my %collseq;
|
|
for my $num (0 .. $#collseq) {
|
|
my $chr = $collseq[$num];
|
|
die "duplicate character '$chr' in collation sequence"
|
|
if exists $collseq{$chr};
|
|
$collseq{$chr} = $num;
|
|
}
|
|
|
|
for (my $i = 0 ; $i < $n ; ++$i) {
|
|
my $chr = substr($str, $i, 1);
|
|
die "input character '$chr' does not exist in collation sequence"
|
|
unless exists $collseq{$chr};
|
|
$x = $class -> _mul($x, $base);
|
|
my $num = $class -> _new($collseq{$chr});
|
|
$x = $class -> _add($x, $num);
|
|
}
|
|
|
|
return $x;
|
|
}
|
|
|
|
##############################################################################
|
|
# special modulus functions
|
|
|
|
sub _modinv {
|
|
# modular multiplicative inverse
|
|
my ($class, $x, $y) = @_;
|
|
|
|
# modulo zero
|
|
if ($class -> _is_zero($y)) {
|
|
return (undef, undef);
|
|
}
|
|
|
|
# modulo one
|
|
if ($class -> _is_one($y)) {
|
|
return ($class -> _zero(), '+');
|
|
}
|
|
|
|
my $u = $class -> _zero();
|
|
my $v = $class -> _one();
|
|
my $a = $class -> _copy($y);
|
|
my $b = $class -> _copy($x);
|
|
|
|
# Euclid's Algorithm for bgcd().
|
|
|
|
my $q;
|
|
my $sign = 1;
|
|
{
|
|
($a, $q, $b) = ($b, $class -> _div($a, $b));
|
|
last if $class -> _is_zero($b);
|
|
|
|
my $vq = $class -> _mul($class -> _copy($v), $q);
|
|
my $t = $class -> _add($vq, $u);
|
|
$u = $v;
|
|
$v = $t;
|
|
$sign = -$sign;
|
|
redo;
|
|
}
|
|
|
|
# if the gcd is not 1, there exists no modular multiplicative inverse
|
|
return (undef, undef) unless $class -> _is_one($a);
|
|
|
|
($v, $sign == 1 ? '+' : '-');
|
|
}
|
|
|
|
sub _modpow {
|
|
# modulus of power ($x ** $y) % $z
|
|
my ($class, $num, $exp, $mod) = @_;
|
|
|
|
# a^b (mod 1) = 0 for all a and b
|
|
if ($class -> _is_one($mod)) {
|
|
return $class -> _zero();
|
|
}
|
|
|
|
# 0^a (mod m) = 0 if m != 0, a != 0
|
|
# 0^0 (mod m) = 1 if m != 0
|
|
if ($class -> _is_zero($num)) {
|
|
return $class -> _is_zero($exp) ? $class -> _one()
|
|
: $class -> _zero();
|
|
}
|
|
|
|
# $num = $class -> _mod($num, $mod); # this does not make it faster
|
|
|
|
my $acc = $class -> _copy($num);
|
|
my $t = $class -> _one();
|
|
|
|
my $expbin = $class -> _as_bin($exp);
|
|
$expbin =~ s/^0b//;
|
|
my $len = length($expbin);
|
|
|
|
while (--$len >= 0) {
|
|
if (substr($expbin, $len, 1) eq '1') {
|
|
$t = $class -> _mul($t, $acc);
|
|
$t = $class -> _mod($t, $mod);
|
|
}
|
|
$acc = $class -> _mul($acc, $acc);
|
|
$acc = $class -> _mod($acc, $mod);
|
|
}
|
|
return $t;
|
|
}
|
|
|
|
sub _gcd {
|
|
# Greatest common divisor.
|
|
|
|
my ($class, $x, $y) = @_;
|
|
|
|
# gcd(0, 0) = 0
|
|
# gcd(0, a) = a, if a != 0
|
|
|
|
if ($class -> _acmp($x, $y) == 0) {
|
|
return $class -> _copy($x);
|
|
}
|
|
|
|
if ($class -> _is_zero($x)) {
|
|
if ($class -> _is_zero($y)) {
|
|
return $class -> _zero();
|
|
} else {
|
|
return $class -> _copy($y);
|
|
}
|
|
} else {
|
|
if ($class -> _is_zero($y)) {
|
|
return $class -> _copy($x);
|
|
} else {
|
|
|
|
# Until $y is zero ...
|
|
|
|
$x = $class -> _copy($x);
|
|
until ($class -> _is_zero($y)) {
|
|
|
|
# Compute remainder.
|
|
|
|
$x = $class -> _mod($x, $y);
|
|
|
|
# Swap $x and $y.
|
|
|
|
my $tmp = $x;
|
|
$x = $class -> _copy($y);
|
|
$y = $tmp;
|
|
}
|
|
|
|
return $x;
|
|
}
|
|
}
|
|
}
|
|
|
|
sub _lcm {
|
|
# Least common multiple.
|
|
|
|
my ($class, $x, $y) = @_;
|
|
|
|
# lcm(0, x) = 0 for all x
|
|
|
|
return $class -> _zero()
|
|
if ($class -> _is_zero($x) ||
|
|
$class -> _is_zero($y));
|
|
|
|
my $gcd = $class -> _gcd($class -> _copy($x), $y);
|
|
$x = $class -> _div($x, $gcd);
|
|
$x = $class -> _mul($x, $y);
|
|
return $x;
|
|
}
|
|
|
|
sub _lucas {
|
|
my ($class, $n) = @_;
|
|
|
|
$n = $class -> _num($n) if ref $n;
|
|
|
|
# In list context, use lucas(n) = lucas(n-1) + lucas(n-2)
|
|
|
|
if (wantarray) {
|
|
my @y;
|
|
|
|
push @y, $class -> _two();
|
|
return @y if $n == 0;
|
|
|
|
push @y, $class -> _one();
|
|
return @y if $n == 1;
|
|
|
|
for (my $i = 2 ; $i <= $n ; ++ $i) {
|
|
$y[$i] = $class -> _add($class -> _copy($y[$i - 1]), $y[$i - 2]);
|
|
}
|
|
|
|
return @y;
|
|
}
|
|
|
|
require Scalar::Util;
|
|
|
|
# In scalar context use that lucas(n) = fib(n-1) + fib(n+1).
|
|
#
|
|
# Remember that _fib() behaves differently in scalar context and list
|
|
# context, so we must add scalar() to get the desired behaviour.
|
|
|
|
return $class -> _two() if $n == 0;
|
|
|
|
return $class -> _add(scalar $class -> _fib($n - 1),
|
|
scalar $class -> _fib($n + 1));
|
|
}
|
|
|
|
sub _fib {
|
|
my ($class, $n) = @_;
|
|
|
|
$n = $class -> _num($n) if ref $n;
|
|
|
|
# In list context, use fib(n) = fib(n-1) + fib(n-2)
|
|
|
|
if (wantarray) {
|
|
my @y;
|
|
|
|
push @y, $class -> _zero();
|
|
return @y if $n == 0;
|
|
|
|
push @y, $class -> _one();
|
|
return @y if $n == 1;
|
|
|
|
for (my $i = 2 ; $i <= $n ; ++ $i) {
|
|
$y[$i] = $class -> _add($class -> _copy($y[$i - 1]), $y[$i - 2]);
|
|
}
|
|
|
|
return @y;
|
|
}
|
|
|
|
# In scalar context use a fast algorithm that is much faster than the
|
|
# recursive algorith used in list context.
|
|
|
|
my $cache = {};
|
|
my $two = $class -> _two();
|
|
my $fib;
|
|
|
|
$fib = sub {
|
|
my $n = shift;
|
|
return $class -> _zero() if $n <= 0;
|
|
return $class -> _one() if $n <= 2;
|
|
return $cache -> {$n} if exists $cache -> {$n};
|
|
|
|
my $k = int($n / 2);
|
|
my $a = $fib -> ($k + 1);
|
|
my $b = $fib -> ($k);
|
|
my $y;
|
|
|
|
if ($n % 2 == 1) {
|
|
# a*a + b*b
|
|
$y = $class -> _add($class -> _mul($class -> _copy($a), $a),
|
|
$class -> _mul($class -> _copy($b), $b));
|
|
} else {
|
|
# (2*a - b)*b
|
|
$y = $class -> _mul($class -> _sub($class -> _mul(
|
|
$class -> _copy($two), $a), $b), $b);
|
|
}
|
|
|
|
$cache -> {$n} = $y;
|
|
return $y;
|
|
};
|
|
|
|
return $fib -> ($n);
|
|
}
|
|
|
|
##############################################################################
|
|
##############################################################################
|
|
|
|
1;
|
|
|
|
__END__
|
|
|
|
=pod
|
|
|
|
=head1 NAME
|
|
|
|
Math::BigInt::Lib - virtual parent class for Math::BigInt libraries
|
|
|
|
=head1 SYNOPSIS
|
|
|
|
# In the backend library for Math::BigInt et al.
|
|
|
|
package Math::BigInt::MyBackend;
|
|
|
|
use Math::BigInt::lib;
|
|
our @ISA = qw< Math::BigInt::lib >;
|
|
|
|
sub _new { ... }
|
|
sub _str { ... }
|
|
sub _add { ... }
|
|
str _sub { ... }
|
|
...
|
|
|
|
# In your main program.
|
|
|
|
use Math::BigInt lib => 'MyBackend';
|
|
|
|
=head1 DESCRIPTION
|
|
|
|
This module provides support for big integer calculations. It is not intended
|
|
to be used directly, but rather as a parent class for backend libraries used by
|
|
Math::BigInt, Math::BigFloat, Math::BigRat, and related modules.
|
|
|
|
Other backend libraries include Math::BigInt::Calc, Math::BigInt::FastCalc,
|
|
Math::BigInt::GMP, and Math::BigInt::Pari.
|
|
|
|
In order to allow for multiple big integer libraries, Math::BigInt was
|
|
rewritten to use a plug-in library for core math routines. Any module which
|
|
conforms to the API can be used by Math::BigInt by using this in your program:
|
|
|
|
use Math::BigInt lib => 'libname';
|
|
|
|
'libname' is either the long name, like 'Math::BigInt::Pari', or only the short
|
|
version, like 'Pari'.
|
|
|
|
=head2 General Notes
|
|
|
|
A library only needs to deal with unsigned big integers. Testing of input
|
|
parameter validity is done by the caller, so there is no need to worry about
|
|
underflow (e.g., in C<_sub()> and C<_dec()>) or about division by zero (e.g.,
|
|
in C<_div()> and C<_mod()>)) or similar cases.
|
|
|
|
Some libraries use methods that don't modify their argument, and some libraries
|
|
don't even use objects, but rather unblessed references. Because of this,
|
|
liberary methods are always called as class methods, not instance methods:
|
|
|
|
$x = Class -> method($x, $y); # like this
|
|
$x = $x -> method($y); # not like this ...
|
|
$x -> method($y); # ... or like this
|
|
|
|
And with boolean methods
|
|
|
|
$bool = Class -> method($x, $y); # like this
|
|
$bool = $x -> method($y); # not like this
|
|
|
|
Return values are always objects, strings, Perl scalars, or true/false for
|
|
comparison routines.
|
|
|
|
=head3 API version
|
|
|
|
=over 4
|
|
|
|
=item CLASS-E<gt>api_version()
|
|
|
|
This method is no longer used and can be omitted. Methods that are not
|
|
implemented by a subclass will be inherited from this class.
|
|
|
|
=back
|
|
|
|
=head3 Constructors
|
|
|
|
The following methods are mandatory: _new(), _str(), _add(), and _sub().
|
|
However, computations will be very slow without _mul() and _div().
|
|
|
|
=over 4
|
|
|
|
=item CLASS-E<gt>_new(STR)
|
|
|
|
Convert a string representing an unsigned decimal number to an object
|
|
representing the same number. The input is normalized, i.e., it matches
|
|
C<^(0|[1-9]\d*)$>.
|
|
|
|
=item CLASS-E<gt>_zero()
|
|
|
|
Return an object representing the number zero.
|
|
|
|
=item CLASS-E<gt>_one()
|
|
|
|
Return an object representing the number one.
|
|
|
|
=item CLASS-E<gt>_two()
|
|
|
|
Return an object representing the number two.
|
|
|
|
=item CLASS-E<gt>_ten()
|
|
|
|
Return an object representing the number ten.
|
|
|
|
=item CLASS-E<gt>_from_bin(STR)
|
|
|
|
Return an object given a string representing a binary number. The input has a
|
|
'0b' prefix and matches the regular expression C<^0[bB](0|1[01]*)$>.
|
|
|
|
=item CLASS-E<gt>_from_oct(STR)
|
|
|
|
Return an object given a string representing an octal number. The input has a
|
|
'0' prefix and matches the regular expression C<^0[1-7]*$>.
|
|
|
|
=item CLASS-E<gt>_from_hex(STR)
|
|
|
|
Return an object given a string representing a hexadecimal number. The input
|
|
has a '0x' prefix and matches the regular expression
|
|
C<^0x(0|[1-9a-fA-F][\da-fA-F]*)$>.
|
|
|
|
=item CLASS-E<gt>_from_bytes(STR)
|
|
|
|
Returns an object given a byte string representing the number. The byte string
|
|
is in big endian byte order, so the two-byte input string "\x01\x00" should
|
|
give an output value representing the number 256.
|
|
|
|
=item CLASS-E<gt>_from_base(STR, BASE, COLLSEQ)
|
|
|
|
Returns an object given a string STR, a base BASE, and a collation sequence
|
|
COLLSEQ. Each character in STR represents a numerical value identical to the
|
|
character's position in COLLSEQ. All characters in STR must be present in
|
|
COLLSEQ.
|
|
|
|
If BASE is less than or equal to 94, and a collation sequence is not specified,
|
|
the following default collation sequence is used. It contains of all the 94
|
|
printable ASCII characters except space/blank:
|
|
|
|
0123456789 # ASCII 48 to 57
|
|
ABCDEFGHIJKLMNOPQRSTUVWXYZ # ASCII 65 to 90
|
|
abcdefghijklmnopqrstuvwxyz # ASCII 97 to 122
|
|
!"#$%&'()*+,-./ # ASCII 33 to 47
|
|
:;<=>?@ # ASCII 58 to 64
|
|
[\]^_` # ASCII 91 to 96
|
|
{|}~ # ASCII 123 to 126
|
|
|
|
If the default collation sequence is used, and the BASE is less than or equal
|
|
to 36, the letter case in STR is ignored.
|
|
|
|
For instance, with base 3 and collation sequence "-/|", the character "-"
|
|
represents 0, "/" represents 1, and "|" represents 2. So if STR is "/|-", the
|
|
output is 1 * 3**2 + 2 * 3**1 + 0 * 3**0 = 15.
|
|
|
|
The following examples show standard binary, octal, decimal, and hexadecimal
|
|
conversion. All examples return 250.
|
|
|
|
$x = $class -> _from_base("11111010", 2)
|
|
$x = $class -> _from_base("372", 8)
|
|
$x = $class -> _from_base("250", 10)
|
|
$x = $class -> _from_base("FA", 16)
|
|
|
|
Some more examples, all returning 250:
|
|
|
|
$x = $class -> _from_base("100021", 3)
|
|
$x = $class -> _from_base("3322", 4)
|
|
$x = $class -> _from_base("2000", 5)
|
|
$x = $class -> _from_base("caaa", 5, "abcde")
|
|
$x = $class -> _from_base("42", 62)
|
|
$x = $class -> _from_base("2!", 94)
|
|
|
|
=back
|
|
|
|
=head3 Mathematical functions
|
|
|
|
=over 4
|
|
|
|
=item CLASS-E<gt>_add(OBJ1, OBJ2)
|
|
|
|
Returns the result of adding OBJ2 to OBJ1.
|
|
|
|
=item CLASS-E<gt>_mul(OBJ1, OBJ2)
|
|
|
|
Returns the result of multiplying OBJ2 and OBJ1.
|
|
|
|
=item CLASS-E<gt>_div(OBJ1, OBJ2)
|
|
|
|
In scalar context, returns the quotient after dividing OBJ1 by OBJ2 and
|
|
truncating the result to an integer. In list context, return the quotient and
|
|
the remainder.
|
|
|
|
=item CLASS-E<gt>_sub(OBJ1, OBJ2, FLAG)
|
|
|
|
=item CLASS-E<gt>_sub(OBJ1, OBJ2)
|
|
|
|
Returns the result of subtracting OBJ2 by OBJ1. If C<flag> is false or omitted,
|
|
OBJ1 might be modified. If C<flag> is true, OBJ2 might be modified.
|
|
|
|
=item CLASS-E<gt>_dec(OBJ)
|
|
|
|
Returns the result after decrementing OBJ by one.
|
|
|
|
=item CLASS-E<gt>_inc(OBJ)
|
|
|
|
Returns the result after incrementing OBJ by one.
|
|
|
|
=item CLASS-E<gt>_mod(OBJ1, OBJ2)
|
|
|
|
Returns OBJ1 modulo OBJ2, i.e., the remainder after dividing OBJ1 by OBJ2.
|
|
|
|
=item CLASS-E<gt>_sqrt(OBJ)
|
|
|
|
Returns the square root of OBJ, truncated to an integer.
|
|
|
|
=item CLASS-E<gt>_root(OBJ, N)
|
|
|
|
Returns the Nth root of OBJ, truncated to an integer.
|
|
|
|
=item CLASS-E<gt>_fac(OBJ)
|
|
|
|
Returns the factorial of OBJ, i.e., the product of all positive integers up to
|
|
and including OBJ.
|
|
|
|
=item CLASS-E<gt>_dfac(OBJ)
|
|
|
|
Returns the double factorial of OBJ. If OBJ is an even integer, returns the
|
|
product of all positive, even integers up to and including OBJ, i.e.,
|
|
2*4*6*...*OBJ. If OBJ is an odd integer, returns the product of all positive,
|
|
odd integers, i.e., 1*3*5*...*OBJ.
|
|
|
|
=item CLASS-E<gt>_pow(OBJ1, OBJ2)
|
|
|
|
Returns OBJ1 raised to the power of OBJ2. By convention, 0**0 = 1.
|
|
|
|
=item CLASS-E<gt>_modinv(OBJ1, OBJ2)
|
|
|
|
Returns the modular multiplicative inverse, i.e., return OBJ3 so that
|
|
|
|
(OBJ3 * OBJ1) % OBJ2 = 1 % OBJ2
|
|
|
|
The result is returned as two arguments. If the modular multiplicative inverse
|
|
does not exist, both arguments are undefined. Otherwise, the arguments are a
|
|
number (object) and its sign ("+" or "-").
|
|
|
|
The output value, with its sign, must either be a positive value in the range
|
|
1,2,...,OBJ2-1 or the same value subtracted OBJ2. For instance, if the input
|
|
arguments are objects representing the numbers 7 and 5, the method must either
|
|
return an object representing the number 3 and a "+" sign, since (3*7) % 5 = 1
|
|
% 5, or an object representing the number 2 and a "-" sign, since (-2*7) % 5 = 1
|
|
% 5.
|
|
|
|
=item CLASS-E<gt>_modpow(OBJ1, OBJ2, OBJ3)
|
|
|
|
Returns the modular exponentiation, i.e., (OBJ1 ** OBJ2) % OBJ3.
|
|
|
|
=item CLASS-E<gt>_rsft(OBJ, N, B)
|
|
|
|
Returns the result after shifting OBJ N digits to thee right in base B. This is
|
|
equivalent to performing integer division by B**N and discarding the remainder,
|
|
except that it might be much faster.
|
|
|
|
For instance, if the object $obj represents the hexadecimal number 0xabcde,
|
|
then C<_rsft($obj, 2, 16)> returns an object representing the number 0xabc. The
|
|
"remainer", 0xde, is discarded and not returned.
|
|
|
|
=item CLASS-E<gt>_lsft(OBJ, N, B)
|
|
|
|
Returns the result after shifting OBJ N digits to the left in base B. This is
|
|
equivalent to multiplying by B**N, except that it might be much faster.
|
|
|
|
=item CLASS-E<gt>_log_int(OBJ, B)
|
|
|
|
Returns the logarithm of OBJ to base BASE truncted to an integer. This method
|
|
has two output arguments, the OBJECT and a STATUS. The STATUS is Perl scalar;
|
|
it is 1 if OBJ is the exact result, 0 if the result was truncted to give OBJ,
|
|
and undef if it is unknown whether OBJ is the exact result.
|
|
|
|
=item CLASS-E<gt>_gcd(OBJ1, OBJ2)
|
|
|
|
Returns the greatest common divisor of OBJ1 and OBJ2.
|
|
|
|
=item CLASS-E<gt>_lcm(OBJ1, OBJ2)
|
|
|
|
Return the least common multiple of OBJ1 and OBJ2.
|
|
|
|
=item CLASS-E<gt>_fib(OBJ)
|
|
|
|
In scalar context, returns the nth Fibonacci number: _fib(0) returns 0, _fib(1)
|
|
returns 1, _fib(2) returns 1, _fib(3) returns 2 etc. In list context, returns
|
|
the Fibonacci numbers from F(0) to F(n): 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
|
|
|
|
=item CLASS-E<gt>_lucas(OBJ)
|
|
|
|
In scalar context, returns the nth Lucas number: _lucas(0) returns 2, _lucas(1)
|
|
returns 1, _lucas(2) returns 3, etc. In list context, returns the Lucas numbers
|
|
from L(0) to L(n): 2, 1, 3, 4, 7, 11, 18, 29,47, 76, ...
|
|
|
|
=back
|
|
|
|
=head3 Bitwise operators
|
|
|
|
=over 4
|
|
|
|
=item CLASS-E<gt>_and(OBJ1, OBJ2)
|
|
|
|
Returns bitwise and.
|
|
|
|
=item CLASS-E<gt>_or(OBJ1, OBJ2)
|
|
|
|
Returns bitwise or.
|
|
|
|
=item CLASS-E<gt>_xor(OBJ1, OBJ2)
|
|
|
|
Returns bitwise exclusive or.
|
|
|
|
=item CLASS-E<gt>_sand(OBJ1, OBJ2, SIGN1, SIGN2)
|
|
|
|
Returns bitwise signed and.
|
|
|
|
=item CLASS-E<gt>_sor(OBJ1, OBJ2, SIGN1, SIGN2)
|
|
|
|
Returns bitwise signed or.
|
|
|
|
=item CLASS-E<gt>_sxor(OBJ1, OBJ2, SIGN1, SIGN2)
|
|
|
|
Returns bitwise signed exclusive or.
|
|
|
|
=back
|
|
|
|
=head3 Boolean operators
|
|
|
|
=over 4
|
|
|
|
=item CLASS-E<gt>_is_zero(OBJ)
|
|
|
|
Returns a true value if OBJ is zero, and false value otherwise.
|
|
|
|
=item CLASS-E<gt>_is_one(OBJ)
|
|
|
|
Returns a true value if OBJ is one, and false value otherwise.
|
|
|
|
=item CLASS-E<gt>_is_two(OBJ)
|
|
|
|
Returns a true value if OBJ is two, and false value otherwise.
|
|
|
|
=item CLASS-E<gt>_is_ten(OBJ)
|
|
|
|
Returns a true value if OBJ is ten, and false value otherwise.
|
|
|
|
=item CLASS-E<gt>_is_even(OBJ)
|
|
|
|
Return a true value if OBJ is an even integer, and a false value otherwise.
|
|
|
|
=item CLASS-E<gt>_is_odd(OBJ)
|
|
|
|
Return a true value if OBJ is an even integer, and a false value otherwise.
|
|
|
|
=item CLASS-E<gt>_acmp(OBJ1, OBJ2)
|
|
|
|
Compare OBJ1 and OBJ2 and return -1, 0, or 1, if OBJ1 is numerically less than,
|
|
equal to, or larger than OBJ2, respectively.
|
|
|
|
=back
|
|
|
|
=head3 String conversion
|
|
|
|
=over 4
|
|
|
|
=item CLASS-E<gt>_str(OBJ)
|
|
|
|
Returns a string representing OBJ in decimal notation. The returned string
|
|
should have no leading zeros, i.e., it should match C<^(0|[1-9]\d*)$>.
|
|
|
|
=item CLASS-E<gt>_to_bin(OBJ)
|
|
|
|
Returns the binary string representation of OBJ.
|
|
|
|
=item CLASS-E<gt>_to_oct(OBJ)
|
|
|
|
Returns the octal string representation of the number.
|
|
|
|
=item CLASS-E<gt>_to_hex(OBJ)
|
|
|
|
Returns the hexadecimal string representation of the number.
|
|
|
|
=item CLASS-E<gt>_to_bytes(OBJ)
|
|
|
|
Returns a byte string representation of OBJ. The byte string is in big endian
|
|
byte order, so if OBJ represents the number 256, the output should be the
|
|
two-byte string "\x01\x00".
|
|
|
|
=item CLASS-E<gt>_to_base(OBJ, BASE, COLLSEQ)
|
|
|
|
Returns a string representation of OBJ in base BASE with collation sequence
|
|
COLLSEQ.
|
|
|
|
$val = $class -> _new("210");
|
|
$str = $class -> _to_base($val, 10, "xyz") # $str is "zyx"
|
|
|
|
$val = $class -> _new("32");
|
|
$str = $class -> _to_base($val, 2, "-|") # $str is "|-----"
|
|
|
|
See _from_base() for more information.
|
|
|
|
=item CLASS-E<gt>_as_bin(OBJ)
|
|
|
|
Like C<_to_bin()> but with a '0b' prefix.
|
|
|
|
=item CLASS-E<gt>_as_oct(OBJ)
|
|
|
|
Like C<_to_oct()> but with a '0' prefix.
|
|
|
|
=item CLASS-E<gt>_as_hex(OBJ)
|
|
|
|
Like C<_to_hex()> but with a '0x' prefix.
|
|
|
|
=item CLASS-E<gt>_as_bytes(OBJ)
|
|
|
|
This is an alias to C<_to_bytes()>.
|
|
|
|
=back
|
|
|
|
=head3 Numeric conversion
|
|
|
|
=over 4
|
|
|
|
=item CLASS-E<gt>_num(OBJ)
|
|
|
|
Returns a Perl scalar number representing the number OBJ as close as
|
|
possible. Since Perl scalars have limited precision, the returned value might
|
|
not be exactly the same as OBJ.
|
|
|
|
=back
|
|
|
|
=head3 Miscellaneous
|
|
|
|
=over 4
|
|
|
|
=item CLASS-E<gt>_copy(OBJ)
|
|
|
|
Returns a true copy OBJ.
|
|
|
|
=item CLASS-E<gt>_len(OBJ)
|
|
|
|
Returns the number of the decimal digits in OBJ. The output is a Perl scalar.
|
|
|
|
=item CLASS-E<gt>_zeros(OBJ)
|
|
|
|
Returns the number of trailing decimal zeros. The output is a Perl scalar. The
|
|
number zero has no trailing decimal zeros.
|
|
|
|
=item CLASS-E<gt>_digit(OBJ, N)
|
|
|
|
Returns the Nth digit in OBJ as a Perl scalar. N is a Perl scalar, where zero
|
|
refers to the rightmost (least significant) digit, and negative values count
|
|
from the left (most significant digit). If $obj represents the number 123, then
|
|
|
|
CLASS->_digit($obj, 0) # returns 3
|
|
CLASS->_digit($obj, 1) # returns 2
|
|
CLASS->_digit($obj, 2) # returns 1
|
|
CLASS->_digit($obj, -1) # returns 1
|
|
|
|
=item CLASS-E<gt>_digitsum(OBJ)
|
|
|
|
Returns the sum of the base 10 digits.
|
|
|
|
=item CLASS-E<gt>_check(OBJ)
|
|
|
|
Returns true if the object is invalid and false otherwise. Preferably, the true
|
|
value is a string describing the problem with the object. This is a check
|
|
routine to test the internal state of the object for corruption.
|
|
|
|
=item CLASS-E<gt>_set(OBJ)
|
|
|
|
xxx
|
|
|
|
=back
|
|
|
|
=head2 API version 2
|
|
|
|
The following methods are required for an API version of 2 or greater.
|
|
|
|
=head3 Constructors
|
|
|
|
=over 4
|
|
|
|
=item CLASS-E<gt>_1ex(N)
|
|
|
|
Return an object representing the number 10**N where N E<gt>= 0 is a Perl
|
|
scalar.
|
|
|
|
=back
|
|
|
|
=head3 Mathematical functions
|
|
|
|
=over 4
|
|
|
|
=item CLASS-E<gt>_nok(OBJ1, OBJ2)
|
|
|
|
Return the binomial coefficient OBJ1 over OBJ1.
|
|
|
|
=back
|
|
|
|
=head3 Miscellaneous
|
|
|
|
=over 4
|
|
|
|
=item CLASS-E<gt>_alen(OBJ)
|
|
|
|
Return the approximate number of decimal digits of the object. The output is a
|
|
Perl scalar.
|
|
|
|
=back
|
|
|
|
=head1 WRAP YOUR OWN
|
|
|
|
If you want to port your own favourite C library for big numbers to the
|
|
Math::BigInt interface, you can take any of the already existing modules as a
|
|
rough guideline. You should really wrap up the latest Math::BigInt and
|
|
Math::BigFloat testsuites with your module, and replace in them any of the
|
|
following:
|
|
|
|
use Math::BigInt;
|
|
|
|
by this:
|
|
|
|
use Math::BigInt lib => 'yourlib';
|
|
|
|
This way you ensure that your library really works 100% within Math::BigInt.
|
|
|
|
=head1 BUGS
|
|
|
|
Please report any bugs or feature requests to
|
|
C<bug-math-bigint at rt.cpan.org>, or through the web interface at
|
|
L<https://rt.cpan.org/Ticket/Create.html?Queue=Math-BigInt>
|
|
(requires login).
|
|
We will be notified, and then you'll automatically be notified of progress on
|
|
your bug as I make changes.
|
|
|
|
=head1 SUPPORT
|
|
|
|
You can find documentation for this module with the perldoc command.
|
|
|
|
perldoc Math::BigInt::Calc
|
|
|
|
You can also look for information at:
|
|
|
|
=over 4
|
|
|
|
=item * RT: CPAN's request tracker
|
|
|
|
L<https://rt.cpan.org/Public/Dist/Display.html?Name=Math-BigInt>
|
|
|
|
=item * AnnoCPAN: Annotated CPAN documentation
|
|
|
|
L<http://annocpan.org/dist/Math-BigInt>
|
|
|
|
=item * CPAN Ratings
|
|
|
|
L<https://cpanratings.perl.org/dist/Math-BigInt>
|
|
|
|
=item * MetaCPAN
|
|
|
|
L<https://metacpan.org/release/Math-BigInt>
|
|
|
|
=item * CPAN Testers Matrix
|
|
|
|
L<http://matrix.cpantesters.org/?dist=Math-BigInt>
|
|
|
|
=item * The Bignum mailing list
|
|
|
|
=over 4
|
|
|
|
=item * Post to mailing list
|
|
|
|
C<bignum at lists.scsys.co.uk>
|
|
|
|
=item * View mailing list
|
|
|
|
L<http://lists.scsys.co.uk/pipermail/bignum/>
|
|
|
|
=item * Subscribe/Unsubscribe
|
|
|
|
L<http://lists.scsys.co.uk/cgi-bin/mailman/listinfo/bignum>
|
|
|
|
=back
|
|
|
|
=back
|
|
|
|
=head1 LICENSE
|
|
|
|
This program is free software; you may redistribute it and/or modify it under
|
|
the same terms as Perl itself.
|
|
|
|
=head1 AUTHOR
|
|
|
|
Peter John Acklam, E<lt>pjacklam@online.noE<gt>
|
|
|
|
Code and documentation based on the Math::BigInt::Calc module by Tels
|
|
E<lt>nospam-abuse@bloodgate.comE<gt>
|
|
|
|
=head1 SEE ALSO
|
|
|
|
L<Math::BigInt>, L<Math::BigInt::Calc>, L<Math::BigInt::GMP>,
|
|
L<Math::BigInt::FastCalc> and L<Math::BigInt::Pari>.
|
|
|
|
=cut
|