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-rwxr-xr-xMaster/tlpkg/tlperl/lib/Math/BigInt/Lib.pm1715
1 files changed, 1715 insertions, 0 deletions
diff --git a/Master/tlpkg/tlperl/lib/Math/BigInt/Lib.pm b/Master/tlpkg/tlperl/lib/Math/BigInt/Lib.pm
new file mode 100755
index 00000000000..dea1b0c0a38
--- /dev/null
+++ b/Master/tlpkg/tlperl/lib/Math/BigInt/Lib.pm
@@ -0,0 +1,1715 @@
+package Math::BigInt::Lib;
+
+use 5.006001;
+use strict;
+use warnings;
+
+our $VERSION = '1.999806';
+
+use Carp;
+
+use overload
+
+ # overload key: with_assign
+
+ '+' => sub {
+ my $class = ref $_[0];
+ my $x = $class -> _copy($_[0]);
+ my $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ return $class -> _add($x, $y);
+ },
+
+ '-' => sub {
+ my $class = ref $_[0];
+ my ($x, $y);
+ if ($_[2]) { # if swapped
+ $y = $_[0];
+ $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ } else {
+ $x = $class -> _copy($_[0]);
+ $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ }
+ return $class -> _sub($x, $y);
+ },
+
+ '*' => sub {
+ my $class = ref $_[0];
+ my $x = $class -> _copy($_[0]);
+ my $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ return $class -> _mul($x, $y);
+ },
+
+ '/' => sub {
+ my $class = ref $_[0];
+ my ($x, $y);
+ if ($_[2]) { # if swapped
+ $y = $_[0];
+ $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ } else {
+ $x = $class -> _copy($_[0]);
+ $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ }
+ return $class -> _div($x, $y);
+ },
+
+ '%' => sub {
+ my $class = ref $_[0];
+ my ($x, $y);
+ if ($_[2]) { # if swapped
+ $y = $_[0];
+ $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ } else {
+ $x = $class -> _copy($_[0]);
+ $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ }
+ return $class -> _mod($x, $y);
+ },
+
+ '**' => sub {
+ my $class = ref $_[0];
+ my ($x, $y);
+ if ($_[2]) { # if swapped
+ $y = $_[0];
+ $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ } else {
+ $x = $class -> _copy($_[0]);
+ $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ }
+ return $class -> _pow($x, $y);
+ },
+
+ '<<' => sub {
+ my $class = ref $_[0];
+ my ($x, $y);
+ if ($_[2]) { # if swapped
+ $y = $class -> _num($_[0]);
+ $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ } else {
+ $x = $_[0];
+ $y = ref($_[1]) ? $class -> _num($_[1]) : $_[1];
+ }
+ return $class -> _blsft($x, $y);
+ },
+
+ '>>' => sub {
+ my $class = ref $_[0];
+ my ($x, $y);
+ if ($_[2]) { # if swapped
+ $y = $_[0];
+ $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ } else {
+ $x = $class -> _copy($_[0]);
+ $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ }
+ return $class -> _brsft($x, $y);
+ },
+
+ # overload key: num_comparison
+
+ '<' => sub {
+ my $class = ref $_[0];
+ my ($x, $y);
+ if ($_[2]) { # if swapped
+ $y = $_[0];
+ $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ } else {
+ $x = $class -> _copy($_[0]);
+ $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ }
+ return $class -> _acmp($x, $y) < 0;
+ },
+
+ '<=' => sub {
+ my $class = ref $_[0];
+ my ($x, $y);
+ if ($_[2]) { # if swapped
+ $y = $_[0];
+ $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ } else {
+ $x = $class -> _copy($_[0]);
+ $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ }
+ return $class -> _acmp($x, $y) <= 0;
+ },
+
+ '>' => sub {
+ my $class = ref $_[0];
+ my ($x, $y);
+ if ($_[2]) { # if swapped
+ $y = $_[0];
+ $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ } else {
+ $x = $class -> _copy($_[0]);
+ $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ }
+ return $class -> _acmp($x, $y) > 0;
+ },
+
+ '>=' => sub {
+ my $class = ref $_[0];
+ my ($x, $y);
+ if ($_[2]) { # if swapped
+ $y = $_[0];
+ $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ } else {
+ $x = $class -> _copy($_[0]);
+ $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ }
+ return $class -> _acmp($x, $y) >= 0;
+ },
+
+ '==' => sub {
+ my $class = ref $_[0];
+ my $x = $class -> _copy($_[0]);
+ my $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ return $class -> _acmp($x, $y) == 0;
+ },
+
+ '!=' => sub {
+ my $class = ref $_[0];
+ my $x = $class -> _copy($_[0]);
+ my $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ return $class -> _acmp($x, $y) != 0;
+ },
+
+ # overload key: 3way_comparison
+
+ '<=>' => sub {
+ my $class = ref $_[0];
+ my ($x, $y);
+ if ($_[2]) { # if swapped
+ $y = $_[0];
+ $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ } else {
+ $x = $class -> _copy($_[0]);
+ $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ }
+ return $class -> _acmp($x, $y);
+ },
+
+ # overload key: binary
+
+ '&' => sub {
+ my $class = ref $_[0];
+ my ($x, $y);
+ if ($_[2]) { # if swapped
+ $y = $_[0];
+ $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ } else {
+ $x = $class -> _copy($_[0]);
+ $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ }
+ return $class -> _and($x, $y);
+ },
+
+ '|' => sub {
+ my $class = ref $_[0];
+ my ($x, $y);
+ if ($_[2]) { # if swapped
+ $y = $_[0];
+ $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ } else {
+ $x = $class -> _copy($_[0]);
+ $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ }
+ return $class -> _or($x, $y);
+ },
+
+ '^' => sub {
+ my $class = ref $_[0];
+ my ($x, $y);
+ if ($_[2]) { # if swapped
+ $y = $_[0];
+ $x = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ } else {
+ $x = $class -> _copy($_[0]);
+ $y = ref($_[1]) ? $_[1] : $class -> _new($_[1]);
+ }
+ return $class -> _xor($x, $y);
+ },
+
+ # overload key: func
+
+ 'abs' => sub { $_[0] },
+
+ 'sqrt' => sub {
+ my $class = ref $_[0];
+ return $class -> _sqrt($class -> _copy($_[0]));
+ },
+
+ 'int' => sub { $_[0] -> copy() -> bint(); },
+
+ # overload key: conversion
+
+ 'bool' => sub { ref($_[0]) -> _is_zero($_[0]) ? '' : 1; },
+
+ '""' => sub { ref($_[0]) -> _str($_[0]); },
+
+ '0+' => sub { ref($_[0]) -> _num($_[0]); },
+
+ '=' => sub { ref($_[0]) -> _copy($_[0]); },
+
+ ;
+
+# Do we need api_version() at all, now that we have a virtual parent class that
+# will provide any missing methods? Fixme!
+
+sub api_version () {
+ croak "@{[(caller 0)[3]]} method not implemented";
+}
+
+sub _new {
+ croak "@{[(caller 0)[3]]} method not implemented";
+}
+
+sub _zero {
+ my $class = shift;
+ return $class -> _new("0");
+}
+
+sub _one {
+ my $class = shift;
+ return $class -> _new("1");
+}
+
+sub _two {
+ my $class = shift;
+ return $class -> _new("2");
+
+}
+sub _ten {
+ my $class = shift;
+ return $class -> _new("10");
+}
+
+sub _1ex {
+ my ($class, $exp) = @_;
+ $exp = $class -> _num($exp) if ref($exp);
+ return $class -> _new("1" . ("0" x $exp));
+}
+
+sub _copy {
+ my ($class, $x) = @_;
+ return $class -> _new($class -> _str($x));
+}
+
+# catch and throw away
+sub import { }
+
+##############################################################################
+# convert back to string and number
+
+sub _str {
+ # Convert number from internal base 1eN format to string format. Internal
+ # format is always normalized, i.e., no leading zeros.
+ croak "@{[(caller 0)[3]]} method not implemented";
+}
+
+sub _num {
+ my ($class, $x) = @_;
+ 0 + $class -> _str($x);
+}
+
+##############################################################################
+# actual math code
+
+sub _add {
+ croak "@{[(caller 0)[3]]} method not implemented";
+}
+
+sub _sub {
+ croak "@{[(caller 0)[3]]} method not implemented";
+}
+
+sub _mul {
+ my ($class, $x, $y) = @_;
+ my $sum = $class -> _zero();
+ my $i = $class -> _zero();
+ while ($class -> _acmp($i, $y) < 0) {
+ $sum = $class -> _add($sum, $x);
+ $i = $class -> _inc($i);
+ }
+ return $sum;
+}
+
+sub _div {
+ my ($class, $x, $y) = @_;
+
+ croak "@{[(caller 0)[3]]} requires non-zero divisor"
+ if $class -> _is_zero($y);
+
+ my $r = $class -> _copy($x);
+ my $q = $class -> _zero();
+ while ($class -> _acmp($r, $y) >= 0) {
+ $q = $class -> _inc($q);
+ $r = $class -> _sub($r, $y);
+ }
+
+ return $q, $r if wantarray;
+ return $q;
+}
+
+sub _inc {
+ my ($class, $x) = @_;
+ $class -> _add($x, $class -> _one());
+}
+
+sub _dec {
+ my ($class, $x) = @_;
+ $class -> _sub($x, $class -> _one());
+}
+
+##############################################################################
+# testing
+
+sub _acmp {
+ # Compare two (absolute) values. Return -1, 0, or 1.
+ my ($class, $x, $y) = @_;
+ my $xstr = $class -> _str($x);
+ my $ystr = $class -> _str($y);
+
+ length($xstr) <=> length($ystr) || $xstr cmp $ystr;
+}
+
+sub _len {
+ my ($class, $x) = @_;
+ CORE::length($class -> _str($x));
+}
+
+sub _alen {
+ my ($class, $x) = @_;
+ $class -> _len($x);
+}
+
+sub _digit {
+ my ($class, $x, $n) = @_;
+ substr($class ->_str($x), -($n+1), 1);
+}
+
+sub _zeros {
+ my ($class, $x) = @_;
+ my $str = $class -> _str($x);
+ $str =~ /[^0](0*)\z/;
+ CORE::length($1);
+}
+
+##############################################################################
+# _is_* routines
+
+sub _is_zero {
+ # return true if arg is zero
+ my ($class, $x) = @_;
+ $class -> _str($x) == 0;
+}
+
+sub _is_even {
+ # return true if arg is even
+ my ($class, $x) = @_;
+ substr($class -> _str($x), -1, 1) % 2 == 0;
+}
+
+sub _is_odd {
+ # return true if arg is odd
+ my ($class, $x) = @_;
+ substr($class -> _str($x), -1, 1) % 2 != 0;
+}
+
+sub _is_one {
+ # return true if arg is one
+ my ($class, $x) = @_;
+ $class -> _str($x) == 1;
+}
+
+sub _is_two {
+ # return true if arg is two
+ my ($class, $x) = @_;
+ $class -> _str($x) == 2;
+}
+
+sub _is_ten {
+ # return true if arg is ten
+ my ($class, $x) = @_;
+ $class -> _str($x) == 10;
+}
+
+###############################################################################
+# check routine to test internal state for corruptions
+
+sub _check {
+ # used by the test suite
+ my ($class, $x) = @_;
+ return "Input is undefined" unless defined $x;
+ return "$x is not a reference" unless ref($x);
+ return 0;
+}
+
+###############################################################################
+
+sub _mod {
+ # modulus
+ my ($class, $x, $y) = @_;
+
+ croak "@{[(caller 0)[3]]} requires non-zero second operand"
+ if $class -> _is_zero($y);
+
+ my $r = $class -> _copy($x);
+ while ($class -> _acmp($r, $y) >= 0) {
+ $r = $class -> _sub($r, $y);
+ }
+
+ return $r;
+}
+
+##############################################################################
+# shifts
+
+sub _rsft {
+ my ($class, $x, $n, $b) = @_;
+ $b = $class -> _new($b) unless ref $b;
+ return scalar $class -> _div($x, $class -> _pow($class -> _copy($b), $n));
+}
+
+sub _lsft {
+ my ($class, $x, $n, $b) = @_;
+ $b = $class -> _new($b) unless ref $b;
+ return $class -> _mul($x, $class -> _pow($class -> _copy($b), $n));
+}
+
+sub _pow {
+ # power of $x to $y
+ # ref to array, ref to array, return ref to array
+ my ($class, $x, $y) = @_;
+
+ if ($class -> _is_zero($y)) {
+ return $class -> _one(); # y == 0 => x => 1
+ }
+
+ if (($class -> _is_one($x)) || # x == 1
+ ($class -> _is_one($y))) # or y == 1
+ {
+ return $x;
+ }
+
+ if ($class -> _is_zero($x)) {
+ return $class -> _zero(); # 0 ** y => 0 (if not y <= 0)
+ }
+
+ my $pow2 = $class -> _one();
+
+ my $y_bin = $class -> _as_bin($y);
+ $y_bin =~ s/^0b//;
+ my $len = length($y_bin);
+
+ while (--$len > 0) {
+ $pow2 = $class -> _mul($pow2, $x) if substr($y_bin, $len, 1) eq '1';
+ $x = $class -> _mul($x, $x);
+ }
+
+ $x = $class -> _mul($x, $pow2);
+ return $x;
+}
+
+sub _nok {
+ # Return binomial coefficient (n over k).
+ # Given refs to arrays, return ref to array.
+ # First input argument is modified.
+
+ my ($class, $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 = $class -> _mul($class -> _two(), $class -> _copy($k));
+ if ($class -> _acmp($twok, $n) > 0) {
+ $k = $class -> _sub($class -> _copy($n), $k);
+ }
+ }
+
+ # Example:
+ #
+ # / 7 \ 7! 1*2*3*4 * 5*6*7 5 * 6 * 7 6 7
+ # | | = --------- = --------------- = --------- = 5 * - * -
+ # \ 3 / (7-3)! 3! 1*2*3*4 * 1*2*3 1 * 2 * 3 2 3
+
+ if ($class -> _is_zero($k)) {
+ return $class -> _one();
+ }
+
+ # Make a copy of the original n, since we'll be modifying 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);
+ $class -> _inc($f);
+
+ my $d = $class -> _two();
+
+ # while f <= n (the original n, that is) ...
+
+ while ($class -> _acmp($f, $n_orig) <= 0) {
+
+ # n = (n * f / d) == 5 * 6 / 2 (cf. example above)
+
+ $n = $class -> _mul($n, $f);
+ $n = $class -> _div($n, $d);
+
+ # f = 7, d = 3 (cf. example above)
+
+ $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 _log_int {
+ # 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 $x in place
+ my ($class, $x) = @_;
+
+ return $x if $class -> _is_zero($x);
+
+ my $x_str = $class -> _str($x);
+ my $x_len = length($x_str);
+
+ # Compute the guess $y.
+
+ my $ym;
+ my $ye;
+ if ($x_len % 2 == 0) {
+ $ym = sqrt("." . $x_str);
+ $ye = $x_len / 2;
+ $ym = sprintf "%.0f", int($ym * 1e15);
+ $ye -= 15;
+ } else {
+ $ym = sqrt(".0" . $x_str);
+ $ye = ($x_len + 1) / 2;
+ $ym = sprintf "%.0f", int($ym * 1e16);
+ $ye -= 16;
+ }
+
+ my $y;
+ if ($ye < 0) {
+ $y = substr $ym, 0, length($ym) + $ye;
+ } else {
+ $y = $ym . ("0" x $ye);
+ }
+
+ $y = $class -> _new($y);
+
+ # Newton's method for computing square root of x. Generally, the algorithm
+ # below should undershoot.
+ #
+ # y(i+1) = y(i) - f(y(i)) / f'(y(i))
+ # = y(i) - (y(i)^2 - x) / (2 * y(i))
+ # = y(i) + (x - y(i)^2) / (2 * y(i))
+
+ my $two = $class -> _two();
+ my $zero = $class -> _zero();
+ my $over;
+ my $acmp;
+
+ {
+ my $ysq = $class -> _mul($class -> _copy($y), $y); # y(i)^2
+ $acmp = $class -> _acmp($x, $ysq); # x <=> y(i)^2
+ last if $acmp == 0;
+ if ($acmp < 0) { # if we overshot
+ $over = 1;
+ last;
+ }
+
+ my $num = $class -> _sub($class -> _copy($x), $ysq); # x - y(i)^2
+ my $den = $class -> _mul($class -> _copy($two), $y); # 2 * y(i)
+
+ my $delta = $class -> _div($num, $den);
+ last if $class -> _acmp($delta, $zero) == 0;
+ $y = $class -> _add($y, $delta);
+ redo;
+ }
+
+ # If we did overshoot, adjust now.
+
+ while ($acmp < 0) {
+ $class -> _dec($y);
+ my $ysq = $class -> _mul($class -> _copy($y), $y); # y(i)^2
+ $acmp = $class -> _acmp($x, $ysq); # x <=> y(i)^2
+ }
+
+ return $y;
+}
+
+sub _root {
+ my ($class, $x, $n) = @_;
+
+ return undef if $class -> _is_zero($n);
+
+ return $x if $class -> _is_zero($x) || $class -> _is_one($x) ||
+ $class -> _is_one($n);
+
+ my $x_str = $class -> _str($x);
+ my $x_len = length($x_str);
+
+ return $class -> _one() if $class -> _acmp($x, $n) <= 0;
+
+ # Compute the guess $y.
+
+ my $n_num = $class -> _num($n);
+ my $p = int(($x_len - 1) / $n_num);
+ my $q = $x_len - $p * $n_num;
+
+ my $DEBUG = 0;
+
+ if ($DEBUG) {
+ print "\n";
+ print substr($x_str, 0, $p), " ", "0" x $q, "\n";
+ print "\n";
+ }
+
+ my $ymant = substr($x_str, 0, $q) ** (1 / $n_num);
+ my $yexpo = $p;
+
+ my $y = (1 + int $ymant) . ("0" x $p);
+ $y = $class -> _new($y);
+
+ if ($DEBUG) {
+ print "\n";
+ print "p = $p\n";
+ print "q = $q\n";
+ print "\n";
+ print "ym = $ymant\n";
+ print "ye = $yexpo\n";
+ print "\n";
+ print "y = $y (initial guess)\n";
+ print "\n";
+ }
+
+ # Newton's method for computing n'th root of x. Generally, the algorithm
+ # below should undershoot.
+ #
+ # y(i+1) = y(i) - f(y(i)) / f'(y(i))
+ # = y(i) - (y(i)^n - x) / (n * y(i)^(n-1))
+ # = y(i) + (x - y(i)^n) / (n * y(i)^(n-1))
+
+ my $nm1 = $class -> _dec($class -> _copy($n)); # n - 1
+ my $zero = $class -> _zero();
+ my $over;
+ my $acmp;
+
+ {
+ my $ypowm1 = $class -> _pow($class -> _copy($y), $nm1); # y(i)^(n-1)
+ my $ypow = $class -> _mul($class -> _copy($ypowm1), $y); # y(i)^n
+ $acmp = $class -> _acmp($x, $ypow); # x <=> y(i)^n
+ last if $acmp == 0;
+
+ my $num = $acmp > 0
+ ? $class -> _sub($class -> _copy($x), $ypow) # x - y(i)^n
+ : $class -> _sub($ypow, $class -> _copy($x)); # y(i)^n - x
+ my $den = $class -> _mul($class -> _copy($n), $ypowm1); # n * y(i)^(n-1)
+ my $delta = $class -> _div($num, $den);
+ last if $class -> _acmp($delta, $zero) == 0;
+
+ $y = $acmp > 0
+ ? $class -> _add($y, $delta)
+ : $class -> _sub($y, $delta);
+
+ if ($DEBUG) {
+ print "y = $y\n";
+ }
+
+ redo;
+ }
+
+ # Never overestimate. The output should always be exact or truncated.
+
+ while ($acmp < 0) {
+ $class -> _dec($y);
+ if ($DEBUG) {
+ print "y = $y\n";
+ }
+ my $ypow = $class -> _pow($class -> _copy($y), $n); # y(i)^n
+ $acmp = $class -> _acmp($x, $ypow); # x <=> y(i)^2
+ }
+
+ if ($DEBUG) {
+ print "\n";
+ }
+
+ return $y;
+}
+
+##############################################################################
+# binary stuff
+
+sub _and {
+ my ($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 $z;
+}
+
+sub _as_hex {
+ # convert a decimal number to hex
+ my ($class, $x) = @_;
+ my $str = '';
+ my $tmp = $class -> _copy($x);
+ my $zero = $class -> _zero();
+ my $base = $class -> _new("16");
+ my $rem;
+ while ($tmp > $zero) {
+ ($tmp, $rem) = $class -> _div($tmp, $base);
+ $str = sprintf("%0x", $rem) . $str;
+ }
+ $str = '0' if length($str) == 0;
+ return '0x' . $str;
+}
+
+sub _as_bin {
+ # convert a decimal number to bin
+ my ($class, $x) = @_;
+ my $str = '';
+ my $tmp = $class -> _copy($x);
+ my $zero = $class -> _zero();
+ my $base = $class -> _new("2");
+ my $rem;
+ while ($tmp > $zero) {
+ ($tmp, $rem) = $class -> _div($tmp, $base);
+ $str = ($class -> _is_zero($rem) ? '0' : '1') . $str;
+ }
+ $str = '0' if length($str) == 0;
+ return '0b' . $str;
+}
+
+sub _as_oct {
+ # convert a decimal number to octal
+ my ($class, $x) = @_;
+ my $str = '';
+ my $tmp = $class -> _copy($x);
+ my $zero = $class -> _zero();
+ my $base = $class -> _new("8");
+ my $rem;
+ while ($tmp > $zero) {
+ ($tmp, $rem) = $class -> _div($tmp, $base);
+ $str = sprintf("%0o", $rem) . $str;
+ }
+ $str = '0' if length($str) == 0;
+ return '0' . $str; # yes, 0 becomes "00".
+}
+
+sub _as_bytes {
+ # convert a decimal number to a byte string
+ my ($class, $x) = @_;
+ my $str = '';
+ my $tmp = $class -> _copy($x);
+ my $base = $class -> _new("256");
+ my $rem;
+ until ($class -> _is_zero($tmp)) {
+ ($tmp, $rem) = $class -> _div($tmp, $base);
+ my $byte = pack 'C', $rem;
+ $str = $byte . $str;
+ }
+ return "\x00" unless length($str);
+ return $str;
+}
+
+sub _from_oct {
+ # convert a octal string to a decimal number
+ my ($class, $str) = @_;
+ $str =~ s/^0+//;
+ my $x = $class -> _zero();
+ my $base = $class -> _new("8");
+ my $n = length($str);
+ for (my $i = 0 ; $i < $n ; ++$i) {
+ $x = $class -> _mul($x, $base);
+ $x = $class -> _add($x, $class -> _new(substr($str, $i, 1)));
+ }
+ return $x;
+}
+
+sub _from_hex {
+ # convert a hexadecimal string to a decimal number
+ my ($class, $str) = @_;
+ $str =~ s/^0[Xx]//;
+ my $x = $class -> _zero();
+ my $base = $class -> _new("16");
+ my $n = length($str);
+ for (my $i = 0 ; $i < $n ; ++$i) {
+ $x = $class -> _mul($x, $base);
+ $x = $class -> _add($x, $class -> _new(hex substr($str, $i, 1)));
+ }
+ return $x;
+}
+
+sub _from_bin {
+ # convert a binary string to a decimal number
+ my ($class, $str) = @_;
+ $str =~ s/^0[Bb]//;
+ my $x = $class -> _zero();
+ my $base = $class -> _new("2");
+ my $n = length($str);
+ for (my $i = 0 ; $i < $n ; ++$i) {
+ $x = $class -> _mul($x, $base);
+ $x = $class -> _add($x, $class -> _new(substr($str, $i, 1)));
+ }
+ return $x;
+}
+
+sub _from_bytes {
+ # convert a byte string to a decimal number
+ my ($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 $x;
+}
+
+##############################################################################
+# special modulus functions
+
+sub _modinv {
+ # modular multiplicative inverse
+ my ($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, then return NaN
+ 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;
+}
+
+##############################################################################
+##############################################################################
+
+1;
+
+__END__
+
+=pod
+
+=head1 NAME
+
+Math::BigInt::Lib - virtual parent class for Math::BigInt libraries
+
+=head1 SYNOPSIS
+
+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. Backend
+libraries include Math::BigInt::Calc, Math::BigInt::FastCalc,
+Math::BigInt::GMP, Math::BigInt::Pari and others.
+
+=head1 DESCRIPTION
+
+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()>) nor about division by zero (e.g.,
+in C<_div()>) or similar cases.
+
+Some libraries use methods that don't modify their argument, and some libraries
+don't even use objects. 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 api_version()
+
+Return API version as a Perl scalar, 1 for Math::BigInt v1.70, 2 for
+Math::BigInt v1.83.
+
+This method is no longer used. 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 _new(STR)
+
+Convert a string representing an unsigned decimal number to an object
+representing the same number. The input is normalize, i.e., it matches
+C<^(0|[1-9]\d*)$>.
+
+=item _zero()
+
+Return an object representing the number zero.
+
+=item _one()
+
+Return an object representing the number one.
+
+=item _two()
+
+Return an object representing the number two.
+
+=item _ten()
+
+Return an object representing the number ten.
+
+=item _from_bin(STR)
+
+Return an object given a string representing a binary number. The input has a
+'0b' prefix and matches the regular expression C<^0[bB](0|1[01]*)$>.
+
+=item _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]*$>.
+
+=item _from_hex(STR)
+
+Return an object given a string representing a hexadecimal number. The input
+has a '0x' prefix and matches the regular expression
+C<^0x(0|[1-9a-fA-F][\da-fA-F]*)$>.
+
+=item _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.
+
+=back
+
+=head3 Mathematical functions
+
+=over 4
+
+=item _add(OBJ1, OBJ2)
+
+Returns the result of adding OBJ2 to OBJ1.
+
+=item _mul(OBJ1, OBJ2)
+
+Returns the result of multiplying OBJ2 and OBJ1.
+
+=item _div(OBJ1, OBJ2)
+
+Returns the result of dividing OBJ1 by OBJ2 and truncating the result to an
+integer.
+
+=item _sub(OBJ1, OBJ2, FLAG)
+
+=item _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 _dec(OBJ)
+
+Decrement OBJ by one.
+
+=item _inc(OBJ)
+
+Increment OBJ by one.
+
+=item _mod(OBJ1, OBJ2)
+
+Return OBJ1 modulo OBJ2, i.e., the remainder after dividing OBJ1 by OBJ2.
+
+=item _sqrt(OBJ)
+
+Return the square root of the object, truncated to integer.
+
+=item _root(OBJ, N)
+
+Return Nth root of the object, truncated to int. N is E<gt>= 3.
+
+=item _fac(OBJ)
+
+Return factorial of object (1*2*3*4*...).
+
+=item _pow(OBJ1, OBJ2)
+
+Return OBJ1 to the power of OBJ2. By convention, 0**0 = 1.
+
+=item _modinv(OBJ1, OBJ2)
+
+Return 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 "-" sign,
+since (-2*7) % 5 = 1 % 5.
+
+=item _modpow(OBJ1, OBJ2, OBJ3)
+
+Return modular exponentiation, (OBJ1 ** OBJ2) % OBJ3.
+
+=item _rsft(OBJ, N, B)
+
+Shift object N digits right in base B and return the resulting object. This is
+equivalent to performing integer division by B**N and discarding the remainder,
+except that it might be much faster, depending on how the number is represented
+internally.
+
+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 _lsft(OBJ, N, B)
+
+Shift the object N digits left in base B. This is equivalent to multiplying by
+B**N, except that it might be much faster, depending on how the number is
+represented internally.
+
+=item _log_int(OBJ, B)
+
+Return integer log of OBJ to base BASE. 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 _gcd(OBJ1, OBJ2)
+
+Return the greatest common divisor of OBJ1 and OBJ2.
+
+=item _lcm(OBJ1, OBJ2)
+
+Return the least common multiple of OBJ1 and OBJ2.
+
+=back
+
+=head3 Bitwise operators
+
+Each of these methods may modify the first input argument.
+
+=over 4
+
+=item _and(OBJ1, OBJ2)
+
+Return bitwise and. If necessary, the smallest number is padded with leading
+zeros.
+
+=item _or(OBJ1, OBJ2)
+
+Return bitwise or. If necessary, the smallest number is padded with leading
+zeros.
+
+=item _xor(OBJ1, OBJ2)
+
+Return bitwise exclusive or. If necessary, the smallest number is padded
+with leading zeros.
+
+=back
+
+=head3 Boolean operators
+
+=over 4
+
+=item _is_zero(OBJ)
+
+Returns a true value if OBJ is zero, and false value otherwise.
+
+=item _is_one(OBJ)
+
+Returns a true value if OBJ is one, and false value otherwise.
+
+=item _is_two(OBJ)
+
+Returns a true value if OBJ is two, and false value otherwise.
+
+=item _is_ten(OBJ)
+
+Returns a true value if OBJ is ten, and false value otherwise.
+
+=item _is_even(OBJ)
+
+Return a true value if OBJ is an even integer, and a false value otherwise.
+
+=item _is_odd(OBJ)
+
+Return a true value if OBJ is an even integer, and a false value otherwise.
+
+=item _acmp(OBJ1, OBJ2)
+
+Compare OBJ1 and OBJ2 and return -1, 0, or 1, if OBJ1 is less than, equal
+to, or larger than OBJ2, respectively.
+
+=back
+
+=head3 String conversion
+
+=over 4
+
+=item _str(OBJ)
+
+Return a string representing the object. The returned string should have no
+leading zeros, i.e., it should match C<^(0|[1-9]\d*)$>.
+
+=item _as_bin(OBJ)
+
+Return the binary string representation of the number. The string must have a
+'0b' prefix.
+
+=item _as_oct(OBJ)
+
+Return the octal string representation of the number. The string must have
+a '0x' prefix.
+
+Note: This method was required from Math::BigInt version 1.78, but the required
+API version number was not incremented, so there are older libraries that
+support API version 1, but do not support C<_as_oct()>.
+
+=item _as_hex(OBJ)
+
+Return the hexadecimal string representation of the number. The string must
+have a '0x' prefix.
+
+=item _as_bytes(OBJ)
+
+Return a byte string representation of the number. The byte string is in big
+endian byte order, so if the object represents the number 256, the output
+should be the two-byte string "\x01\x00".
+
+=back
+
+=head3 Numeric conversion
+
+=over 4
+
+=item _num(OBJ)
+
+Given an object, return a Perl scalar number (int/float) representing this
+number.
+
+=back
+
+=head3 Miscellaneous
+
+=over 4
+
+=item _copy(OBJ)
+
+Return a true copy of the object.
+
+=item _len(OBJ)
+
+Returns the number of the decimal digits in the number. The output is a
+Perl scalar.
+
+=item _zeros(OBJ)
+
+Return the number of trailing decimal zeros. The output is a Perl scalar.
+
+=item _digit(OBJ, N)
+
+Return the Nth digit 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 $obj represents the number 123, then
+I<$obj->_digit(0)> is 3 and I<_digit(123, -1)> is 1.
+
+=item _check(OBJ)
+
+Return 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.
+
+=back
+
+=head2 API version 2
+
+The following methods are required for an API version of 2 or greater.
+
+=head3 Constructors
+
+=over 4
+
+=item _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 _nok(OBJ1, OBJ2)
+
+Return the binomial coefficient OBJ1 over OBJ1.
+
+=back
+
+=head3 Miscellaneous
+
+=over 4
+
+=item _alen(OBJ)
+
+Return the approximate number of decimal digits of the object. The output is a
+Perl scalar.
+
+=back
+
+=head2 API optional methods
+
+The following methods are optional, and can be defined if the underlying lib
+has a fast way to do them. If undefined, Math::BigInt will use pure Perl (hence
+slow) fallback routines to emulate these:
+
+=head3 Signed bitwise operators.
+
+=over 4
+
+=item _signed_or(OBJ1, OBJ2, SIGN1, SIGN2)
+
+Return the signed bitwise or.
+
+=item _signed_and(OBJ1, OBJ2, SIGN1, SIGN2)
+
+Return the signed bitwise and.
+
+=item _signed_xor(OBJ1, OBJ2, SIGN1, SIGN2)
+
+Return the signed bitwise exclusive or.
+
+=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<http://cpanratings.perl.org/dist/Math-BigInt>
+
+=item * Search CPAN
+
+L<http://search.cpan.org/dist/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