diff options
Diffstat (limited to 'Master/tlpkg/tlperl/lib/Math/BigInt/Lib.pm')
-rwxr-xr-x | Master/tlpkg/tlperl/lib/Math/BigInt/Lib.pm | 1715 |
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 |