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authorNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
committerNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
commite0c6872cf40896c7be36b11dcc744620f10adf1d (patch)
tree60335e10d2f4354b0674ec22d7b53f0f8abee672 /systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt
Initial commit
Diffstat (limited to 'systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt')
-rw-r--r--systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/Calc.pm2530
-rw-r--r--systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/CalcEmu.pm394
-rw-r--r--systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/FastCalc.pm168
-rw-r--r--systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/Lib.pm2070
-rw-r--r--systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/Trace.pm48
5 files changed, 5210 insertions, 0 deletions
diff --git a/systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/Calc.pm b/systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/Calc.pm
new file mode 100644
index 0000000000..571006963f
--- /dev/null
+++ b/systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/Calc.pm
@@ -0,0 +1,2530 @@
+package Math::BigInt::Calc;
+
+use 5.006001;
+use strict;
+use warnings;
+
+use Carp;
+use Math::BigInt::Lib;
+
+our $VERSION = '1.999811';
+
+our @ISA = ('Math::BigInt::Lib');
+
+# Package to store unsigned big integers in decimal and do math with them
+
+# Internally the numbers are stored in an array with at least 1 element, no
+# leading zero parts (except the first) and in base 1eX where X is determined
+# automatically at loading time to be the maximum possible value
+
+# todo:
+# - fully remove funky $# stuff in div() (maybe - that code scares me...)
+
+# USE_MUL: due to problems on certain os (os390, posix-bc) "* 1e-5" is used
+# instead of "/ 1e5" at some places, (marked with USE_MUL). Other platforms
+# BS2000, some Crays need USE_DIV instead.
+# The BEGIN block is used to determine which of the two variants gives the
+# correct result.
+
+# Beware of things like:
+# $i = $i * $y + $car; $car = int($i / $BASE); $i = $i % $BASE;
+# This works on x86, but fails on ARM (SA1100, iPAQ) due to who knows what
+# reasons. So, use this instead (slower, but correct):
+# $i = $i * $y + $car; $car = int($i / $BASE); $i -= $BASE * $car;
+
+##############################################################################
+# global constants, flags and accessory
+
+# announce that we are compatible with MBI v1.83 and up
+sub api_version () { 2; }
+
+# constants for easier life
+my ($BASE, $BASE_LEN, $RBASE, $MAX_VAL);
+my ($AND_BITS, $XOR_BITS, $OR_BITS);
+my ($AND_MASK, $XOR_MASK, $OR_MASK);
+
+sub _base_len {
+ # Set/get the BASE_LEN and assorted other, related values.
+ # Used only by the testsuite, the set variant is used only by the BEGIN
+ # block below:
+
+ my ($class, $b, $int) = @_;
+ if (defined $b) {
+ # avoid redefinitions
+ undef &_mul;
+ undef &_div;
+
+ if ($] >= 5.008 && $int && $b > 7) {
+ $BASE_LEN = $b;
+ *_mul = \&_mul_use_div_64;
+ *_div = \&_div_use_div_64;
+ $BASE = int("1e" . $BASE_LEN);
+ $MAX_VAL = $BASE-1;
+ return $BASE_LEN unless wantarray;
+ return ($BASE_LEN, $BASE, $AND_BITS, $XOR_BITS, $OR_BITS, $BASE_LEN, $MAX_VAL);
+ }
+
+ # find whether we can use mul or div in mul()/div()
+ $BASE_LEN = $b + 1;
+ my $caught = 0;
+ while (--$BASE_LEN > 5) {
+ $BASE = int("1e" . $BASE_LEN);
+ $RBASE = abs('1e-' . $BASE_LEN); # see USE_MUL
+ $caught = 0;
+ $caught += 1 if (int($BASE * $RBASE) != 1); # should be 1
+ $caught += 2 if (int($BASE / $BASE) != 1); # should be 1
+ last if $caught != 3;
+ }
+ $BASE = int("1e" . $BASE_LEN);
+ $RBASE = abs('1e-' . $BASE_LEN); # see USE_MUL
+ $MAX_VAL = $BASE-1;
+
+ # ($caught & 1) != 0 => cannot use MUL
+ # ($caught & 2) != 0 => cannot use DIV
+ if ($caught == 2) # 2
+ {
+ # must USE_MUL since we cannot use DIV
+ *_mul = \&_mul_use_mul;
+ *_div = \&_div_use_mul;
+ } else # 0 or 1
+ {
+ # can USE_DIV instead
+ *_mul = \&_mul_use_div;
+ *_div = \&_div_use_div;
+ }
+ }
+ return $BASE_LEN unless wantarray;
+ return ($BASE_LEN, $BASE, $AND_BITS, $XOR_BITS, $OR_BITS, $BASE_LEN, $MAX_VAL);
+}
+
+sub _new {
+ # Given a string representing an integer, returns a reference to an array
+ # of integers, where each integer represents a chunk of the original input
+ # integer.
+
+ my ($class, $str) = @_;
+ #unless ($str =~ /^([1-9]\d*|0)\z/) {
+ # require Carp;
+ # Carp::croak("Invalid input string '$str'");
+ #}
+
+ my $input_len = length($str) - 1;
+
+ # Shortcut for small numbers.
+ return bless [ $str ], $class if $input_len < $BASE_LEN;
+
+ my $format = "a" . (($input_len % $BASE_LEN) + 1);
+ $format .= $] < 5.008 ? "a$BASE_LEN" x int($input_len / $BASE_LEN)
+ : "(a$BASE_LEN)*";
+
+ my $self = [ reverse(map { 0 + $_ } unpack($format, $str)) ];
+ return bless $self, $class;
+}
+
+BEGIN {
+ # from Daniel Pfeiffer: determine largest group of digits that is precisely
+ # multipliable with itself plus carry
+ # Test now changed to expect the proper pattern, not a result off by 1 or 2
+ my ($e, $num) = 3; # lowest value we will use is 3+1-1 = 3
+ do {
+ $num = '9' x ++$e;
+ $num *= $num + 1;
+ } while $num =~ /9{$e}0{$e}/; # must be a certain pattern
+ $e--; # last test failed, so retract one step
+ # the limits below brush the problems with the test above under the rug:
+ # the test should be able to find the proper $e automatically
+ $e = 5 if $^O =~ /^uts/; # UTS get's some special treatment
+ $e = 5 if $^O =~ /^unicos/; # unicos is also problematic (6 seems to work
+ # there, but we play safe)
+
+ my $int = 0;
+ if ($e > 7) {
+ use integer;
+ my $e1 = 7;
+ $num = 7;
+ do {
+ $num = ('9' x ++$e1) + 0;
+ $num *= $num + 1;
+ } while ("$num" =~ /9{$e1}0{$e1}/); # must be a certain pattern
+ $e1--; # last test failed, so retract one step
+ if ($e1 > 7) {
+ $int = 1;
+ $e = $e1;
+ }
+ }
+
+ __PACKAGE__ -> _base_len($e, $int); # set and store
+
+ use integer;
+ # find out how many bits _and, _or and _xor can take (old default = 16)
+ # I don't think anybody has yet 128 bit scalars, so let's play safe.
+ local $^W = 0; # don't warn about 'nonportable number'
+ $AND_BITS = 15;
+ $XOR_BITS = 15;
+ $OR_BITS = 15;
+
+ # find max bits, we will not go higher than numberofbits that fit into $BASE
+ # to make _and etc simpler (and faster for smaller, slower for large numbers)
+ my $max = 16;
+ while (2 ** $max < $BASE) {
+ $max++;
+ }
+ {
+ no integer;
+ $max = 16 if $] < 5.006; # older Perls might not take >16 too well
+ }
+ my ($x, $y, $z);
+
+ do {
+ $AND_BITS++;
+ $x = CORE::oct('0b' . '1' x $AND_BITS);
+ $y = $x & $x;
+ $z = (2 ** $AND_BITS) - 1;
+ } while ($AND_BITS < $max && $x == $z && $y == $x);
+ $AND_BITS --; # retreat one step
+
+ do {
+ $XOR_BITS++;
+ $x = CORE::oct('0b' . '1' x $XOR_BITS);
+ $y = $x ^ 0;
+ $z = (2 ** $XOR_BITS) - 1;
+ } while ($XOR_BITS < $max && $x == $z && $y == $x);
+ $XOR_BITS --; # retreat one step
+
+ do {
+ $OR_BITS++;
+ $x = CORE::oct('0b' . '1' x $OR_BITS);
+ $y = $x | $x;
+ $z = (2 ** $OR_BITS) - 1;
+ } while ($OR_BITS < $max && $x == $z && $y == $x);
+ $OR_BITS--; # retreat one step
+
+ $AND_MASK = __PACKAGE__->_new(( 2 ** $AND_BITS ));
+ $XOR_MASK = __PACKAGE__->_new(( 2 ** $XOR_BITS ));
+ $OR_MASK = __PACKAGE__->_new(( 2 ** $OR_BITS ));
+
+ # We can compute the approximate length no faster than the real length:
+ *_alen = \&_len;
+}
+
+###############################################################################
+
+sub _zero {
+ # create a zero
+ my $class = shift;
+ return bless [ 0 ], $class;
+}
+
+sub _one {
+ # create a one
+ my $class = shift;
+ return bless [ 1 ], $class;
+}
+
+sub _two {
+ # create a two
+ my $class = shift;
+ return bless [ 2 ], $class;
+}
+
+sub _ten {
+ # create a 10
+ my $class = shift;
+ bless [ 10 ], $class;
+}
+
+sub _1ex {
+ # create a 1Ex
+ my $class = shift;
+
+ my $rem = $_[0] % $BASE_LEN; # remainder
+ my $parts = $_[0] / $BASE_LEN; # parts
+
+ # 000000, 000000, 100
+ bless [ (0) x $parts, '1' . ('0' x $rem) ], $class;
+}
+
+sub _copy {
+ # make a true copy
+ my $class = shift;
+ return bless [ @{ $_[0] } ], $class;
+}
+
+# catch and throw away
+sub import { }
+
+##############################################################################
+# convert back to string and number
+
+sub _str {
+ # Convert number from internal base 1eN format to string format. Internal
+ # format is always normalized, i.e., no leading zeros.
+
+ my $ary = $_[1];
+ my $idx = $#$ary; # index of last element
+
+ if ($idx < 0) { # should not happen
+ require Carp;
+ Carp::croak("$_[1] has no elements");
+ }
+
+ # Handle first one differently, since it should not have any leading zeros.
+ my $ret = int($ary->[$idx]);
+ if ($idx > 0) {
+ # Interestingly, the pre-padd method uses more time.
+ # The old grep variant takes longer (14 vs. 10 sec).
+ my $z = '0' x ($BASE_LEN - 1);
+ while (--$idx >= 0) {
+ $ret .= substr($z . $ary->[$idx], -$BASE_LEN);
+ }
+ }
+ $ret;
+}
+
+sub _num {
+ # Make a Perl scalar number (int/float) from a BigInt object.
+ my $x = $_[1];
+
+ return $x->[0] if @$x == 1; # below $BASE
+
+ # Start with the most significant element and work towards the least
+ # significant element. Avoid multiplying "inf" (which happens if the number
+ # overflows) with "0" (if there are zero elements in $x) since this gives
+ # "nan" which propagates to the output.
+
+ my $num = 0;
+ for (my $i = $#$x ; $i >= 0 ; --$i) {
+ $num *= $BASE;
+ $num += $x -> [$i];
+ }
+ return $num;
+}
+
+##############################################################################
+# actual math code
+
+sub _add {
+ # (ref to int_num_array, ref to int_num_array)
+ #
+ # Routine to add two base 1eX numbers stolen from Knuth Vol 2 Algorithm A
+ # pg 231. There are separate routines to add and sub as per Knuth pg 233.
+ # This routine modifies array x, but not y.
+
+ my ($c, $x, $y) = @_;
+
+ # $x + 0 => $x
+
+ return $x if @$y == 1 && $y->[0] == 0;
+
+ # 0 + $y => $y->copy
+
+ if (@$x == 1 && $x->[0] == 0) {
+ @$x = @$y;
+ return $x;
+ }
+
+ # For each in Y, add Y to X and carry. If after that, something is left in
+ # X, foreach in X add carry to X and then return X, carry. Trades one
+ # "$j++" for having to shift arrays.
+ my $i;
+ my $car = 0;
+ my $j = 0;
+ for $i (@$y) {
+ $x->[$j] -= $BASE if $car = (($x->[$j] += $i + $car) >= $BASE) ? 1 : 0;
+ $j++;
+ }
+ while ($car != 0) {
+ $x->[$j] -= $BASE if $car = (($x->[$j] += $car) >= $BASE) ? 1 : 0;
+ $j++;
+ }
+ $x;
+}
+
+sub _inc {
+ # (ref to int_num_array, ref to int_num_array)
+ # Add 1 to $x, modify $x in place
+ my ($c, $x) = @_;
+
+ for my $i (@$x) {
+ return $x if ($i += 1) < $BASE; # early out
+ $i = 0; # overflow, next
+ }
+ push @$x, 1 if $x->[-1] == 0; # last overflowed, so extend
+ $x;
+}
+
+sub _dec {
+ # (ref to int_num_array, ref to int_num_array)
+ # Sub 1 from $x, modify $x in place
+ my ($c, $x) = @_;
+
+ my $MAX = $BASE - 1; # since MAX_VAL based on BASE
+ for my $i (@$x) {
+ last if ($i -= 1) >= 0; # early out
+ $i = $MAX; # underflow, next
+ }
+ pop @$x if $x->[-1] == 0 && @$x > 1; # last underflowed (but leave 0)
+ $x;
+}
+
+sub _sub {
+ # (ref to int_num_array, ref to int_num_array, swap)
+ #
+ # Subtract base 1eX numbers -- stolen from Knuth Vol 2 pg 232, $x > $y
+ # subtract Y from X by modifying x in place
+ my ($c, $sx, $sy, $s) = @_;
+
+ my $car = 0;
+ my $i;
+ my $j = 0;
+ if (!$s) {
+ for $i (@$sx) {
+ last unless defined $sy->[$j] || $car;
+ $i += $BASE if $car = (($i -= ($sy->[$j] || 0) + $car) < 0);
+ $j++;
+ }
+ # might leave leading zeros, so fix that
+ return __strip_zeros($sx);
+ }
+ for $i (@$sx) {
+ # We can't do an early out if $x < $y, since we need to copy the high
+ # chunks from $y. Found by Bob Mathews.
+ #last unless defined $sy->[$j] || $car;
+ $sy->[$j] += $BASE
+ if $car = ($sy->[$j] = $i - ($sy->[$j] || 0) - $car) < 0;
+ $j++;
+ }
+ # might leave leading zeros, so fix that
+ __strip_zeros($sy);
+}
+
+sub _mul_use_mul {
+ # (ref to int_num_array, ref to int_num_array)
+ # multiply two numbers in internal representation
+ # modifies first arg, second need not be different from first
+ my ($c, $xv, $yv) = @_;
+
+ if (@$yv == 1) {
+ # shortcut for two very short numbers (improved by Nathan Zook)
+ # works also if xv and yv are the same reference, and handles also $x == 0
+ if (@$xv == 1) {
+ if (($xv->[0] *= $yv->[0]) >= $BASE) {
+ $xv->[0] = $xv->[0] - ($xv->[1] = int($xv->[0] * $RBASE)) * $BASE;
+ }
+ ;
+ return $xv;
+ }
+ # $x * 0 => 0
+ if ($yv->[0] == 0) {
+ @$xv = (0);
+ return $xv;
+ }
+ # multiply a large number a by a single element one, so speed up
+ my $y = $yv->[0];
+ my $car = 0;
+ foreach my $i (@$xv) {
+ $i = $i * $y + $car;
+ $car = int($i * $RBASE);
+ $i -= $car * $BASE;
+ }
+ push @$xv, $car if $car != 0;
+ return $xv;
+ }
+ # shortcut for result $x == 0 => result = 0
+ return $xv if @$xv == 1 && $xv->[0] == 0;
+
+ # since multiplying $x with $x fails, make copy in this case
+ $yv = [ @$xv ] if $xv == $yv; # same references?
+
+ my @prod = ();
+ my ($prod, $car, $cty, $xi, $yi);
+
+ for $xi (@$xv) {
+ $car = 0;
+ $cty = 0;
+
+ # slow variant
+ # for $yi (@$yv)
+ # {
+ # $prod = $xi * $yi + ($prod[$cty] || 0) + $car;
+ # $prod[$cty++] =
+ # $prod - ($car = int($prod * RBASE)) * $BASE; # see USE_MUL
+ # }
+ # $prod[$cty] += $car if $car; # need really to check for 0?
+ # $xi = shift @prod;
+
+ # faster variant
+ # looping through this if $xi == 0 is silly - so optimize it away!
+ $xi = (shift @prod || 0), next if $xi == 0;
+ for $yi (@$yv) {
+ $prod = $xi * $yi + ($prod[$cty] || 0) + $car;
+ ## this is actually a tad slower
+ ## $prod = $prod[$cty]; $prod += ($car + $xi * $yi); # no ||0 here
+ $prod[$cty++] =
+ $prod - ($car = int($prod * $RBASE)) * $BASE; # see USE_MUL
+ }
+ $prod[$cty] += $car if $car; # need really to check for 0?
+ $xi = shift @prod || 0; # || 0 makes v5.005_3 happy
+ }
+ push @$xv, @prod;
+ # can't have leading zeros
+ # __strip_zeros($xv);
+ $xv;
+}
+
+sub _mul_use_div_64 {
+ # (ref to int_num_array, ref to int_num_array)
+ # multiply two numbers in internal representation
+ # modifies first arg, second need not be different from first
+ # works for 64 bit integer with "use integer"
+ my ($c, $xv, $yv) = @_;
+
+ use integer;
+ if (@$yv == 1) {
+ # shortcut for two small numbers, also handles $x == 0
+ if (@$xv == 1) {
+ # shortcut for two very short numbers (improved by Nathan Zook)
+ # works also if xv and yv are the same reference, and handles also $x == 0
+ if (($xv->[0] *= $yv->[0]) >= $BASE) {
+ $xv->[0] =
+ $xv->[0] - ($xv->[1] = $xv->[0] / $BASE) * $BASE;
+ }
+ return $xv;
+ }
+ # $x * 0 => 0
+ if ($yv->[0] == 0) {
+ @$xv = (0);
+ return $xv;
+ }
+ # multiply a large number a by a single element one, so speed up
+ my $y = $yv->[0];
+ my $car = 0;
+ foreach my $i (@$xv) {
+ #$i = $i * $y + $car; $car = $i / $BASE; $i -= $car * $BASE;
+ $i = $i * $y + $car;
+ $i -= ($car = $i / $BASE) * $BASE;
+ }
+ push @$xv, $car if $car != 0;
+ return $xv;
+ }
+ # shortcut for result $x == 0 => result = 0
+ return $xv if ( ((@$xv == 1) && ($xv->[0] == 0)) );
+
+ # since multiplying $x with $x fails, make copy in this case
+ $yv = $c->_copy($xv) if $xv == $yv; # same references?
+
+ my @prod = ();
+ my ($prod, $car, $cty, $xi, $yi);
+ for $xi (@$xv) {
+ $car = 0;
+ $cty = 0;
+ # looping through this if $xi == 0 is silly - so optimize it away!
+ $xi = (shift @prod || 0), next if $xi == 0;
+ for $yi (@$yv) {
+ $prod = $xi * $yi + ($prod[$cty] || 0) + $car;
+ $prod[$cty++] = $prod - ($car = $prod / $BASE) * $BASE;
+ }
+ $prod[$cty] += $car if $car; # need really to check for 0?
+ $xi = shift @prod || 0; # || 0 makes v5.005_3 happy
+ }
+ push @$xv, @prod;
+ $xv;
+}
+
+sub _mul_use_div {
+ # (ref to int_num_array, ref to int_num_array)
+ # multiply two numbers in internal representation
+ # modifies first arg, second need not be different from first
+ my ($c, $xv, $yv) = @_;
+
+ if (@$yv == 1) {
+ # shortcut for two small numbers, also handles $x == 0
+ if (@$xv == 1) {
+ # shortcut for two very short numbers (improved by Nathan Zook)
+ # works also if xv and yv are the same reference, and handles also $x == 0
+ if (($xv->[0] *= $yv->[0]) >= $BASE) {
+ $xv->[0] =
+ $xv->[0] - ($xv->[1] = int($xv->[0] / $BASE)) * $BASE;
+ }
+ ;
+ return $xv;
+ }
+ # $x * 0 => 0
+ if ($yv->[0] == 0) {
+ @$xv = (0);
+ return $xv;
+ }
+ # multiply a large number a by a single element one, so speed up
+ my $y = $yv->[0];
+ my $car = 0;
+ foreach my $i (@$xv) {
+ $i = $i * $y + $car;
+ $car = int($i / $BASE);
+ $i -= $car * $BASE;
+ # This (together with use integer;) does not work on 32-bit Perls
+ #$i = $i * $y + $car; $i -= ($car = $i / $BASE) * $BASE;
+ }
+ push @$xv, $car if $car != 0;
+ return $xv;
+ }
+ # shortcut for result $x == 0 => result = 0
+ return $xv if ( ((@$xv == 1) && ($xv->[0] == 0)) );
+
+ # since multiplying $x with $x fails, make copy in this case
+ $yv = $c->_copy($xv) if $xv == $yv; # same references?
+
+ my @prod = ();
+ my ($prod, $car, $cty, $xi, $yi);
+ for $xi (@$xv) {
+ $car = 0;
+ $cty = 0;
+ # looping through this if $xi == 0 is silly - so optimize it away!
+ $xi = (shift @prod || 0), next if $xi == 0;
+ for $yi (@$yv) {
+ $prod = $xi * $yi + ($prod[$cty] || 0) + $car;
+ $prod[$cty++] = $prod - ($car = int($prod / $BASE)) * $BASE;
+ }
+ $prod[$cty] += $car if $car; # need really to check for 0?
+ $xi = shift @prod || 0; # || 0 makes v5.005_3 happy
+ }
+ push @$xv, @prod;
+ # can't have leading zeros
+ # __strip_zeros($xv);
+ $xv;
+}
+
+sub _div_use_mul {
+ # ref to array, ref to array, modify first array and return remainder if
+ # in list context
+
+ # see comments in _div_use_div() for more explanations
+
+ my ($c, $x, $yorg) = @_;
+
+ # the general div algorithm here is about O(N*N) and thus quite slow, so
+ # we first check for some special cases and use shortcuts to handle them.
+
+ # This works, because we store the numbers in a chunked format where each
+ # element contains 5..7 digits (depending on system).
+
+ # if both numbers have only one element:
+ if (@$x == 1 && @$yorg == 1) {
+ # shortcut, $yorg and $x are two small numbers
+ if (wantarray) {
+ my $rem = [ $x->[0] % $yorg->[0] ];
+ bless $rem, $c;
+ $x->[0] = int($x->[0] / $yorg->[0]);
+ return ($x, $rem);
+ } else {
+ $x->[0] = int($x->[0] / $yorg->[0]);
+ return $x;
+ }
+ }
+
+ # if x has more than one, but y has only one element:
+ if (@$yorg == 1) {
+ my $rem;
+ $rem = $c->_mod($c->_copy($x), $yorg) if wantarray;
+
+ # shortcut, $y is < $BASE
+ my $j = @$x;
+ my $r = 0;
+ my $y = $yorg->[0];
+ my $b;
+ while ($j-- > 0) {
+ $b = $r * $BASE + $x->[$j];
+ $x->[$j] = int($b/$y);
+ $r = $b % $y;
+ }
+ pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero
+ return ($x, $rem) if wantarray;
+ return $x;
+ }
+
+ # now x and y have more than one element
+
+ # check whether y has more elements than x, if yet, the result will be 0
+ if (@$yorg > @$x) {
+ my $rem;
+ $rem = $c->_copy($x) if wantarray; # make copy
+ @$x = 0; # set to 0
+ return ($x, $rem) if wantarray; # including remainder?
+ return $x; # only x, which is [0] now
+ }
+ # check whether the numbers have the same number of elements, in that case
+ # the result will fit into one element and can be computed efficiently
+ if (@$yorg == @$x) {
+
+ # if $yorg has more digits than $x (it's leading element is longer than
+ # the one from $x), the result will also be 0:
+ if (length(int($yorg->[-1])) > length(int($x->[-1]))) {
+ my $rem = $c->_copy($x) if wantarray; # make copy
+ @$x = 0; # set to 0
+ return ($x, $rem) if wantarray; # including remainder?
+ return $x;
+ }
+ # now calculate $x / $yorg
+ if (length(int($yorg->[-1])) == length(int($x->[-1]))) {
+ # same length, so make full compare
+
+ my $a = 0;
+ my $j = @$x - 1;
+ # manual way (abort if unequal, good for early ne)
+ while ($j >= 0) {
+ last if ($a = $x->[$j] - $yorg->[$j]);
+ $j--;
+ }
+ # $a contains the result of the compare between X and Y
+ # a < 0: x < y, a == 0: x == y, a > 0: x > y
+ if ($a <= 0) {
+ # a = 0 => x == y => rem 0
+ # a < 0 => x < y => rem = x
+ my $rem = $a == 0 ? $c->_zero() : $c->_copy($x);
+ @$x = 0; # if $a < 0
+ $x->[0] = 1 if $a == 0; # $x == $y
+ return ($x, $rem) if wantarray;
+ return $x;
+ }
+ # $x >= $y, so proceed normally
+ }
+ }
+
+ # all other cases:
+
+ my $y = $c->_copy($yorg); # always make copy to preserve
+
+ my ($car, $bar, $prd, $dd, $xi, $yi, @q, $v2, $v1, @d, $tmp, $q, $u2, $u1, $u0);
+
+ $car = $bar = $prd = 0;
+ if (($dd = int($BASE / ($y->[-1] + 1))) != 1) {
+ for $xi (@$x) {
+ $xi = $xi * $dd + $car;
+ $xi -= ($car = int($xi * $RBASE)) * $BASE; # see USE_MUL
+ }
+ push(@$x, $car);
+ $car = 0;
+ for $yi (@$y) {
+ $yi = $yi * $dd + $car;
+ $yi -= ($car = int($yi * $RBASE)) * $BASE; # see USE_MUL
+ }
+ } else {
+ push(@$x, 0);
+ }
+ @q = ();
+ ($v2, $v1) = @$y[-2, -1];
+ $v2 = 0 unless $v2;
+ while ($#$x > $#$y) {
+ ($u2, $u1, $u0) = @$x[-3 .. -1];
+ $u2 = 0 unless $u2;
+ #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n"
+ # if $v1 == 0;
+ $q = (($u0 == $v1) ? $MAX_VAL : int(($u0 * $BASE + $u1) / $v1));
+ --$q while ($v2 * $q > ($u0 * $BASE + $u1 - $q * $v1) * $BASE + $u2);
+ if ($q) {
+ ($car, $bar) = (0, 0);
+ for ($yi = 0, $xi = $#$x - $#$y-1; $yi <= $#$y; ++$yi, ++$xi) {
+ $prd = $q * $y->[$yi] + $car;
+ $prd -= ($car = int($prd * $RBASE)) * $BASE; # see USE_MUL
+ $x->[$xi] += $BASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0));
+ }
+ if ($x->[-1] < $car + $bar) {
+ $car = 0;
+ --$q;
+ for ($yi = 0, $xi = $#$x - $#$y-1; $yi <= $#$y; ++$yi, ++$xi) {
+ $x->[$xi] -= $BASE
+ if ($car = (($x->[$xi] += $y->[$yi] + $car) >= $BASE));
+ }
+ }
+ }
+ pop(@$x);
+ unshift(@q, $q);
+ }
+ if (wantarray) {
+ my $d = bless [], $c;
+ if ($dd != 1) {
+ $car = 0;
+ for $xi (reverse @$x) {
+ $prd = $car * $BASE + $xi;
+ $car = $prd - ($tmp = int($prd / $dd)) * $dd; # see USE_MUL
+ unshift(@$d, $tmp);
+ }
+ } else {
+ @$d = @$x;
+ }
+ @$x = @q;
+ __strip_zeros($x);
+ __strip_zeros($d);
+ return ($x, $d);
+ }
+ @$x = @q;
+ __strip_zeros($x);
+ $x;
+}
+
+sub _div_use_div_64 {
+ # ref to array, ref to array, modify first array and return remainder if
+ # in list context
+ # This version works on 64 bit integers
+ my ($c, $x, $yorg) = @_;
+
+ use integer;
+ # the general div algorithm here is about O(N*N) and thus quite slow, so
+ # we first check for some special cases and use shortcuts to handle them.
+
+ # This works, because we store the numbers in a chunked format where each
+ # element contains 5..7 digits (depending on system).
+
+ # if both numbers have only one element:
+ if (@$x == 1 && @$yorg == 1) {
+ # shortcut, $yorg and $x are two small numbers
+ if (wantarray) {
+ my $rem = [ $x->[0] % $yorg->[0] ];
+ bless $rem, $c;
+ $x->[0] = int($x->[0] / $yorg->[0]);
+ return ($x, $rem);
+ } else {
+ $x->[0] = int($x->[0] / $yorg->[0]);
+ return $x;
+ }
+ }
+ # if x has more than one, but y has only one element:
+ if (@$yorg == 1) {
+ my $rem;
+ $rem = $c->_mod($c->_copy($x), $yorg) if wantarray;
+
+ # shortcut, $y is < $BASE
+ my $j = @$x;
+ my $r = 0;
+ my $y = $yorg->[0];
+ my $b;
+ while ($j-- > 0) {
+ $b = $r * $BASE + $x->[$j];
+ $x->[$j] = int($b/$y);
+ $r = $b % $y;
+ }
+ pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero
+ return ($x, $rem) if wantarray;
+ return $x;
+ }
+ # now x and y have more than one element
+
+ # check whether y has more elements than x, if yet, the result will be 0
+ if (@$yorg > @$x) {
+ my $rem;
+ $rem = $c->_copy($x) if wantarray; # make copy
+ @$x = 0; # set to 0
+ return ($x, $rem) if wantarray; # including remainder?
+ return $x; # only x, which is [0] now
+ }
+ # check whether the numbers have the same number of elements, in that case
+ # the result will fit into one element and can be computed efficiently
+ if (@$yorg == @$x) {
+ my $rem;
+ # if $yorg has more digits than $x (it's leading element is longer than
+ # the one from $x), the result will also be 0:
+ if (length(int($yorg->[-1])) > length(int($x->[-1]))) {
+ $rem = $c->_copy($x) if wantarray; # make copy
+ @$x = 0; # set to 0
+ return ($x, $rem) if wantarray; # including remainder?
+ return $x;
+ }
+ # now calculate $x / $yorg
+
+ if (length(int($yorg->[-1])) == length(int($x->[-1]))) {
+ # same length, so make full compare
+
+ my $a = 0;
+ my $j = @$x - 1;
+ # manual way (abort if unequal, good for early ne)
+ while ($j >= 0) {
+ last if ($a = $x->[$j] - $yorg->[$j]);
+ $j--;
+ }
+ # $a contains the result of the compare between X and Y
+ # a < 0: x < y, a == 0: x == y, a > 0: x > y
+ if ($a <= 0) {
+ $rem = $c->_zero(); # a = 0 => x == y => rem 0
+ $rem = $c->_copy($x) if $a != 0; # a < 0 => x < y => rem = x
+ @$x = 0; # if $a < 0
+ $x->[0] = 1 if $a == 0; # $x == $y
+ return ($x, $rem) if wantarray; # including remainder?
+ return $x;
+ }
+ # $x >= $y, so proceed normally
+ }
+ }
+
+ # all other cases:
+
+ my $y = $c->_copy($yorg); # always make copy to preserve
+
+ my ($car, $bar, $prd, $dd, $xi, $yi, @q, $v2, $v1, @d, $tmp, $q, $u2, $u1, $u0);
+
+ $car = $bar = $prd = 0;
+ if (($dd = int($BASE / ($y->[-1] + 1))) != 1) {
+ for $xi (@$x) {
+ $xi = $xi * $dd + $car;
+ $xi -= ($car = int($xi / $BASE)) * $BASE;
+ }
+ push(@$x, $car);
+ $car = 0;
+ for $yi (@$y) {
+ $yi = $yi * $dd + $car;
+ $yi -= ($car = int($yi / $BASE)) * $BASE;
+ }
+ } else {
+ push(@$x, 0);
+ }
+
+ # @q will accumulate the final result, $q contains the current computed
+ # part of the final result
+
+ @q = ();
+ ($v2, $v1) = @$y[-2, -1];
+ $v2 = 0 unless $v2;
+ while ($#$x > $#$y) {
+ ($u2, $u1, $u0) = @$x[-3..-1];
+ $u2 = 0 unless $u2;
+ #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n"
+ # if $v1 == 0;
+ $q = (($u0 == $v1) ? $MAX_VAL : int(($u0 * $BASE + $u1) / $v1));
+ --$q while ($v2 * $q > ($u0 * $BASE +$ u1- $q*$v1) * $BASE + $u2);
+ if ($q) {
+ ($car, $bar) = (0, 0);
+ for ($yi = 0, $xi = $#$x - $#$y - 1; $yi <= $#$y; ++$yi, ++$xi) {
+ $prd = $q * $y->[$yi] + $car;
+ $prd -= ($car = int($prd / $BASE)) * $BASE;
+ $x->[$xi] += $BASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0));
+ }
+ if ($x->[-1] < $car + $bar) {
+ $car = 0;
+ --$q;
+ for ($yi = 0, $xi = $#$x - $#$y - 1; $yi <= $#$y; ++$yi, ++$xi) {
+ $x->[$xi] -= $BASE
+ if ($car = (($x->[$xi] += $y->[$yi] + $car) >= $BASE));
+ }
+ }
+ }
+ pop(@$x);
+ unshift(@q, $q);
+ }
+ if (wantarray) {
+ my $d = bless [], $c;
+ if ($dd != 1) {
+ $car = 0;
+ for $xi (reverse @$x) {
+ $prd = $car * $BASE + $xi;
+ $car = $prd - ($tmp = int($prd / $dd)) * $dd;
+ unshift(@$d, $tmp);
+ }
+ } else {
+ @$d = @$x;
+ }
+ @$x = @q;
+ __strip_zeros($x);
+ __strip_zeros($d);
+ return ($x, $d);
+ }
+ @$x = @q;
+ __strip_zeros($x);
+ $x;
+}
+
+sub _div_use_div {
+ # ref to array, ref to array, modify first array and return remainder if
+ # in list context
+ my ($c, $x, $yorg) = @_;
+
+ # the general div algorithm here is about O(N*N) and thus quite slow, so
+ # we first check for some special cases and use shortcuts to handle them.
+
+ # This works, because we store the numbers in a chunked format where each
+ # element contains 5..7 digits (depending on system).
+
+ # if both numbers have only one element:
+ if (@$x == 1 && @$yorg == 1) {
+ # shortcut, $yorg and $x are two small numbers
+ if (wantarray) {
+ my $rem = [ $x->[0] % $yorg->[0] ];
+ bless $rem, $c;
+ $x->[0] = int($x->[0] / $yorg->[0]);
+ return ($x, $rem);
+ } else {
+ $x->[0] = int($x->[0] / $yorg->[0]);
+ return $x;
+ }
+ }
+ # if x has more than one, but y has only one element:
+ if (@$yorg == 1) {
+ my $rem;
+ $rem = $c->_mod($c->_copy($x), $yorg) if wantarray;
+
+ # shortcut, $y is < $BASE
+ my $j = @$x;
+ my $r = 0;
+ my $y = $yorg->[0];
+ my $b;
+ while ($j-- > 0) {
+ $b = $r * $BASE + $x->[$j];
+ $x->[$j] = int($b/$y);
+ $r = $b % $y;
+ }
+ pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero
+ return ($x, $rem) if wantarray;
+ return $x;
+ }
+ # now x and y have more than one element
+
+ # check whether y has more elements than x, if yet, the result will be 0
+ if (@$yorg > @$x) {
+ my $rem;
+ $rem = $c->_copy($x) if wantarray; # make copy
+ @$x = 0; # set to 0
+ return ($x, $rem) if wantarray; # including remainder?
+ return $x; # only x, which is [0] now
+ }
+ # check whether the numbers have the same number of elements, in that case
+ # the result will fit into one element and can be computed efficiently
+ if (@$yorg == @$x) {
+ my $rem;
+ # if $yorg has more digits than $x (it's leading element is longer than
+ # the one from $x), the result will also be 0:
+ if (length(int($yorg->[-1])) > length(int($x->[-1]))) {
+ $rem = $c->_copy($x) if wantarray; # make copy
+ @$x = 0; # set to 0
+ return ($x, $rem) if wantarray; # including remainder?
+ return $x;
+ }
+ # now calculate $x / $yorg
+
+ if (length(int($yorg->[-1])) == length(int($x->[-1]))) {
+ # same length, so make full compare
+
+ my $a = 0;
+ my $j = @$x - 1;
+ # manual way (abort if unequal, good for early ne)
+ while ($j >= 0) {
+ last if ($a = $x->[$j] - $yorg->[$j]);
+ $j--;
+ }
+ # $a contains the result of the compare between X and Y
+ # a < 0: x < y, a == 0: x == y, a > 0: x > y
+ if ($a <= 0) {
+ $rem = $c->_zero(); # a = 0 => x == y => rem 0
+ $rem = $c->_copy($x) if $a != 0; # a < 0 => x < y => rem = x
+ @$x = 0;
+ $x->[0] = 0; # if $a < 0
+ $x->[0] = 1 if $a == 0; # $x == $y
+ return ($x, $rem) if wantarray; # including remainder?
+ return $x;
+ }
+ # $x >= $y, so proceed normally
+
+ }
+ }
+
+ # all other cases:
+
+ my $y = $c->_copy($yorg); # always make copy to preserve
+
+ my ($car, $bar, $prd, $dd, $xi, $yi, @q, $v2, $v1, @d, $tmp, $q, $u2, $u1, $u0);
+
+ $car = $bar = $prd = 0;
+ if (($dd = int($BASE / ($y->[-1] + 1))) != 1) {
+ for $xi (@$x) {
+ $xi = $xi * $dd + $car;
+ $xi -= ($car = int($xi / $BASE)) * $BASE;
+ }
+ push(@$x, $car);
+ $car = 0;
+ for $yi (@$y) {
+ $yi = $yi * $dd + $car;
+ $yi -= ($car = int($yi / $BASE)) * $BASE;
+ }
+ } else {
+ push(@$x, 0);
+ }
+
+ # @q will accumulate the final result, $q contains the current computed
+ # part of the final result
+
+ @q = ();
+ ($v2, $v1) = @$y[-2, -1];
+ $v2 = 0 unless $v2;
+ while ($#$x > $#$y) {
+ ($u2, $u1, $u0) = @$x[-3..-1];
+ $u2 = 0 unless $u2;
+ #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n"
+ # if $v1 == 0;
+ $q = (($u0 == $v1) ? $MAX_VAL : int(($u0 * $BASE + $u1) / $v1));
+ --$q while ($v2 * $q > ($u0 * $BASE + $u1 - $q * $v1) * $BASE + $u2);
+ if ($q) {
+ ($car, $bar) = (0, 0);
+ for ($yi = 0, $xi = $#$x - $#$y - 1; $yi <= $#$y; ++$yi, ++$xi) {
+ $prd = $q * $y->[$yi] + $car;
+ $prd -= ($car = int($prd / $BASE)) * $BASE;
+ $x->[$xi] += $BASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0));
+ }
+ if ($x->[-1] < $car + $bar) {
+ $car = 0;
+ --$q;
+ for ($yi = 0, $xi = $#$x - $#$y - 1; $yi <= $#$y; ++$yi, ++$xi) {
+ $x->[$xi] -= $BASE
+ if ($car = (($x->[$xi] += $y->[$yi] + $car) >= $BASE));
+ }
+ }
+ }
+ pop(@$x);
+ unshift(@q, $q);
+ }
+ if (wantarray) {
+ my $d = bless [], $c;
+ if ($dd != 1) {
+ $car = 0;
+ for $xi (reverse @$x) {
+ $prd = $car * $BASE + $xi;
+ $car = $prd - ($tmp = int($prd / $dd)) * $dd;
+ unshift(@$d, $tmp);
+ }
+ } else {
+ @$d = @$x;
+ }
+ @$x = @q;
+ __strip_zeros($x);
+ __strip_zeros($d);
+ return ($x, $d);
+ }
+ @$x = @q;
+ __strip_zeros($x);
+ $x;
+}
+
+##############################################################################
+# testing
+
+sub _acmp {
+ # Internal absolute post-normalized compare (ignore signs)
+ # ref to array, ref to array, return <0, 0, >0
+ # Arrays must have at least one entry; this is not checked for.
+ my ($c, $cx, $cy) = @_;
+
+ # shortcut for short numbers
+ return (($cx->[0] <=> $cy->[0]) <=> 0)
+ if @$cx == 1 && @$cy == 1;
+
+ # fast comp based on number of array elements (aka pseudo-length)
+ my $lxy = (@$cx - @$cy)
+ # or length of first element if same number of elements (aka difference 0)
+ ||
+ # need int() here because sometimes the last element is '00018' vs '18'
+ (length(int($cx->[-1])) - length(int($cy->[-1])));
+
+ return -1 if $lxy < 0; # already differs, ret
+ return 1 if $lxy > 0; # ditto
+
+ # manual way (abort if unequal, good for early ne)
+ my $a;
+ my $j = @$cx;
+ while (--$j >= 0) {
+ last if $a = $cx->[$j] - $cy->[$j];
+ }
+ $a <=> 0;
+}
+
+sub _len {
+ # compute number of digits in base 10
+
+ # int() because add/sub sometimes leaves strings (like '00005') instead of
+ # '5' in this place, thus causing length() to report wrong length
+ my $cx = $_[1];
+
+ (@$cx - 1) * $BASE_LEN + length(int($cx->[-1]));
+}
+
+sub _digit {
+ # Return the nth digit. Zero is rightmost, so _digit(123, 0) gives 3.
+ # Negative values count from the left, so _digit(123, -1) gives 1.
+ my ($c, $x, $n) = @_;
+
+ my $len = _len('', $x);
+
+ $n += $len if $n < 0; # -1 last, -2 second-to-last
+
+ # Math::BigInt::Calc returns 0 if N is out of range, but this is not done
+ # by the other backend libraries.
+
+ return "0" if $n < 0 || $n >= $len; # return 0 for digits out of range
+
+ my $elem = int($n / $BASE_LEN); # index of array element
+ my $digit = $n % $BASE_LEN; # index of digit within the element
+ substr("0" x $BASE_LEN . "$x->[$elem]", -1 - $digit, 1);
+}
+
+sub _zeros {
+ # Return number of trailing zeros in decimal.
+ # Check each array element for having 0 at end as long as elem == 0
+ # Upon finding a elem != 0, stop.
+
+ my $x = $_[1];
+
+ return 0 if @$x == 1 && $x->[0] == 0;
+
+ my $zeros = 0;
+ foreach my $elem (@$x) {
+ if ($elem != 0) {
+ $elem =~ /[^0](0*)\z/;
+ $zeros += length($1); # count trailing zeros
+ last; # early out
+ }
+ $zeros += $BASE_LEN;
+ }
+ $zeros;
+}
+
+##############################################################################
+# _is_* routines
+
+sub _is_zero {
+ # return true if arg is zero
+ @{$_[1]} == 1 && $_[1]->[0] == 0 ? 1 : 0;
+}
+
+sub _is_even {
+ # return true if arg is even
+ $_[1]->[0] & 1 ? 0 : 1;
+}
+
+sub _is_odd {
+ # return true if arg is odd
+ $_[1]->[0] & 1 ? 1 : 0;
+}
+
+sub _is_one {
+ # return true if arg is one
+ @{$_[1]} == 1 && $_[1]->[0] == 1 ? 1 : 0;
+}
+
+sub _is_two {
+ # return true if arg is two
+ @{$_[1]} == 1 && $_[1]->[0] == 2 ? 1 : 0;
+}
+
+sub _is_ten {
+ # return true if arg is ten
+ @{$_[1]} == 1 && $_[1]->[0] == 10 ? 1 : 0;
+}
+
+sub __strip_zeros {
+ # Internal normalization function that strips leading zeros from the array.
+ # Args: ref to array
+ my $x = shift;
+
+ push @$x, 0 if @$x == 0; # div might return empty results, so fix it
+ return $x if @$x == 1; # early out
+
+ #print "strip: cnt $cnt i $i\n";
+ # '0', '3', '4', '0', '0',
+ # 0 1 2 3 4
+ # cnt = 5, i = 4
+ # i = 4
+ # i = 3
+ # => fcnt = cnt - i (5-2 => 3, cnt => 5-1 = 4, throw away from 4th pos)
+ # >= 1: skip first part (this can be zero)
+
+ my $i = $#$x;
+ while ($i > 0) {
+ last if $x->[$i] != 0;
+ $i--;
+ }
+ $i++;
+ splice(@$x, $i) if $i < @$x;
+ $x;
+}
+
+###############################################################################
+# check routine to test internal state for corruptions
+
+sub _check {
+ # used by the test suite
+ my ($class, $x) = @_;
+
+ my $msg = $class -> SUPER::_check($x);
+ return $msg if $msg;
+
+ my $n;
+ eval { $n = @$x };
+ return "Not an array reference" unless $@ eq '';
+
+ return "Reference to an empty array" unless $n > 0;
+
+ # The following fails with Math::BigInt::FastCalc because a
+ # Math::BigInt::FastCalc "object" is an unblessed array ref.
+ #
+ #return 0 unless ref($x) eq $class;
+
+ for (my $i = 0 ; $i <= $#$x ; ++ $i) {
+ my $e = $x -> [$i];
+
+ return "Element at index $i is undefined"
+ unless defined $e;
+
+ return "Element at index $i is a '" . ref($e) .
+ "', which is not a scalar"
+ unless ref($e) eq "";
+
+ # It would be better to use the regex /^([1-9]\d*|0)\z/, but that fails
+ # in Math::BigInt::FastCalc, because it sometimes creates array
+ # elements like "000000".
+ return "Element at index $i is '$e', which does not look like an" .
+ " normal integer" unless $e =~ /^\d+\z/;
+
+ return "Element at index $i is '$e', which is not smaller than" .
+ " the base '$BASE'" if $e >= $BASE;
+
+ return "Element at index $i (last element) is zero"
+ if $#$x > 0 && $i == $#$x && $e == 0;
+ }
+
+ return 0;
+}
+
+###############################################################################
+
+sub _mod {
+ # if possible, use mod shortcut
+ my ($c, $x, $yo) = @_;
+
+ # slow way since $y too big
+ if (@$yo > 1) {
+ my ($xo, $rem) = $c->_div($x, $yo);
+ @$x = @$rem;
+ return $x;
+ }
+
+ my $y = $yo->[0];
+
+ # if both are single element arrays
+ if (@$x == 1) {
+ $x->[0] %= $y;
+ return $x;
+ }
+
+ # if @$x has more than one element, but @$y is a single element
+ my $b = $BASE % $y;
+ if ($b == 0) {
+ # when BASE % Y == 0 then (B * BASE) % Y == 0
+ # (B * BASE) % $y + A % Y => A % Y
+ # so need to consider only last element: O(1)
+ $x->[0] %= $y;
+ } elsif ($b == 1) {
+ # else need to go through all elements in @$x: O(N), but loop is a bit
+ # simplified
+ my $r = 0;
+ foreach (@$x) {
+ $r = ($r + $_) % $y; # not much faster, but heh...
+ #$r += $_ % $y; $r %= $y;
+ }
+ $r = 0 if $r == $y;
+ $x->[0] = $r;
+ } else {
+ # else need to go through all elements in @$x: O(N)
+ my $r = 0;
+ my $bm = 1;
+ foreach (@$x) {
+ $r = ($_ * $bm + $r) % $y;
+ $bm = ($bm * $b) % $y;
+
+ #$r += ($_ % $y) * $bm;
+ #$bm *= $b;
+ #$bm %= $y;
+ #$r %= $y;
+ }
+ $r = 0 if $r == $y;
+ $x->[0] = $r;
+ }
+ @$x = $x->[0]; # keep one element of @$x
+ return $x;
+}
+
+##############################################################################
+# shifts
+
+sub _rsft {
+ my ($c, $x, $y, $n) = @_;
+
+ if ($n != 10) {
+ $n = $c->_new($n);
+ return scalar $c->_div($x, $c->_pow($n, $y));
+ }
+
+ # shortcut (faster) for shifting by 10)
+ # multiples of $BASE_LEN
+ my $dst = 0; # destination
+ my $src = $c->_num($y); # as normal int
+ my $xlen = (@$x - 1) * $BASE_LEN + length(int($x->[-1]));
+ if ($src >= $xlen or ($src == $xlen and !defined $x->[1])) {
+ # 12345 67890 shifted right by more than 10 digits => 0
+ splice(@$x, 1); # leave only one element
+ $x->[0] = 0; # set to zero
+ return $x;
+ }
+ my $rem = $src % $BASE_LEN; # remainder to shift
+ $src = int($src / $BASE_LEN); # source
+ if ($rem == 0) {
+ splice(@$x, 0, $src); # even faster, 38.4 => 39.3
+ } else {
+ my $len = @$x - $src; # elems to go
+ my $vd;
+ my $z = '0' x $BASE_LEN;
+ $x->[ @$x ] = 0; # avoid || 0 test inside loop
+ while ($dst < $len) {
+ $vd = $z . $x->[$src];
+ $vd = substr($vd, -$BASE_LEN, $BASE_LEN - $rem);
+ $src++;
+ $vd = substr($z . $x->[$src], -$rem, $rem) . $vd;
+ $vd = substr($vd, -$BASE_LEN, $BASE_LEN) if length($vd) > $BASE_LEN;
+ $x->[$dst] = int($vd);
+ $dst++;
+ }
+ splice(@$x, $dst) if $dst > 0; # kill left-over array elems
+ pop @$x if $x->[-1] == 0 && @$x > 1; # kill last element if 0
+ } # else rem == 0
+ $x;
+}
+
+sub _lsft {
+ my ($c, $x, $n, $b) = @_;
+
+ return $x if $c->_is_zero($x);
+
+ # Handle the special case when the base is a power of 10. Don't check
+ # whether log($b)/log(10) is an integer, because log(1000)/log(10) is not
+ # exactly 3.
+
+ my $log10 = sprintf "%.0f", log($b) / log(10);
+ if ($b == 10 ** $log10) {
+ $b = 10;
+ $n = $c->_mul($n, $c->_new($log10));
+
+ # shortcut (faster) for shifting by 10) since we are in base 10eX
+ # multiples of $BASE_LEN:
+ my $src = @$x; # source
+ my $len = $c->_num($n); # shift-len as normal int
+ my $rem = $len % $BASE_LEN; # remainder to shift
+ my $dst = $src + int($len / $BASE_LEN); # destination
+ my $vd; # further speedup
+ $x->[$src] = 0; # avoid first ||0 for speed
+ my $z = '0' x $BASE_LEN;
+ while ($src >= 0) {
+ $vd = $x->[$src];
+ $vd = $z . $vd;
+ $vd = substr($vd, -$BASE_LEN + $rem, $BASE_LEN - $rem);
+ $vd .= $src > 0 ? substr($z . $x->[$src - 1], -$BASE_LEN, $rem)
+ : '0' x $rem;
+ $vd = substr($vd, -$BASE_LEN, $BASE_LEN) if length($vd) > $BASE_LEN;
+ $x->[$dst] = int($vd);
+ $dst--;
+ $src--;
+ }
+ # set lowest parts to 0
+ while ($dst >= 0) {
+ $x->[$dst--] = 0;
+ }
+ # fix spurious last zero element
+ splice @$x, -1 if $x->[-1] == 0;
+ return $x;
+ } else {
+ $b = $c->_new($b);
+ #print $c->_str($b);
+ return $c->_mul($x, $c->_pow($b, $n));
+ }
+}
+
+sub _pow {
+ # power of $x to $y
+ # ref to array, ref to array, return ref to array
+ my ($c, $cx, $cy) = @_;
+
+ if (@$cy == 1 && $cy->[0] == 0) {
+ splice(@$cx, 1);
+ $cx->[0] = 1; # y == 0 => x => 1
+ return $cx;
+ }
+
+ if ((@$cx == 1 && $cx->[0] == 1) || # x == 1
+ (@$cy == 1 && $cy->[0] == 1)) # or y == 1
+ {
+ return $cx;
+ }
+
+ if (@$cx == 1 && $cx->[0] == 0) {
+ splice (@$cx, 1);
+ $cx->[0] = 0; # 0 ** y => 0 (if not y <= 0)
+ return $cx;
+ }
+
+ my $pow2 = $c->_one();
+
+ my $y_bin = $c->_as_bin($cy);
+ $y_bin =~ s/^0b//;
+ my $len = length($y_bin);
+ while (--$len > 0) {
+ $c->_mul($pow2, $cx) if substr($y_bin, $len, 1) eq '1'; # is odd?
+ $c->_mul($cx, $cx);
+ }
+
+ $c->_mul($cx, $pow2);
+ $cx;
+}
+
+sub _nok {
+ # Return binomial coefficient (n over k).
+ # Given refs to arrays, return ref to array.
+ # First input argument is modified.
+
+ my ($c, $n, $k) = @_;
+
+ # If k > n/2, or, equivalently, 2*k > n, compute nok(n, k) as
+ # nok(n, n-k), to minimize the number if iterations in the loop.
+
+ {
+ my $twok = $c->_mul($c->_two(), $c->_copy($k)); # 2 * k
+ if ($c->_acmp($twok, $n) > 0) { # if 2*k > n
+ $k = $c->_sub($c->_copy($n), $k); # k = n - k
+ }
+ }
+
+ # Example:
+ #
+ # / 7 \ 7! 1*2*3*4 * 5*6*7 5 * 6 * 7 6 7
+ # | | = --------- = --------------- = --------- = 5 * - * -
+ # \ 3 / (7-3)! 3! 1*2*3*4 * 1*2*3 1 * 2 * 3 2 3
+
+ if ($c->_is_zero($k)) {
+ @$n = 1;
+ } else {
+
+ # Make a copy of the original n, since we'll be modifying n in-place.
+
+ my $n_orig = $c->_copy($n);
+
+ # n = 5, f = 6, d = 2 (cf. example above)
+
+ $c->_sub($n, $k);
+ $c->_inc($n);
+
+ my $f = $c->_copy($n);
+ $c->_inc($f);
+
+ my $d = $c->_two();
+
+ # while f <= n (the original n, that is) ...
+
+ while ($c->_acmp($f, $n_orig) <= 0) {
+
+ # n = (n * f / d) == 5 * 6 / 2 (cf. example above)
+
+ $c->_mul($n, $f);
+ $c->_div($n, $d);
+
+ # f = 7, d = 3 (cf. example above)
+
+ $c->_inc($f);
+ $c->_inc($d);
+ }
+
+ }
+
+ return $n;
+}
+
+my @factorials = (
+ 1,
+ 1,
+ 2,
+ 2*3,
+ 2*3*4,
+ 2*3*4*5,
+ 2*3*4*5*6,
+ 2*3*4*5*6*7,
+ );
+
+sub _fac {
+ # factorial of $x
+ # ref to array, return ref to array
+ my ($c, $cx) = @_;
+
+ if ((@$cx == 1) && ($cx->[0] <= 7)) {
+ $cx->[0] = $factorials[$cx->[0]]; # 0 => 1, 1 => 1, 2 => 2 etc.
+ return $cx;
+ }
+
+ if ((@$cx == 1) && # we do this only if $x >= 12 and $x <= 7000
+ ($cx->[0] >= 12 && $cx->[0] < 7000)) {
+
+ # Calculate (k-j) * (k-j+1) ... k .. (k+j-1) * (k + j)
+ # See http://blogten.blogspot.com/2007/01/calculating-n.html
+ # The above series can be expressed as factors:
+ # k * k - (j - i) * 2
+ # We cache k*k, and calculate (j * j) as the sum of the first j odd integers
+
+ # This will not work when N exceeds the storage of a Perl scalar, however,
+ # in this case the algorithm would be way too slow to terminate, anyway.
+
+ # As soon as the last element of $cx is 0, we split it up and remember
+ # how many zeors we got so far. The reason is that n! will accumulate
+ # zeros at the end rather fast.
+ my $zero_elements = 0;
+
+ # If n is even, set n = n -1
+ my $k = $c->_num($cx);
+ my $even = 1;
+ if (($k & 1) == 0) {
+ $even = $k;
+ $k --;
+ }
+ # set k to the center point
+ $k = ($k + 1) / 2;
+ # print "k $k even: $even\n";
+ # now calculate k * k
+ my $k2 = $k * $k;
+ my $odd = 1;
+ my $sum = 1;
+ my $i = $k - 1;
+ # keep reference to x
+ my $new_x = $c->_new($k * $even);
+ @$cx = @$new_x;
+ if ($cx->[0] == 0) {
+ $zero_elements ++;
+ shift @$cx;
+ }
+ # print STDERR "x = ", $c->_str($cx), "\n";
+ my $BASE2 = int(sqrt($BASE))-1;
+ my $j = 1;
+ while ($j <= $i) {
+ my $m = ($k2 - $sum);
+ $odd += 2;
+ $sum += $odd;
+ $j++;
+ while ($j <= $i && ($m < $BASE2) && (($k2 - $sum) < $BASE2)) {
+ $m *= ($k2 - $sum);
+ $odd += 2;
+ $sum += $odd;
+ $j++;
+ # print STDERR "\n k2 $k2 m $m sum $sum odd $odd\n"; sleep(1);
+ }
+ if ($m < $BASE) {
+ $c->_mul($cx, [$m]);
+ } else {
+ $c->_mul($cx, $c->_new($m));
+ }
+ if ($cx->[0] == 0) {
+ $zero_elements ++;
+ shift @$cx;
+ }
+ # print STDERR "Calculate $k2 - $sum = $m (x = ", $c->_str($cx), ")\n";
+ }
+ # multiply in the zeros again
+ unshift @$cx, (0) x $zero_elements;
+ return $cx;
+ }
+
+ # go forward until $base is exceeded limit is either $x steps (steps == 100
+ # means a result always too high) or $base.
+ my $steps = 100;
+ $steps = $cx->[0] if @$cx == 1;
+ my $r = 2;
+ my $cf = 3;
+ my $step = 2;
+ my $last = $r;
+ while ($r * $cf < $BASE && $step < $steps) {
+ $last = $r;
+ $r *= $cf++;
+ $step++;
+ }
+ if ((@$cx == 1) && $step == $cx->[0]) {
+ # completely done, so keep reference to $x and return
+ $cx->[0] = $r;
+ return $cx;
+ }
+
+ # now we must do the left over steps
+ my $n; # steps still to do
+ if (@$cx == 1) {
+ $n = $cx->[0];
+ } else {
+ $n = $c->_copy($cx);
+ }
+
+ # Set $cx to the last result below $BASE (but keep ref to $x)
+ $cx->[0] = $last;
+ splice (@$cx, 1);
+ # As soon as the last element of $cx is 0, we split it up and remember
+ # how many zeors we got so far. The reason is that n! will accumulate
+ # zeros at the end rather fast.
+ my $zero_elements = 0;
+
+ # do left-over steps fit into a scalar?
+ if (ref $n eq 'ARRAY') {
+ # No, so use slower inc() & cmp()
+ # ($n is at least $BASE here)
+ my $base_2 = int(sqrt($BASE)) - 1;
+ #print STDERR "base_2: $base_2\n";
+ while ($step < $base_2) {
+ if ($cx->[0] == 0) {
+ $zero_elements ++;
+ shift @$cx;
+ }
+ my $b = $step * ($step + 1);
+ $step += 2;
+ $c->_mul($cx, [$b]);
+ }
+ $step = [$step];
+ while ($c->_acmp($step, $n) <= 0) {
+ if ($cx->[0] == 0) {
+ $zero_elements ++;
+ shift @$cx;
+ }
+ $c->_mul($cx, $step);
+ $c->_inc($step);
+ }
+ } else {
+ # Yes, so we can speed it up slightly
+
+ # print "# left over steps $n\n";
+
+ my $base_4 = int(sqrt(sqrt($BASE))) - 2;
+ #print STDERR "base_4: $base_4\n";
+ my $n4 = $n - 4;
+ while ($step < $n4 && $step < $base_4) {
+ if ($cx->[0] == 0) {
+ $zero_elements ++;
+ shift @$cx;
+ }
+ my $b = $step * ($step + 1);
+ $step += 2;
+ $b *= $step * ($step + 1);
+ $step += 2;
+ $c->_mul($cx, [$b]);
+ }
+ my $base_2 = int(sqrt($BASE)) - 1;
+ my $n2 = $n - 2;
+ #print STDERR "base_2: $base_2\n";
+ while ($step < $n2 && $step < $base_2) {
+ if ($cx->[0] == 0) {
+ $zero_elements ++;
+ shift @$cx;
+ }
+ my $b = $step * ($step + 1);
+ $step += 2;
+ $c->_mul($cx, [$b]);
+ }
+ # do what's left over
+ while ($step <= $n) {
+ $c->_mul($cx, [$step]);
+ $step++;
+ if ($cx->[0] == 0) {
+ $zero_elements ++;
+ shift @$cx;
+ }
+ }
+ }
+ # multiply in the zeros again
+ unshift @$cx, (0) x $zero_elements;
+ $cx; # return result
+}
+
+sub _log_int {
+ # calculate integer log of $x to base $base
+ # ref to array, ref to array - return ref to array
+ my ($c, $x, $base) = @_;
+
+ # X == 0 => NaN
+ return if @$x == 1 && $x->[0] == 0;
+
+ # BASE 0 or 1 => NaN
+ return if @$base == 1 && $base->[0] < 2;
+
+ # X == 1 => 0 (is exact)
+ if (@$x == 1 && $x->[0] == 1) {
+ @$x = 0;
+ return $x, 1;
+ }
+
+ my $cmp = $c->_acmp($x, $base);
+
+ # X == BASE => 1 (is exact)
+ if ($cmp == 0) {
+ @$x = 1;
+ return $x, 1;
+ }
+
+ # 1 < X < BASE => 0 (is truncated)
+ if ($cmp < 0) {
+ @$x = 0;
+ return $x, 0;
+ }
+
+ my $x_org = $c->_copy($x); # preserve x
+
+ # Compute a guess for the result based on:
+ # $guess = int ( length_in_base_10(X) / ( log(base) / log(10) ) )
+ my $len = $c->_len($x_org);
+ my $log = log($base->[-1]) / log(10);
+
+ # for each additional element in $base, we add $BASE_LEN to the result,
+ # based on the observation that log($BASE, 10) is BASE_LEN and
+ # log(x*y) == log(x) + log(y):
+ $log += (@$base - 1) * $BASE_LEN;
+
+ # calculate now a guess based on the values obtained above:
+ my $res = int($len / $log);
+
+ @$x = $res;
+ my $trial = $c->_pow($c->_copy($base), $x);
+ my $acmp = $c->_acmp($trial, $x_org);
+
+ # Did we get the exact result?
+
+ return $x, 1 if $acmp == 0;
+
+ # Too small?
+
+ while ($acmp < 0) {
+ $c->_mul($trial, $base);
+ $c->_inc($x);
+ $acmp = $c->_acmp($trial, $x_org);
+ }
+
+ # Too big?
+
+ while ($acmp > 0) {
+ $c->_div($trial, $base);
+ $c->_dec($x);
+ $acmp = $c->_acmp($trial, $x_org);
+ }
+
+ return $x, 1 if $acmp == 0; # result is exact
+ return $x, 0; # result is too small
+}
+
+# for debugging:
+use constant DEBUG => 0;
+my $steps = 0;
+sub steps { $steps };
+
+sub _sqrt {
+ # square-root of $x in place
+ # Compute a guess of the result (by rule of thumb), then improve it via
+ # Newton's method.
+ my ($c, $x) = @_;
+
+ if (@$x == 1) {
+ # fits into one Perl scalar, so result can be computed directly
+ $x->[0] = int(sqrt($x->[0]));
+ return $x;
+ }
+ my $y = $c->_copy($x);
+ # hopefully _len/2 is < $BASE, the -1 is to always undershot the guess
+ # since our guess will "grow"
+ my $l = int(($c->_len($x)-1) / 2);
+
+ my $lastelem = $x->[-1]; # for guess
+ my $elems = @$x - 1;
+ # not enough digits, but could have more?
+ if ((length($lastelem) <= 3) && ($elems > 1)) {
+ # right-align with zero pad
+ my $len = length($lastelem) & 1;
+ print "$lastelem => " if DEBUG;
+ $lastelem .= substr($x->[-2] . '0' x $BASE_LEN, 0, $BASE_LEN);
+ # former odd => make odd again, or former even to even again
+ $lastelem = $lastelem / 10 if (length($lastelem) & 1) != $len;
+ print "$lastelem\n" if DEBUG;
+ }
+
+ # construct $x (instead of $c->_lsft($x, $l, 10)
+ my $r = $l % $BASE_LEN; # 10000 00000 00000 00000 ($BASE_LEN=5)
+ $l = int($l / $BASE_LEN);
+ print "l = $l " if DEBUG;
+
+ splice @$x, $l; # keep ref($x), but modify it
+
+ # we make the first part of the guess not '1000...0' but int(sqrt($lastelem))
+ # that gives us:
+ # 14400 00000 => sqrt(14400) => guess first digits to be 120
+ # 144000 000000 => sqrt(144000) => guess 379
+
+ print "$lastelem (elems $elems) => " if DEBUG;
+ $lastelem = $lastelem / 10 if ($elems & 1 == 1); # odd or even?
+ my $g = sqrt($lastelem);
+ $g =~ s/\.//; # 2.345 => 2345
+ $r -= 1 if $elems & 1 == 0; # 70 => 7
+
+ # padd with zeros if result is too short
+ $x->[$l--] = int(substr($g . '0' x $r, 0, $r+1));
+ print "now ", $x->[-1] if DEBUG;
+ print " would have been ", int('1' . '0' x $r), "\n" if DEBUG;
+
+ # If @$x > 1, we could compute the second elem of the guess, too, to create
+ # an even better guess. Not implemented yet. Does it improve performance?
+ $x->[$l--] = 0 while ($l >= 0); # all other digits of guess are zero
+
+ print "start x= ", $c->_str($x), "\n" if DEBUG;
+ my $two = $c->_two();
+ my $last = $c->_zero();
+ my $lastlast = $c->_zero();
+ $steps = 0 if DEBUG;
+ while ($c->_acmp($last, $x) != 0 && $c->_acmp($lastlast, $x) != 0) {
+ $steps++ if DEBUG;
+ $lastlast = $c->_copy($last);
+ $last = $c->_copy($x);
+ $c->_add($x, $c->_div($c->_copy($y), $x));
+ $c->_div($x, $two );
+ print " x= ", $c->_str($x), "\n" if DEBUG;
+ }
+ print "\nsteps in sqrt: $steps, " if DEBUG;
+ $c->_dec($x) if $c->_acmp($y, $c->_mul($c->_copy($x), $x)) < 0; # overshot?
+ print " final ", $x->[-1], "\n" if DEBUG;
+ $x;
+}
+
+sub _root {
+ # Take n'th root of $x in place.
+
+ my ($c, $x, $n) = @_;
+
+ # Small numbers.
+
+ if (@$x == 1 && @$n == 1) {
+ # Result can be computed directly. Adjust initial result for numerical
+ # errors, e.g., int(1000**(1/3)) is 2, not 3.
+ my $y = int($x->[0] ** (1 / $n->[0]));
+ my $yp1 = $y + 1;
+ $y = $yp1 if $yp1 ** $n->[0] == $x->[0];
+ $x->[0] = $y;
+ return $x;
+ }
+
+ # If x <= n, the result is always (truncated to) 1.
+
+ if ((@$x > 1 || $x -> [0] > 0) && # if x is non-zero ...
+ $c -> _acmp($x, $n) <= 0) # ... and x <= n
+ {
+ my $one = $x -> _one();
+ @$x = @$one;
+ return $x;
+ }
+
+ # If $n is a power of two, take sqrt($x) repeatedly, e.g., root($x, 4) =
+ # sqrt(sqrt($x)), root($x, 8) = sqrt(sqrt(sqrt($x))).
+
+ my $b = $c -> _as_bin($n);
+ if ($b =~ /0b1(0+)$/) {
+ my $count = length($1); # 0b100 => len('00') => 2
+ my $cnt = $count; # counter for loop
+ unshift @$x, 0; # add one element, together with one
+ # more below in the loop this makes 2
+ while ($cnt-- > 0) {
+ # 'Inflate' $x by adding one element, basically computing
+ # $x * $BASE * $BASE. This gives us more $BASE_LEN digits for
+ # result since len(sqrt($X)) approx == len($x) / 2.
+ unshift @$x, 0;
+ # Calculate sqrt($x), $x is now one element to big, again. In the
+ # next round we make that two, again.
+ $c -> _sqrt($x);
+ }
+
+ # $x is now one element too big, so truncate result by removing it.
+ shift @$x;
+
+ return $x;
+ }
+
+ my $DEBUG = 0;
+
+ # Now the general case. This works by finding an initial guess. If this
+ # guess is incorrect, a relatively small delta is chosen. This delta is
+ # used to find a lower and upper limit for the correct value. The delta is
+ # doubled in each iteration. When a lower and upper limit is found,
+ # bisection is applied to narrow down the region until we have the correct
+ # value.
+
+ # Split x into mantissa and exponent in base 10, so that
+ #
+ # x = xm * 10^xe, where 0 < xm < 1 and xe is an integer
+
+ my $x_str = $c -> _str($x);
+ my $xm = "." . $x_str;
+ my $xe = length($x_str);
+
+ # From this we compute the base 10 logarithm of x
+ #
+ # log_10(x) = log_10(xm) + log_10(xe^10)
+ # = log(xm)/log(10) + xe
+ #
+ # and then the base 10 logarithm of y, where y = x^(1/n)
+ #
+ # log_10(y) = log_10(x)/n
+
+ my $log10x = log($xm) / log(10) + $xe;
+ my $log10y = $log10x / $c -> _num($n);
+
+ # And from this we compute ym and ye, the mantissa and exponent (in
+ # base 10) of y, where 1 < ym <= 10 and ye is an integer.
+
+ my $ye = int $log10y;
+ my $ym = 10 ** ($log10y - $ye);
+
+ # Finally, we scale the mantissa and exponent to incraese the integer
+ # part of ym, before building the string representing our guess of y.
+
+ if ($DEBUG) {
+ print "\n";
+ print "xm = $xm\n";
+ print "xe = $xe\n";
+ print "log10x = $log10x\n";
+ print "log10y = $log10y\n";
+ print "ym = $ym\n";
+ print "ye = $ye\n";
+ print "\n";
+ }
+
+ my $d = $ye < 15 ? $ye : 15;
+ $ym *= 10 ** $d;
+ $ye -= $d;
+
+ my $y_str = sprintf('%.0f', $ym) . "0" x $ye;
+ my $y = $c -> _new($y_str);
+
+ if ($DEBUG) {
+ print "ym = $ym\n";
+ print "ye = $ye\n";
+ print "\n";
+ print "y_str = $y_str (initial guess)\n";
+ print "\n";
+ }
+
+ # See if our guess y is correct.
+
+ my $trial = $c -> _pow($c -> _copy($y), $n);
+ my $acmp = $c -> _acmp($trial, $x);
+
+ if ($acmp == 0) {
+ @$x = @$y;
+ return $x;
+ }
+
+ # Find a lower and upper limit for the correct value of y. Start off with a
+ # delta value that is approximately the size of the accuracy of the guess.
+
+ my $lower;
+ my $upper;
+
+ my $delta = $c -> _new("1" . ("0" x $ye));
+ my $two = $c -> _two();
+
+ if ($acmp < 0) {
+ $lower = $y;
+ while ($acmp < 0) {
+ $upper = $c -> _add($c -> _copy($lower), $delta);
+
+ if ($DEBUG) {
+ print "lower = $lower\n";
+ print "upper = $upper\n";
+ print "delta = $delta\n";
+ print "\n";
+ }
+ $acmp = $c -> _acmp($c -> _pow($c -> _copy($upper), $n), $x);
+ if ($acmp == 0) {
+ @$x = @$upper;
+ return $x;
+ }
+ $delta = $c -> _mul($delta, $two);
+ }
+ }
+
+ elsif ($acmp > 0) {
+ $upper = $y;
+ my $zero = $c -> _zero();
+ while ($acmp > 0) {
+ if ($c -> _acmp($upper, $delta) <= 0) {
+ $lower = $c -> _zero();
+ last;
+ }
+ $lower = $c -> _sub($c -> _copy($upper), $delta);
+
+ if ($DEBUG) {
+ print "lower = $lower\n";
+ print "upper = $upper\n";
+ print "delta = $delta\n";
+ print "\n";
+ }
+ $acmp = $c -> _acmp($c -> _pow($c -> _copy($lower), $n), $x);
+ if ($acmp == 0) {
+ @$x = @$lower;
+ return $x;
+ }
+ $delta = $c -> _mul($delta, $two);
+ }
+ }
+
+ # Use bisection to narrow down the interval.
+
+ my $one = $c -> _one();
+ {
+
+ $delta = $c -> _sub($c -> _copy($upper), $lower);
+ if ($c -> _acmp($delta, $one) <= 0) {
+ @$x = @$lower;
+ return $x;
+ }
+
+ if ($DEBUG) {
+ print "lower = $lower\n";
+ print "upper = $upper\n";
+ print "delta = $delta\n";
+ print "\n";
+ }
+
+ $delta = $c -> _div($delta, $two);
+ my $middle = $c -> _add($c -> _copy($lower), $delta);
+
+ $acmp = $c -> _acmp($c -> _pow($c -> _copy($middle), $n), $x);
+ if ($acmp < 0) {
+ $lower = $middle;
+ } elsif ($acmp > 0) {
+ $upper = $middle;
+ } else {
+ @$x = @$middle;
+ return $x;
+ }
+
+ redo;
+ }
+
+ $x;
+}
+
+##############################################################################
+# binary stuff
+
+sub _and {
+ my ($c, $x, $y) = @_;
+
+ # the shortcut makes equal, large numbers _really_ fast, and makes only a
+ # very small performance drop for small numbers (e.g. something with less
+ # than 32 bit) Since we optimize for large numbers, this is enabled.
+ return $x if $c->_acmp($x, $y) == 0; # shortcut
+
+ my $m = $c->_one();
+ my ($xr, $yr);
+ my $mask = $AND_MASK;
+
+ my $x1 = $c->_copy($x);
+ my $y1 = $c->_copy($y);
+ my $z = $c->_zero();
+
+ use integer;
+ until ($c->_is_zero($x1) || $c->_is_zero($y1)) {
+ ($x1, $xr) = $c->_div($x1, $mask);
+ ($y1, $yr) = $c->_div($y1, $mask);
+
+ $c->_add($z, $c->_mul([ 0 + $xr->[0] & 0 + $yr->[0] ], $m));
+ $c->_mul($m, $mask);
+ }
+
+ @$x = @$z;
+ return $x;
+}
+
+sub _xor {
+ my ($c, $x, $y) = @_;
+
+ return $c->_zero() if $c->_acmp($x, $y) == 0; # shortcut (see -and)
+
+ my $m = $c->_one();
+ my ($xr, $yr);
+ my $mask = $XOR_MASK;
+
+ my $x1 = $c->_copy($x);
+ my $y1 = $c->_copy($y); # make copy
+ my $z = $c->_zero();
+
+ use integer;
+ until ($c->_is_zero($x1) || $c->_is_zero($y1)) {
+ ($x1, $xr) = $c->_div($x1, $mask);
+ ($y1, $yr) = $c->_div($y1, $mask);
+ # make ints() from $xr, $yr (see _and())
+ #$b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; }
+ #$b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; }
+ #$c->_add($x, $c->_mul($c->_new($xrr ^ $yrr)), $m) );
+
+ $c->_add($z, $c->_mul([ 0 + $xr->[0] ^ 0 + $yr->[0] ], $m));
+ $c->_mul($m, $mask);
+ }
+ # the loop stops when the shorter of the two numbers is exhausted
+ # the remainder of the longer one will survive bit-by-bit, so we simple
+ # multiply-add it in
+ $c->_add($z, $c->_mul($x1, $m) ) if !$c->_is_zero($x1);
+ $c->_add($z, $c->_mul($y1, $m) ) if !$c->_is_zero($y1);
+
+ @$x = @$z;
+ return $x;
+}
+
+sub _or {
+ my ($c, $x, $y) = @_;
+
+ return $x if $c->_acmp($x, $y) == 0; # shortcut (see _and)
+
+ my $m = $c->_one();
+ my ($xr, $yr);
+ my $mask = $OR_MASK;
+
+ my $x1 = $c->_copy($x);
+ my $y1 = $c->_copy($y); # make copy
+ my $z = $c->_zero();
+
+ use integer;
+ until ($c->_is_zero($x1) || $c->_is_zero($y1)) {
+ ($x1, $xr) = $c->_div($x1, $mask);
+ ($y1, $yr) = $c->_div($y1, $mask);
+ # make ints() from $xr, $yr (see _and())
+ # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; }
+ # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; }
+ # $c->_add($x, $c->_mul(_new( $c, ($xrr | $yrr) ), $m) );
+
+ $c->_add($z, $c->_mul([ 0 + $xr->[0] | 0 + $yr->[0] ], $m));
+ $c->_mul($m, $mask);
+ }
+ # the loop stops when the shorter of the two numbers is exhausted
+ # the remainder of the longer one will survive bit-by-bit, so we simple
+ # multiply-add it in
+ $c->_add($z, $c->_mul($x1, $m) ) if !$c->_is_zero($x1);
+ $c->_add($z, $c->_mul($y1, $m) ) if !$c->_is_zero($y1);
+
+ @$x = @$z;
+ return $x;
+}
+
+sub _as_hex {
+ # convert a decimal number to hex (ref to array, return ref to string)
+ my ($c, $x) = @_;
+
+ # fits into one element (handle also 0x0 case)
+ return sprintf("0x%x", $x->[0]) if @$x == 1;
+
+ my $x1 = $c->_copy($x);
+
+ my $es = '';
+ my ($xr, $h, $x10000);
+ if ($] >= 5.006) {
+ $x10000 = [ 0x10000 ];
+ $h = 'h4';
+ } else {
+ $x10000 = [ 0x1000 ];
+ $h = 'h3';
+ }
+ while (@$x1 != 1 || $x1->[0] != 0) # _is_zero()
+ {
+ ($x1, $xr) = $c->_div($x1, $x10000);
+ $es .= unpack($h, pack('V', $xr->[0]));
+ }
+ $es = reverse $es;
+ $es =~ s/^[0]+//; # strip leading zeros
+ '0x' . $es; # return result prepended with 0x
+}
+
+sub _as_bin {
+ # convert a decimal number to bin (ref to array, return ref to string)
+ my ($c, $x) = @_;
+
+ # fits into one element (and Perl recent enough), handle also 0b0 case
+ # handle zero case for older Perls
+ if ($] <= 5.005 && @$x == 1 && $x->[0] == 0) {
+ my $t = '0b0';
+ return $t;
+ }
+ if (@$x == 1 && $] >= 5.006) {
+ my $t = sprintf("0b%b", $x->[0]);
+ return $t;
+ }
+ my $x1 = $c->_copy($x);
+
+ my $es = '';
+ my ($xr, $b, $x10000);
+ if ($] >= 5.006) {
+ $x10000 = [ 0x10000 ];
+ $b = 'b16';
+ } else {
+ $x10000 = [ 0x1000 ];
+ $b = 'b12';
+ }
+ while (!(@$x1 == 1 && $x1->[0] == 0)) # _is_zero()
+ {
+ ($x1, $xr) = $c->_div($x1, $x10000);
+ $es .= unpack($b, pack('v', $xr->[0]));
+ }
+ $es = reverse $es;
+ $es =~ s/^[0]+//; # strip leading zeros
+ '0b' . $es; # return result prepended with 0b
+}
+
+sub _as_oct {
+ # convert a decimal number to octal (ref to array, return ref to string)
+ my ($c, $x) = @_;
+
+ # fits into one element (handle also 0 case)
+ return sprintf("0%o", $x->[0]) if @$x == 1;
+
+ my $x1 = $c->_copy($x);
+
+ my $es = '';
+ my $xr;
+ my $x1000 = [ 0100000 ];
+ while (@$x1 != 1 || $x1->[0] != 0) # _is_zero()
+ {
+ ($x1, $xr) = $c->_div($x1, $x1000);
+ $es .= reverse sprintf("%05o", $xr->[0]);
+ }
+ $es = reverse $es;
+ $es =~ s/^0+//; # strip leading zeros
+ '0' . $es; # return result prepended with 0
+}
+
+sub _from_oct {
+ # convert a octal number to decimal (string, return ref to array)
+ my ($c, $os) = @_;
+
+ # for older Perls, play safe
+ my $m = [ 0100000 ];
+ my $d = 5; # 5 digits at a time
+
+ my $mul = $c->_one();
+ my $x = $c->_zero();
+
+ my $len = int((length($os) - 1) / $d); # $d digit parts, w/o the '0'
+ my $val;
+ my $i = -$d;
+ while ($len >= 0) {
+ $val = substr($os, $i, $d); # get oct digits
+ $val = CORE::oct($val);
+ $i -= $d;
+ $len --;
+ my $adder = [ $val ];
+ $c->_add($x, $c->_mul($adder, $mul)) if $val != 0;
+ $c->_mul($mul, $m) if $len >= 0; # skip last mul
+ }
+ $x;
+}
+
+sub _from_hex {
+ # convert a hex number to decimal (string, return ref to array)
+ my ($c, $hs) = @_;
+
+ my $m = $c->_new(0x10000000); # 28 bit at a time (<32 bit!)
+ my $d = 7; # 7 digits at a time
+ my $mul = $c->_one();
+ my $x = $c->_zero();
+
+ my $len = int((length($hs) - 2) / $d); # $d digit parts, w/o the '0x'
+ my $val;
+ my $i = -$d;
+ while ($len >= 0) {
+ $val = substr($hs, $i, $d); # get hex digits
+ $val =~ s/^0x// if $len == 0; # for last part only because
+ $val = CORE::hex($val); # hex does not like wrong chars
+ $i -= $d;
+ $len --;
+ my $adder = [ $val ];
+ # if the resulting number was to big to fit into one element, create a
+ # two-element version (bug found by Mark Lakata - Thanx!)
+ if (CORE::length($val) > $BASE_LEN) {
+ $adder = $c->_new($val);
+ }
+ $c->_add($x, $c->_mul($adder, $mul)) if $val != 0;
+ $c->_mul($mul, $m) if $len >= 0; # skip last mul
+ }
+ $x;
+}
+
+sub _from_bin {
+ # convert a hex number to decimal (string, return ref to array)
+ my ($c, $bs) = @_;
+
+ # instead of converting X (8) bit at a time, it is faster to "convert" the
+ # number to hex, and then call _from_hex.
+
+ my $hs = $bs;
+ $hs =~ s/^[+-]?0b//; # remove sign and 0b
+ my $l = length($hs); # bits
+ $hs = '0' x (8 - ($l % 8)) . $hs if ($l % 8) != 0; # padd left side w/ 0
+ my $h = '0x' . unpack('H*', pack ('B*', $hs)); # repack as hex
+
+ $c->_from_hex($h);
+}
+
+##############################################################################
+# special modulus functions
+
+sub _modinv {
+ # modular multiplicative inverse
+ my ($c, $x, $y) = @_;
+
+ # modulo zero
+ if ($c->_is_zero($y)) {
+ return undef, undef;
+ }
+
+ # modulo one
+ if ($c->_is_one($y)) {
+ return $c->_zero(), '+';
+ }
+
+ my $u = $c->_zero();
+ my $v = $c->_one();
+ my $a = $c->_copy($y);
+ my $b = $c->_copy($x);
+
+ # Euclid's Algorithm for bgcd(), only that we calc bgcd() ($a) and the result
+ # ($u) at the same time. See comments in BigInt for why this works.
+ my $q;
+ my $sign = 1;
+ {
+ ($a, $q, $b) = ($b, $c->_div($a, $b)); # step 1
+ last if $c->_is_zero($b);
+
+ my $t = $c->_add( # step 2:
+ $c->_mul($c->_copy($v), $q), # t = v * q
+ $u); # + u
+ $u = $v; # u = v
+ $v = $t; # v = t
+ $sign = -$sign;
+ redo;
+ }
+
+ # if the gcd is not 1, then return NaN
+ return (undef, undef) unless $c->_is_one($a);
+
+ ($v, $sign == 1 ? '+' : '-');
+}
+
+sub _modpow {
+ # modulus of power ($x ** $y) % $z
+ my ($c, $num, $exp, $mod) = @_;
+
+ # a^b (mod 1) = 0 for all a and b
+ if ($c->_is_one($mod)) {
+ @$num = 0;
+ return $num;
+ }
+
+ # 0^a (mod m) = 0 if m != 0, a != 0
+ # 0^0 (mod m) = 1 if m != 0
+ if ($c->_is_zero($num)) {
+ if ($c->_is_zero($exp)) {
+ @$num = 1;
+ } else {
+ @$num = 0;
+ }
+ return $num;
+ }
+
+ # $num = $c->_mod($num, $mod); # this does not make it faster
+
+ my $acc = $c->_copy($num);
+ my $t = $c->_one();
+
+ my $expbin = $c->_as_bin($exp);
+ $expbin =~ s/^0b//;
+ my $len = length($expbin);
+ while (--$len >= 0) {
+ if (substr($expbin, $len, 1) eq '1') { # is_odd
+ $t = $c->_mul($t, $acc);
+ $t = $c->_mod($t, $mod);
+ }
+ $acc = $c->_mul($acc, $acc);
+ $acc = $c->_mod($acc, $mod);
+ }
+ @$num = @$t;
+ $num;
+}
+
+sub _gcd {
+ # Greatest common divisor.
+
+ my ($c, $x, $y) = @_;
+
+ # gcd(0, 0) = 0
+ # gcd(0, a) = a, if a != 0
+
+ if (@$x == 1 && $x->[0] == 0) {
+ if (@$y == 1 && $y->[0] == 0) {
+ @$x = 0;
+ } else {
+ @$x = @$y;
+ }
+ return $x;
+ }
+
+ # Until $y is zero ...
+
+ until (@$y == 1 && $y->[0] == 0) {
+
+ # Compute remainder.
+
+ $c->_mod($x, $y);
+
+ # Swap $x and $y.
+
+ my $tmp = $c->_copy($x);
+ @$x = @$y;
+ $y = $tmp; # no deref here; that would modify input $y
+ }
+
+ return $x;
+}
+
+1;
+
+=pod
+
+=head1 NAME
+
+Math::BigInt::Calc - Pure Perl module to support Math::BigInt
+
+=head1 SYNOPSIS
+
+ # to use it with Math::BigInt
+ use Math::BigInt lib => 'Calc';
+
+ # to use it with Math::BigFloat
+ use Math::BigFloat lib => 'Calc';
+
+ # to use it with Math::BigRat
+ use Math::BigRat lib => 'Calc';
+
+=head1 DESCRIPTION
+
+Math::BigInt::Calc inherits from Math::BigInt::Lib.
+
+In this library, the numbers are represented in base B = 10**N, where N is the
+largest possible value that does not cause overflow in the intermediate
+computations. The base B elements are stored in an array, with the least
+significant element stored in array element zero. There are no leading zero
+elements, except a single zero element when the number is zero.
+
+For instance, if B = 10000, the number 1234567890 is represented internally
+as [7890, 3456, 12].
+
+=head1 SEE ALSO
+
+L<Math::BigInt::Lib> for a description of the API.
+
+Alternative libraries L<Math::BigInt::FastCalc>, L<Math::BigInt::GMP>, and
+L<Math::BigInt::Pari>.
+
+Some of the modules that use these libraries L<Math::BigInt>,
+L<Math::BigFloat>, and L<Math::BigRat>.
+
+=cut
diff --git a/systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/CalcEmu.pm b/systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/CalcEmu.pm
new file mode 100644
index 0000000000..69c02caffe
--- /dev/null
+++ b/systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/CalcEmu.pm
@@ -0,0 +1,394 @@
+package Math::BigInt::CalcEmu;
+
+use 5.006001;
+use strict;
+use warnings;
+
+our $VERSION = '1.999811';
+
+package Math::BigInt;
+
+# See SYNOPSIS below.
+
+my $CALC_EMU;
+
+BEGIN
+ {
+ $CALC_EMU = Math::BigInt->config('lib');
+ # register us with MBI to get notified of future lib changes
+ Math::BigInt::_register_callback( __PACKAGE__, sub { $CALC_EMU = $_[0]; } );
+ }
+
+sub __emu_band
+ {
+ my ($self,$x,$y,$sx,$sy,@r) = @_;
+
+ return $x->bzero(@r) if $y->is_zero() || $x->is_zero();
+
+ my $sign = 0; # sign of result
+ $sign = 1 if $sx == -1 && $sy == -1;
+
+ my ($bx,$by);
+
+ if ($sx == -1) # if x is negative
+ {
+ # two's complement: inc and flip all "bits" in $bx
+ $bx = $x->binc()->as_hex(); # -1 => 0, -2 => 1, -3 => 2 etc
+ $bx =~ s/-?0x//;
+ $bx =~ tr/0123456789abcdef/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/;
+ }
+ else
+ {
+ $bx = $x->as_hex(); # 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 == -1) # if y is negative
+ {
+ # two's complement: inc and flip all "bits" in $by
+ $by = $y->copy()->binc()->as_hex(); # -1 => 0, -2 => 1, -3 => 2 etc
+ $by =~ s/-?0x//;
+ $by =~ tr/0123456789abcdef/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/;
+ }
+ else
+ {
+ $by = $y->as_hex(); # 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 == -1;
+ my $yy = "\x00"; $yy = "\x0f" if $sy == -1;
+ 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 $x or $y was negative, so need to flip bits in the result.
+ # In both cases (one or two of them negative, or both positive) we need
+ # to get the characters back.
+ if ($sign == 1)
+ {
+ $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;
+ $x->{value} = $CALC_EMU->_from_hex( $bx );
+
+ # calculate sign of result
+ $x->{sign} = '+';
+ $x->{sign} = '-' if $sign == 1 && !$x->is_zero();
+
+ $x->bdec() if $sign == 1;
+
+ $x->round(@r);
+ }
+
+sub __emu_bior
+ {
+ my ($self,$x,$y,$sx,$sy,@r) = @_;
+
+ return $x->round(@r) if $y->is_zero();
+
+ my $sign = 0; # sign of result
+ $sign = 1 if ($sx == -1) || ($sy == -1);
+
+ my ($bx,$by);
+
+ if ($sx == -1) # if x is negative
+ {
+ # two's complement: inc and flip all "bits" in $bx
+ $bx = $x->binc()->as_hex(); # -1 => 0, -2 => 1, -3 => 2 etc
+ $bx =~ s/-?0x//;
+ $bx =~ tr/0123456789abcdef/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/;
+ }
+ else
+ {
+ $bx = $x->as_hex(); # 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 == -1) # if y is negative
+ {
+ # two's complement: inc and flip all "bits" in $by
+ $by = $y->copy()->binc()->as_hex(); # -1 => 0, -2 => 1, -3 => 2 etc
+ $by =~ s/-?0x//;
+ $by =~ tr/0123456789abcdef/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/;
+ }
+ else
+ {
+ $by = $y->as_hex(); # 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 == -1;
+ my $yy = "\x00"; $yy = "\x0f" if $sy == -1;
+ my $diff = CORE::length($bx) - CORE::length($by);
+ if ($diff > 0)
+ {
+ $by .= $yy x $diff;
+ }
+ elsif ($diff < 0)
+ {
+ $bx .= $xx x abs($diff);
+ }
+
+ # or the strings together
+ my $r = $bx | $by;
+
+ # and reverse the result again
+ $bx = reverse $r;
+
+ # one of $x or $y was negative, so need to flip bits in the result
+ # in both cases (one or two of them negative, or both positive) we need
+ # to get the characters back.
+ if ($sign == 1)
+ {
+ $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;
+ $x->{value} = $CALC_EMU->_from_hex( $bx );
+
+ # calculate sign of result
+ $x->{sign} = '+';
+ $x->{sign} = '-' if $sign == 1 && !$x->is_zero();
+
+ # if one of X or Y was negative, we need to decrement result
+ $x->bdec() if $sign == 1;
+
+ $x->round(@r);
+ }
+
+sub __emu_bxor
+ {
+ my ($self,$x,$y,$sx,$sy,@r) = @_;
+
+ return $x->round(@r) if $y->is_zero();
+
+ my $sign = 0; # sign of result
+ $sign = 1 if $x->{sign} ne $y->{sign};
+
+ my ($bx,$by);
+
+ if ($sx == -1) # if x is negative
+ {
+ # two's complement: inc and flip all "bits" in $bx
+ $bx = $x->binc()->as_hex(); # -1 => 0, -2 => 1, -3 => 2 etc
+ $bx =~ s/-?0x//;
+ $bx =~ tr/0123456789abcdef/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/;
+ }
+ else
+ {
+ $bx = $x->as_hex(); # 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 == -1) # if y is negative
+ {
+ # two's complement: inc and flip all "bits" in $by
+ $by = $y->copy()->binc()->as_hex(); # -1 => 0, -2 => 1, -3 => 2 etc
+ $by =~ s/-?0x//;
+ $by =~ tr/0123456789abcdef/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/;
+ }
+ else
+ {
+ $by = $y->as_hex(); # 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 == -1;
+ my $yy = "\x00"; $yy = "\x0f" if $sy == -1;
+ my $diff = CORE::length($bx) - CORE::length($by);
+ if ($diff > 0)
+ {
+ $by .= $yy x $diff;
+ }
+ elsif ($diff < 0)
+ {
+ $bx .= $xx x abs($diff);
+ }
+
+ # xor the strings together
+ my $r = $bx ^ $by;
+
+ # and reverse the result again
+ $bx = reverse $r;
+
+ # one of $x or $y was negative, so need to flip bits in the result
+ # in both cases (one or two of them negative, or both positive) we need
+ # to get the characters back.
+ if ($sign == 1)
+ {
+ $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;
+ $x->{value} = $CALC_EMU->_from_hex( $bx );
+
+ # calculate sign of result
+ $x->{sign} = '+';
+ $x->{sign} = '-' if $sx != $sy && !$x->is_zero();
+
+ $x->bdec() if $sign == 1;
+
+ $x->round(@r);
+ }
+
+##############################################################################
+##############################################################################
+
+1;
+
+__END__
+
+=pod
+
+=head1 NAME
+
+Math::BigInt::CalcEmu - Emulate low-level math with BigInt code
+
+=head1 SYNOPSIS
+
+ use Math::BigInt::CalcEmu;
+
+=head1 DESCRIPTION
+
+Contains routines that emulate low-level math functions in BigInt, e.g.
+optional routines the low-level math package does not provide on its own.
+
+Will be loaded on demand and called automatically by BigInt.
+
+Stuff here is really low-priority to optimize, since it is far better to
+implement the operation in the low-level math library directly, possible even
+using a call to the native lib.
+
+=head1 METHODS
+
+=over
+
+=item __emu_bxor
+
+=item __emu_band
+
+=item __emu_bior
+
+=back
+
+=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::CalcEmu
+
+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 AUTHORS
+
+(c) Tels http://bloodgate.com 2003, 2004 - based on BigInt code by
+Tels from 2001-2003.
+
+=head1 SEE ALSO
+
+L<Math::BigInt>, L<Math::BigFloat>,
+L<Math::BigInt::GMP> and L<Math::BigInt::Pari>.
+
+=cut
diff --git a/systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/FastCalc.pm b/systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/FastCalc.pm
new file mode 100644
index 0000000000..8d0ba4097a
--- /dev/null
+++ b/systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/FastCalc.pm
@@ -0,0 +1,168 @@
+package Math::BigInt::FastCalc;
+
+use 5.006;
+use strict;
+use warnings;
+
+use Math::BigInt::Calc 1.999801;
+
+our @ISA = qw< Math::BigInt::Calc >;
+
+our $VERSION = '0.5006';
+
+##############################################################################
+# global constants, flags and accessory
+
+# announce that we are compatible with MBI v1.83 and up
+sub api_version () { 2; }
+
+# use Calc to override the methods that we do not provide in XS
+
+require XSLoader;
+XSLoader::load(__PACKAGE__, $VERSION, Math::BigInt::Calc->_base_len());
+
+##############################################################################
+##############################################################################
+
+1;
+
+__END__
+
+=pod
+
+=head1 NAME
+
+Math::BigInt::FastCalc - Math::BigInt::Calc with some XS for more speed
+
+=head1 SYNOPSIS
+
+ # to use it with Math::BigInt
+ use Math::BigInt lib => 'FastCalc';
+
+ # to use it with Math::BigFloat
+ use Math::BigFloat lib => 'FastCalc';
+
+ # to use it with Math::BigRat
+ use Math::BigRat lib => 'FastCalc';
+
+=head1 DESCRIPTION
+
+Math::BigInt::FastCalc inherits from Math::BigInt::Calc.
+
+Provides support for big integer calculations. Not intended to be used by
+other modules. Other modules which sport the same functions can also be used
+to support Math::BigInt, like L<Math::BigInt::GMP> or L<Math::BigInt::Pari>.
+
+In order to allow for multiple big integer libraries, Math::BigInt was
+rewritten to use library modules for core math routines. Any module which
+follows the same API as this can be used instead by using the following:
+
+ use Math::BigInt lib => 'libname';
+
+'libname' is either the long name ('Math::BigInt::Pari'), or only the short
+version like 'Pari'. To use this library:
+
+ use Math::BigInt lib => 'FastCalc';
+
+=head1 STORAGE
+
+Math::BigInt::FastCalc works exactly like Math::BigInt::Calc. Numbers are
+stored in decimal form chopped into parts.
+
+=head1 METHODS
+
+The following functions are now implemented in FastCalc.xs:
+
+ _is_odd _is_even _is_one _is_zero
+ _is_two _is_ten
+ _zero _one _two _ten
+ _acmp _len
+ _inc _dec
+ __strip_zeros _copy
+
+=head1 BUGS
+
+Please report any bugs or feature requests to
+C<bug-math-bigint-fastcalc at rt.cpan.org>, or through the web interface at
+L<https://rt.cpan.org/Ticket/Create.html?Queue=Math-BigInt-FastCalc>
+(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::FastCalc
+
+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-FastCalc>
+
+=item * AnnoCPAN: Annotated CPAN documentation
+
+L<http://annocpan.org/dist/Math-BigInt-FastCalc>
+
+=item * CPAN Ratings
+
+L<http://cpanratings.perl.org/dist/Math-BigInt-FastCalc>
+
+=item * Search CPAN
+
+L<http://search.cpan.org/dist/Math-BigInt-FastCalc/>
+
+=item * CPAN Testers Matrix
+
+L<http://matrix.cpantesters.org/?dist=Math-BigInt-FastCalc>
+
+=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 AUTHORS
+
+Original math code by Mark Biggar, rewritten by Tels L<http://bloodgate.com/>
+in late 2000.
+Separated from BigInt and shaped API with the help of John Peacock.
+
+Fixed, sped-up and enhanced by Tels http://bloodgate.com 2001-2003.
+Further streamlining (api_version 1 etc.) by Tels 2004-2007.
+
+Bug-fixing by Peter John Acklam E<lt>pjacklam@online.noE<gt> 2010-2016.
+
+=head1 SEE ALSO
+
+L<Math::BigInt::Lib> for a description of the API.
+
+Alternative libraries L<Math::BigInt::Calc>, L<Math::BigInt::GMP>, and
+L<Math::BigInt::Pari>.
+
+Some of the modules that use these libraries L<Math::BigInt>,
+L<Math::BigFloat>, and L<Math::BigRat>.
+
+=cut
diff --git a/systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/Lib.pm b/systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/Lib.pm
new file mode 100644
index 0000000000..23a44aa955
--- /dev/null
+++ b/systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/Lib.pm
@@ -0,0 +1,2070 @@
+package Math::BigInt::Lib;
+
+use 5.006001;
+use strict;
+use warnings;
+
+our $VERSION = '1.999811';
+
+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 { $_[0] },
+
+ # overload key: conversion
+
+ 'bool' => sub { ref($_[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($1) : 0;
+}
+
+##############################################################################
+# _is_* routines
+
+sub _is_zero {
+ # return true if arg is zero
+ my ($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);
+
+ if ($class -> can('_div')) {
+ $x = $class -> _copy($x);
+ my ($q, $r) = $class -> _div($x, $y);
+ return $r;
+ } else {
+ 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 $y
+ my ($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 $x;
+}
+
+sub _nok {
+ # Return binomial coefficient (n over k).
+ my ($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), $k);
+ }
+ }
+
+ # Example:
+ #
+ # / 7 \ 7! 1*2*3*4 * 5*6*7 5 * 6 * 7
+ # | | = --------- = --------------- = --------- = ((5 * 6) / 2 * 7) / 3
+ # \ 3 / (7-3)! 3! 1*2*3*4 * 1*2*3 1 * 2 * 3
+ #
+ # Equivalently, _nok(11, 5) is computed as
+ #
+ # (((((((7 * 8) / 2) * 9) / 3) * 10) / 4) * 11) / 5
+
+ if ($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 $z;
+}
+
+sub _to_bin {
+ # convert the number to a string of binary digits without prefix
+ my ($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) ? $str : "\x00";
+}
+
+*_as_bytes = \&_to_bytes;
+
+sub _from_hex {
+ # Convert a string of hexadecimal digits to a number.
+
+ my ($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 $x;
+}
+
+##############################################################################
+# special modulus functions
+
+sub _modinv {
+ # modular multiplicative inverse
+ my ($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()
+
+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 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.
+
+=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)
+
+Return bitwise or.
+
+=item CLASS-E<gt>_xor(OBJ1, OBJ2)
+
+Return bitwise 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>_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>_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
+
+=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 CLASS-E<gt>_signed_or(OBJ1, OBJ2, SIGN1, SIGN2)
+
+Return the signed bitwise or.
+
+=item CLASS-E<gt>_signed_and(OBJ1, OBJ2, SIGN1, SIGN2)
+
+Return the signed bitwise and.
+
+=item CLASS-E<gt>_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
diff --git a/systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/Trace.pm b/systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/Trace.pm
new file mode 100644
index 0000000000..5f83c79210
--- /dev/null
+++ b/systems/texlive/tlnet/tlpkg/tlperl/lib/Math/BigInt/Trace.pm
@@ -0,0 +1,48 @@
+#!perl
+
+package Math::BigInt::Trace;
+
+require 5.010;
+use strict;
+use warnings;
+
+use Exporter;
+use Math::BigInt;
+
+our ($accuracy, $precision, $round_mode, $div_scale);
+
+our @ISA = qw(Exporter Math::BigInt);
+
+our $VERSION = '0.49';
+
+use overload; # inherit overload from Math::BigInt
+
+# Globals
+$accuracy = $precision = undef;
+$round_mode = 'even';
+$div_scale = 40;
+
+sub new {
+ my $proto = shift;
+ my $class = ref($proto) || $proto;
+
+ my $value = shift;
+ my $a = $accuracy;
+ $a = $_[0] if defined $_[0];
+ my $p = $precision;
+ $p = $_[1] if defined $_[1];
+ my $self = Math::BigInt->new($value, $a, $p, $round_mode);
+ bless $self, $class;
+ print "MBI new '$value' => '$self' (", ref($self), ")";
+ return $self;
+}
+
+sub import {
+ print "MBI import ", join(' ', @_);
+ my $self = shift;
+ Math::BigInt::import($self, @_); # need it for subclasses
+# $self->export_to_level(1, $self, @_); # need this ?
+ @_ = ();
+}
+
+1;