X-Git-Url: http://git.shadowcat.co.uk/gitweb/gitweb.cgi?a=blobdiff_plain;f=lib%2FMath%2FBigInt%2FCalc.pm;h=694bdd57b4c058316a6a23c72f4da51f230c0e40;hb=c38b2de23cd38f4082b61dab277e518dbd2807a0;hp=24a664049baea1fe9f98c2e8da2184b1434f3800;hpb=e745a66c2cdc9fa9305c81afdec93bddfb34aefd;p=p5sagit%2Fp5-mst-13.2.git
diff --git a/lib/Math/BigInt/Calc.pm b/lib/Math/BigInt/Calc.pm
index 24a6640..694bdd5 100644
--- a/lib/Math/BigInt/Calc.pm
+++ b/lib/Math/BigInt/Calc.pm
@@ -8,12 +8,13 @@ require Exporter;
use vars qw/@ISA $VERSION/;
@ISA = qw(Exporter);
-$VERSION = '0.14';
+$VERSION = '0.36';
# 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 100000
+# 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 (maybe)
@@ -24,41 +25,80 @@ $VERSION = '0.14';
# 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 / $MBASE); $i = $i % $MBASE;
+# This works on x86, but fails on ARM (SA1100, iPAQ) due to whoknows what
+# reasons. So, use this instead (slower, but correct):
+# $i = $i * $y + $car; $car = int($i / $MBASE); $i -= $MBASE * $car;
+
##############################################################################
# global constants, flags and accessory
# constants for easier life
my $nan = 'NaN';
-my ($BASE,$RBASE,$BASE_LEN,$MAX_VAL);
+my ($MBASE,$BASE,$RBASE,$BASE_LEN,$MAX_VAL,$BASE_LEN2,$BASE_LEN_SMALL);
+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, connected values
# used only be the testsuite, set is used only by the BEGIN block below
+ shift;
+
my $b = shift;
if (defined $b)
{
- $b = 8 if $b > 8; # cap, for VMS, OS/390 and other 64 bit
- $BASE_LEN = $b;
+ # find whether we can use mul or div or none in mul()/div()
+ # (in last case reduce BASE_LEN_SMALL)
+ $BASE_LEN_SMALL = $b+1;
+ my $caught = 0;
+ while (--$BASE_LEN_SMALL > 5)
+ {
+ $MBASE = int("1e".$BASE_LEN_SMALL);
+ $RBASE = abs('1e-'.$BASE_LEN_SMALL); # see USE_MUL
+ $caught = 0;
+ $caught += 1 if (int($MBASE * $RBASE) != 1); # should be 1
+ $caught += 2 if (int($MBASE / $MBASE) != 1); # should be 1
+ last if $caught != 3;
+ }
+ # BASE_LEN is used for anything else than mul()/div()
+ $BASE_LEN = $BASE_LEN_SMALL;
+ $BASE_LEN = shift if (defined $_[0]); # one more arg?
$BASE = int("1e".$BASE_LEN);
- $RBASE = abs('1e-'.$BASE_LEN); # see USE_MUL
- $MAX_VAL = $BASE-1;
- # print "BASE_LEN: $BASE_LEN MAX_VAL: $MAX_VAL\n";
- # print "int: ",int($BASE * $RBASE),"\n";
- if (int($BASE * $RBASE) == 0) # should be 1
+
+ $BASE_LEN2 = int($BASE_LEN_SMALL / 2); # for mul shortcut
+ $MBASE = int("1e".$BASE_LEN_SMALL);
+ $RBASE = abs('1e-'.$BASE_LEN_SMALL); # see USE_MUL
+ $MAX_VAL = $MBASE-1;
+
+ #print "BASE_LEN: $BASE_LEN MAX_VAL: $MAX_VAL BASE: $BASE RBASE: $RBASE ";
+ #print "BASE_LEN_SMALL: $BASE_LEN_SMALL MBASE: $MBASE\n";
+
+ undef &_mul;
+ undef &_div;
+
+ # $caught & 1 != 0 => cannot use MUL
+ # $caught & 2 != 0 => cannot use DIV
+ # The parens around ($caught & 1) were important, indeed, if we would use
+ # & here.
+ if ($caught == 2) # 2
{
- # must USE_MUL
+ # print "# use mul\n";
+ # must USE_MUL since we cannot use DIV
*{_mul} = \&_mul_use_mul;
*{_div} = \&_div_use_mul;
}
- else
+ else # 0 or 1
{
+ # print "# use div\n";
# can USE_DIV instead
*{_mul} = \&_mul_use_div;
*{_div} = \&_div_use_div;
}
}
- $BASE_LEN;
+ return $BASE_LEN unless wantarray;
+ return ($BASE_LEN, $AND_BITS, $XOR_BITS, $OR_BITS, $BASE_LEN_SMALL, $MAX_VAL);
}
BEGIN
@@ -70,44 +110,123 @@ BEGIN
do
{
$num = ('9' x ++$e) + 0;
- $num *= $num + 1;
- # print "$num $e\n";
- } while ("$num" =~ /9{$e}0{$e}/); # must be a certain pattern
- # last test failed, so retract one step:
- _base_len($e-1);
+ $num *= $num + 1.0;
+ } while ("$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)
+ $e = 5 if $] < 5.006; # cap, for older Perls
+ $e = 7 if $e > 7; # cap, for VMS, OS/390 and other 64 bit systems
+ # 8 fails inside random testsuite, so take 7
+
+ # determine how many digits fit into an integer and can be safely added
+ # together plus carry w/o causing an overflow
+
+ use integer;
+
+ ############################################################################
+ # the next block is no longer important
+
+ ## this below detects 15 on a 64 bit system, because after that it becomes
+ ## 1e16 and not 1000000 :/ I can make it detect 18, but then I get a lot of
+ ## test failures. Ugh! (Tomake detect 18: uncomment lines marked with *)
+
+ #my $bi = 5; # approx. 16 bit
+ #$num = int('9' x $bi);
+ ## $num = 99999; # *
+ ## while ( ($num+$num+1) eq '1' . '9' x $bi) # *
+ #while ( int($num+$num+1) eq '1' . '9' x $bi)
+ # {
+ # $bi++; $num = int('9' x $bi);
+ # # $bi++; $num *= 10; $num += 9; # *
+ # }
+ #$bi--; # back off one step
+ # by setting them equal, we ignore the findings and use the default
+ # one-size-fits-all approach from former versions
+ my $bi = $e; # XXX, this should work always
+
+ __PACKAGE__->_base_len($e,$bi); # set and store
+
+ # 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 = 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 = 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 = 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
+
}
-##############################################################################
-# create objects from various representations
+###############################################################################
sub _new
{
- # (string) return ref to num_array
- # Convert a number from string format to internal base 100000 format.
- # Assumes normalized value as input.
+ # (ref to string) return ref to num_array
+ # Convert a number from string format (without sign) to internal base
+ # 1ex format. Assumes normalized value as input.
my $d = $_[1];
- # print "_new $d $$d\n";
- my $il = CORE::length($$d)-1;
- # these leaves '00000' instead of int 0 and will be corrected after any op
- return [ reverse(unpack("a" . ($il % $BASE_LEN+1)
+ my $il = length($$d)-1;
+ # this leaves '00000' instead of int 0 and will be corrected after any op
+ [ reverse(unpack("a" . ($il % $BASE_LEN+1)
. ("a$BASE_LEN" x ($il / $BASE_LEN)), $$d)) ];
}
+
+BEGIN
+ {
+ $AND_MASK = __PACKAGE__->_new( \( 2 ** $AND_BITS ));
+ $XOR_MASK = __PACKAGE__->_new( \( 2 ** $XOR_BITS ));
+ $OR_MASK = __PACKAGE__->_new( \( 2 ** $OR_BITS ));
+ }
sub _zero
{
# create a zero
- return [ 0 ];
+ [ 0 ];
}
sub _one
{
# create a one
- return [ 1 ];
+ [ 1 ];
+ }
+
+sub _two
+ {
+ # create a two (used internally for shifting)
+ [ 2 ];
}
sub _copy
{
- return [ @{$_[1]} ];
+ [ @{$_[1]} ];
}
# catch and throw away
@@ -123,11 +242,13 @@ sub _str
# internal format is always normalized (no leading zeros, "-0" => "+0")
my $ar = $_[1];
my $ret = "";
- my $l = scalar @$ar; # number of parts
- return $nan if $l < 1; # should not happen
+
+ my $l = scalar @$ar; # number of parts
+ return $nan if $l < 1; # should not happen
+
# handle first one different to strip leading zeros from it (there are no
# leading zero parts in internal representation)
- $l --; $ret .= $ar->[$l]; $l--;
+ $l --; $ret .= int($ar->[$l]); $l--;
# Interestingly, the pre-padd method uses more time
# the old grep variant takes longer (14 to 10 sec)
my $z = '0' x ($BASE_LEN-1);
@@ -136,7 +257,7 @@ sub _str
$ret .= substr($z.$ar->[$l],-$BASE_LEN); # fastest way I could think of
$l--;
}
- return \$ret;
+ \$ret;
}
sub _num
@@ -150,7 +271,7 @@ sub _num
{
$num += $fac*$_; $fac *= $BASE;
}
- return $num;
+ $num;
}
##############################################################################
@@ -165,6 +286,13 @@ sub _add
# This routine clobbers up array x, but not y.
my ($c,$x,$y) = @_;
+
+ return $x if (@$y == 1) && $y->[0] == 0; # $x + 0 => $x
+ if ((@$x == 1) && $x->[0] == 0) # 0 + $y => $y->copy
+ {
+ # twice as slow as $x = [ @$y ], but necc. to retain $x as ref :(
+ @$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
@@ -180,49 +308,47 @@ sub _add
{
$x->[$j] -= $BASE if $car = (($x->[$j] += $car) >= $BASE) ? 1 : 0; $j++;
}
- return $x;
+ $x;
}
sub _inc
{
# (ref to int_num_array, ref to int_num_array)
# routine to add 1 to a base 1eX numbers
- # This routine clobbers up array x, but not y.
+ # This routine modifies array x
my ($c,$x) = @_;
for my $i (@$x)
{
return $x if (($i += 1) < $BASE); # early out
- $i -= $BASE;
- }
- if ($x->[-1] == 0) # last overflowed
- {
- push @$x,1; # extend
+ $i = 0; # overflow, next
}
- return $x;
+ push @$x,1 if ($x->[-1] == 0); # last overflowed, so extend
+ $x;
}
sub _dec
{
# (ref to int_num_array, ref to int_num_array)
# routine to add 1 to a base 1eX numbers
- # This routine clobbers up array x, but not y.
+ # This routine modifies array x
my ($c,$x) = @_;
+ my $MAX = $BASE-1; # since MAX_VAL based on MBASE
for my $i (@$x)
{
last if (($i -= 1) >= 0); # early out
- $i = $MAX_VAL;
+ $i = $MAX; # overflow, next
}
pop @$x if $x->[-1] == 0 && @$x > 1; # last overflowed (but leave 0)
- return $x;
+ $x;
}
sub _sub
{
- # (ref to int_num_array, ref to int_num_array)
+ # (ref to int_num_array, ref to int_num_array, swap)
# subtract base 1eX numbers -- stolen from Knuth Vol 2 pg 232, $x > $y
- # subtract Y from X (X is always greater/equal!) by modifying x in place
+ # subtract Y from X by modifying x in place
my ($c,$sx,$sy,$s) = @_;
my $car = 0; my $i; my $j = 0;
@@ -232,42 +358,67 @@ sub _sub
for $i (@$sx)
{
last unless defined $sy->[$j] || $car;
- #print "x: $i y: $sy->[$j] c: $car\n";
$i += $BASE if $car = (($i -= ($sy->[$j] || 0) + $car) < 0); $j++;
- #print "x: $i y: $sy->[$j-1] c: $car\n";
}
# might leave leading zeros, so fix that
- __strip_zeros($sx);
- return $sx;
+ return __strip_zeros($sx);
}
- else
+ #print "case 1 (swap)\n";
+ for $i (@$sx)
{
- #print "case 1 (swap)\n";
- for $i (@$sx)
- {
- last unless defined $sy->[$j] || $car;
- #print "$sy->[$j] $i $car => $sx->[$j]\n";
- $sy->[$j] += $BASE
- if $car = (($sy->[$j] = $i-($sy->[$j]||0) - $car) < 0);
- #print "$sy->[$j] $i $car => $sy->[$j]\n";
- $j++;
- }
- # might leave leading zeros, so fix that
- __strip_zeros($sy);
- return $sy;
+ # we can't do an early out if $x is < than $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
{
- # (BINT, BINT) return nothing
+ # (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,$yv) = @_;
- my @prod = (); my ($prod,$car,$cty,$xi,$yi);
+ 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]) >= $MBASE)
+ {
+ $xv->[0] = $xv->[0] - ($xv->[1] = int($xv->[0] * $RBASE)) * $MBASE;
+ };
+ 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 * $MBASE;
+ }
+ push @$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 "$xv" eq "$yv"; # same references?
+ $yv = [@$xv] if $xv == $yv; # same references?
+
+ my @prod = (); my ($prod,$car,$cty,$xi,$yi);
+
for $xi (@$xv)
{
$car = 0; $cty = 0;
@@ -277,7 +428,7 @@ sub _mul_use_mul
# {
# $prod = $xi * $yi + ($prod[$cty] || 0) + $car;
# $prod[$cty++] =
-# $prod - ($car = int($prod * RBASE)) * $BASE; # see USE_MUL
+# $prod - ($car = int($prod * RBASE)) * $MBASE; # see USE_MUL
# }
# $prod[$cty] += $car if $car; # need really to check for 0?
# $xi = shift @prod;
@@ -291,27 +442,59 @@ sub _mul_use_mul
## 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
+ $prod - ($car = int($prod * $RBASE)) * $MBASE; # see USE_MUL
}
$prod[$cty] += $car if $car; # need really to check for 0?
- $xi = shift @prod;
+ $xi = shift @prod || 0; # || 0 makes v5.005_3 happy
}
push @$xv, @prod;
__strip_zeros($xv);
- # normalize (handled last to save check for $y->is_zero()
- return $xv;
+ $xv;
}
sub _mul_use_div
{
- # (BINT, BINT) return nothing
+ # (ref to int_num_array, ref to int_num_array)
# multiply two numbers in internal representation
# modifies first arg, second need not be different from first
my ($c,$xv,$yv) = @_;
- my @prod = (); my ($prod,$car,$cty,$xi,$yi);
+ if (@$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]) >= $MBASE)
+ {
+ $xv->[0] =
+ $xv->[0] - ($xv->[1] = int($xv->[0] / $MBASE)) * $MBASE;
+ };
+ 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 / $MBASE); $i -= $car * $MBASE;
+ }
+ push @$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 "$xv" eq "$yv"; # same references?
+ $yv = [@$xv] if $xv == $yv; # same references?
+
+ my @prod = (); my ($prod,$car,$cty,$xi,$yi);
for $xi (@$xv)
{
$car = 0; $cty = 0;
@@ -321,42 +504,148 @@ sub _mul_use_div
{
$prod = $xi * $yi + ($prod[$cty] || 0) + $car;
$prod[$cty++] =
- $prod - ($car = int($prod / $BASE)) * $BASE;
+ $prod - ($car = int($prod / $MBASE)) * $MBASE;
}
$prod[$cty] += $car if $car; # need really to check for 0?
- $xi = shift @prod;
+ $xi = shift @prod || 0; # || 0 makes v5.005_3 happy
}
push @$xv, @prod;
__strip_zeros($xv);
- # normalize (handled last to save check for $y->is_zero()
- return $xv;
+ $xv;
}
sub _div_use_mul
{
# ref to array, ref to array, modify first array and return remainder if
# in list context
- # no longer handles sign
+
+ # see comments in _div_use_div() for more explanations
+
my ($c,$x,$yorg) = @_;
- my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1);
+
+ # the general div algorithmn 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).
- my (@d,$tmp,$q,$u2,$u1,$u0);
+ # if both numbers have only one element:
+ if (@$x == 1 && @$yorg == 1)
+ {
+ # shortcut, $yorg and $x are two small numbers
+ if (wantarray)
+ {
+ my $r = [ $x->[0] % $yorg->[0] ];
+ $x->[0] = int($x->[0] / $yorg->[0]);
+ return ($x,$r);
+ }
+ 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;
+ $rem = _mod($c,[ @$x ],$yorg) if wantarray;
+
+ # shortcut, $y is < $BASE
+ my $j = scalar @$x; my $r = 0;
+ my $y = $yorg->[0]; my $b;
+ while ($j-- > 0)
+ {
+ $b = $r * $MBASE + $x->[$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;
+ $rem = [@$x] if wantarray; # make copy
+ splice (@$x,1); # keep ref to original array
+ $x->[0] = 0; # set to 0
+ return ($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])))
+ {
+ $rem = [@$x] if wantarray; # make copy
+ splice (@$x,1); # keep ref to org array
+ $x->[0] = 0; # set to 0
+ return ($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, and if equal, return 1
+ # hm, same lengths, but same contents? So we need to check all parts:
+ my $a = 0; my $j = scalar @$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)
+ {
+ if (wantarray)
+ {
+ $rem = [ 0 ]; # a = 0 => x == y => rem 1
+ $rem = [@$x] if $a != 0; # a < 0 => x < y => rem = x
+ }
+ splice(@$x,1); # keep single element
+ $x->[0] = 0; # if $a < 0
+ if ($a == 0)
+ {
+ # $x == $y
+ $x->[0] = 1;
+ }
+ return ($x,$rem) if wantarray;
+ return $x;
+ }
+ # $x >= $y, proceed normally
+ }
+ }
+
+ # all other cases:
+
+ my $y = [ @$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;
-
- my $y = [ @$yorg ];
- if (($dd = int($BASE/($y->[-1]+1))) != 1)
+ if (($dd = int($MBASE/($y->[-1]+1))) != 1)
{
for $xi (@$x)
{
$xi = $xi * $dd + $car;
- $xi -= ($car = int($xi * $RBASE)) * $BASE; # see USE_MUL
+ $xi -= ($car = int($xi * $RBASE)) * $MBASE; # see USE_MUL
}
push(@$x, $car); $car = 0;
for $yi (@$y)
{
$yi = $yi * $dd + $car;
- $yi -= ($car = int($yi * $RBASE)) * $BASE; # see USE_MUL
+ $yi -= ($car = int($yi * $RBASE)) * $MBASE; # see USE_MUL
}
}
else
@@ -371,29 +660,29 @@ sub _div_use_mul
$u2 = 0 unless $u2;
#warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n"
# if $v1 == 0;
- # $q = (($u0 == $v1) ? 99999 : int(($u0*$BASE+$u1)/$v1));
- $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$BASE+$u1)/$v1));
- --$q while ($v2*$q > ($u0*$BASE+$u1-$q*$v1)*$BASE+$u2);
+ $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$MBASE+$u1)/$v1));
+ --$q while ($v2*$q > ($u0*$MBASE+$u1-$q*$v1)*$MBASE+$u2);
if ($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));
+ $prd -= ($car = int($prd * $RBASE)) * $MBASE; # see USE_MUL
+ $x->[$xi] += $MBASE if ($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));
+ $x->[$xi] -= $MBASE
+ if ($car = (($x->[$xi] += $y->[$yi] + $car) >= $MBASE));
}
}
}
- pop(@$x); unshift(@q, $q);
+ pop(@$x);
+ unshift(@q, $q);
}
if (wantarray)
{
@@ -403,7 +692,7 @@ sub _div_use_mul
$car = 0;
for $xi (reverse @$x)
{
- $prd = $car * $BASE + $xi;
+ $prd = $car * $MBASE + $xi;
$car = $prd - ($tmp = int($prd / $dd)) * $dd; # see USE_MUL
unshift(@d, $tmp);
}
@@ -413,46 +702,153 @@ sub _div_use_mul
@d = @$x;
}
@$x = @q;
- __strip_zeros($x);
- __strip_zeros(\@d);
- return ($x,\@d);
+ my $d = \@d;
+ __strip_zeros($x);
+ __strip_zeros($d);
+ return ($x,$d);
}
@$x = @q;
- __strip_zeros($x);
- return $x;
+ __strip_zeros($x);
+ $x;
}
sub _div_use_div
{
# ref to array, ref to array, modify first array and return remainder if
# in list context
- # no longer handles sign
my ($c,$x,$yorg) = @_;
- my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1);
- my (@d,$tmp,$q,$u2,$u1,$u0);
+ # the general div algorithmn here is about O(N*N) and thus quite slow, so
+ # we first check for some special cases and use shortcuts to handle them.
+
+ # This works, because we store the numbers in a chunked format where each
+ # element contains 5..7 digits (depending on system).
+
+ # if both numbers have only one element:
+ if (@$x == 1 && @$yorg == 1)
+ {
+ # shortcut, $yorg and $x are two small numbers
+ if (wantarray)
+ {
+ my $r = [ $x->[0] % $yorg->[0] ];
+ $x->[0] = int($x->[0] / $yorg->[0]);
+ return ($x,$r);
+ }
+ 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;
+ $rem = _mod($c,[ @$x ],$yorg) if wantarray;
+
+ # shortcut, $y is < $BASE
+ my $j = scalar @$x; my $r = 0;
+ my $y = $yorg->[0]; my $b;
+ while ($j-- > 0)
+ {
+ $b = $r * $MBASE + $x->[$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;
+ $rem = [@$x] if wantarray; # make copy
+ splice (@$x,1); # keep ref to original array
+ $x->[0] = 0; # set to 0
+ return ($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])))
+ {
+ $rem = [@$x] if wantarray; # make copy
+ splice (@$x,1); # keep ref to org array
+ $x->[0] = 0; # set to 0
+ return ($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, and if equal, return 1
+ # hm, same lengths, but same contents? So we need to check all parts:
+ my $a = 0; my $j = scalar @$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)
+ {
+ if (wantarray)
+ {
+ $rem = [ 0 ]; # a = 0 => x == y => rem 1
+ $rem = [@$x] if $a != 0; # a < 0 => x < y => rem = x
+ }
+ splice(@$x,1); # keep single element
+ $x->[0] = 0; # if $a < 0
+ if ($a == 0)
+ {
+ # $x == $y
+ $x->[0] = 1;
+ }
+ return ($x,$rem) if wantarray;
+ return $x;
+ }
+ # $x >= $y, so proceed normally
+ }
+ }
+
+ # all other cases:
+
+ my $y = [ @$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;
-
- my $y = [ @$yorg ];
- if (($dd = int($BASE/($y->[-1]+1))) != 1)
+ if (($dd = int($MBASE/($y->[-1]+1))) != 1)
{
for $xi (@$x)
{
$xi = $xi * $dd + $car;
- $xi -= ($car = int($xi / $BASE)) * $BASE;
+ $xi -= ($car = int($xi / $MBASE)) * $MBASE;
}
push(@$x, $car); $car = 0;
for $yi (@$y)
{
$yi = $yi * $dd + $car;
- $yi -= ($car = int($yi / $BASE)) * $BASE;
+ $yi -= ($car = int($yi / $MBASE)) * $MBASE;
}
}
else
{
push(@$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)
@@ -461,29 +857,28 @@ sub _div_use_div
$u2 = 0 unless $u2;
#warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n"
# if $v1 == 0;
- # $q = (($u0 == $v1) ? 99999 : int(($u0*$BASE+$u1)/$v1));
- $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$BASE+$u1)/$v1));
- --$q while ($v2*$q > ($u0*$BASE+$u1-$q*$v1)*$BASE+$u2);
+ $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$MBASE+$u1)/$v1));
+ --$q while ($v2*$q > ($u0*$MBASE+$u1-$q*$v1)*$MBASE+$u2);
if ($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));
+ $prd -= ($car = int($prd / $MBASE)) * $MBASE;
+ $x->[$xi] += $MBASE if ($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));
+ $x->[$xi] -= $MBASE
+ if ($car = (($x->[$xi] += $y->[$yi] + $car) >= $MBASE));
}
}
}
- pop(@$x); unshift(@q, $q);
+ pop(@$x); unshift(@q, $q);
}
if (wantarray)
{
@@ -493,7 +888,7 @@ sub _div_use_div
$car = 0;
for $xi (reverse @$x)
{
- $prd = $car * $BASE + $xi;
+ $prd = $car * $MBASE + $xi;
$car = $prd - ($tmp = int($prd / $dd)) * $dd;
unshift(@d, $tmp);
}
@@ -503,156 +898,13 @@ sub _div_use_div
@d = @$x;
}
@$x = @q;
- __strip_zeros($x);
- __strip_zeros(\@d);
- return ($x,\@d);
+ my $d = \@d;
+ __strip_zeros($x);
+ __strip_zeros($d);
+ return ($x,$d);
}
@$x = @q;
- __strip_zeros($x);
- return $x;
- }
-
-sub _mod
- {
- # if possible, use mod shortcut
- my ($c,$x,$yo) = @_;
-
- # slow way since $y to big
- if (scalar @$yo > 1)
- {
- my ($xo,$rem) = _div($c,$x,$yo);
- return $rem;
- }
- my $y = $yo->[0];
- # both are single element
- if (scalar @$x == 1)
- {
- $x->[0] %= $y;
- return $x;
- }
-
- 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] %= $y;
- }
- else
- {
- # else need to go trough all elemens: O(N)
- # XXX not ready yet
- my ($xo,$rem) = _div($c,$x,$yo);
- return $rem;
-
-# my $i = 0; my $r = 1;
-# print "Multi: ";
-# foreach (@$x)
-# {
-# print "$_ $r $b $y\n";
-# print "\$_ % \$y = ",$_ % $y,"\n";
-# print "\$_ % \$y * \$b = ",($_ % $y) * $b,"\n";
-# $r += ($_ % $y) * $b;
-# print "$r $b $y =>";
-# $r %= $y if $r > $y;
-# print " $r\n";
-# }
-# $x->[0] = $r;
- }
- splice (@$x,1);
- return $x;
- }
-
-##############################################################################
-# shifts
-
-sub _rsft
- {
- my ($c,$x,$y,$n) = @_;
-
- if ($n != 10)
- {
- return; # we cant do this here, due to now _pow, so signal failure
- }
- else
- {
- # shortcut (faster) for shifting by 10)
- # multiples of $BASE_LEN
- my $dst = 0; # destination
- my $src = _num($c,$y); # as normal int
- 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 $len = scalar @$x - $src; # elems to go
- my $vd; my $z = '0'x $BASE_LEN;
- $x->[scalar @$x] = 0; # avoid || 0 test inside loop
- while ($dst < $len)
- {
- $vd = $z.$x->[$src];
- #print "$dst $src '$vd' ";
- $vd = substr($vd,-$BASE_LEN,$BASE_LEN-$rem);
- #print "'$vd' ";
- $src++;
- $vd = substr($z.$x->[$src],-$rem,$rem) . $vd;
- #print "'$vd1' ";
- #print "'$vd'\n";
- $vd = substr($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; # kill last element if 0
- } # else rem == 0
- }
- $x;
- }
-
-sub _lsft
- {
- my ($c,$x,$y,$n) = @_;
-
- if ($n != 10)
- {
- return; # we cant do this here, due to now _pow, so signal failure
- }
- else
- {
- # shortcut (faster) for shifting by 10) since we are in base 10eX
- # multiples of $BASE_LEN:
- my $src = scalar @$x; # source
- my $len = _num($c,$y); # 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
- #print "src $src:",$x->[$src]||0," dst $dst:",$v->[$dst]||0," rem $rem\n";
- $x->[$src] = 0; # avoid first ||0 for speed
- my $z = '0' x $BASE_LEN;
- while ($src >= 0)
- {
- $vd = $x->[$src]; $vd = $z.$vd;
- #print "s $src d $dst '$vd' ";
- $vd = substr($vd,-$BASE_LEN+$rem,$BASE_LEN-$rem);
- #print "'$vd' ";
- $vd .= $src > 0 ? substr($z.$x->[$src-1],-$BASE_LEN,$rem) : '0' x $rem;
- #print "'$vd' ";
- $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN;
- #print "'$vd'\n";
- $x->[$dst] = int($vd);
- $dst--; $src--;
- }
- # set lowest parts to 0
- while ($dst >= 0) { $x->[$dst--] = 0; }
- # fix spurios last zero element
- splice @$x,-1 if $x->[-1] == 0;
- #print "elems: "; my $i = 0;
- #foreach (reverse @$v) { print "$i $_ "; $i++; } print "\n";
- }
+ __strip_zeros($x);
$x;
}
@@ -665,46 +917,41 @@ sub _acmp
# ref to array, ref to array, return <0, 0, >0
# arrays must have at least one entry; this is not checked for
- my ($c,$cx, $cy) = @_;
+ my ($c,$cx,$cy) = @_;
- #print "$cx $cy\n";
- my ($i,$a,$x,$y,$k);
- # calculate length based on digits, not parts
- $x = _len('',$cx); $y = _len('',$cy);
- # print "length: ",($x-$y),"\n";
- my $lxy = $x - $y; # if different in length
+ # fast comp based on number of array elements (aka pseudo-length)
+ my $lxy = scalar @$cx - scalar @$cy;
+ return -1 if $lxy < 0; # already differs, ret
+ return 1 if $lxy > 0; # ditto
+
+ # now calculate length based on digits, not parts
+ # we need only the length of the last element, since both array have the
+ # same number of parts
+ $lxy = length(int($cx->[-1])) - length(int($cy->[-1]));
return -1 if $lxy < 0;
return 1 if $lxy > 0;
- #print "full compare\n";
- $i = 0; $a = 0;
- # first way takes 5.49 sec instead of 4.87, but has the early out advantage
- # so grep is slightly faster, but more inflexible. hm. $_ instead of $k
- # yields 5.6 instead of 5.5 sec huh?
+
+ # hm, same lengths, but same contents? So we need to check all parts:
+ my $a; my $j = scalar @$cx - 1;
# manual way (abort if unequal, good for early ne)
- my $j = scalar @$cx - 1;
while ($j >= 0)
- {
- # print "$cx->[$j] $cy->[$j] $a",$cx->[$j]-$cy->[$j],"\n";
- last if ($a = $cx->[$j] - $cy->[$j]); $j--;
- }
+ {
+ last if ($a = $cx->[$j] - $cy->[$j]); $j--;
+ }
return 1 if $a > 0;
return -1 if $a < 0;
- return 0; # equal
- # while it early aborts, it is even slower than the manual variant
- #grep { return $a if ($a = $_ - $cy->[$i++]); } @$cx;
- # grep way, go trough all (bad for early ne)
- #grep { $a = $_ - $cy->[$i++]; } @$cx;
- #return $a;
+ 0; # numbers are equal
}
sub _len
{
- # compute number of digits in bigint, minus the sign
+ # compute number of digits
+
# int() because add/sub sometimes leaves strings (like '00005') instead of
- # int ('5') in this place, thus causing length() to report wrong length
+ # '5' in this place, thus causing length() to report wrong length
my $cx = $_[1];
- return (@$cx-1)*$BASE_LEN+length(int($cx->[-1]));
+ (@$cx-1)*$BASE_LEN+length(int($cx->[-1]));
}
sub _digit
@@ -722,7 +969,7 @@ sub _digit
my $elem = int($n / $BASE_LEN); # which array element
my $digit = $n % $BASE_LEN; # which digit in this element
$elem = '0000'.@$x[$elem]; # get element padded with 0's
- return substr($elem,-$digit-1,1);
+ substr($elem,-$digit-1,1);
}
sub _zeros
@@ -739,12 +986,12 @@ sub _zeros
$elem = "$e"; # preserve x
$elem =~ s/.*?(0*$)/$1/; # strip anything not zero
$zeros *= $BASE_LEN; # elems * 5
- $zeros += CORE::length($elem); # count trailing zeros
+ $zeros += length($elem); # count trailing zeros
last; # early out
}
$zeros ++; # real else branch: 50% slower!
}
- return $zeros;
+ $zeros;
}
##############################################################################
@@ -754,28 +1001,31 @@ sub _is_zero
{
# return true if arg (BINT or num_str) is zero (array '+', '0')
my $x = $_[1];
- return (((scalar @$x == 1) && ($x->[0] == 0))) <=> 0;
+
+ (((scalar @$x == 1) && ($x->[0] == 0))) <=> 0;
}
sub _is_even
{
# return true if arg (BINT or num_str) is even
my $x = $_[1];
- return (!($x->[0] & 1)) <=> 0;
+ (!($x->[0] & 1)) <=> 0;
}
sub _is_odd
{
# return true if arg (BINT or num_str) is even
my $x = $_[1];
- return (($x->[0] & 1)) <=> 0;
+
+ (($x->[0] & 1)) <=> 0;
}
sub _is_one
{
# return true if arg (BINT or num_str) is one (array '+', '1')
my $x = $_[1];
- return (scalar @$x == 1) && ($x->[0] == 1) <=> 0;
+
+ (scalar @$x == 1) && ($x->[0] == 1) <=> 0;
}
sub __strip_zeros
@@ -786,6 +1036,10 @@ sub __strip_zeros
my $cnt = scalar @$s; # get count of parts
my $i = $cnt-1;
+ push @$s,0 if $i < 0; # div might return empty results, so fix it
+
+ return $s if @$s == 1; # early out
+
#print "strip: cnt $cnt i $i\n";
# '0', '3', '4', '0', '0',
# 0 1 2 3 4
@@ -796,7 +1050,7 @@ sub __strip_zeros
# >= 1: skip first part (this can be zero)
while ($i > 0) { last if $s->[$i] != 0; $i--; }
$i++; splice @$s,$i if ($i < $cnt); # $i cant be 0
- return $s;
+ $s;
}
###############################################################################
@@ -832,6 +1086,697 @@ sub _check
return 0;
}
+
+###############################################################################
+###############################################################################
+# some optional routines to make BigInt faster
+
+sub _mod
+ {
+ # if possible, use mod shortcut
+ my ($c,$x,$yo) = @_;
+
+ # slow way since $y to big
+ if (scalar @$yo > 1)
+ {
+ my ($xo,$rem) = _div($c,$x,$yo);
+ return $rem;
+ }
+
+ my $y = $yo->[0];
+ # both are single element arrays
+ if (scalar @$x == 1)
+ {
+ $x->[0] %= $y;
+ return $x;
+ }
+
+ # @y is a single element, but @x has more than one 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 trough all elements: 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 trough all elements: 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] = $r;
+ }
+ splice (@$x,1);
+ $x;
+ }
+
+##############################################################################
+# shifts
+
+sub _rsft
+ {
+ my ($c,$x,$y,$n) = @_;
+
+ if ($n != 10)
+ {
+ $n = _new($c,\$n); return _div($c,$x, _pow($c,$n,$y));
+ }
+
+ # shortcut (faster) for shifting by 10)
+ # multiples of $BASE_LEN
+ my $dst = 0; # destination
+ my $src = _num($c,$y); # as normal int
+ my $xlen = (@$x-1)*$BASE_LEN+length(int($x->[-1])); # len of x in digits
+ 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 $len = scalar @$x - $src; # elems to go
+ my $vd; my $z = '0'x $BASE_LEN;
+ $x->[scalar @$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,$x,$y,$n) = @_;
+
+ if ($n != 10)
+ {
+ $n = _new($c,\$n); return _mul($c,$x, _pow($c,$n,$y));
+ }
+
+ # shortcut (faster) for shifting by 10) since we are in base 10eX
+ # multiples of $BASE_LEN:
+ my $src = scalar @$x; # source
+ my $len = _num($c,$y); # 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 spurios last zero element
+ splice @$x,-1 if $x->[-1] == 0;
+ $x;
+ }
+
+sub _pow
+ {
+ # power of $x to $y
+ # ref to array, ref to array, return ref to array
+ my ($c,$cx,$cy) = @_;
+
+ my $pow2 = _one();
+
+ my $y_bin = ${_as_bin($c,$cy)}; $y_bin =~ s/^0b//;
+ my $len = length($y_bin);
+ while (--$len > 0)
+ {
+ _mul($c,$pow2,$cx) if substr($y_bin,$len,1) eq '1'; # is odd?
+ _mul($c,$cx,$cx);
+ }
+
+ _mul($c,$cx,$pow2);
+ $cx;
+ }
+
+sub _fac
+ {
+ # factorial of $x
+ # ref to array, return ref to array
+ my ($c,$cx) = @_;
+
+ if ((@$cx == 1) && ($cx->[0] <= 2))
+ {
+ $cx->[0] = 1 * ($cx->[0]||1); # 0,1 => 1, 2 => 2
+ return $cx;
+ }
+
+ # go forward until $base is exceeded
+ # limit is either $x or $base (x == 100 means as result too high)
+ my $steps = 100; $steps = $cx->[0] if @$cx == 1;
+ my $r = 2; my $cf = 3; my $step = 1; my $last = $r;
+ while ($r < $BASE && $step < $steps)
+ {
+ $last = $r; $r *= $cf++; $step++;
+ }
+ if ((@$cx == 1) && ($step == $cx->[0]))
+ {
+ # completely done
+ $cx = [$last];
+ return $cx;
+ }
+ # now we must do the left over steps
+
+ # do so as long as n has more than one element
+ my $n = $cx->[0];
+ # 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;
+ $cx = [$last];
+ if (scalar @$cx == 1)
+ {
+ my $n = _copy($c,$cx);
+ # no need to test for $steps, since $steps is a scalar and we stop before
+ while (scalar @$n != 1)
+ {
+ if ($cx->[0] == 0)
+ {
+ $zero_elements ++; shift @$cx;
+ }
+ _mul($c,$cx,$n); _dec($c,$n);
+ }
+ $n = $n->[0]; # "convert" to scalar
+ }
+
+ # the left over steps will fit into a scalar, so we can speed it up
+ while ($n != $step)
+ {
+ if ($cx->[0] == 0)
+ {
+ $zero_elements ++; shift @$cx;
+ }
+ _mul($c,$cx,[$n]); $n--;
+ }
+ # multiply in the zeros again
+ while ($zero_elements-- > 0)
+ {
+ unshift @$cx, 0;
+ }
+ $cx;
+ }
+
+# 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,$x) = @_;
+
+ if (scalar @$x == 1)
+ {
+ # fit's into one Perl scalar, so result can be computed directly
+ $x->[0] = int(sqrt($x->[0]));
+ return $x;
+ }
+ my $y = _copy($c,$x);
+ # hopefully _len/2 is < $BASE, the -1 is to always undershot the guess
+ # since our guess will "grow"
+ my $l = int((_len($c,$x)-1) / 2);
+
+ my $lastelem = $x->[-1]; # for guess
+ my $elems = scalar @$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 $x (instead of _lsft($c,$x,$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 ($l >= 0); # all other digits of guess are zero
+
+ print "start x= ",${_str($c,$x)},"\n" if DEBUG;
+ my $two = _two();
+ my $last = _zero();
+ my $lastlast = _zero();
+ $steps = 0 if DEBUG;
+ while (_acmp($c,$last,$x) != 0 && _acmp($c,$lastlast,$x) != 0)
+ {
+ $steps++ if DEBUG;
+ $lastlast = _copy($c,$last);
+ $last = _copy($c,$x);
+ _add($c,$x, _div($c,_copy($c,$y),$x));
+ _div($c,$x, $two );
+ print " x= ",${_str($c,$x)},"\n" if DEBUG;
+ }
+ print "\nsteps in sqrt: $steps, " if DEBUG;
+ _dec($c,$x) if _acmp($c,$y,_mul($c,_copy($c,$x),$x)) < 0; # overshot?
+ print " final ",$x->[-1],"\n" if DEBUG;
+ $x;
+ }
+
+sub _root
+ {
+ # take n'th root of $x in place (n >= 3)
+ my ($c,$x,$n) = @_;
+
+ if (scalar @$x == 1)
+ {
+ if (scalar @$n > 1)
+ {
+ # result will always be smaller than 2 so trunc to 1 at once
+ $x->[0] = 1;
+ }
+ else
+ {
+ # fit's into one Perl scalar, so result can be computed directly
+ $x->[0] = int( $x->[0] ** (1 / $n->[0]) );
+ }
+ return $x;
+ }
+
+ # X is more than one element
+ # if $n is a power of two, we can repeatedly take sqrt($X) and find the
+ # proper result, because sqrt(sqrt($x)) == root($x,4)
+ my $b = _as_bin($c,$n);
+ if ($$b =~ /0b1(0+)/)
+ {
+ my $count = CORE::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), $x is now one element to big, again. In the next
+ # round we make that two, again.
+ _sqrt($c,$x);
+ }
+ # $x is now one element to big, so truncate result by removing it
+ splice (@$x,0,1);
+ }
+ else
+ {
+ # Should compute a guess of the result (by rule of thumb), then improve it
+ # via Newton's method or something similiar.
+ # XXX TODO
+ warn ('_root() not fully implemented in Calc.');
+ }
+ $x;
+ }
+
+##############################################################################
+# binary stuff
+
+sub _and
+ {
+ my ($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 $x if _acmp($c,$x,$y) == 0; # shortcut
+
+ my $m = _one(); my ($xr,$yr);
+ my $mask = $AND_MASK;
+
+ my $x1 = $x;
+ my $y1 = _copy($c,$y); # make copy
+ $x = _zero();
+ my ($b,$xrr,$yrr);
+ use integer;
+ while (!_is_zero($c,$x1) && !_is_zero($c,$y1))
+ {
+ ($x1, $xr) = _div($c,$x1,$mask);
+ ($y1, $yr) = _div($c,$y1,$mask);
+
+ # make ints() from $xr, $yr
+ # this is when the AND_BITS are greater tahn $BASE and is slower for
+ # small (<256 bits) numbers, but faster for large numbers. Disabled
+ # due to KISS principle
+
+# $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; }
+# $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; }
+# _add($c,$x, _mul($c, _new( $c, \($xrr & $yrr) ), $m) );
+
+ # 0+ due to '&' doesn't work in strings
+ _add($c,$x, _mul($c, [ 0+$xr->[0] & 0+$yr->[0] ], $m) );
+ _mul($c,$m,$mask);
+ }
+ $x;
+ }
+
+sub _xor
+ {
+ my ($c,$x,$y) = @_;
+
+ return _zero() if _acmp($c,$x,$y) == 0; # shortcut (see -and)
+
+ my $m = _one(); my ($xr,$yr);
+ my $mask = $XOR_MASK;
+
+ my $x1 = $x;
+ my $y1 = _copy($c,$y); # make copy
+ $x = _zero();
+ my ($b,$xrr,$yrr);
+ use integer;
+ while (!_is_zero($c,$x1) && !_is_zero($c,$y1))
+ {
+ ($x1, $xr) = _div($c,$x1,$mask);
+ ($y1, $yr) = _div($c,$y1,$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 *= $BASE; }
+ #_add($c,$x, _mul($c, _new( $c, \($xrr ^ $yrr) ), $m) );
+
+ # 0+ due to '^' doesn't work in strings
+ _add($c,$x, _mul($c, [ 0+$xr->[0] ^ 0+$yr->[0] ], $m) );
+ _mul($c,$m,$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
+ _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1);
+ _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1);
+
+ $x;
+ }
+
+sub _or
+ {
+ my ($c,$x,$y) = @_;
+
+ return $x if _acmp($c,$x,$y) == 0; # shortcut (see _and)
+
+ my $m = _one(); my ($xr,$yr);
+ my $mask = $OR_MASK;
+
+ my $x1 = $x;
+ my $y1 = _copy($c,$y); # make copy
+ $x = _zero();
+ my ($b,$xrr,$yrr);
+ use integer;
+ while (!_is_zero($c,$x1) && !_is_zero($c,$y1))
+ {
+ ($x1, $xr) = _div($c,$x1,$mask);
+ ($y1, $yr) = _div($c,$y1,$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 *= $BASE; }
+# _add($c,$x, _mul($c, _new( $c, \($xrr | $yrr) ), $m) );
+
+ # 0+ due to '|' doesn't work in strings
+ _add($c,$x, _mul($c, [ 0+$xr->[0] | 0+$yr->[0] ], $m) );
+ _mul($c,$m,$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
+ _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1);
+ _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1);
+
+ $x;
+ }
+
+sub _as_hex
+ {
+ # convert a decimal number to hex (ref to array, return ref to string)
+ my ($c,$x) = @_;
+
+ # fit's into one element
+ if (@$x == 1)
+ {
+ my $t = '0x' . sprintf("%x",$x->[0]);
+ return \$t;
+ }
+
+ my $x1 = _copy($c,$x);
+
+ my $es = '';
+ my ($xr, $h, $x10000);
+ if ($] >= 5.006)
+ {
+ $x10000 = [ 0x10000 ]; $h = 'h4';
+ }
+ else
+ {
+ $x10000 = [ 0x1000 ]; $h = 'h3';
+ }
+ while (! _is_zero($c,$x1))
+ {
+ ($x1, $xr) = _div($c,$x1,$x10000);
+ $es .= unpack($h,pack('v',$xr->[0]));
+ }
+ $es = reverse $es;
+ $es =~ s/^[0]+//; # strip leading zeros
+ $es = '0x' . $es;
+ \$es;
+ }
+
+sub _as_bin
+ {
+ # convert a decimal number to bin (ref to array, return ref to string)
+ my ($c,$x) = @_;
+
+ # fit's into one element
+ if (@$x == 1)
+ {
+ my $t = '0b' . sprintf("%b",$x->[0]);
+ return \$t;
+ }
+ my $x1 = _copy($c,$x);
+
+ my $es = '';
+ my ($xr, $b, $x10000);
+ if ($] >= 5.006)
+ {
+ $x10000 = [ 0x10000 ]; $b = 'b16';
+ }
+ else
+ {
+ $x10000 = [ 0x1000 ]; $b = 'b12';
+ }
+ while (! _is_zero($c,$x1))
+ {
+ ($x1, $xr) = _div($c,$x1,$x10000);
+ $es .= unpack($b,pack('v',$xr->[0]));
+ }
+ $es = reverse $es;
+ $es =~ s/^[0]+//; # strip leading zeros
+ $es = '0b' . $es;
+ \$es;
+ }
+
+sub _from_hex
+ {
+ # convert a hex number to decimal (ref to string, return ref to array)
+ my ($c,$hs) = @_;
+
+ my $mul = _one();
+ my $m = [ 0x10000 ]; # 16 bit at a time
+ my $x = _zero();
+
+ my $len = length($$hs)-2;
+ $len = int($len/4); # 4-digit parts, w/o '0x'
+ my $val; my $i = -4;
+ while ($len >= 0)
+ {
+ $val = substr($$hs,$i,4);
+ $val =~ s/^[+-]?0x// if $len == 0; # for last part only because
+ $val = hex($val); # hex does not like wrong chars
+ $i -= 4; $len --;
+ _add ($c, $x, _mul ($c, [ $val ], $mul ) ) if $val != 0;
+ _mul ($c, $mul, $m ) if $len >= 0; # skip last mul
+ }
+ $x;
+ }
+
+sub _from_bin
+ {
+ # convert a hex number to decimal (ref to string, return ref to array)
+ my ($c,$bs) = @_;
+
+ # instead of converting 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 = unpack('H*', pack ('B*', $hs)); # repack as hex
+ return $c->_from_hex(\('0x'.$h));
+
+ my $mul = _one();
+ my $m = [ 0x100 ]; # 8 bit at a time
+ my $x = _zero();
+
+ my $len = length($$bs)-2;
+ $len = int($len/8); # 4-digit parts, w/o '0x'
+ my $val; my $i = -8;
+ while ($len >= 0)
+ {
+ $val = substr($$bs,$i,8);
+ $val =~ s/^[+-]?0b// if $len == 0; # for last part only
+
+ $val = ord(pack('B8',substr('00000000'.$val,-8,8)));
+
+ $i -= 8; $len --;
+ _add ($c, $x, _mul ($c, [ $val ], $mul ) ) if $val != 0;
+ _mul ($c, $mul, $m ) if $len >= 0; # skip last mul
+ }
+ $x;
+ }
+
+##############################################################################
+# special modulus functions
+
+sub _modinv
+ {
+ # modular inverse
+ my ($c,$x,$y) = @_;
+
+ my $u = _zero($c); my $u1 = _one($c);
+ my $a = _copy($c,$y); my $b = _copy($c,$x);
+
+ # Euclid's Algorithm for bgcd(), only that we calc bgcd() ($a) and the
+ # result ($u) at the same time. See comments in BigInt for why this works.
+ my $q;
+ ($a, $q, $b) = ($b, _div($c,$a,$b)); # step 1
+ my $sign = 1;
+ while (!_is_zero($c,$b))
+ {
+ my $t = _add($c, # step 2:
+ _mul($c,_copy($c,$u1), $q) , # t = u1 * q
+ $u ); # + u
+ $u = $u1; # u = u1, u1 = t
+ $u1 = $t;
+ $sign = -$sign;
+ ($a, $q, $b) = ($b, _div($c,$a,$b)); # step 1
+ }
+
+ # if the gcd is not 1, then return NaN
+ return (undef,undef) unless _is_one($c,$a);
+
+ $sign = $sign == 1 ? '+' : '-';
+ ($u1,$sign);
+ }
+
+sub _modpow
+ {
+ # modulus of power ($x ** $y) % $z
+ my ($c,$num,$exp,$mod) = @_;
+
+ # in the trivial case,
+ if (_is_one($c,$mod))
+ {
+ splice @$num,0,1; $num->[0] = 0;
+ return $num;
+ }
+ if ((scalar @$num == 1) && (($num->[0] == 0) || ($num->[0] == 1)))
+ {
+ $num->[0] = 1;
+ return $num;
+ }
+
+# $num = _mod($c,$num,$mod); # this does not make it faster
+
+ my $acc = _copy($c,$num); my $t = _one();
+
+ my $expbin = ${_as_bin($c,$exp)}; $expbin =~ s/^0b//;
+ my $len = length($expbin);
+ while (--$len >= 0)
+ {
+ if ( substr($expbin,$len,1) eq '1') # is_odd
+ {
+ _mul($c,$t,$acc);
+ $t = _mod($c,$t,$mod);
+ }
+ _mul($c,$acc,$acc);
+ $acc = _mod($c,$acc,$mod);
+ }
+ @$num = @$t;
+ $num;
+ }
+
+##############################################################################
+##############################################################################
+
1;
__END__
@@ -843,21 +1788,25 @@ Math::BigInt::Calc - Pure Perl module to support Math::BigInt
Provides support for big integer calculations. Not intended to be used by other
modules (except Math::BigInt::Cached). Other modules which sport the same
-functions can also be used to support Math::Bigint, like Math::BigInt::Pari.
+functions can also be used to support Math::BigInt, like Math::BigInt::Pari.
=head1 DESCRIPTION
-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 call:
+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';
-=head1 EXPORT
+'libname' is either the long name ('Math::BigInt::Pari'), or only the short
+version like 'Pari'.
-The following functions MUST be defined in order to support
-the use by Math::BigInt:
+=head1 STORAGE
+
+=head1 METHODS
+
+The following functions MUST be defined in order to support the use by
+Math::BigInt:
_new(string) return ref to new object from ref to decimal string
_zero() return a new object with value 0
@@ -875,6 +1824,8 @@ the use by Math::BigInt:
In list context, returns (result,remainder).
NOTE: this is integer math, so no
fractional part will be returned.
+ The second operand will be not be 0, so no need to
+ check for that.
_sub(obj,obj) Simple subtraction of 1 object from another
a third, optional parameter indicates that the params
are swapped. In this case, the first param needs to
@@ -900,8 +1851,8 @@ the use by Math::BigInt:
return 0 for ok, otherwise error message as string
The following functions are optional, and can be defined if the underlying lib
-has a fast way to do them. If undefined, Math::BigInt will use a pure, but
-slow, Perl way as fallback to emulate these:
+has a fast way to do them. If undefined, Math::BigInt will use pure Perl (hence
+slow) fallback routines to emulate these:
_from_hex(str) return ref to new object from ref to hexadecimal string
_from_bin(str) return ref to new object from ref to binary string
@@ -924,28 +1875,36 @@ slow, Perl way as fallback to emulate these:
_or(obj1,obj2) OR (bit-wise) object 1 with object 2
_mod(obj,obj) Return remainder of div of the 1st by the 2nd object
- _sqrt(obj) return the square root of object
+ _sqrt(obj) return the square root of object (truncated to int)
+ _root(obj) return the n'th (n >= 3) root of obj (truncated to int)
+ _fac(obj) return factorial of object 1 (1*2*3*4..)
_pow(obj,obj) return object 1 to the power of object 2
_gcd(obj,obj) return Greatest Common Divisor of two objects
_zeros(obj) return number of trailing decimal zeros
+ _modinv return inverse modulus
+ _modpow return modulus of power ($x ** $y) % $z
Input strings come in as unsigned but with prefix (i.e. as '123', '0xabc'
or '0b1101').
-Testing of input parameter validity is done by the caller, so you need not
-worry about underflow (f.i. in C<_sub()>, C<_dec()>) nor about division by
-zero or similar cases.
+So the library needs only to deal with unsigned big integers. Testing of input
+parameter validity is done by the caller, so you need not worry about
+underflow (f.i. in C<_sub()>, C<_dec()>) nor about division by zero or similar
+cases.
The first parameter can be modified, that includes the possibility that you
return a reference to a completely different object instead. Although keeping
the reference and just changing it's contents is prefered over creating and
returning a different reference.
-Return values are always references to objects or strings. Exceptions are
-C<_lsft()> and C<_rsft()>, which return undef if they can not shift the
-argument. This is used to delegate shifting of bases different than 10 back
-to Math::BigInt, which will use some generic code to calculate the result.
+Return values are always references to objects, strings, or true/false for
+comparisation routines.
+
+Exceptions are C<_lsft()> and C<_rsft()>, which return undef if they can not
+shift the argument. This is used to delegate shifting of bases different than
+the one you can support back to Math::BigInt, which will use some generic code
+to calculate the result.
=head1 WRAP YOUR OWN
@@ -970,12 +1929,13 @@ the same terms as Perl itself.
=head1 AUTHORS
Original math code by Mark Biggar, rewritten by Tels L
-in late 2000, 2001.
+in late 2000.
Seperated from BigInt and shaped API with the help of John Peacock.
+Fixed/enhanced by Tels 2001-2002.
=head1 SEE ALSO
L, L, L,
-L, L and L.
+L, L and L.
=cut