Calc.pm 78.9 KB
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package Math::BigInt::Calc;

use 5.006002;
use strict;
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use warnings;
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our $VERSION = '1.999709';
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# 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;
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# This works on x86, but fails on ARM (SA1100, iPAQ) due to who knows what
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# 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, connected values.
  # Used only by the testsuite, the set variant is used only by the BEGIN
  # block below:
  shift;

  my ($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 $BASE_LEN unless wantarray;
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      return ($BASE_LEN, $BASE, $AND_BITS, $XOR_BITS, $OR_BITS, $BASE_LEN, $MAX_VAL,);
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      }

    # 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 $BASE_LEN unless wantarray;
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  return ($BASE_LEN, $BASE, $AND_BITS, $XOR_BITS, $OR_BITS, $BASE_LEN, $MAX_VAL);
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  }

sub _new
  {
  # (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 $il = length($_[1])-1;

  # < BASE_LEN due len-1 above
  return [ int($_[1]) ] if $il < $BASE_LEN;	# shortcut for short numbers

  # 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)), $_[1])) ];
  }                                                                             

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) + 0;
    $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)

  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 ));

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  # We can compute the approximate length no faster than the real length:
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  *_alen = \&_len;
  }

###############################################################################

sub _zero
  {
  # create a zero
  [ 0 ];
  }

sub _one
  {
  # create a one
  [ 1 ];
  }

sub _two
  {
  # create a two (used internally for shifting)
  [ 2 ];
  }

sub _ten
  {
  # create a 10 (used internally for shifting)
  [ 10 ];
  }

sub _1ex
  {
  # create a 1Ex
  my $rem = $_[1] % $BASE_LEN;		# remainder
  my $parts = $_[1] / $BASE_LEN;	# parts

  # 000000, 000000, 100 
  [ (0) x $parts, '1' . ('0' x $rem) ];
  }

sub _copy
  {
  # make a true copy
  [ @{$_[1]} ];
  }

# catch and throw away
sub import { }

##############################################################################
# convert back to string and number

sub _str
  {
  # (ref to BINT) return num_str
  # Convert number from internal base 100000 format to string format.
  # internal format is always normalized (no leading zeros, "-0" => "+0")
  my $ar = $_[1];

  my $l = scalar @$ar;				# number of parts
  if ($l < 1)					# should not happen
    {
    require Carp;
    Carp::croak("$_[1] has no elements");
    }

  my $ret = "";
  # handle first one different to strip leading zeros from it (there are no
  # leading zero parts in internal representation)
  $l --; $ret .= int($ar->[$l]); $l--;
  # 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 ($l >= 0)
    {
    $ret .= substr($z.$ar->[$l],-$BASE_LEN); # fastest way I could think of
    $l--;
    }
  $ret;
  }                                                                             

sub _num
  {
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    # Make a Perl scalar number (int/float) from a BigInt object.
    my $x = $_[1];
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    return 0 + $x->[0] if scalar @$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];
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    }
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    return $num;
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  }

##############################################################################
# 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
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  # This routine modifies array x, but not y.
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  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 nec. 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
  # 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) == 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 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
  {
  # (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
  $yv = [@$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
  $yv = [@$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) = @_;
  
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  # the general div algorithm here is about O(N*N) and thus quite slow, so
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  # 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 * $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;
    $rem = [@$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

      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)
        {
        $rem = [ 0 ];                   # a = 0 => x == y => rem 0
        $rem = [@$x] if $a != 0;        # a < 0 => x < y => rem = x
        splice(@$x,1);                  # keep single element
        $x->[0] = 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 $y = [ @$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, $q);
    }
  if (wantarray) 
    {
    @d = ();
    if ($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;
    my $d = \@d; 
    __strip_zeros($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;
793
  # the general div algorithm here is about O(N*N) and thus quite slow, so
794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983
  # 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 * $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;
    $rem = [@$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

      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)
        {
        $rem = [ 0 ];			# a = 0 => x == y => rem 0
        $rem = [@$x] if $a != 0;	# a < 0 => x < y => rem = x
        splice(@$x,1);			# keep single element
        $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 $y = [ @$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, $q);
    }
  if (wantarray) 
    {
    @d = ();
    if ($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;
    my $d = \@d; 
    __strip_zeros($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) = @_;

984
  # the general div algorithm here is about O(N*N) and thus quite slow, so
985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213
  # 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 * $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;
    $rem = [@$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

      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)
        {
        $rem = [ 0 ];			# a = 0 => x == y => rem 0
        $rem = [@$x] if $a != 0;	# a < 0 => x < y => rem = x
        splice(@$x,1);			# keep single element
        $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 $y = [ @$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, $q);
    }
  if (wantarray) 
    {
    @d = ();
    if ($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;
    my $d = \@d; 
    __strip_zeros($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] <=> $cy->[0]) <=> 0) 
   if scalar @$cx == scalar @$cy && scalar @$cx == 1;

  # fast comp based on number of array elements (aka pseudo-length)
  my $lxy = (scalar @$cx - scalar @$cy)
  # or length of first element if same number of elements (aka difference 0)
    ||
  # need int() here because sometimes the last element is '00018' vs '18'
   (length(int($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 $j = scalar @$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
  {
1214 1215
  # 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.
1216 1217 1218 1219
  my ($c,$x,$n) = @_;

  my $len = _len('',$x);

1220 1221 1222 1223 1224 1225
  $n += $len if $n < 0;                 # -1 last, -2 second-to-last
  return "0" if $n < 0 || $n >= $len;   # return 0 for digits out of range

  my $elem = int($n / $BASE_LEN);       # which array element
  my $digit = $n % $BASE_LEN;           # which digit in this element
  substr("$x->[$elem]", -$digit-1, 1);
1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270
  }

sub _zeros
  {
  # return amount of trailing zeros in decimal
  # check each array elem in _m for having 0 at end as long as elem == 0
  # Upon finding a elem != 0, stop
  my $x = $_[1];

  return 0 if scalar @$x == 1 && $x->[0] == 0;

  my $zeros = 0; my $elem;
  foreach my $e (@$x)
    {
    if ($e != 0)
      {
      $elem = "$e";				# preserve x
      $elem =~ s/.*?(0*$)/$1/;			# strip anything not zero
      $zeros *= $BASE_LEN;			# elems * 5
      $zeros += length($elem);			# count trailing zeros
      last;					# early out
      }
    $zeros ++;					# real else branch: 50% slower!
    }
  $zeros;
  }

##############################################################################
# _is_* routines

sub _is_zero
  {
  # return true if arg is zero 
  (((scalar @{$_[1]} == 1) && ($_[1]->[0] == 0))) <=> 0;
  }

sub _is_even
  {
  # return true if arg is even
  (!($_[1]->[0] & 1)) <=> 0; 
  }

sub _is_odd
  {
  # return true if arg is odd
1271
  (($_[1]->[0] & 1)) <=> 0;
1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357
  }

sub _is_one
  {
  # return true if arg is one
  (scalar @{$_[1]} == 1) && ($_[1]->[0] == 1) <=> 0; 
  }

sub _is_two
  {
  # return true if arg is two 
  (scalar @{$_[1]} == 1) && ($_[1]->[0] == 2) <=> 0; 
  }

sub _is_ten
  {
  # return true if arg is ten 
  (scalar @{$_[1]} == 1) && ($_[1]->[0] == 10) <=> 0; 
  }

sub __strip_zeros
  {
  # internal normalization function that strips leading zeros from the array
  # args: ref to array
  my $s = shift;
 
  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 $i\n";
  # '0', '3', '4', '0', '0',
  #  0    1    2    3    4
  # cnt = 5, i = 4
  # i = 4
  # i = 3
  # => fcnt = cnt - i (5-2 => 3, cnt => 5-1 = 4, throw away from 4th pos)
  # >= 1: skip first part (this can be zero)
  while ($i > 0) { last if $s->[$i] != 0; $i--; }
  $i++; splice @$s,$i if ($i < $cnt); # $i cant be 0
  $s;                                                                    
  }                                                                             

###############################################################################
# check routine to test internal state for corruptions

sub _check
  {
  # used by the test suite
  my $x = $_[1];

  return "$x is not a reference" if !ref($x);

  # are all parts are valid?
  my $i = 0; my $j = scalar @$x; my ($e,$try);
  while ($i < $j)
    {
    $e = $x->[$i]; $e = 'undef' unless defined $e;
    $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e)";
    last if $e !~ /^[+]?[0-9]+$/;
    $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (stringify)";
    last if "$e" !~ /^[+]?[0-9]+$/;
    $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (cat-stringify)";
    last if '' . "$e" !~ /^[+]?[0-9]+$/;
    $try = ' < 0 || >= $BASE; '."($x, $e)";
    last if $e <0 || $e >= $BASE;
    # this test is disabled, since new/bnorm and certain ops (like early out
    # in add/sub) are allowed/expected to leave '00000' in some elements
    #$try = '=~ /^00+/; '."($x, $e)";
    #last if $e =~ /^00+/;
    $i++;
    }
  return "Illegal part '$e' at pos $i (tested: $try)" if $i < $j;
  0;
  }


###############################################################################

sub _mod
  {
  # if possible, use mod shortcut
  my ($c,$x,$yo) = @_;

1358
  # slow way since $y too big
1359 1360 1361
  if (scalar @$yo > 1)
    {
    my ($xo,$rem) = _div($c,$x,$yo);
1362 1363
    @$x = @$rem;
    return $x;
1364 1365 1366
    }

  my $y = $yo->[0];
1367 1368

  # if both are single element arrays
1369 1370 1371 1372 1373 1374
  if (scalar @$x == 1)
    {
    $x->[0] %= $y;
    return $x;
    }

1375
  # if @$x has more than one element, but @$y is a single element
1376 1377 1378 1379 1380 1381 1382 1383 1384 1385
  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)
    {
1386 1387
    # else need to go through all elements in @$x: O(N), but loop is a bit
    # simplified
1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398
    my $r = 0;
    foreach (@$x)
      {
      $r = ($r + $_) % $y;		# not much faster, but heh...
      #$r += $_ % $y; $r %= $y;
      }
    $r = 0 if $r == $y;
    $x->[0] = $r;
    }
  else
    {
1399 1400 1401
    # else need to go through all elements in @$x: O(N)
    my $r = 0;
    my $bm = 1;
1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414
    foreach (@$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] = $r;
    }
1415 1416
  @$x = $x->[0];		# keep one element of @$x
  return $x;
1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498
  }

##############################################################################
# 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; }
1499
  # fix spurious last zero element
1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 1537 1538 1539
  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) = @_;

  if (scalar @$cy == 1 && $cy->[0] == 0)
    {
    splice (@$cx,1); $cx->[0] = 1;		# y == 0 => x => 1
    return $cx;
    }
  if ((scalar @$cx == 1 && $cx->[0] == 1) ||	#    x == 1
      (scalar @$cy == 1 && $cy->[0] == 1))	# or y == 1
    {
    return $cx;
    }
  if (scalar @$cx == 1 && $cx->[0] == 0)
    {
    splice (@$cx,1); $cx->[0] = 0;		# 0 ** y => 0 (if not y <= 0)
    return $cx;
    }

  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;
  }

1540 1541 1542 1543 1544 1545
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) = @_;
1546

1547 1548
    # 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.
1549 1550

    {
1551 1552 1553 1554
        my $twok = _mul($c, _two($c), _copy($c, $k));   # 2 * k
        if (_acmp($c, $twok, $n) > 0) {                 # if 2*k > n
            $k = _sub($c, _copy($c, $n), $k);           # k = n - k
        }
1555
    }
1556 1557 1558 1559 1560 1561 1562 1563 1564

    # 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 (_is_zero($c, $k)) {
        @$n = 1;
1565
    }
1566 1567 1568

    else {

1569
        # Make a copy of the original n, since we'll be modifying n in-place.
1570 1571 1572 1573 1574 1575 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601

        my $n_orig = _copy($c, $n);

        # n = 5, f = 6, d = 2 (cf. example above)

        _sub($c, $n, $k);
        _inc($c, $n);

        my $f = _copy($c, $n);
        _inc($c, $f);

        my $d = _two($c);

        # while f <= n (the original n, that is) ...

        while (_acmp($c, $f, $n_orig) <= 0) {

            # n = (n * f / d) == 5 * 6 / 2 (cf. example above)

            _mul($c, $n, $f);
            _div($c, $n, $d);

            # f = 7, d = 3 (cf. example above)

            _inc($c, $f);
            _inc($c, $d);
        }

    }

    return $n;
}
1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650 1651 1652 1653 1654 1655 1656 1657 1658 1659 1660 1661 1662 1663 1664 1665 1666 1667 1668 1669 1670 1671 1672 1673 1674 1675 1676 1677 1678 1679 1680 1681 1682 1683 1684 1685 1686 1687 1688 1689 1690 1691 1692 1693 1694 1695 1696 1697 1698 1699 1700 1701 1702 1703 1704 1705 1706 1707 1708 1709 1710 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720 1721 1722 1723 1724 1725 1726 1727 1728 1729 1730 1731 1732 1733 1734 1735 1736 1737 1738 1739 1740 1741 1742 1743 1744 1745 1746 1747 1748 1749 1750 1751 1752 1753 1754 1755 1756 1757 1758 1759 1760 1761 1762 1763 1764 1765 1766 1767 1768 1769 1770 1771 1772 1773 1774 1775 1776 1777 1778 1779 1780 1781 1782 1783 1784 1785 1786 1787 1788 1789 1790 1791 1792 1793 1794 1795 1796 1797 1798 1799 1800 1801 1802 1803 1804 1805 1806 1807 1808 1809 1810 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846 1847 1848 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 2033 2034 2035 2036 2037 2038 2039 2040 2041 2042 2043 2044 2045 2046 2047 2048 2049 2050 2051 2052 2053 2054 2055 2056 2057 2058 2059 2060 2061 2062 2063 2064 2065 2066 2067

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 to 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 $k = _num($c,$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 $new_x = _new($c, $k * $even);
  @$cx = @$new_x;
  if ($cx->[0] == 0)
    {
    $zero_elements ++; shift @$cx;
    }
#  print STDERR "x = ", _str($c,$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 < $BASE)
      {
      _mul($c,$cx,[$m]);
      }
    else
      {
      _mul($c,$cx,$c->_new($m));
      }
    if ($cx->[0] == 0)
      {
      $zero_elements ++; shift @$cx;
      }
#    print STDERR "Calculate $k2 - $sum = $m (x = ", _str($c,$cx),")\n";
    }
  # multiply in the zeros again
  unshift @$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 $n;					# steps still to do
  if (scalar @$cx == 1)
    {
    $n = $cx->[0];
    }
  else
    {
    $n = _copy($c,$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); $step += 2;
      _mul($c,$cx,[$b]);
      }
    $step = [$step];
    while (_acmp($c,$step,$n) <= 0)
      {
      if ($cx->[0] == 0)
        {
        $zero_elements ++; shift @$cx;
        }
      _mul($c,$cx,$step); _inc($c,$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); $step += 2;
      _mul($c,$cx,[$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); $step += 2;
      _mul($c,$cx,[$b]);
      }
    # do what's left over
    while ($step <= $n)
      {
      _mul($c,$cx,[$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 (scalar @$x == 1 && $x->[0] == 0);
  # BASE 0 or 1 => NaN
  return if (scalar @$base == 1 && $base->[0] < 2);
  my $cmp = _acmp($c,$x,$base); # X == BASE => 1
  if ($cmp == 0)
    {
    splice (@$x,1); $x->[0] = 1;
    return ($x,1)
    }
  # X < BASE
  if ($cmp < 0)
    {
    splice (@$x,1); $x->[0] = 0;
    return ($x,undef);
    }

  my $x_org = _copy($c,$x);		# preserve x
  splice(@$x,1); $x->[0] = 1;		# keep ref to $x

  # Compute a guess for the result based on:
  # $guess = int ( length_in_base_10(X) / ( log(base) / log(10) ) )
  my $len = _len($c,$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):
  $log += ((scalar @$base)-1) * $BASE_LEN;

  # calculate now a guess based on the values obtained above:
  my $res = int($len / $log);

  $x->[0] = $res;
  my $trial = _pow ($c, _copy($c, $base), $x);
  my $a = _acmp($c,$trial,$x_org);

#  print STDERR "# trial ", _str($c,$x)," was: $a (0 = exact, -1 too small, +1 too big)\n";

  # found an exact result?
  return ($x,1) if $a == 0;

  if ($a > 0)
    {
    # or too big
    _div($c,$trial,$base); _dec($c, $x);
    while (($a = _acmp($c,$trial,$x_org)) > 0)
      {
#      print STDERR "# big _log_int at ", _str($c,$x), "\n"; 
      _div($c,$trial,$base); _dec($c, $x);
      }
    # result is now exact (a == 0), or too small (a < 0)
    return ($x, $a == 0 ? 1 : 0);
    }

  # else: result was to small
  _mul($c,$trial,$base);

  # did we now get the right result?
  $a = _acmp($c,$trial,$x_org);

  if ($a == 0)				# yes, exactly
    {
    _inc($c, $x);
    return ($x,1); 
    }
  return ($x,0) if $a > 0;  

  # Result still too small (we should come here only if the estimate above
  # was very off base):
 
  # Now let the normal trial run obtain the real result
  # Simple loop that increments $x by 2 in each step, possible overstepping
  # the real result

  my $base_mul = _mul($c, _copy($c,$base), $base);	# $base * $base

  while (($a = _acmp($c,$trial,$x_org)) < 0)
    {
#    print STDERR "# small _log_int at ", _str($c,$x), "\n"; 
    _mul($c,$trial,$base_mul); _add($c, $x, [2]);
    }

  my $exact = 1;
  if ($a > 0)
    {
    # overstepped the result
    _dec($c, $x);
    _div($c,$trial,$base);
    $a = _acmp($c,$trial,$x_org);
    if ($a > 0)
      {
      _dec($c, $x);
      }
    $exact = 0 if $a != 0;		# a = -1 => not exact result, a = 0 => exact
    }
  
  ($x,$exact);				# return result
  }

# 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)
    {
    # fits 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
      $x->[0] = 1;
      }
    else
      {
      # fits into one Perl scalar, so result can be computed directly
      # cannot use int() here, because it rounds wrongly (try 
      # (81 ** 3) ** (1/3) to see what I mean)
      #$x->[0] = int( $x->[0] ** (1 / $n->[0]) );
      # round to 8 digits, then truncate result to integer
      $x->[0] = int ( sprintf ("%.8f", $x->[0] ** (1 / $n->[0]) ) );
      }
    return $x;
    } 

  # we know now that X is more than one element long

  # 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
    {
    # trial computation by starting with 2,4,8,16 etc until we overstep
    my $step;
    my $trial = _two();

    # while still to do more than X steps
    do
      {
      $step = _two();
      while (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) < 0)
        {
        _mul ($c, $step, [2]);
        _add ($c, $trial, $step);
        }

      # hit exactly?
      if (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) == 0)
        {
        @$x = @$trial;			# make copy while preserving ref to $x
        return $x;
        }
      # overstepped, so go back on step
      _sub($c, $trial, $step);
      } while (scalar @$step > 1 || $step->[0] > 128);

    # reset step to 2
    $step = _two();
    # add two, because $trial cannot be exactly the result (otherwise we would
2068
    # already have found it)
2069 2070 2071 2072 2073 2074 2075 2076 2077 2078 2079 2080 2081 2082 2083 2084 2085 2086 2087 2088 2089 2090 2091 2092 2093 2094 2095 2096 2097 2098 2099 2100 2101 2102 2103 2104 2105 2106 2107 2108 2109 2110 2111 2112 2113 2114 2115 2116 2117 2118 2119 2120 2121 2122 2123 2124 2125 2126 2127 2128 2129 2130 2131 2132 2133 2134 2135 2136 2137 2138 2139 2140 2141 2142 2143 2144 2145 2146 2147 2148 2149 2150 2151 2152 2153 2154 2155 2156 2157 2158 2159 2160 2161 2162 2163 2164 2165 2166 2167 2168 2169 2170 2171 2172 2173 2174 2175 2176 2177 2178 2179 2180 2181 2182 2183 2184 2185 2186 2187 2188 2189 2190 2191 2192 2193 2194 2195 2196 2197 2198 2199 2200 2201 2202 2203 2204 2205 2206 2207 2208 2209 2210 2211 2212 2213 2214 2215 2216 2217 2218 2219 2220 2221 2222 2223 2224 2225 2226 2227 2228 2229 2230 2231 2232 2233 2234 2235 2236 2237 2238 2239 2240 2241 2242 2243 2244 2245 2246 2247 2248 2249 2250 2251 2252 2253 2254 2255 2256 2257 2258 2259 2260 2261 2262 2263 2264 2265 2266 2267 2268 2269 2270 2271 2272 2273 2274 2275 2276 2277 2278 2279 2280 2281 2282 2283 2284 2285 2286 2287 2288 2289 2290 2291 2292 2293 2294 2295 2296 2297 2298 2299 2300 2301 2302 2303 2304 2305 2306 2307 2308 2309 2310 2311 2312 2313 2314 2315 2316 2317 2318 2319 2320 2321 2322 2323 2324 2325 2326 2327 2328 2329 2330 2331 2332 2333 2334 2335 2336 2337 2338 2339 2340 2341 2342 2343 2344 2345 2346 2347 2348 2349 2350 2351 2352 2353 2354 2355 2356 2357 2358 2359 2360 2361 2362 2363 2364 2365 2366 2367 2368 2369 2370 2371 2372 2373 2374 2375 2376 2377 2378 2379 2380 2381 2382 2383 2384 2385
    _add($c, $trial, $step);
 
    # and now add more and more (2,4,6,8,10 etc)
    while (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) < 0)
      {
      _add ($c, $trial, $step);
      }

    # hit not exactly? (overstepped)
    if (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) > 0)
      {
      _dec($c,$trial);
      }

    # hit not exactly? (overstepped)
    # 80 too small, 81 slightly too big, 82 too big
    if (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) > 0)
      {
      _dec ($c, $trial); 
      }

    @$x = @$trial;			# make copy while preserving ref to $x
    return $x;
    }
  $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 than $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) = @_;

  # fits into one element (handle also 0x0 case)
  return sprintf("0x%x",$x->[0]) if @$x == 1;

  my $x1 = _copy($c,$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, $xr) = _div($c,$x1,$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 $x1 = _copy($c,$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, $xr) = _div($c,$x1,$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 $x1 = _copy($c,$x);

  my $es = '';
  my $xr;
  my $x1000 = [ 0100000 ];
  while (@$x1 != 1 || $x1->[0] != 0)		# _is_zero()
    {
    ($x1, $xr) = _div($c,$x1,$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 = _one();
  my $x = _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 = [ $val ];
    _add ($c, $x, _mul ($c, $adder, $mul ) ) if $val != 0;
    _mul ($c, $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 $m = _new($c, 0x10000000);			# 28 bit at a time (<32 bit!)
  my $d = 7;					# 7 digits at a time
  if ($] <= 5.006)
    {
    # for older Perls, play safe
    $m = [ 0x10000 ];				# 16 bit at a time (<32 bit!)
    $d = 4;					# 4 digits at a time
    }

  my $mul = _one();
  my $x = _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)
      {
      $adder = _new($c,$val);
      }
    _add ($c, $x, _mul ($c, $adder, $mul ) ) if $val != 0;
    _mul ($c, $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
  {
2386
  # modular multiplicative inverse
2387 2388
  my ($c,$x,$y) = @_;

2389 2390 2391 2392
  # modulo zero
  if (_is_zero($c, $y)) {
      return (undef, undef);
  }
2393

2394 2395 2396 2397 2398 2399 2400 2401 2402 2403 2404 2405
  # modulo one
  if (_is_one($c, $y)) {
      return (_zero($c), '+');
  }

  my $u = _zero($c);
  my $v = _one($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.
2406 2407
  my $q;
  my $sign = 1;
2408 2409 2410 2411 2412 2413 2414 2415 2416 2417 2418 2419
  {
      ($a, $q, $b) = ($b, _div($c, $a, $b));        # step 1
      last if _is_zero($c, $b);

      my $t = _add($c,                              # step 2:
                   _mul($c, _copy($c, $v), $q) ,    #  t =   v * q
                   $u );                            #      + u
      $u = $v;                                      #  u = v
      $v = $t;                                      #  v = t
      $sign = -$sign;
      redo;
  }
2420 2421

  # if the gcd is not 1, then return NaN
2422 2423 2424
  return (undef, undef) unless _is_one($c, $a);

  ($v, $sign == 1 ? '+' : '-');
2425 2426 2427 2428 2429 2430 2431
  }

sub _modpow
  {
  # modulus of power ($x ** $y) % $z
  my ($c,$num,$exp,$mod) = @_;

2432
  # a^b (mod 1) = 0 for all a and b
2433 2434
  if (_is_one($c,$mod))
    {
2435 2436
        @$num = 0;
        return $num;
2437 2438
    }

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  # 0^a (mod m) = 0 if m != 0, a != 0
  # 0^0 (mod m) = 1 if m != 0
  if (_is_zero($c, $num)) {
      if (_is_zero($c, $exp)) {
          @$num = 1;
      } else {
          @$num = 0;
      }
      return $num;
  }

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#  $num = _mod($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;
  }

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sub _gcd {
    # Greatest common divisor.
2472

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    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;
2485
    }
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    # Until $y is zero ...

    until (@$y == 1 && $y->[0] == 0) {

        # Compute remainder.

        _mod($c, $x, $y);

        # Swap $x and $y.

        my $tmp = [ @$x ];
        @$x = @$y;
        $y = $tmp;      # no deref here; that would modify input $y
    }

    return $x;
}
2504 2505 2506 2507 2508 2509 2510

##############################################################################
##############################################################################

1;
__END__

2511 2512
=pod

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=head1 NAME

Math::BigInt::Calc - Pure Perl module to support Math::BigInt

=head1 SYNOPSIS

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This library provides support for big integer calculations. It is not
intended to be used by other modules. Other modules which support the same
API (see below) can also be used to support Math::BigInt, like
Math::BigInt::GMP and Math::BigInt::Pari.
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=head1 DESCRIPTION

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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 [3456, 7890, 12].

=head1 THE Math::BigInt API

2537
In order to allow for multiple big integer libraries, Math::BigInt was
2538 2539
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:
2540 2541 2542

	use Math::BigInt lib => 'libname';

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