package Math::BigInt::Calc;

use 5.006002;
use strict;
# use warnings;	# dont use warnings for older Perls

our $VERSION = '0.56';

# Package to store unsigned big integers in decimal and do math with them

# Internally the numbers are stored in an array with at least 1 element, no
# leading zero parts (except the first) and in base 1eX where X is determined
# automatically at loading time to be the maximum possible value

# todo:
# - fully remove funky $# stuff in div() (maybe - that code scares me...)

# USE_MUL: due to problems on certain os (os390, posix-bc) "* 1e-5" is used
# instead of "/ 1e5" at some places, (marked with USE_MUL). Other platforms
# BS2000, some Crays need USE_DIV instead.
# The BEGIN block is used to determine which of the two variants gives the
# correct result.

# Beware of things like:
# $i = $i * $y + $car; $car = int($i / $BASE); $i = $i % $BASE;
# This works on x86, but fails on ARM (SA1100, iPAQ) due to whoknows what
# reasons. So, use this instead (slower, but correct):
# $i = $i * $y + $car; $car = int($i / $BASE); $i -= $BASE * $car;

##############################################################################
# global constants, flags and accessory

# announce that we are compatible with MBI v1.83 and up
sub api_version () { 2; }
 
# constants for easier life
my ($BASE,$BASE_LEN,$RBASE,$MAX_VAL);
my ($AND_BITS,$XOR_BITS,$OR_BITS);
my ($AND_MASK,$XOR_MASK,$OR_MASK);

sub _base_len 
  {
  # Set/get the BASE_LEN and assorted other, 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 $BASE_LEN unless wantarray;
      return ($BASE_LEN, $BASE, $AND_BITS, $XOR_BITS, $OR_BITS, $BASE_LEN, $MAX_VAL,);
      }

    # find whether we can use mul or div in mul()/div()
    $BASE_LEN = $b+1;
    my $caught = 0;
    while (--$BASE_LEN > 5)
      {
      $BASE = int("1e".$BASE_LEN);
      $RBASE = abs('1e-'.$BASE_LEN);			# see USE_MUL
      $caught = 0;
      $caught += 1 if (int($BASE * $RBASE) != 1);	# should be 1
      $caught += 2 if (int($BASE / $BASE) != 1);	# should be 1
      last if $caught != 3;
      }
    $BASE = int("1e".$BASE_LEN);
    $RBASE = abs('1e-'.$BASE_LEN);			# see USE_MUL
    $MAX_VAL = $BASE-1;
   
    # ($caught & 1) != 0 => cannot use MUL
    # ($caught & 2) != 0 => cannot use DIV
    if ($caught == 2)				# 2
      {
      # must USE_MUL since we cannot use DIV
      *_mul = \&_mul_use_mul;
      *_div = \&_div_use_mul;
      }
    else					# 0 or 1
      {
      # can USE_DIV instead
      *_mul = \&_mul_use_div;
      *_div = \&_div_use_div;
      }
    }
  return $BASE_LEN unless wantarray;
  return ($BASE_LEN, $BASE, $AND_BITS, $XOR_BITS, $OR_BITS, $BASE_LEN, $MAX_VAL);
  }

sub _new
  {
  # (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 ));

  # We can compute the approximate lenght no faster than the real length:
  *_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
  {
  # Make a number (scalar int/float) from a BigInt object 
  my $x = $_[1];

  return 0+$x->[0] if scalar @$x == 1;  # below $BASE
  my $fac = 1;
  my $num = 0;
  foreach (@$x)
    {
    $num += $fac*$_; $fac *= $BASE;
    }
  $num; 
  }

##############################################################################
# actual math code

sub _add
  {
  # (ref to int_num_array, ref to int_num_array)
  # routine to add two base 1eX numbers
  # stolen from Knuth Vol 2 Algorithm A pg 231
  # there are separate routines to add and sub as per Knuth pg 233
  # This routine 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 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) = @_;
  
  # 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 * $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;
  # 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 * $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) = @_;

  # 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 * $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
  {
  # return the nth digit, negative values count backward
  # zero is rightmost, so _digit(123,0) will give 3
  my ($c,$x,$n) = @_;

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

  $n = $len+$n if $n < 0;		# -1 last, -2 second-to-last
  $n = abs($n);				# if negative was too big
  $len--; $n = $len if $n > $len;	# n to big?
  
  my $elem = int($n / $BASE_LEN);	# which array element
  my $digit = $n % $BASE_LEN;		# which digit in this element
  $elem = '0' x $BASE_LEN . @$x[$elem];	# get element padded with 0's
  substr($elem,-$digit-1,1);
  }

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
  (($_[1]->[0] & 1)) <=> 0; 
  }

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) = @_;

  # 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 through 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 through 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);		# keep one element of $x
  $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) = @_;

  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;
  }

sub _nok
  {
  # n over k
  # ref to array, return ref to array
  my ($c,$n,$k) = @_;

  # ( 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))
    {
  my $x = _copy($c,$n);
  _sub($c,$n,$k);
    _inc($c,$n);
    my $f = _copy($c,$n); _inc($c,$f);		# n = 5, f = 6, d = 2
    my $d = _two($c);
    while (_acmp($c,$f,$x) <= 0)		# f <= n ?
      {
      # n = (n * f / d) == 5 * 6 / 2
      $n = _mul($c,$n,$f); $n = _div($c,$n,$d);
      # f = 7, d = 3
      _inc($c,$f); _inc($c,$d);
      }
    }
  else 
    {
    # keep ref to $n and set it to 1
    splice (@$n,1); $n->[0] = 1;
    }
  $n;
  }

my @factorials = (
  1,
  1,
  2,
  2*3,
  2*3*4,
  2*3*4*5,
  2*3*4*5*6,
  2*3*4*5*6*7,
);

sub _fac
  {
  # factorial of $x
  # ref to array, return ref to array
  my ($c,$cx) = @_;

  if ((@$cx == 1) && ($cx->[0] <= 7))
    {
    $cx->[0] = $factorials[$cx->[0]];		# 0 => 1, 1 => 1, 2 => 2 etc.
    return $cx;
    }

  if ((@$cx == 1) && 		# we do this only if $x >= 12 and $x <= 7000
      ($cx->[0] >= 12 && $cx->[0] < 7000))
    {

  # Calculate (k-j) * (k-j+1) ... k .. (k+j-1) * (k + j)
  # See http://blogten.blogspot.com/2007/01/calculating-n.html
  # The above series can be expressed as factors:
  #   k * k - (j - i) * 2
  # We cache k*k, and calculate (j * j) as the sum of the first j odd integers

  # This will not work when N exceeds the storage of a Perl scalar, however,
  # in this case the algorithm would be way 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…