#line 1 "Math/BigInt/Calc.pm"
package Math::BigInt::Calc;

use 5.006001;
use strict;
use warnings;

our $VERSION = '1.999722';
$VERSION = eval $VERSION;

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

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

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

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

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

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

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

sub _base_len 
  {
  # Set/get the BASE_LEN and assorted other, 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 {
    # Given a string representing an integer, returns a reference to an array
    # of integers, where each integer represents a chunk of the original input
    # integer. Assumes normalized value as input.

    my ($proto, $str) = @_;

    my $input_len = length($str) - 1;

    # Shortcut for small numbers.
    return [ int($str) ] if $input_len < $BASE_LEN;

    my $format = "a" . (($input_len % $BASE_LEN) + 1);
    $format .= $] < 5.008 ? "a$BASE_LEN" x int($input_len / $BASE_LEN)
                          : "(a$BASE_LEN)*";

    [ reverse(map { 0 + $_ } unpack($format, $str)) ];
}

BEGIN
  {
  # from Daniel Pfeiffer: determine largest group of digits that is precisely
  # multipliable with itself plus carry
  # Test now changed to expect the proper pattern, not a result off by 1 or 2
  my ($e, $num) = 3;	# lowest value we will use is 3+1-1 = 3
  do {
      $num = '9' x ++$e;
      $num *= $num + 1;
  } while $num =~ /9{$e}0{$e}/;		# must be a certain pattern
  $e--; 				# last test failed, so retract one step
  # the limits below brush the problems with the test above under the rug:
  # the test should be able to find the proper $e automatically
  $e = 5 if $^O =~ /^uts/;	# UTS get's some special treatment
  $e = 5 if $^O =~ /^unicos/;	# unicos is also problematic (6 seems to work
				# there, but we play safe)

  my $int = 0;
  if ($e > 7)
    {
    use integer;
    my $e1 = 7;
    $num = 7;
    do 
      {
      $num = ('9' x ++$e1) + 0;
      $num *= $num + 1;
      } while ("$num" =~ /9{$e1}0{$e1}/);	# must be a certain pattern
    $e1--; 					# last test failed, so retract one step
    if ($e1 > 7)
      { 
      $int = 1; $e = $e1; 
      }
    }
 
  __PACKAGE__->_base_len($e,$int);	# set and store

  use integer;
  # find out how many bits _and, _or and _xor can take (old default = 16)
  # I don't think anybody has yet 128 bit scalars, so let's play safe.
  local $^W = 0;	# don't warn about 'nonportable number'
  $AND_BITS = 15; $XOR_BITS = 15; $OR_BITS = 15;

  # find max bits, we will not go higher than numberofbits that fit into $BASE
  # to make _and etc simpler (and faster for smaller, slower for large numbers)
  my $max = 16;
  while (2 ** $max < $BASE) { $max++; }
  {
    no integer;
    $max = 16 if $] < 5.006;	# older Perls might not take >16 too well
  }
  my ($x,$y,$z);
  do {
    $AND_BITS++;
    $x = CORE::oct('0b' . '1' x $AND_BITS); $y = $x & $x;
    $z = (2 ** $AND_BITS) - 1;
    } while ($AND_BITS < $max && $x == $z && $y == $x);
  $AND_BITS --;						# retreat one step
  do {
    $XOR_BITS++;
    $x = CORE::oct('0b' . '1' x $XOR_BITS); $y = $x ^ 0;
    $z = (2 ** $XOR_BITS) - 1;
    } while ($XOR_BITS < $max && $x == $z && $y == $x);
  $XOR_BITS --;						# retreat one step
  do {
    $OR_BITS++;
    $x = CORE::oct('0b' . '1' x $OR_BITS); $y = $x | $x;
    $z = (2 ** $OR_BITS) - 1;
    } while ($OR_BITS < $max && $x == $z && $y == $x);
  $OR_BITS --;						# retreat one step
  
  $AND_MASK = __PACKAGE__->_new( ( 2 ** $AND_BITS ));
  $XOR_MASK = __PACKAGE__->_new( ( 2 ** $XOR_BITS ));
  $OR_MASK = __PACKAGE__->_new( ( 2 ** $OR_BITS ));

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

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

sub _zero
  {
  # create a zero
  [ 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 {
    # Convert number from internal base 1eN format to string format. Internal
    # format is always normalized, i.e., no leading zeros.

    my $ary = $_[1];
    my $idx = $#$ary;             # index of last element

    if ($idx < 0) {               # should not happen
        require Carp;
        Carp::croak("$_[1] has no elements");
    }

    # Handle first one differently, since it should not have any leading zeros.
    my $ret = int($ary->[$idx]);
    if ($idx > 0) {
        $idx--;
        # Interestingly, the pre-padd method uses more time
        # the old grep variant takes longer (14 vs. 10 sec)
        my $z = '0' x ($BASE_LEN - 1);
        while ($idx >= 0) {
            $ret .= substr($z . $ary->[$idx], -$BASE_LEN);
            $idx--;
        }
    }
    $ret;
}

sub _num
  {
    # Make a Perl scalar number (int/float) from a BigInt object.
    my $x = $_[1];

    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];
    }
    return $num;
  }

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

sub _add {
    # (ref to int_num_array, ref to int_num_array)
    #
    # Routine to add two base 1eX numbers stolen from Knuth Vol 2 Algorithm A
    # pg 231. There are separate routines to add and sub as per Knuth pg 233.
    # This routine modifies array x, but not y.

    my ($c, $x, $y) = @_;

    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 necessary to modify $x in-place.
        @$x = @$y;
        return $x;
    }

    # For each in Y, add Y to X and carry. If after that, something is left in
    # X, foreach in X add carry to X and then return X, carry. Trades one
    # "$j++" for having to shift arrays.
    my $i;
    my $car = 0;
    my $j = 0;
    for $i (@$y) {
        $x->[$j] -= $BASE if $car = (($x->[$j] += $i + $car) >= $BASE) ? 1 : 0;
        $j++;
    }
    while ($car != 0) {
        $x->[$j] -= $BASE if $car = (($x->[$j] += $car) >= $BASE) ? 1 : 0;
        $j++;
    }
    $x;
}

sub _inc {
    # (ref to int_num_array, ref to int_num_array)
    # Add 1 to $x, modify $x in place
    my ($c, $x) = @_;

    for my $i (@$x) {
        return $x if ($i += 1) < $BASE;		# early out
        $i = 0;					# overflow, next
    }
    push @$x, 1 if $x->[-1] == 0;		# last overflowed, so extend
    $x;
}

sub _dec {
    # (ref to int_num_array, ref to int_num_array)
    # Sub 1 from $x, modify $x in place
    my ($c, $x) = @_;

    my $MAX = $BASE - 1;			# since MAX_VAL based on BASE
    for my $i (@$x) {
        last if ($i -= 1) >= 0;			# early out
        $i = $MAX;				# underflow, next
    }
    pop @$x if $x->[-1] == 0 && @$x > 1;	# last underflowed (but leave 0)
    $x;
}

sub _sub {
    # (ref to int_num_array, ref to int_num_array, swap)
    #
    # Subtract base 1eX numbers -- stolen from Knuth Vol 2 pg 232, $x > $y
    # subtract Y from X by modifying x in place
    my ($c, $sx, $sy, $s) = @_;

    my $car = 0;
    my $i;
    my $j = 0;
    if (!$s) {
        for $i (@$sx) {
            last unless defined $sy->[$j] || $car;
            $i += $BASE if $car = (($i -= ($sy->[$j] || 0) + $car) < 0);
            $j++;
        }
        # might leave leading zeros, so fix that
        return __strip_zeros($sx);
    }
    for $i (@$sx) {
        # We can't do an early out if $x < $y, since we need to copy the high
        # chunks from $y. Found by Bob Mathews.
        #last unless defined $sy->[$j] || $car;
        $sy->[$j] += $BASE
          if $car = ($sy->[$j] = $i - ($sy->[$j] || 0) - $car) < 0;
        $j++;
    }
    # might leave leading zeros, so fix that
    __strip_zeros($sy);
}

sub _mul_use_mul
  {
  # (ref to int_num_array, ref to int_num_array)
  # multiply two numbers in internal representation
  # modifies first arg, second need not be different from first
  my ($c,$xv,$yv) = @_;

  if (@$yv == 1)
    {
    # shortcut for two very short numbers (improved by Nathan Zook)
    # works also if xv and yv are the same reference, and handles also $x == 0
    if (@$xv == 1)
      {
      if (($xv->[0] *= $yv->[0]) >= $BASE)
         {
         $xv->[0] = $xv->[0] - ($xv->[1] = int($xv->[0] * $RBASE)) * $BASE;
         };
      return $xv;
      }
    # $x * 0 => 0
    if ($yv->[0] == 0)
      {
      @$xv = (0);
      return $xv;
      }
    # multiply a large number a by a single element one, so speed up
    my $y = $yv->[0]; my $car = 0;
    foreach my $i (@$xv)
      {
      $i = $i * $y + $car; $car = int($i * $RBASE); $i -= $car * $BASE;
      }
    push @$xv, $car if $car != 0;
    return $xv;
    }
  # shortcut for result $x == 0 => result = 0
  return $xv if ( ((@$xv == 1) && ($xv->[0] == 0)) ); 

  # since multiplying $x with $x fails, make copy in this case
  $yv = [@$xv] if $xv == $yv;	# same references?

  my @prod = (); my ($prod,$car,$cty,$xi,$yi);

  for $xi (@$xv)
    {
    $car = 0; $cty = 0;

    # slow variant
#    for $yi (@$yv)
#      {
#      $prod = $xi * $yi + ($prod[$cty] || 0) + $car;
#      $prod[$cty++] =
#       $prod - ($car = int($prod * RBASE)) * $BASE;  # see USE_MUL
#      }
#    $prod[$cty] += $car if $car; # need really to check for 0?
#    $xi = shift @prod;

    # faster variant
    # looping through this if $xi == 0 is silly - so optimize it away!
    $xi = (shift @prod || 0), next if $xi == 0;
    for $yi (@$yv)
      {
      $prod = $xi * $yi + ($prod[$cty] || 0) + $car;
##     this is actually a tad slower
##        $prod = $prod[$cty]; $prod += ($car + $xi * $yi);	# no ||0 here
      $prod[$cty++] =
       $prod - ($car = int($prod * $RBASE)) * $BASE;  # see USE_MUL
      }
    $prod[$cty] += $car if $car; # need really to check for 0?
    $xi = shift @prod || 0;	# || 0 makes v5.005_3 happy
    }
  push @$xv, @prod;
  # can't have leading zeros
#  __strip_zeros($xv);
  $xv;
  }                                                                             

sub _mul_use_div_64
  {
  # (ref to int_num_array, ref to int_num_array)
  # multiply two numbers in internal representation
  # modifies first arg, second need not be different from first
  # works for 64 bit integer with "use integer"
  my ($c,$xv,$yv) = @_;

  use integer;
  if (@$yv == 1)
    {
    # shortcut for two small numbers, also handles $x == 0
    if (@$xv == 1)
      {
      # shortcut for two very short numbers (improved by Nathan Zook)
      # works also if xv and yv are the same reference, and handles also $x == 0
      if (($xv->[0] *= $yv->[0]) >= $BASE)
          {
          $xv->[0] =
              $xv->[0] - ($xv->[1] = $xv->[0] / $BASE) * $BASE;
          };
      return $xv;
      }
    # $x * 0 => 0
    if ($yv->[0] == 0)
      {
      @$xv = (0);
      return $xv;
      }
    # multiply a large number a by a single element one, so speed up
    my $y = $yv->[0]; my $car = 0;
    foreach my $i (@$xv)
      {
      #$i = $i * $y + $car; $car = $i / $BASE; $i -= $car * $BASE;
      $i = $i * $y + $car; $i -= ($car = $i / $BASE) * $BASE;
      }
    push @$xv, $car if $car != 0;
    return $xv;
    }
  # shortcut for result $x == 0 => result = 0
  return $xv if ( ((@$xv == 1) && ($xv->[0] == 0)) ); 

  # since multiplying $x with $x fails, make copy in this case
  $yv = [@$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 algorithm here is about O(N*N) and thus quite slow, so
  # we first check for some special cases and use shortcuts to handle them.

  # This works, because we store the numbers in a chunked format where each
  # element contains 5..7 digits (depending on system).

  # if both numbers have only one element:
  if (@$x == 1 && @$yorg == 1)
    {
    # shortcut, $yorg and $x are two small numbers
    if (wantarray)
      {
      my $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 algorithm here is about O(N*N) and thus quite slow, so
  # we first check for some special cases and use shortcuts to handle them.

  # This works, because we store the numbers in a chunked format where each
  # element contains 5..7 digits (depending on system).

  # if both numbers have only one element:
  if (@$x == 1 && @$yorg == 1)
    {
    # shortcut, $yorg and $x are two small numbers
    if (wantarray)
      {
      my $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 algorithm here is about O(N*N) and thus quite slow, so
  # we first check for some special cases and use shortcuts to handle them.

  # This works, because we store the numbers in a chunked format where each
  # element contains 5..7 digits (depending on system).

  # if both numbers have only one element:
  if (@$x == 1 && @$yorg == 1)
    {
    # shortcut, $yorg and $x are two small numbers
    if (wantarray)
      {
      my $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] <=> $cy->[0]) <=> 0)
      if @$cx == @$cy && @$cx == 1;

    # fast comp based on number of array elements (aka pseudo-length)
    my $lxy = (@$cx - @$cy)
      # or length of first element if same number of elements (aka difference 0)
      ||
        # need int() here because sometimes the last element is '00018' vs '18'
        (length(int($cx->[-1])) - length(int($cy->[-1])));

    return -1 if $lxy < 0;      # already differs, ret
    return  1 if $lxy > 0;      # ditto

    # manual way (abort if unequal, good for early ne)
    my $a;
    my $j = @$cx;
    while (--$j >= 0) {
        last if $a = $cx->[$j] - $cy->[$j];
    }
    $a <=> 0;
}

sub _len {
    # compute number of digits in base 10

    # int() because add/sub sometimes leaves strings (like '00005') instead of
    # '5' in this place, thus causing length() to report wrong length
    my $cx = $_[1];

    (@$cx - 1) * $BASE_LEN + length(int($cx->[-1]));
}

sub _digit {
    # Return the nth digit. Zero is rightmost, so _digit(123,0) gives 3.
    # Negative values count from the left, so _digit(123, -1) gives 1.
    my ($c, $x, $n) = @_;

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

    $n += $len if $n < 0;               # -1 last, -2 second-to-last
    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);
}

sub _zeros {
    # Return number of trailing zeros in decimal.
    # Check each array element for having 0 at end as long as elem == 0
    # Upon finding a elem != 0, stop.

    my $x = $_[1];

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

    my $zeros = 0;
    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 
    @{$_[1]} == 1 && $_[1]->[0] == 0 ? 1 : 0;
}

sub _is_even {
    # return true if arg is even
    $_[1]->[0] & 1 ? 0 : 1;
}

sub _is_odd {
    # return true if arg is odd
    $_[1]->[0] & 1 ? 1 : 0;
}

sub _is_one {
    # return true if arg is one
    @{$_[1]} == 1 && $_[1]->[0] == 1 ? 1 : 0;
}

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

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

sub __strip_zeros {
    # Internal normalization function that strips leading zeros from the array.
    # Args: ref to array
    my $s = shift;

    my $cnt = @$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 $j = @$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 $y too big
    if (@$yo > 1) {
        my ($xo, $rem) = _div($c, $x, $yo);
        @$x = @$rem;
        return $x;
    }

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

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

    # if @$x has more than one element, but @$y is a single element
    my $b = $BASE % $y;
    if ($b == 0) {
        # when BASE % Y == 0 then (B * BASE) % Y == 0
        # (B * BASE) % $y + A % Y => A % Y
        # so need to consider only last element: O(1)
        $x->[0] %= $y;
    } elsif ($b == 1) {
        # else need to go through all elements in @$x: O(N), but loop is a bit
        # simplified
        my $r = 0;
        foreach (@$x) {
            $r = ($r + $_) % $y;        # not much faster, but heh...
            #$r += $_ % $y; $r %= $y;
        }
        $r = 0 if $r == $y;
        $x->[0] = $r;
    } else {
        # else need to go through all elements in @$x: O(N)
        my $r = 0;
        my $bm = 1;
        foreach (@$x) {
            $r = ($_ * $bm + $r) % $y;
            $bm = ($bm * $b) % $y;

            #$r += ($_ % $y) * $bm;
            #$bm *= $b;
            #$bm %= $y;
            #$r %= $y;
        }
        $r = 0 if $r == $y;
        $x->[0] = $r;
    }
    @$x = $x->[0];		# keep one element of @$x
    return $x;
}

##############################################################################
# shifts

sub _rsft {
    my ($c, $x, $y, $n) = @_;

    if ($n != 10) {
        $n = _new($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]));
    if ($src >= $xlen or ($src == $xlen and !defined $x->[1])) {
        # 12345 67890 shifted right by more than 10 digits => 0
        splice(@$x, 1);                 # leave only one element
        $x->[0] = 0;                    # set to zero
        return $x;
    }
    my $rem = $src % $BASE_LEN;         # remainder to shift
    $src = int($src / $BASE_LEN);       # source
    if ($rem == 0) {
        splice(@$x, 0, $src);           # even faster, 38.4 => 39.3
    } else {
        my $len = @$x - $src;           # elems to go
        my $vd;
        my $z = '0' x $BASE_LEN;
        $x->[@$x] = 0;                  # avoid || 0 test inside loop
        while ($dst < $len) {
            $vd = $z . $x->[$src];
            $vd = substr($vd, -$BASE_LEN, $BASE_LEN - $rem);
            $src++;
            $vd = substr($z . $x->[$src], -$rem, $rem) . $vd;
            $vd = substr($vd, -$BASE_LEN, $BASE_LEN) if length($vd) > $BASE_LEN;
            $x->[$dst] = int($vd);
            $dst++;
        }
        splice(@$x, $dst) if $dst > 0;          # kill left-over array elems
        pop @$x if $x->[-1] == 0 && @$x > 1;    # kill last element if 0
    }                                           # else rem == 0
    $x;
}

sub _lsft {
    my ($c, $x, $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 $src = @$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 spurious 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 (@$cy == 1 && $cy->[0] == 0) {
        splice(@$cx, 1);
        $cx->[0] = 1;                   # y == 0 => x => 1
        return $cx;
    }

    if ((@$cx == 1 && $cx->[0] == 1) ||     #    x == 1
        (@$cy == 1 && $cy->[0] == 1))       # or y == 1
    {
        return $cx;
    }

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

    my $pow2 = _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 {
    # Return binomial coefficient (n over k).
    # Given refs to arrays, return ref to array.
    # First input argument is modified.

    my ($c, $n, $k) = @_;

    # If k > n/2, or, equivalently, 2*k > n, compute nok(n, k) as
    # nok(n, n-k), to minimize the number if iterations in the loop.

    {
        my $twok = _mul($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
        }
    }

    # 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;
    }

    else {

        # Make a copy of the original n, since we'll be modifying n in-place.

        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;
}

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 $trial cannot be exactly the result (otherwise we would
    # already have found it)
    _add($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
  {
  # modular multiplicative inverse
  my ($c,$x,$y) = @_;

  # modulo zero
  if (_is_zero($c, $y)) {
      return (undef, undef);
  }

  # 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.
  my $q;
  my $sign = 1;
  {
      ($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;
  }

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

  ($v, $sign == 1 ? '+' : '-');
  }

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

  # a^b (mod 1) = 0 for all a and b
  if (_is_one($c,$mod))
    {
        @$num = 0;
        return $num;
    }

  # 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;
  }

#  $num = _mod($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;
  }

sub _gcd {
    # Greatest common divisor.

    my ($c, $x, $y) = @_;

    # gcd(0,0) = 0
    # gcd(0,a) = a, if a != 0

    if (@$x == 1 && $x->[0] == 0) {
        if (@$y == 1 && $y->[0] == 0) {
            @$x = 0;
        } else {
            @$x = @$y;
        }
        return $x;
    }

    # Until $y is zero ...

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

        # Compute remainder.

        _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;
}

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

1;
__END__

#line 3010
