| package Math::BigInt::Calc; |
| |
| use 5.006002; |
| use strict; |
| # use warnings; # dont use warnings for older Perls |
| |
| our $VERSION = '1.997'; |
| |
| # 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 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 |
| { |
| # (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 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 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 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] <=> $cy->[0]) <=> 0) |
| if scalar @$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. 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 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 $y too big |
| if (scalar @$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])); # 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 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 (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); |
| $cx; |
| } |
| |
| sub _nok { |
| # Return binomial coefficient (n over k). |
| # Given refs to arrays, return ref to array. |
| # First input argument is modified. |
| |
| my ($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 modifing 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__ |
| |
| =pod |
| |
| =head1 NAME |
| |
| Math::BigInt::Calc - Pure Perl module to support Math::BigInt |
| |
| =head1 SYNOPSIS |
| |
| This library provides support for big integer calculations. It is not |
| intended to be used by other modules. Other modules which support the same |
| API (see below) can also be used to support Math::BigInt, like |
| Math::BigInt::GMP and Math::BigInt::Pari. |
| |
| =head1 DESCRIPTION |
| |
| In this library, the numbers are represented in base B = 10**N, where N is |
| the largest possible value that does not cause overflow in the intermediate |
| computations. The base B elements are stored in an array, with the least |
| significant element stored in array element zero. There are no leading zero |
| elements, except a single zero element when the number is zero. |
| |
| For instance, if B = 10000, the number 1234567890 is represented internally |
| as [3456, 7890, 12]. |
| |
| =head1 THE Math::BigInt API |
| |
| In order to allow for multiple big integer libraries, Math::BigInt was |
| rewritten to use a plug-in library for core math routines. Any module which |
| conforms to the API can be used by Math::BigInt by using this in your program: |
| |
| use Math::BigInt lib => 'libname'; |
| |
| 'libname' is either the long name, like 'Math::BigInt::Pari', or only the short |
| version, like 'Pari'. |
| |
| =head2 General Notes |
| |
| A library only needs to deal with unsigned big integers. Testing of input |
| parameter validity is done by the caller, so there is no need to worry about |
| underflow (e.g., in C<_sub()> and C<_dec()>) nor about division by zero (e.g., |
| in C<_div()>) or similar cases. |
| |
| For some methods, the first parameter can be modified. That includes the |
| possibility that you return a reference to a completely different object |
| instead. Although keeping the reference and just changing its contents is |
| preferred over creating and returning a different reference. |
| |
| Return values are always objects, strings, Perl scalars, or true/false for |
| comparison routines. |
| |
| =head2 API version 1 |
| |
| The following methods must be defined in order to support the use by |
| Math::BigInt v1.70 or later. |
| |
| =head3 API version |
| |
| =over 4 |
| |
| =item I<api_version()> |
| |
| Return API version as a Perl scalar, 1 for Math::BigInt v1.70, 2 for |
| Math::BigInt v1.83. |
| |
| =back |
| |
| =head3 Constructors |
| |
| =over 4 |
| |
| =item I<_new(STR)> |
| |
| Convert a string representing an unsigned decimal number to an object |
| representing the same number. The input is normalize, i.e., it matches |
| C<^(0|[1-9]\d*)$>. |
| |
| =item I<_zero()> |
| |
| Return an object representing the number zero. |
| |
| =item I<_one()> |
| |
| Return an object representing the number one. |
| |
| =item I<_two()> |
| |
| Return an object representing the number two. |
| |
| =item I<_ten()> |
| |
| Return an object representing the number ten. |
| |
| =item I<_from_bin(STR)> |
| |
| Return an object given a string representing a binary number. The input has a |
| '0b' prefix and matches the regular expression C<^0[bB](0|1[01]*)$>. |
| |
| =item I<_from_oct(STR)> |
| |
| Return an object given a string representing an octal number. The input has a |
| '0' prefix and matches the regular expression C<^0[1-7]*$>. |
| |
| =item I<_from_hex(STR)> |
| |
| Return an object given a string representing a hexadecimal number. The input |
| has a '0x' prefix and matches the regular expression |
| C<^0x(0|[1-9a-fA-F][\da-fA-F]*)$>. |
| |
| =back |
| |
| =head3 Mathematical functions |
| |
| Each of these methods may modify the first input argument, except I<_bgcd()>, |
| which shall not modify any input argument, and I<_sub()> which may modify the |
| second input argument. |
| |
| =over 4 |
| |
| =item I<_add(OBJ1, OBJ2)> |
| |
| Returns the result of adding OBJ2 to OBJ1. |
| |
| =item I<_mul(OBJ1, OBJ2)> |
| |
| Returns the result of multiplying OBJ2 and OBJ1. |
| |
| =item I<_div(OBJ1, OBJ2)> |
| |
| Returns the result of dividing OBJ1 by OBJ2 and truncating the result to an |
| integer. |
| |
| =item I<_sub(OBJ1, OBJ2, FLAG)> |
| |
| =item I<_sub(OBJ1, OBJ2)> |
| |
| Returns the result of subtracting OBJ2 by OBJ1. If C<flag> is false or omitted, |
| OBJ1 might be modified. If C<flag> is true, OBJ2 might be modified. |
| |
| =item I<_dec(OBJ)> |
| |
| Decrement OBJ by one. |
| |
| =item I<_inc(OBJ)> |
| |
| Increment OBJ by one. |
| |
| =item I<_mod(OBJ1, OBJ2)> |
| |
| Return OBJ1 modulo OBJ2, i.e., the remainder after dividing OBJ1 by OBJ2. |
| |
| =item I<_sqrt(OBJ)> |
| |
| Return the square root of the object, truncated to integer. |
| |
| =item I<_root(OBJ, N)> |
| |
| Return Nth root of the object, truncated to int. N is E<gt>= 3. |
| |
| =item I<_fac(OBJ)> |
| |
| Return factorial of object (1*2*3*4*...). |
| |
| =item I<_pow(OBJ1, OBJ2)> |
| |
| Return OBJ1 to the power of OBJ2. By convention, 0**0 = 1. |
| |
| =item I<_modinv(OBJ1, OBJ2)> |
| |
| Return modular multiplicative inverse, i.e., return OBJ3 so that |
| |
| (OBJ3 * OBJ1) % OBJ2 = 1 % OBJ2 |
| |
| The result is returned as two arguments. If the modular multiplicative |
| inverse does not exist, both arguments are undefined. Otherwise, the |
| arguments are a number (object) and its sign ("+" or "-"). |
| |
| The output value, with its sign, must either be a positive value in the |
| range 1,2,...,OBJ2-1 or the same value subtracted OBJ2. For instance, if the |
| input arguments are objects representing the numbers 7 and 5, the method |
| must either return an object representing the number 3 and a "+" sign, since |
| (3*7) % 5 = 1 % 5, or an object representing the number 2 and "-" sign, |
| since (-2*7) % 5 = 1 % 5. |
| |
| =item I<_modpow(OBJ1, OBJ2, OBJ3)> |
| |
| Return modular exponentiation, (OBJ1 ** OBJ2) % OBJ3. |
| |
| =item I<_rsft(OBJ, N, B)> |
| |
| Shift object N digits right in base B and return the resulting object. This is |
| equivalent to performing integer division by B**N and discarding the remainder, |
| except that it might be much faster, depending on how the number is represented |
| internally. |
| |
| For instance, if the object $obj represents the hexadecimal number 0xabcde, |
| then C<_rsft($obj, 2, 16)> returns an object representing the number 0xabc. The |
| "remainer", 0xde, is discarded and not returned. |
| |
| =item I<_lsft(OBJ, N, B)> |
| |
| Shift the object N digits left in base B. This is equivalent to multiplying by |
| B**N, except that it might be much faster, depending on how the number is |
| represented internally. |
| |
| =item I<_log_int(OBJ, B)> |
| |
| Return integer log of OBJ to base BASE. This method has two output arguments, |
| the OBJECT and a STATUS. The STATUS is Perl scalar; it is 1 if OBJ is the exact |
| result, 0 if the result was truncted to give OBJ, and undef if it is unknown |
| whether OBJ is the exact result. |
| |
| =item I<_gcd(OBJ1, OBJ2)> |
| |
| Return the greatest common divisor of OBJ1 and OBJ2. |
| |
| =back |
| |
| =head3 Bitwise operators |
| |
| Each of these methods may modify the first input argument. |
| |
| =over 4 |
| |
| =item I<_and(OBJ1, OBJ2)> |
| |
| Return bitwise and. If necessary, the smallest number is padded with leading |
| zeros. |
| |
| =item I<_or(OBJ1, OBJ2)> |
| |
| Return bitwise or. If necessary, the smallest number is padded with leading |
| zeros. |
| |
| =item I<_xor(OBJ1, OBJ2)> |
| |
| Return bitwise exclusive or. If necessary, the smallest number is padded |
| with leading zeros. |
| |
| =back |
| |
| =head3 Boolean operators |
| |
| =over 4 |
| |
| =item I<_is_zero(OBJ)> |
| |
| Returns a true value if OBJ is zero, and false value otherwise. |
| |
| =item I<_is_one(OBJ)> |
| |
| Returns a true value if OBJ is one, and false value otherwise. |
| |
| =item I<_is_two(OBJ)> |
| |
| Returns a true value if OBJ is two, and false value otherwise. |
| |
| =item I<_is_ten(OBJ)> |
| |
| Returns a true value if OBJ is ten, and false value otherwise. |
| |
| =item I<_is_even(OBJ)> |
| |
| Return a true value if OBJ is an even integer, and a false value otherwise. |
| |
| =item I<_is_odd(OBJ)> |
| |
| Return a true value if OBJ is an even integer, and a false value otherwise. |
| |
| =item I<_acmp(OBJ1, OBJ2)> |
| |
| Compare OBJ1 and OBJ2 and return -1, 0, or 1, if OBJ1 is less than, equal |
| to, or larger than OBJ2, respectively. |
| |
| =back |
| |
| =head3 String conversion |
| |
| =over 4 |
| |
| =item I<_str(OBJ)> |
| |
| Return a string representing the object. The returned string should have no |
| leading zeros, i.e., it should match C<^(0|[1-9]\d*)$>. |
| |
| =item I<_as_bin(OBJ)> |
| |
| Return the binary string representation of the number. The string must have a |
| '0b' prefix. |
| |
| =item I<_as_oct(OBJ)> |
| |
| Return the octal string representation of the number. The string must have |
| a '0x' prefix. |
| |
| Note: This method was required from Math::BigInt version 1.78, but the required |
| API version number was not incremented, so there are older libraries that |
| support API version 1, but do not support C<_as_oct()>. |
| |
| =item I<_as_hex(OBJ)> |
| |
| Return the hexadecimal string representation of the number. The string must |
| have a '0x' prefix. |
| |
| =back |
| |
| =head3 Numeric conversion |
| |
| =over 4 |
| |
| =item I<_num(OBJ)> |
| |
| Given an object, return a Perl scalar number (int/float) representing this |
| number. |
| |
| =back |
| |
| =head3 Miscellaneous |
| |
| =over 4 |
| |
| =item I<_copy(OBJ)> |
| |
| Return a true copy of the object. |
| |
| =item I<_len(OBJ)> |
| |
| Returns the number of the decimal digits in the number. The output is a |
| Perl scalar. |
| |
| =item I<_zeros(OBJ)> |
| |
| Return the number of trailing decimal zeros. The output is a Perl scalar. |
| |
| =item I<_digit(OBJ, N)> |
| |
| Return the Nth digit as a Perl scalar. N is a Perl scalar, where zero refers to |
| the rightmost (least significant) digit, and negative values count from the |
| left (most significant digit). If $obj represents the number 123, then |
| I<_digit($obj, 0)> is 3 and I<_digit(123, -1)> is 1. |
| |
| =item I<_check(OBJ)> |
| |
| Return a true value if the object is OK, and a false value otherwise. This is a |
| check routine to test the internal state of the object for corruption. |
| |
| =back |
| |
| =head2 API version 2 |
| |
| The following methods are required for an API version of 2 or greater. |
| |
| =head3 Constructors |
| |
| =over 4 |
| |
| =item I<_1ex(N)> |
| |
| Return an object representing the number 10**N where N E<gt>= 0 is a Perl |
| scalar. |
| |
| =back |
| |
| =head3 Mathematical functions |
| |
| =over 4 |
| |
| =item I<_nok(OBJ1, OBJ2)> |
| |
| Return the binomial coefficient OBJ1 over OBJ1. |
| |
| =back |
| |
| =head3 Miscellaneous |
| |
| =over 4 |
| |
| =item I<_alen(OBJ)> |
| |
| Return the approximate number of decimal digits of the object. The |
| output is one Perl scalar. This estimate must be greater than or equal |
| to what C<_len()> returns. |
| |
| =back |
| |
| =head2 API optional methods |
| |
| The following methods are optional, and can be defined if the underlying lib |
| has a fast way to do them. If undefined, Math::BigInt will use pure Perl (hence |
| slow) fallback routines to emulate these: |
| |
| =head3 Signed bitwise operators. |
| |
| Each of these methods may modify the first input argument. |
| |
| =over 4 |
| |
| =item I<_signed_or(OBJ1, OBJ2, SIGN1, SIGN2)> |
| |
| Return the signed bitwise or. |
| |
| =item I<_signed_and(OBJ1, OBJ2, SIGN1, SIGN2)> |
| |
| Return the signed bitwise and. |
| |
| =item I<_signed_xor(OBJ1, OBJ2, SIGN1, SIGN2)> |
| |
| Return the signed bitwise exclusive or. |
| |
| =back |
| |
| =head1 WRAP YOUR OWN |
| |
| If you want to port your own favourite c-lib for big numbers to the |
| Math::BigInt interface, you can take any of the already existing modules as |
| a rough guideline. You should really wrap up the latest BigInt and BigFloat |
| testsuites with your module, and replace in them any of the following: |
| |
| use Math::BigInt; |
| |
| by this: |
| |
| use Math::BigInt lib => 'yourlib'; |
| |
| This way you ensure that your library really works 100% within Math::BigInt. |
| |
| =head1 LICENSE |
| |
| This program is free software; you may redistribute it and/or modify it under |
| the same terms as Perl itself. |
| |
| =head1 AUTHORS |
| |
| =over 4 |
| |
| =item * |
| |
| Original math code by Mark Biggar, rewritten by Tels L<http://bloodgate.com/> |
| in late 2000. |
| |
| =item * |
| |
| Separated from BigInt and shaped API with the help of John Peacock. |
| |
| =item * |
| |
| Fixed, speed-up, streamlined and enhanced by Tels 2001 - 2007. |
| |
| =item * |
| |
| API documentation corrected and extended by Peter John Acklam, |
| E<lt>pjacklam@online.noE<gt> |
| |
| =back |
| |
| =head1 SEE ALSO |
| |
| L<Math::BigInt>, L<Math::BigFloat>, |
| L<Math::BigInt::GMP>, L<Math::BigInt::FastCalc> and L<Math::BigInt::Pari>. |
| |
| =cut |
