1 package Math::BigInt::Calc;
5 # use warnings; # dont use warnings for older Perls
8 use vars qw/@ISA $VERSION/;
13 # Package to store unsigned big integers in decimal and do math with them
15 # Internally the numbers are stored in an array with at least 1 element, no
16 # leading zero parts (except the first) and in base 1eX where X is determined
17 # automatically at loading time to be the maximum possible value
20 # - fully remove funky $# stuff (maybe)
22 # USE_MUL: due to problems on certain os (os390, posix-bc) "* 1e-5" is used
23 # instead of "/ 1e5" at some places, (marked with USE_MUL). Other platforms
24 # BS2000, some Crays need USE_DIV instead.
25 # The BEGIN block is used to determine which of the two variants gives the
28 ##############################################################################
29 # global constants, flags and accessory
31 # constants for easier life
33 my ($MBASE,$BASE,$RBASE,$BASE_LEN,$MAX_VAL,$BASE_LEN2,$BASE_LEN_SMALL);
34 my ($AND_BITS,$XOR_BITS,$OR_BITS);
35 my ($AND_MASK,$XOR_MASK,$OR_MASK);
40 # set/get the BASE_LEN and assorted other, connected values
41 # used only be the testsuite, set is used only by the BEGIN block below
47 # find whether we can use mul or div or none in mul()/div()
48 # (in last case reduce BASE_LEN_SMALL)
49 $BASE_LEN_SMALL = $b+1;
51 while (--$BASE_LEN_SMALL > 5)
53 $MBASE = int("1e".$BASE_LEN_SMALL);
54 $RBASE = abs('1e-'.$BASE_LEN_SMALL); # see USE_MUL
56 $caught += 1 if (int($MBASE * $RBASE) != 1); # should be 1
57 $caught += 2 if (int($MBASE / $MBASE) != 1); # should be 1
60 # BASE_LEN is used for anything else than mul()/div()
61 $BASE_LEN = $BASE_LEN_SMALL;
62 $BASE_LEN = shift if (defined $_[0]); # one more arg?
63 $BASE = int("1e".$BASE_LEN);
65 $BASE_LEN2 = int($BASE_LEN_SMALL / 2); # for mul shortcut
66 $MBASE = int("1e".$BASE_LEN_SMALL);
67 $RBASE = abs('1e-'.$BASE_LEN_SMALL); # see USE_MUL
70 $LEN_CONVERT = 1 if $BASE_LEN_SMALL != $BASE_LEN;
72 #print "BASE_LEN: $BASE_LEN MAX_VAL: $MAX_VAL BASE: $BASE RBASE: $RBASE ";
73 #print "BASE_LEN_SMALL: $BASE_LEN_SMALL MBASE: $MBASE\n";
78 *{_mul} = \&_mul_use_mul;
79 *{_div} = \&_div_use_mul;
81 else # $caught must be 2, since it can't be 1 nor 3
84 *{_mul} = \&_mul_use_div;
85 *{_div} = \&_div_use_div;
88 return $BASE_LEN unless wantarray;
89 return ($BASE_LEN, $AND_BITS, $XOR_BITS, $OR_BITS, $BASE_LEN_SMALL, $MAX_VAL);
94 # from Daniel Pfeiffer: determine largest group of digits that is precisely
95 # multipliable with itself plus carry
96 # Test now changed to expect the proper pattern, not a result off by 1 or 2
97 my ($e, $num) = 3; # lowest value we will use is 3+1-1 = 3
100 $num = ('9' x ++$e) + 0;
102 } while ("$num" =~ /9{$e}0{$e}/); # must be a certain pattern
103 $e--; # last test failed, so retract one step
104 # the limits below brush the problems with the test above under the rug:
105 # the test should be able to find the proper $e automatically
106 $e = 5 if $^O =~ /^uts/; # UTS get's some special treatment
107 $e = 5 if $^O =~ /^unicos/; # unicos is also problematic (6 seems to work
108 # there, but we play safe)
109 $e = 8 if $e > 8; # cap, for VMS, OS/390 and other 64 bit systems
111 # determine how many digits fit into an integer and can be safely added
112 # together plus carry w/o causing an overflow
114 # this below detects 15 on a 64 bit system, because after that it becomes
115 # 1e16 and not 1000000 :/ I can make it detect 18, but then I get a lot of
116 # test failures. Ugh! (Tomake detect 18: uncomment lines marked with *)
118 my $bi = 5; # approx. 16 bit
119 $num = int('9' x $bi);
121 # while ( ($num+$num+1) eq '1' . '9' x $bi) # *
122 while ( int($num+$num+1) eq '1' . '9' x $bi)
124 $bi++; $num = int('9' x $bi);
125 # $bi++; $num *= 10; $num += 9; # *
127 $bi--; # back off one step
128 # by setting them equal, we ignore the findings and use the default
129 # one-size-fits-all approach from former versions
130 $bi = $e; # XXX, this should work always
132 __PACKAGE__->_base_len($e,$bi); # set and store
134 # find out how many bits _and, _or and _xor can take (old default = 16)
135 # I don't think anybody has yet 128 bit scalars, so let's play safe.
136 local $^W = 0; # don't warn about 'nonportable number'
137 $AND_BITS = 15; $XOR_BITS = 15; $OR_BITS = 15;
139 # find max bits, we will not go higher than numberofbits that fit into $BASE
140 # to make _and etc simpler (and faster for smaller, slower for large numbers)
142 while (2 ** $max < $BASE) { $max++; }
146 $x = oct('0b' . '1' x $AND_BITS); $y = $x & $x;
147 $z = (2 ** $AND_BITS) - 1;
148 } while ($AND_BITS < $max && $x == $z && $y == $x);
149 $AND_BITS --; # retreat one step
152 $x = oct('0b' . '1' x $XOR_BITS); $y = $x ^ 0;
153 $z = (2 ** $XOR_BITS) - 1;
154 } while ($XOR_BITS < $max && $x == $z && $y == $x);
155 $XOR_BITS --; # retreat one step
158 $x = oct('0b' . '1' x $OR_BITS); $y = $x | $x;
159 $z = (2 ** $OR_BITS) - 1;
160 } while ($OR_BITS < $max && $x == $z && $y == $x);
161 $OR_BITS --; # retreat one step
165 ##############################################################################
166 # convert between the "small" and the "large" representation
170 # take an array in base $BASE_LEN_SMALL and convert it in-place to $BASE_LEN
173 # print "_to_large $BASE_LEN_SMALL => $BASE_LEN\n";
175 return $x if $LEN_CONVERT == 0 || # nothing to converconvertor
176 @$x == 1; # only one element => early out
178 # 12345 67890 12345 67890 contents
180 # 123456 7890123 4567890 contents
183 # my @d; my $str = '';
184 # my $z = '0' x $BASE_LEN_SMALL;
187 # # ... . 04321 . 000321
188 # $str = substr($z.$_,-$BASE_LEN_SMALL,$BASE_LEN_SMALL) . $str;
189 # if (length($str) > $BASE_LEN)
191 # push @d, substr($str,-$BASE_LEN,$BASE_LEN); # extract one piece
192 # substr($str,-$BASE_LEN,$BASE_LEN) = ''; # remove it
195 # push @d, $str if $str !~ /^0*$/; # extract last piece
197 # $x->[-1] = int($x->[-1]); # strip leading zero
201 my $l = scalar @$x; # number of parts
202 $l --; $ret .= int($x->[$l]); $l--;
203 my $z = '0' x ($BASE_LEN_SMALL-1);
206 $ret .= substr($z.$x->[$l],-$BASE_LEN_SMALL);
209 my $str = _new($c,\$ret); # make array
210 @$x = @$str; # clobber contents of $x
211 $x->[-1] = int($x->[-1]); # strip leading zero
216 # take an array in base $BASE_LEN and convert it in-place to $BASE_LEN_SMALL
219 return $x if $LEN_CONVERT == 0; # nothing to do
220 return $x if @$x == 1 && length(int($x->[0])) <= $BASE_LEN_SMALL;
223 my $il = length($$d)-1;
224 ## this leaves '00000' instead of int 0 and will be corrected after any op
225 # clobber contents of $x
226 @$x = reverse(unpack("a" . ($il % $BASE_LEN_SMALL+1)
227 . ("a$BASE_LEN_SMALL" x ($il / $BASE_LEN_SMALL)), $$d));
229 $x->[-1] = int($x->[-1]); # strip leading zero
232 ###############################################################################
236 # (ref to string) return ref to num_array
237 # Convert a number from string format to internal base 100000 format.
238 # Assumes normalized value as input.
240 my $il = length($$d)-1;
241 # this leaves '00000' instead of int 0 and will be corrected after any op
242 [ reverse(unpack("a" . ($il % $BASE_LEN+1)
243 . ("a$BASE_LEN" x ($il / $BASE_LEN)), $$d)) ];
248 $AND_MASK = __PACKAGE__->_new( \( 2 ** $AND_BITS ));
249 $XOR_MASK = __PACKAGE__->_new( \( 2 ** $XOR_BITS ));
250 $OR_MASK = __PACKAGE__->_new( \( 2 ** $OR_BITS ));
267 # create a two (for _pow)
276 # catch and throw away
279 ##############################################################################
280 # convert back to string and number
284 # (ref to BINT) return num_str
285 # Convert number from internal base 100000 format to string format.
286 # internal format is always normalized (no leading zeros, "-0" => "+0")
290 my $l = scalar @$ar; # number of parts
291 return $nan if $l < 1; # should not happen
293 # handle first one different to strip leading zeros from it (there are no
294 # leading zero parts in internal representation)
295 $l --; $ret .= int($ar->[$l]); $l--;
296 # Interestingly, the pre-padd method uses more time
297 # the old grep variant takes longer (14 to 10 sec)
298 my $z = '0' x ($BASE_LEN-1);
301 $ret .= substr($z.$ar->[$l],-$BASE_LEN); # fastest way I could think of
309 # Make a number (scalar int/float) from a BigInt object
311 return $x->[0] if scalar @$x == 1; # below $BASE
316 $num += $fac*$_; $fac *= $BASE;
321 ##############################################################################
326 # (ref to int_num_array, ref to int_num_array)
327 # routine to add two base 1eX numbers
328 # stolen from Knuth Vol 2 Algorithm A pg 231
329 # there are separate routines to add and sub as per Knuth pg 233
330 # This routine clobbers up array x, but not y.
334 return $x if (@$y == 1) && $y->[0] == 0; # $x + 0 => $x
335 if ((@$x == 1) && $x->[0] == 0) # 0 + $y => $y->copy
337 # twice as slow as $x = [ @$y ], but necc. to retain $x as ref :(
338 @$x = @$y; return $x;
341 # for each in Y, add Y to X and carry. If after that, something is left in
342 # X, foreach in X add carry to X and then return X, carry
343 # Trades one "$j++" for having to shift arrays, $j could be made integer
344 # but this would impose a limit to number-length of 2**32.
345 my $i; my $car = 0; my $j = 0;
348 $x->[$j] -= $BASE if $car = (($x->[$j] += $i + $car) >= $BASE) ? 1 : 0;
353 $x->[$j] -= $BASE if $car = (($x->[$j] += $car) >= $BASE) ? 1 : 0; $j++;
360 # (ref to int_num_array, ref to int_num_array)
361 # routine to add 1 to a base 1eX numbers
362 # This routine clobbers up array x, but not y.
367 return $x if (($i += 1) < $BASE); # early out
368 $i = 0; # overflow, next
370 push @$x,1 if ($x->[-1] == 0); # last overflowed, so extend
376 # (ref to int_num_array, ref to int_num_array)
377 # routine to add 1 to a base 1eX numbers
378 # This routine clobbers up array x, but not y.
381 my $MAX = $BASE-1; # since MAX_VAL based on MBASE
384 last if (($i -= 1) >= 0); # early out
385 $i = $MAX; # overflow, next
387 pop @$x if $x->[-1] == 0 && @$x > 1; # last overflowed (but leave 0)
393 # (ref to int_num_array, ref to int_num_array)
394 # subtract base 1eX numbers -- stolen from Knuth Vol 2 pg 232, $x > $y
395 # subtract Y from X (X is always greater/equal!) by modifying x in place
396 my ($c,$sx,$sy,$s) = @_;
398 my $car = 0; my $i; my $j = 0;
404 last unless defined $sy->[$j] || $car;
405 $i += $BASE if $car = (($i -= ($sy->[$j] || 0) + $car) < 0); $j++;
407 # might leave leading zeros, so fix that
408 return __strip_zeros($sx);
410 #print "case 1 (swap)\n";
413 last unless defined $sy->[$j] || $car;
415 if $car = (($sy->[$j] = $i-($sy->[$j]||0) - $car) < 0);
418 # might leave leading zeros, so fix that
424 # (BINT, BINT) return nothing
425 # multiply two numbers in internal representation
426 # modifies first arg, second need not be different from first
427 my ($c,$xv,$yv) = @_;
429 # shortcut for two very short numbers (improved by Nathan Zook)
430 # works also if xv and yv are the same reference
431 if ((@$xv == 1) && (@$yv == 1))
433 if (($xv->[0] *= $yv->[0]) >= $MBASE)
435 $xv->[0] = $xv->[0] - ($xv->[1] = int($xv->[0] * $RBASE)) * $MBASE;
439 # shortcut for result == 0
440 if ( ((@$xv == 1) && ($xv->[0] == 0)) ||
441 ((@$yv == 1) && ($yv->[0] == 0)) )
447 # since multiplying $x with $x fails, make copy in this case
448 $yv = [@$xv] if "$xv" eq "$yv"; # same references?
449 if ($LEN_CONVERT != 0)
451 $c->_to_small($xv); $c->_to_small($yv);
454 my @prod = (); my ($prod,$car,$cty,$xi,$yi);
463 # $prod = $xi * $yi + ($prod[$cty] || 0) + $car;
465 # $prod - ($car = int($prod * RBASE)) * $MBASE; # see USE_MUL
467 # $prod[$cty] += $car if $car; # need really to check for 0?
471 # looping through this if $xi == 0 is silly - so optimize it away!
472 $xi = (shift @prod || 0), next if $xi == 0;
475 $prod = $xi * $yi + ($prod[$cty] || 0) + $car;
476 ## this is actually a tad slower
477 ## $prod = $prod[$cty]; $prod += ($car + $xi * $yi); # no ||0 here
479 $prod - ($car = int($prod * $RBASE)) * $MBASE; # see USE_MUL
481 $prod[$cty] += $car if $car; # need really to check for 0?
482 $xi = shift @prod || 0; # || 0 makes v5.005_3 happy
485 if ($LEN_CONVERT != 0)
499 # (BINT, BINT) return nothing
500 # multiply two numbers in internal representation
501 # modifies first arg, second need not be different from first
502 my ($c,$xv,$yv) = @_;
504 # shortcut for two very short numbers (improved by Nathan Zook)
505 # works also if xv and yv are the same reference
506 if ((@$xv == 1) && (@$yv == 1))
508 if (($xv->[0] *= $yv->[0]) >= $MBASE)
511 $xv->[0] - ($xv->[1] = int($xv->[0] / $MBASE)) * $MBASE;
515 # shortcut for result == 0
516 if ( ((@$xv == 1) && ($xv->[0] == 0)) ||
517 ((@$yv == 1) && ($yv->[0] == 0)) )
524 # since multiplying $x with $x fails, make copy in this case
525 $yv = [@$xv] if "$xv" eq "$yv"; # same references?
526 if ($LEN_CONVERT != 0)
528 $c->_to_small($xv); $c->_to_small($yv);
531 my @prod = (); my ($prod,$car,$cty,$xi,$yi);
535 # looping through this if $xi == 0 is silly - so optimize it away!
536 $xi = (shift @prod || 0), next if $xi == 0;
539 $prod = $xi * $yi + ($prod[$cty] || 0) + $car;
541 $prod - ($car = int($prod / $MBASE)) * $MBASE;
543 $prod[$cty] += $car if $car; # need really to check for 0?
544 $xi = shift @prod || 0; # || 0 makes v5.005_3 happy
547 if ($LEN_CONVERT != 0)
561 # ref to array, ref to array, modify first array and return remainder if
563 my ($c,$x,$yorg) = @_;
565 if (@$x == 1 && @$yorg == 1)
567 # shortcut, $yorg and $x are two small numbers
570 my $r = [ $x->[0] % $yorg->[0] ];
571 $x->[0] = int($x->[0] / $yorg->[0]);
576 $x->[0] = int($x->[0] / $yorg->[0]);
582 # # shortcut, $y is < $BASE
588 if ($LEN_CONVERT != 0)
590 $c->_to_small($x); $c->_to_small($y);
593 my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0);
595 $car = $bar = $prd = 0;
596 if (($dd = int($MBASE/($y->[-1]+1))) != 1)
600 $xi = $xi * $dd + $car;
601 $xi -= ($car = int($xi * $RBASE)) * $MBASE; # see USE_MUL
603 push(@$x, $car); $car = 0;
606 $yi = $yi * $dd + $car;
607 $yi -= ($car = int($yi * $RBASE)) * $MBASE; # see USE_MUL
614 @q = (); ($v2,$v1) = @$y[-2,-1];
618 ($u2,$u1,$u0) = @$x[-3..-1];
620 #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n"
622 $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$MBASE+$u1)/$v1));
623 --$q while ($v2*$q > ($u0*$MBASE+$u1-$q*$v1)*$MBASE+$u2);
626 ($car, $bar) = (0,0);
627 for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi)
629 $prd = $q * $y->[$yi] + $car;
630 $prd -= ($car = int($prd * $RBASE)) * $MBASE; # see USE_MUL
631 $x->[$xi] += $MBASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0));
633 if ($x->[-1] < $car + $bar)
636 for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi)
639 if ($car = (($x->[$xi] += $y->[$yi] + $car) > $MBASE));
643 pop(@$x); unshift(@q, $q);
651 for $xi (reverse @$x)
653 $prd = $car * $MBASE + $xi;
654 $car = $prd - ($tmp = int($prd / $dd)) * $dd; # see USE_MUL
664 if ($LEN_CONVERT != 0)
666 $c->_to_large($x); $c->_to_large($d);
676 if ($LEN_CONVERT != 0)
689 # ref to array, ref to array, modify first array and return remainder if
691 my ($c,$x,$yorg) = @_;
693 if (@$x == 1 && @$yorg == 1)
695 # shortcut, $yorg and $x are two small numbers
698 my $r = [ $x->[0] % $yorg->[0] ];
699 $x->[0] = int($x->[0] / $yorg->[0]);
704 $x->[0] = int($x->[0] / $yorg->[0]);
710 # # shortcut, $y is < $BASE
715 if ($LEN_CONVERT != 0)
717 $c->_to_small($x); $c->_to_small($y);
720 my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0);
722 $car = $bar = $prd = 0;
723 if (($dd = int($MBASE/($y->[-1]+1))) != 1)
727 $xi = $xi * $dd + $car;
728 $xi -= ($car = int($xi / $MBASE)) * $MBASE;
730 push(@$x, $car); $car = 0;
733 $yi = $yi * $dd + $car;
734 $yi -= ($car = int($yi / $MBASE)) * $MBASE;
741 @q = (); ($v2,$v1) = @$y[-2,-1];
745 ($u2,$u1,$u0) = @$x[-3..-1];
747 #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n"
749 $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$MBASE+$u1)/$v1));
750 --$q while ($v2*$q > ($u0*$MBASE+$u1-$q*$v1)*$MBASE+$u2);
753 ($car, $bar) = (0,0);
754 for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi)
756 $prd = $q * $y->[$yi] + $car;
757 $prd -= ($car = int($prd / $MBASE)) * $MBASE;
758 $x->[$xi] += $MBASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0));
760 if ($x->[-1] < $car + $bar)
763 for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi)
766 if ($car = (($x->[$xi] += $y->[$yi] + $car) > $MBASE));
770 pop(@$x); unshift(@q, $q);
778 for $xi (reverse @$x)
780 $prd = $car * $MBASE + $xi;
781 $car = $prd - ($tmp = int($prd / $dd)) * $dd;
791 if ($LEN_CONVERT != 0)
793 $c->_to_large($x); $c->_to_large($d);
803 if ($LEN_CONVERT != 0)
814 ##############################################################################
819 # internal absolute post-normalized compare (ignore signs)
820 # ref to array, ref to array, return <0, 0, >0
821 # arrays must have at least one entry; this is not checked for
823 my ($c,$cx,$cy) = @_;
825 # fast comp based on array elements
826 my $lxy = scalar @$cx - scalar @$cy;
827 return -1 if $lxy < 0; # already differs, ret
828 return 1 if $lxy > 0; # ditto
830 # now calculate length based on digits, not parts
831 $lxy = _len($c,$cx) - _len($c,$cy); # difference
832 return -1 if $lxy < 0;
833 return 1 if $lxy > 0;
835 # hm, same lengths, but same contents?
837 # first way takes 5.49 sec instead of 4.87, but has the early out advantage
838 # so grep is slightly faster, but more inflexible. hm. $_ instead of $k
839 # yields 5.6 instead of 5.5 sec huh?
840 # manual way (abort if unequal, good for early ne)
841 my $j = scalar @$cx - 1;
844 last if ($a = $cx->[$j] - $cy->[$j]); $j--;
850 # while it early aborts, it is even slower than the manual variant
851 #grep { return $a if ($a = $_ - $cy->[$i++]); } @$cx;
852 # grep way, go trough all (bad for early ne)
853 #grep { $a = $_ - $cy->[$i++]; } @$cx;
859 # compute number of digits in bigint, minus the sign
861 # int() because add/sub sometimes leaves strings (like '00005') instead of
862 # '5' in this place, thus causing length() to report wrong length
865 return (@$cx-1)*$BASE_LEN+length(int($cx->[-1]));
870 # return the nth digit, negative values count backward
871 # zero is rightmost, so _digit(123,0) will give 3
874 my $len = _len('',$x);
876 $n = $len+$n if $n < 0; # -1 last, -2 second-to-last
877 $n = abs($n); # if negative was too big
878 $len--; $n = $len if $n > $len; # n to big?
880 my $elem = int($n / $BASE_LEN); # which array element
881 my $digit = $n % $BASE_LEN; # which digit in this element
882 $elem = '0000'.@$x[$elem]; # get element padded with 0's
883 return substr($elem,-$digit-1,1);
888 # return amount of trailing zeros in decimal
889 # check each array elem in _m for having 0 at end as long as elem == 0
890 # Upon finding a elem != 0, stop
892 my $zeros = 0; my $elem;
897 $elem = "$e"; # preserve x
898 $elem =~ s/.*?(0*$)/$1/; # strip anything not zero
899 $zeros *= $BASE_LEN; # elems * 5
900 $zeros += length($elem); # count trailing zeros
903 $zeros ++; # real else branch: 50% slower!
908 ##############################################################################
913 # return true if arg (BINT or num_str) is zero (array '+', '0')
916 (((scalar @$x == 1) && ($x->[0] == 0))) <=> 0;
921 # return true if arg (BINT or num_str) is even
923 (!($x->[0] & 1)) <=> 0;
928 # return true if arg (BINT or num_str) is even
931 (($x->[0] & 1)) <=> 0;
936 # return true if arg (BINT or num_str) is one (array '+', '1')
939 (scalar @$x == 1) && ($x->[0] == 1) <=> 0;
944 # internal normalization function that strips leading zeros from the array
948 my $cnt = scalar @$s; # get count of parts
950 push @$s,0 if $i < 0; # div might return empty results, so fix it
952 return $s if @$s == 1; # early out
954 #print "strip: cnt $cnt i $i\n";
955 # '0', '3', '4', '0', '0',
960 # => fcnt = cnt - i (5-2 => 3, cnt => 5-1 = 4, throw away from 4th pos)
961 # >= 1: skip first part (this can be zero)
962 while ($i > 0) { last if $s->[$i] != 0; $i--; }
963 $i++; splice @$s,$i if ($i < $cnt); # $i cant be 0
967 ###############################################################################
968 # check routine to test internal state of corruptions
972 # used by the test suite
975 return "$x is not a reference" if !ref($x);
977 # are all parts are valid?
978 my $i = 0; my $j = scalar @$x; my ($e,$try);
981 $e = $x->[$i]; $e = 'undef' unless defined $e;
982 $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e)";
983 last if $e !~ /^[+]?[0-9]+$/;
984 $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (stringify)";
985 last if "$e" !~ /^[+]?[0-9]+$/;
986 $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (cat-stringify)";
987 last if '' . "$e" !~ /^[+]?[0-9]+$/;
988 $try = ' < 0 || >= $BASE; '."($x, $e)";
989 last if $e <0 || $e >= $BASE;
990 # this test is disabled, since new/bnorm and certain ops (like early out
991 # in add/sub) are allowed/expected to leave '00000' in some elements
992 #$try = '=~ /^00+/; '."($x, $e)";
993 #last if $e =~ /^00+/;
996 return "Illegal part '$e' at pos $i (tested: $try)" if $i < $j;
1001 ###############################################################################
1002 ###############################################################################
1003 # some optional routines to make BigInt faster
1007 # if possible, use mod shortcut
1008 my ($c,$x,$yo) = @_;
1010 # slow way since $y to big
1011 if (scalar @$yo > 1)
1013 my ($xo,$rem) = _div($c,$x,$yo);
1017 # both are single element arrays
1018 if (scalar @$x == 1)
1024 # @y is single element, but @x has more than one
1028 # when BASE % Y == 0 then (B * BASE) % Y == 0
1029 # (B * BASE) % $y + A % Y => A % Y
1030 # so need to consider only last element: O(1)
1035 # else need to go trough all elements: O(N), but loop is a bit simplified
1047 # else need to go trough all elements: O(N)
1048 my $r = 0; my $bm = 1;
1051 $r += ($_ % $y) * $bm;
1063 ##############################################################################
1068 my ($c,$x,$y,$n) = @_;
1072 $n = _new($c,\$n); return _div($c,$x, _pow($c,$n,$y));
1075 # shortcut (faster) for shifting by 10)
1076 # multiples of $BASE_LEN
1077 my $dst = 0; # destination
1078 my $src = _num($c,$y); # as normal int
1079 my $rem = $src % $BASE_LEN; # remainder to shift
1080 $src = int($src / $BASE_LEN); # source
1083 splice (@$x,0,$src); # even faster, 38.4 => 39.3
1087 my $len = scalar @$x - $src; # elems to go
1088 my $vd; my $z = '0'x $BASE_LEN;
1089 $x->[scalar @$x] = 0; # avoid || 0 test inside loop
1092 $vd = $z.$x->[$src];
1093 $vd = substr($vd,-$BASE_LEN,$BASE_LEN-$rem);
1095 $vd = substr($z.$x->[$src],-$rem,$rem) . $vd;
1096 $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN;
1097 $x->[$dst] = int($vd);
1100 splice (@$x,$dst) if $dst > 0; # kill left-over array elems
1101 pop @$x if $x->[-1] == 0; # kill last element if 0
1108 my ($c,$x,$y,$n) = @_;
1112 $n = _new($c,\$n); return _mul($c,$x, _pow($c,$n,$y));
1115 # shortcut (faster) for shifting by 10) since we are in base 10eX
1116 # multiples of $BASE_LEN:
1117 my $src = scalar @$x; # source
1118 my $len = _num($c,$y); # shift-len as normal int
1119 my $rem = $len % $BASE_LEN; # remainder to shift
1120 my $dst = $src + int($len/$BASE_LEN); # destination
1121 my $vd; # further speedup
1122 $x->[$src] = 0; # avoid first ||0 for speed
1123 my $z = '0' x $BASE_LEN;
1126 $vd = $x->[$src]; $vd = $z.$vd;
1127 $vd = substr($vd,-$BASE_LEN+$rem,$BASE_LEN-$rem);
1128 $vd .= $src > 0 ? substr($z.$x->[$src-1],-$BASE_LEN,$rem) : '0' x $rem;
1129 $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN;
1130 $x->[$dst] = int($vd);
1133 # set lowest parts to 0
1134 while ($dst >= 0) { $x->[$dst--] = 0; }
1135 # fix spurios last zero element
1136 splice @$x,-1 if $x->[-1] == 0;
1143 # ref to array, ref to array, return ref to array
1144 my ($c,$cx,$cy) = @_;
1148 my $y1 = _copy($c,$cy);
1149 while (!_is_one($c,$y1))
1151 _mul($c,$pow2,$cx) if _is_odd($c,$y1);
1155 _mul($c,$cx,$pow2) unless _is_one($c,$pow2);
1162 # ref to array, return ref to array
1165 if ((@$cx == 1) && ($cx->[0] <= 2))
1167 $cx->[0] = 1 * ($cx->[0]||1); # 0,1 => 1, 2 => 2
1171 # go forward until $base is exceeded
1172 # limit is either $x or $base (x == 100 means as result too high)
1173 my $steps = 100; $steps = $cx->[0] if @$cx == 1;
1174 my $r = 2; my $cf = 3; my $step = 1; my $last = $r;
1175 while ($r < $BASE && $step < $steps)
1177 $last = $r; $r *= $cf++; $step++;
1179 if ((@$cx == 1) && ($step == $cx->[0]))
1185 my $n = _copy($c,$cx);
1189 while (!(@$n == 1 && $n->[0] == $step))
1191 _mul($c,$cx,$n); _dec($c,$n);
1196 use constant DEBUG => 0;
1200 sub steps { $steps };
1205 # ref to array, return ref to array
1208 if (scalar @$x == 1)
1210 # fit's into one Perl scalar
1211 $x->[0] = int(sqrt($x->[0]));
1214 my $y = _copy($c,$x);
1215 # hopefully _len/2 is < $BASE, the -1 is to always undershot the guess
1216 # since our guess will "grow"
1217 my $l = int((_len($c,$x)-1) / 2);
1219 my $lastelem = $x->[-1]; # for guess
1220 my $elems = scalar @$x - 1;
1221 # not enough digits, but could have more?
1222 if ((length($lastelem) <= 3) && ($elems > 1))
1224 # right-align with zero pad
1225 my $len = length($lastelem) & 1;
1226 print "$lastelem => " if DEBUG;
1227 $lastelem .= substr($x->[-2] . '0' x $BASE_LEN,0,$BASE_LEN);
1228 # former odd => make odd again, or former even to even again
1229 $lastelem = $lastelem / 10 if (length($lastelem) & 1) != $len;
1230 print "$lastelem\n" if DEBUG;
1233 # construct $x (instead of _lsft($c,$x,$l,10)
1234 my $r = $l % $BASE_LEN; # 10000 00000 00000 00000 ($BASE_LEN=5)
1235 $l = int($l / $BASE_LEN);
1236 print "l = $l " if DEBUG;
1238 splice @$x,$l; # keep ref($x), but modify it
1240 # we make the first part of the guess not '1000...0' but int(sqrt($lastelem))
1242 # 14400 00000 => sqrt(14400) => 120
1243 # 144000 000000 => sqrt(144000) => 379
1245 # $x->[$l--] = int('1' . '0' x $r); # old way of guessing
1246 print "$lastelem (elems $elems) => " if DEBUG;
1247 $lastelem = $lastelem / 10 if ($elems & 1 == 1); # odd or even?
1248 my $g = sqrt($lastelem); $g =~ s/\.//; # 2.345 => 2345
1249 $r -= 1 if $elems & 1 == 0; # 70 => 7
1251 # padd with zeros if result is too short
1252 $x->[$l--] = int(substr($g . '0' x $r,0,$r+1));
1253 print "now ",$x->[-1] if DEBUG;
1254 print " would have been ", int('1' . '0' x $r),"\n" if DEBUG;
1256 # If @$x > 1, we could compute the second elem of the guess, too, to create
1257 # an even better guess. Not implemented yet.
1258 $x->[$l--] = 0 while ($l >= 0); # all other digits of guess are zero
1260 print "start x= ",${_str($c,$x)},"\n" if DEBUG;
1263 my $lastlast = _zero();
1264 $steps = 0 if DEBUG;
1265 while (_acmp($c,$last,$x) != 0 && _acmp($c,$lastlast,$x) != 0)
1268 $lastlast = _copy($c,$last);
1269 $last = _copy($c,$x);
1270 _add($c,$x, _div($c,_copy($c,$y),$x));
1272 print " x= ",${_str($c,$x)},"\n" if DEBUG;
1274 print "\nsteps in sqrt: $steps, " if DEBUG;
1275 _dec($c,$x) if _acmp($c,$y,_mul($c,_copy($c,$x),$x)) < 0; # overshot?
1276 print " final ",$x->[-1],"\n" if DEBUG;
1280 ##############################################################################
1287 # the shortcut makes equal, large numbers _really_ fast, and makes only a
1288 # very small performance drop for small numbers (e.g. something with less
1289 # than 32 bit) Since we optimize for large numbers, this is enabled.
1290 return $x if _acmp($c,$x,$y) == 0; # shortcut
1292 my $m = _one(); my ($xr,$yr);
1293 my $mask = $AND_MASK;
1296 my $y1 = _copy($c,$y); # make copy
1300 while (!_is_zero($c,$x1) && !_is_zero($c,$y1))
1302 ($x1, $xr) = _div($c,$x1,$mask);
1303 ($y1, $yr) = _div($c,$y1,$mask);
1305 # make ints() from $xr, $yr
1306 # this is when the AND_BITS are greater tahn $BASE and is slower for
1307 # small (<256 bits) numbers, but faster for large numbers. Disabled
1308 # due to KISS principle
1310 # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; }
1311 # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; }
1312 # _add($c,$x, _mul($c, _new( $c, \($xrr & $yrr) ), $m) );
1314 # 0+ due to '&' doesn't work in strings
1315 _add($c,$x, _mul($c, [ 0+$xr->[0] & 0+$yr->[0] ], $m) );
1325 return _zero() if _acmp($c,$x,$y) == 0; # shortcut (see -and)
1327 my $m = _one(); my ($xr,$yr);
1328 my $mask = $XOR_MASK;
1331 my $y1 = _copy($c,$y); # make copy
1335 while (!_is_zero($c,$x1) && !_is_zero($c,$y1))
1337 ($x1, $xr) = _div($c,$x1,$mask);
1338 ($y1, $yr) = _div($c,$y1,$mask);
1339 # make ints() from $xr, $yr (see _and())
1340 #$b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; }
1341 #$b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; }
1342 #_add($c,$x, _mul($c, _new( $c, \($xrr ^ $yrr) ), $m) );
1344 # 0+ due to '^' doesn't work in strings
1345 _add($c,$x, _mul($c, [ 0+$xr->[0] ^ 0+$yr->[0] ], $m) );
1348 # the loop stops when the shorter of the two numbers is exhausted
1349 # the remainder of the longer one will survive bit-by-bit, so we simple
1350 # multiply-add it in
1351 _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1);
1352 _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1);
1361 return $x if _acmp($c,$x,$y) == 0; # shortcut (see _and)
1363 my $m = _one(); my ($xr,$yr);
1364 my $mask = $OR_MASK;
1367 my $y1 = _copy($c,$y); # make copy
1371 while (!_is_zero($c,$x1) && !_is_zero($c,$y1))
1373 ($x1, $xr) = _div($c,$x1,$mask);
1374 ($y1, $yr) = _div($c,$y1,$mask);
1375 # make ints() from $xr, $yr (see _and())
1376 # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; }
1377 # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; }
1378 # _add($c,$x, _mul($c, _new( $c, \($xrr | $yrr) ), $m) );
1380 # 0+ due to '|' doesn't work in strings
1381 _add($c,$x, _mul($c, [ 0+$xr->[0] | 0+$yr->[0] ], $m) );
1384 # the loop stops when the shorter of the two numbers is exhausted
1385 # the remainder of the longer one will survive bit-by-bit, so we simple
1386 # multiply-add it in
1387 _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1);
1388 _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1);
1395 # convert a decimal number to hex (ref to array, return ref to string)
1398 my $x1 = _copy($c,$x);
1402 my $x10000 = [ 0x10000 ];
1403 while (! _is_zero($c,$x1))
1405 ($x1, $xr) = _div($c,$x1,$x10000);
1406 $es .= unpack('h4',pack('v',$xr->[0]));
1409 $es =~ s/^[0]+//; # strip leading zeros
1416 # convert a decimal number to bin (ref to array, return ref to string)
1419 my $x1 = _copy($c,$x);
1423 my $x10000 = [ 0x10000 ];
1424 while (! _is_zero($c,$x1))
1426 ($x1, $xr) = _div($c,$x1,$x10000);
1427 $es .= unpack('b16',pack('v',$xr->[0]));
1430 $es =~ s/^[0]+//; # strip leading zeros
1437 # convert a hex number to decimal (ref to string, return ref to array)
1441 my $m = [ 0x10000 ]; # 16 bit at a time
1444 my $len = length($$hs)-2;
1445 $len = int($len/4); # 4-digit parts, w/o '0x'
1446 my $val; my $i = -4;
1449 $val = substr($$hs,$i,4);
1450 $val =~ s/^[+-]?0x// if $len == 0; # for last part only because
1451 $val = hex($val); # hex does not like wrong chars
1453 _add ($c, $x, _mul ($c, [ $val ], $mul ) ) if $val != 0;
1454 _mul ($c, $mul, $m ) if $len >= 0; # skip last mul
1461 # convert a hex number to decimal (ref to string, return ref to array)
1464 # instead of converting 8 bit at a time, it is faster to convert the
1465 # number to hex, and then call _from_hex.
1468 $hs =~ s/^[+-]?0b//; # remove sign and 0b
1469 my $l = length($hs); # bits
1470 $hs = '0' x (8-($l % 8)) . $hs if ($l % 8) != 0; # padd left side w/ 0
1471 my $h = unpack('H*', pack ('B*', $hs)); # repack as hex
1472 return $c->_from_hex(\('0x'.$h));
1475 my $m = [ 0x100 ]; # 8 bit at a time
1478 my $len = length($$bs)-2;
1479 $len = int($len/8); # 4-digit parts, w/o '0x'
1480 my $val; my $i = -8;
1483 $val = substr($$bs,$i,8);
1484 $val =~ s/^[+-]?0b// if $len == 0; # for last part only
1486 $val = ord(pack('B8',substr('00000000'.$val,-8,8)));
1489 _add ($c, $x, _mul ($c, [ $val ], $mul ) ) if $val != 0;
1490 _mul ($c, $mul, $m ) if $len >= 0; # skip last mul
1495 ##############################################################################
1496 ##############################################################################
1503 Math::BigInt::Calc - Pure Perl module to support Math::BigInt
1507 Provides support for big integer calculations. Not intended to be used by other
1508 modules (except Math::BigInt::Cached). Other modules which sport the same
1509 functions can also be used to support Math::Bigint, like Math::BigInt::Pari.
1513 In order to allow for multiple big integer libraries, Math::BigInt was
1514 rewritten to use library modules for core math routines. Any module which
1515 follows the same API as this can be used instead by using the following:
1517 use Math::BigInt lib => 'libname';
1519 'libname' is either the long name ('Math::BigInt::Pari'), or only the short
1520 version like 'Pari'.
1524 The following functions MUST be defined in order to support the use by
1527 _new(string) return ref to new object from ref to decimal string
1528 _zero() return a new object with value 0
1529 _one() return a new object with value 1
1531 _str(obj) return ref to a string representing the object
1532 _num(obj) returns a Perl integer/floating point number
1533 NOTE: because of Perl numeric notation defaults,
1534 the _num'ified obj may lose accuracy due to
1535 machine-dependend floating point size limitations
1537 _add(obj,obj) Simple addition of two objects
1538 _mul(obj,obj) Multiplication of two objects
1539 _div(obj,obj) Division of the 1st object by the 2nd
1540 In list context, returns (result,remainder).
1541 NOTE: this is integer math, so no
1542 fractional part will be returned.
1543 _sub(obj,obj) Simple subtraction of 1 object from another
1544 a third, optional parameter indicates that the params
1545 are swapped. In this case, the first param needs to
1546 be preserved, while you can destroy the second.
1547 sub (x,y,1) => return x - y and keep x intact!
1548 _dec(obj) decrement object by one (input is garant. to be > 0)
1549 _inc(obj) increment object by one
1552 _acmp(obj,obj) <=> operator for objects (return -1, 0 or 1)
1554 _len(obj) returns count of the decimal digits of the object
1555 _digit(obj,n) returns the n'th decimal digit of object
1557 _is_one(obj) return true if argument is +1
1558 _is_zero(obj) return true if argument is 0
1559 _is_even(obj) return true if argument is even (0,2,4,6..)
1560 _is_odd(obj) return true if argument is odd (1,3,5,7..)
1562 _copy return a ref to a true copy of the object
1564 _check(obj) check whether internal representation is still intact
1565 return 0 for ok, otherwise error message as string
1567 The following functions are optional, and can be defined if the underlying lib
1568 has a fast way to do them. If undefined, Math::BigInt will use pure Perl (hence
1569 slow) fallback routines to emulate these:
1571 _from_hex(str) return ref to new object from ref to hexadecimal string
1572 _from_bin(str) return ref to new object from ref to binary string
1574 _as_hex(str) return ref to scalar string containing the value as
1575 unsigned hex string, with the '0x' prepended.
1576 Leading zeros must be stripped.
1577 _as_bin(str) Like as_hex, only as binary string containing only
1578 zeros and ones. Leading zeros must be stripped and a
1579 '0b' must be prepended.
1581 _rsft(obj,N,B) shift object in base B by N 'digits' right
1582 For unsupported bases B, return undef to signal failure
1583 _lsft(obj,N,B) shift object in base B by N 'digits' left
1584 For unsupported bases B, return undef to signal failure
1586 _xor(obj1,obj2) XOR (bit-wise) object 1 with object 2
1587 Note: XOR, AND and OR pad with zeros if size mismatches
1588 _and(obj1,obj2) AND (bit-wise) object 1 with object 2
1589 _or(obj1,obj2) OR (bit-wise) object 1 with object 2
1591 _mod(obj,obj) Return remainder of div of the 1st by the 2nd object
1592 _sqrt(obj) return the square root of object (truncate to int)
1593 _fac(obj) return factorial of object 1 (1*2*3*4..)
1594 _pow(obj,obj) return object 1 to the power of object 2
1595 _gcd(obj,obj) return Greatest Common Divisor of two objects
1597 _zeros(obj) return number of trailing decimal zeros
1599 Input strings come in as unsigned but with prefix (i.e. as '123', '0xabc'
1602 Testing of input parameter validity is done by the caller, so you need not
1603 worry about underflow (f.i. in C<_sub()>, C<_dec()>) nor about division by
1604 zero or similar cases.
1606 The first parameter can be modified, that includes the possibility that you
1607 return a reference to a completely different object instead. Although keeping
1608 the reference and just changing it's contents is prefered over creating and
1609 returning a different reference.
1611 Return values are always references to objects or strings. Exceptions are
1612 C<_lsft()> and C<_rsft()>, which return undef if they can not shift the
1613 argument. This is used to delegate shifting of bases different than the one
1614 you can support back to Math::BigInt, which will use some generic code to
1615 calculate the result.
1617 =head1 WRAP YOUR OWN
1619 If you want to port your own favourite c-lib for big numbers to the
1620 Math::BigInt interface, you can take any of the already existing modules as
1621 a rough guideline. You should really wrap up the latest BigInt and BigFloat
1622 testsuites with your module, and replace in them any of the following:
1628 use Math::BigInt lib => 'yourlib';
1630 This way you ensure that your library really works 100% within Math::BigInt.
1634 This program is free software; you may redistribute it and/or modify it under
1635 the same terms as Perl itself.
1639 Original math code by Mark Biggar, rewritten by Tels L<http://bloodgate.com/>
1641 Seperated from BigInt and shaped API with the help of John Peacock.
1645 L<Math::BigInt>, L<Math::BigFloat>, L<Math::BigInt::BitVect>,
1646 L<Math::BigInt::GMP>, L<Math::BigInt::Cached> and L<Math::BigInt::Pari>.