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, $y is smaller than $x
570 my $r = [ $x->[0] % $yorg->[0] ];
571 $x->[0] = int($x->[0] / $yorg->[0]);
576 $x->[0] = int($x->[0] / $yorg->[0]);
582 if ($LEN_CONVERT != 0)
584 $c->_to_small($x); $c->_to_small($y);
587 my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0);
589 $car = $bar = $prd = 0;
590 if (($dd = int($MBASE/($y->[-1]+1))) != 1)
594 $xi = $xi * $dd + $car;
595 $xi -= ($car = int($xi * $RBASE)) * $MBASE; # see USE_MUL
597 push(@$x, $car); $car = 0;
600 $yi = $yi * $dd + $car;
601 $yi -= ($car = int($yi * $RBASE)) * $MBASE; # see USE_MUL
608 @q = (); ($v2,$v1) = @$y[-2,-1];
612 ($u2,$u1,$u0) = @$x[-3..-1];
614 #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n"
616 $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$MBASE+$u1)/$v1));
617 --$q while ($v2*$q > ($u0*$MBASE+$u1-$q*$v1)*$MBASE+$u2);
620 ($car, $bar) = (0,0);
621 for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi)
623 $prd = $q * $y->[$yi] + $car;
624 $prd -= ($car = int($prd * $RBASE)) * $MBASE; # see USE_MUL
625 $x->[$xi] += $MBASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0));
627 if ($x->[-1] < $car + $bar)
630 for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi)
633 if ($car = (($x->[$xi] += $y->[$yi] + $car) > $MBASE));
637 pop(@$x); unshift(@q, $q);
645 for $xi (reverse @$x)
647 $prd = $car * $MBASE + $xi;
648 $car = $prd - ($tmp = int($prd / $dd)) * $dd; # see USE_MUL
658 if ($LEN_CONVERT != 0)
660 $c->_to_large($x); $c->_to_large($d);
670 if ($LEN_CONVERT != 0)
683 # ref to array, ref to array, modify first array and return remainder if
685 my ($c,$x,$yorg) = @_;
687 if (@$x == 1 && @$yorg == 1)
689 # shortcut, $y is smaller than $x
692 my $r = [ $x->[0] % $yorg->[0] ];
693 $x->[0] = int($x->[0] / $yorg->[0]);
698 $x->[0] = int($x->[0] / $yorg->[0]);
704 if ($LEN_CONVERT != 0)
706 $c->_to_small($x); $c->_to_small($y);
709 my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0);
711 $car = $bar = $prd = 0;
712 if (($dd = int($MBASE/($y->[-1]+1))) != 1)
716 $xi = $xi * $dd + $car;
717 $xi -= ($car = int($xi / $MBASE)) * $MBASE;
719 push(@$x, $car); $car = 0;
722 $yi = $yi * $dd + $car;
723 $yi -= ($car = int($yi / $MBASE)) * $MBASE;
730 @q = (); ($v2,$v1) = @$y[-2,-1];
734 ($u2,$u1,$u0) = @$x[-3..-1];
736 #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n"
738 $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$MBASE+$u1)/$v1));
739 --$q while ($v2*$q > ($u0*$MBASE+$u1-$q*$v1)*$MBASE+$u2);
742 ($car, $bar) = (0,0);
743 for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi)
745 $prd = $q * $y->[$yi] + $car;
746 $prd -= ($car = int($prd / $MBASE)) * $MBASE;
747 $x->[$xi] += $MBASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0));
749 if ($x->[-1] < $car + $bar)
752 for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi)
755 if ($car = (($x->[$xi] += $y->[$yi] + $car) > $MBASE));
759 pop(@$x); unshift(@q, $q);
767 for $xi (reverse @$x)
769 $prd = $car * $MBASE + $xi;
770 $car = $prd - ($tmp = int($prd / $dd)) * $dd;
780 if ($LEN_CONVERT != 0)
782 $c->_to_large($x); $c->_to_large($d);
792 if ($LEN_CONVERT != 0)
803 ##############################################################################
808 # internal absolute post-normalized compare (ignore signs)
809 # ref to array, ref to array, return <0, 0, >0
810 # arrays must have at least one entry; this is not checked for
812 my ($c,$cx,$cy) = @_;
814 # fast comp based on array elements
815 my $lxy = scalar @$cx - scalar @$cy;
816 return -1 if $lxy < 0; # already differs, ret
817 return 1 if $lxy > 0; # ditto
819 # now calculate length based on digits, not parts
820 $lxy = _len($c,$cx) - _len($c,$cy); # difference
821 return -1 if $lxy < 0;
822 return 1 if $lxy > 0;
824 # hm, same lengths, but same contents?
826 # first way takes 5.49 sec instead of 4.87, but has the early out advantage
827 # so grep is slightly faster, but more inflexible. hm. $_ instead of $k
828 # yields 5.6 instead of 5.5 sec huh?
829 # manual way (abort if unequal, good for early ne)
830 my $j = scalar @$cx - 1;
833 last if ($a = $cx->[$j] - $cy->[$j]); $j--;
839 # while it early aborts, it is even slower than the manual variant
840 #grep { return $a if ($a = $_ - $cy->[$i++]); } @$cx;
841 # grep way, go trough all (bad for early ne)
842 #grep { $a = $_ - $cy->[$i++]; } @$cx;
848 # compute number of digits in bigint, minus the sign
850 # int() because add/sub sometimes leaves strings (like '00005') instead of
851 # '5' in this place, thus causing length() to report wrong length
854 return (@$cx-1)*$BASE_LEN+length(int($cx->[-1]));
859 # return the nth digit, negative values count backward
860 # zero is rightmost, so _digit(123,0) will give 3
863 my $len = _len('',$x);
865 $n = $len+$n if $n < 0; # -1 last, -2 second-to-last
866 $n = abs($n); # if negative was too big
867 $len--; $n = $len if $n > $len; # n to big?
869 my $elem = int($n / $BASE_LEN); # which array element
870 my $digit = $n % $BASE_LEN; # which digit in this element
871 $elem = '0000'.@$x[$elem]; # get element padded with 0's
872 return substr($elem,-$digit-1,1);
877 # return amount of trailing zeros in decimal
878 # check each array elem in _m for having 0 at end as long as elem == 0
879 # Upon finding a elem != 0, stop
881 my $zeros = 0; my $elem;
886 $elem = "$e"; # preserve x
887 $elem =~ s/.*?(0*$)/$1/; # strip anything not zero
888 $zeros *= $BASE_LEN; # elems * 5
889 $zeros += length($elem); # count trailing zeros
892 $zeros ++; # real else branch: 50% slower!
897 ##############################################################################
902 # return true if arg (BINT or num_str) is zero (array '+', '0')
905 (((scalar @$x == 1) && ($x->[0] == 0))) <=> 0;
910 # return true if arg (BINT or num_str) is even
912 (!($x->[0] & 1)) <=> 0;
917 # return true if arg (BINT or num_str) is even
920 (($x->[0] & 1)) <=> 0;
925 # return true if arg (BINT or num_str) is one (array '+', '1')
928 (scalar @$x == 1) && ($x->[0] == 1) <=> 0;
933 # internal normalization function that strips leading zeros from the array
937 my $cnt = scalar @$s; # get count of parts
939 push @$s,0 if $i < 0; # div might return empty results, so fix it
941 return $s if @$s == 1; # early out
943 #print "strip: cnt $cnt i $i\n";
944 # '0', '3', '4', '0', '0',
949 # => fcnt = cnt - i (5-2 => 3, cnt => 5-1 = 4, throw away from 4th pos)
950 # >= 1: skip first part (this can be zero)
951 while ($i > 0) { last if $s->[$i] != 0; $i--; }
952 $i++; splice @$s,$i if ($i < $cnt); # $i cant be 0
956 ###############################################################################
957 # check routine to test internal state of corruptions
961 # used by the test suite
964 return "$x is not a reference" if !ref($x);
966 # are all parts are valid?
967 my $i = 0; my $j = scalar @$x; my ($e,$try);
970 $e = $x->[$i]; $e = 'undef' unless defined $e;
971 $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e)";
972 last if $e !~ /^[+]?[0-9]+$/;
973 $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (stringify)";
974 last if "$e" !~ /^[+]?[0-9]+$/;
975 $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (cat-stringify)";
976 last if '' . "$e" !~ /^[+]?[0-9]+$/;
977 $try = ' < 0 || >= $BASE; '."($x, $e)";
978 last if $e <0 || $e >= $BASE;
979 # this test is disabled, since new/bnorm and certain ops (like early out
980 # in add/sub) are allowed/expected to leave '00000' in some elements
981 #$try = '=~ /^00+/; '."($x, $e)";
982 #last if $e =~ /^00+/;
985 return "Illegal part '$e' at pos $i (tested: $try)" if $i < $j;
990 ###############################################################################
991 ###############################################################################
992 # some optional routines to make BigInt faster
996 # if possible, use mod shortcut
999 # slow way since $y to big
1000 if (scalar @$yo > 1)
1002 my ($xo,$rem) = _div($c,$x,$yo);
1006 # both are single element arrays
1007 if (scalar @$x == 1)
1013 # @y is single element, but @x has more than one
1017 # when BASE % Y == 0 then (B * BASE) % Y == 0
1018 # (B * BASE) % $y + A % Y => A % Y
1019 # so need to consider only last element: O(1)
1024 # else need to go trough all elements: O(N), but loop is a bit simplified
1036 # else need to go trough all elements: O(N)
1037 my $r = 0; my $bm = 1;
1040 $r += ($_ % $y) * $bm;
1052 ##############################################################################
1057 my ($c,$x,$y,$n) = @_;
1061 $n = _new($c,\$n); return _div($c,$x, _pow($c,$n,$y));
1064 # shortcut (faster) for shifting by 10)
1065 # multiples of $BASE_LEN
1066 my $dst = 0; # destination
1067 my $src = _num($c,$y); # as normal int
1068 my $rem = $src % $BASE_LEN; # remainder to shift
1069 $src = int($src / $BASE_LEN); # source
1072 splice (@$x,0,$src); # even faster, 38.4 => 39.3
1076 my $len = scalar @$x - $src; # elems to go
1077 my $vd; my $z = '0'x $BASE_LEN;
1078 $x->[scalar @$x] = 0; # avoid || 0 test inside loop
1081 $vd = $z.$x->[$src];
1082 $vd = substr($vd,-$BASE_LEN,$BASE_LEN-$rem);
1084 $vd = substr($z.$x->[$src],-$rem,$rem) . $vd;
1085 $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN;
1086 $x->[$dst] = int($vd);
1089 splice (@$x,$dst) if $dst > 0; # kill left-over array elems
1090 pop @$x if $x->[-1] == 0; # kill last element if 0
1097 my ($c,$x,$y,$n) = @_;
1101 $n = _new($c,\$n); return _mul($c,$x, _pow($c,$n,$y));
1104 # shortcut (faster) for shifting by 10) since we are in base 10eX
1105 # multiples of $BASE_LEN:
1106 my $src = scalar @$x; # source
1107 my $len = _num($c,$y); # shift-len as normal int
1108 my $rem = $len % $BASE_LEN; # remainder to shift
1109 my $dst = $src + int($len/$BASE_LEN); # destination
1110 my $vd; # further speedup
1111 $x->[$src] = 0; # avoid first ||0 for speed
1112 my $z = '0' x $BASE_LEN;
1115 $vd = $x->[$src]; $vd = $z.$vd;
1116 $vd = substr($vd,-$BASE_LEN+$rem,$BASE_LEN-$rem);
1117 $vd .= $src > 0 ? substr($z.$x->[$src-1],-$BASE_LEN,$rem) : '0' x $rem;
1118 $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN;
1119 $x->[$dst] = int($vd);
1122 # set lowest parts to 0
1123 while ($dst >= 0) { $x->[$dst--] = 0; }
1124 # fix spurios last zero element
1125 splice @$x,-1 if $x->[-1] == 0;
1132 # ref to array, ref to array, return ref to array
1133 my ($c,$cx,$cy) = @_;
1137 my $y1 = _copy($c,$cy);
1138 while (!_is_one($c,$y1))
1140 _mul($c,$pow2,$cx) if _is_odd($c,$y1);
1144 _mul($c,$cx,$pow2) unless _is_one($c,$pow2);
1151 # ref to array, return ref to array
1154 if ((@$cx == 1) && ($cx->[0] <= 2))
1156 $cx->[0] = 1 * ($cx->[0]||1); # 0,1 => 1, 2 => 2
1160 # go forward until $base is exceeded
1161 # limit is either $x or $base (x == 100 means as result too high)
1162 my $steps = 100; $steps = $cx->[0] if @$cx == 1;
1163 my $r = 2; my $cf = 3; my $step = 1; my $last = $r;
1164 while ($r < $BASE && $step < $steps)
1166 $last = $r; $r *= $cf++; $step++;
1168 if ((@$cx == 1) && ($step == $cx->[0]))
1174 my $n = _copy($c,$cx);
1178 while (!(@$n == 1 && $n->[0] == $step))
1180 _mul($c,$cx,$n); _dec($c,$n);
1185 use constant DEBUG => 0;
1189 sub steps { $steps };
1194 # ref to array, return ref to array
1197 if (scalar @$x == 1)
1199 # fit's into one Perl scalar
1200 $x->[0] = int(sqrt($x->[0]));
1203 my $y = _copy($c,$x);
1204 # hopefully _len/2 is < $BASE, the -1 is to always undershot the guess
1205 # since our guess will "grow"
1206 my $l = int((_len($c,$x)-1) / 2);
1208 my $lastelem = $x->[-1]; # for guess
1209 my $elems = scalar @$x - 1;
1210 # not enough digits, but could have more?
1211 if ((length($lastelem) <= 3) && ($elems > 1))
1213 # right-align with zero pad
1214 my $len = length($lastelem) & 1;
1215 print "$lastelem => " if DEBUG;
1216 $lastelem .= substr($x->[-2] . '0' x $BASE_LEN,0,$BASE_LEN);
1217 # former odd => make odd again, or former even to even again
1218 $lastelem = $lastelem / 10 if (length($lastelem) & 1) != $len;
1219 print "$lastelem\n" if DEBUG;
1222 # construct $x (instead of _lsft($c,$x,$l,10)
1223 my $r = $l % $BASE_LEN; # 10000 00000 00000 00000 ($BASE_LEN=5)
1224 $l = int($l / $BASE_LEN);
1225 print "l = $l " if DEBUG;
1227 splice @$x,$l; # keep ref($x), but modify it
1229 # we make the first part of the guess not '1000...0' but int(sqrt($lastelem))
1231 # 14400 00000 => sqrt(14400) => 120
1232 # 144000 000000 => sqrt(144000) => 379
1234 # $x->[$l--] = int('1' . '0' x $r); # old way of guessing
1235 print "$lastelem (elems $elems) => " if DEBUG;
1236 $lastelem = $lastelem / 10 if ($elems & 1 == 1); # odd or even?
1237 my $g = sqrt($lastelem); $g =~ s/\.//; # 2.345 => 2345
1238 $r -= 1 if $elems & 1 == 0; # 70 => 7
1240 # padd with zeros if result is too short
1241 $x->[$l--] = int(substr($g . '0' x $r,0,$r+1));
1242 print "now ",$x->[-1] if DEBUG;
1243 print " would have been ", int('1' . '0' x $r),"\n" if DEBUG;
1245 # If @$x > 1, we could compute the second elem of the guess, too, to create
1246 # an even better guess. Not implemented yet.
1247 $x->[$l--] = 0 while ($l >= 0); # all other digits of guess are zero
1249 print "start x= ",${_str($c,$x)},"\n" if DEBUG;
1252 my $lastlast = _zero();
1253 $steps = 0 if DEBUG;
1254 while (_acmp($c,$last,$x) != 0 && _acmp($c,$lastlast,$x) != 0)
1257 $lastlast = _copy($c,$last);
1258 $last = _copy($c,$x);
1259 _add($c,$x, _div($c,_copy($c,$y),$x));
1261 print " x= ",${_str($c,$x)},"\n" if DEBUG;
1263 print "\nsteps in sqrt: $steps, " if DEBUG;
1264 _dec($c,$x) if _acmp($c,$y,_mul($c,_copy($c,$x),$x)) < 0; # overshot?
1265 print " final ",$x->[-1],"\n" if DEBUG;
1269 ##############################################################################
1276 # the shortcut makes equal, large numbers _really_ fast, and makes only a
1277 # very small performance drop for small numbers (e.g. something with less
1278 # than 32 bit) Since we optimize for large numbers, this is enabled.
1279 return $x if _acmp($c,$x,$y) == 0; # shortcut
1281 my $m = _one(); my ($xr,$yr);
1282 my $mask = $AND_MASK;
1285 my $y1 = _copy($c,$y); # make copy
1289 while (!_is_zero($c,$x1) && !_is_zero($c,$y1))
1291 ($x1, $xr) = _div($c,$x1,$mask);
1292 ($y1, $yr) = _div($c,$y1,$mask);
1294 # make ints() from $xr, $yr
1295 # this is when the AND_BITS are greater tahn $BASE and is slower for
1296 # small (<256 bits) numbers, but faster for large numbers. Disabled
1297 # due to KISS principle
1299 # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; }
1300 # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; }
1301 # _add($c,$x, _mul($c, _new( $c, \($xrr & $yrr) ), $m) );
1303 # 0+ due to '&' doesn't work in strings
1304 _add($c,$x, _mul($c, [ 0+$xr->[0] & 0+$yr->[0] ], $m) );
1314 return _zero() if _acmp($c,$x,$y) == 0; # shortcut (see -and)
1316 my $m = _one(); my ($xr,$yr);
1317 my $mask = $XOR_MASK;
1320 my $y1 = _copy($c,$y); # make copy
1324 while (!_is_zero($c,$x1) && !_is_zero($c,$y1))
1326 ($x1, $xr) = _div($c,$x1,$mask);
1327 ($y1, $yr) = _div($c,$y1,$mask);
1328 # make ints() from $xr, $yr (see _and())
1329 #$b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; }
1330 #$b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; }
1331 #_add($c,$x, _mul($c, _new( $c, \($xrr ^ $yrr) ), $m) );
1333 # 0+ due to '^' doesn't work in strings
1334 _add($c,$x, _mul($c, [ 0+$xr->[0] ^ 0+$yr->[0] ], $m) );
1337 # the loop stops when the shorter of the two numbers is exhausted
1338 # the remainder of the longer one will survive bit-by-bit, so we simple
1339 # multiply-add it in
1340 _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1);
1341 _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1);
1350 return $x if _acmp($c,$x,$y) == 0; # shortcut (see _and)
1352 my $m = _one(); my ($xr,$yr);
1353 my $mask = $OR_MASK;
1356 my $y1 = _copy($c,$y); # make copy
1360 while (!_is_zero($c,$x1) && !_is_zero($c,$y1))
1362 ($x1, $xr) = _div($c,$x1,$mask);
1363 ($y1, $yr) = _div($c,$y1,$mask);
1364 # make ints() from $xr, $yr (see _and())
1365 # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; }
1366 # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; }
1367 # _add($c,$x, _mul($c, _new( $c, \($xrr | $yrr) ), $m) );
1369 # 0+ due to '|' doesn't work in strings
1370 _add($c,$x, _mul($c, [ 0+$xr->[0] | 0+$yr->[0] ], $m) );
1373 # the loop stops when the shorter of the two numbers is exhausted
1374 # the remainder of the longer one will survive bit-by-bit, so we simple
1375 # multiply-add it in
1376 _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1);
1377 _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1);
1384 # convert a decimal number to hex (ref to array, return ref to string)
1387 my $x1 = _copy($c,$x);
1391 my $x10000 = [ 0x10000 ];
1392 while (! _is_zero($c,$x1))
1394 ($x1, $xr) = _div($c,$x1,$x10000);
1395 $es .= unpack('h4',pack('v',$xr->[0]));
1398 $es =~ s/^[0]+//; # strip leading zeros
1405 # convert a decimal number to bin (ref to array, return ref to string)
1408 my $x1 = _copy($c,$x);
1412 my $x10000 = [ 0x10000 ];
1413 while (! _is_zero($c,$x1))
1415 ($x1, $xr) = _div($c,$x1,$x10000);
1416 $es .= unpack('b16',pack('v',$xr->[0]));
1419 $es =~ s/^[0]+//; # strip leading zeros
1426 # convert a hex number to decimal (ref to string, return ref to array)
1430 my $m = [ 0x10000 ]; # 16 bit at a time
1433 my $len = length($$hs)-2;
1434 $len = int($len/4); # 4-digit parts, w/o '0x'
1435 my $val; my $i = -4;
1438 $val = substr($$hs,$i,4);
1439 $val =~ s/^[+-]?0x// if $len == 0; # for last part only because
1440 $val = hex($val); # hex does not like wrong chars
1442 _add ($c, $x, _mul ($c, [ $val ], $mul ) ) if $val != 0;
1443 _mul ($c, $mul, $m ) if $len >= 0; # skip last mul
1450 # convert a hex number to decimal (ref to string, return ref to array)
1454 my $m = [ 0x100 ]; # 8 bit at a time
1457 my $len = length($$bs)-2;
1458 $len = int($len/8); # 4-digit parts, w/o '0x'
1459 my $val; my $i = -8;
1462 $val = substr($$bs,$i,8);
1463 $val =~ s/^[+-]?0b// if $len == 0; # for last part only
1465 $val = ord(pack('B8',substr('00000000'.$val,-8,8)));
1468 _add ($c, $x, _mul ($c, [ $val ], $mul ) ) if $val != 0;
1469 _mul ($c, $mul, $m ) if $len >= 0; # skip last mul
1474 ##############################################################################
1475 ##############################################################################
1482 Math::BigInt::Calc - Pure Perl module to support Math::BigInt
1486 Provides support for big integer calculations. Not intended to be used by other
1487 modules (except Math::BigInt::Cached). Other modules which sport the same
1488 functions can also be used to support Math::Bigint, like Math::BigInt::Pari.
1492 In order to allow for multiple big integer libraries, Math::BigInt was
1493 rewritten to use library modules for core math routines. Any module which
1494 follows the same API as this can be used instead by using the following:
1496 use Math::BigInt lib => 'libname';
1498 'libname' is either the long name ('Math::BigInt::Pari'), or only the short
1499 version like 'Pari'.
1503 The following functions MUST be defined in order to support the use by
1506 _new(string) return ref to new object from ref to decimal string
1507 _zero() return a new object with value 0
1508 _one() return a new object with value 1
1510 _str(obj) return ref to a string representing the object
1511 _num(obj) returns a Perl integer/floating point number
1512 NOTE: because of Perl numeric notation defaults,
1513 the _num'ified obj may lose accuracy due to
1514 machine-dependend floating point size limitations
1516 _add(obj,obj) Simple addition of two objects
1517 _mul(obj,obj) Multiplication of two objects
1518 _div(obj,obj) Division of the 1st object by the 2nd
1519 In list context, returns (result,remainder).
1520 NOTE: this is integer math, so no
1521 fractional part will be returned.
1522 _sub(obj,obj) Simple subtraction of 1 object from another
1523 a third, optional parameter indicates that the params
1524 are swapped. In this case, the first param needs to
1525 be preserved, while you can destroy the second.
1526 sub (x,y,1) => return x - y and keep x intact!
1527 _dec(obj) decrement object by one (input is garant. to be > 0)
1528 _inc(obj) increment object by one
1531 _acmp(obj,obj) <=> operator for objects (return -1, 0 or 1)
1533 _len(obj) returns count of the decimal digits of the object
1534 _digit(obj,n) returns the n'th decimal digit of object
1536 _is_one(obj) return true if argument is +1
1537 _is_zero(obj) return true if argument is 0
1538 _is_even(obj) return true if argument is even (0,2,4,6..)
1539 _is_odd(obj) return true if argument is odd (1,3,5,7..)
1541 _copy return a ref to a true copy of the object
1543 _check(obj) check whether internal representation is still intact
1544 return 0 for ok, otherwise error message as string
1546 The following functions are optional, and can be defined if the underlying lib
1547 has a fast way to do them. If undefined, Math::BigInt will use pure Perl (hence
1548 slow) fallback routines to emulate these:
1550 _from_hex(str) return ref to new object from ref to hexadecimal string
1551 _from_bin(str) return ref to new object from ref to binary string
1553 _as_hex(str) return ref to scalar string containing the value as
1554 unsigned hex string, with the '0x' prepended.
1555 Leading zeros must be stripped.
1556 _as_bin(str) Like as_hex, only as binary string containing only
1557 zeros and ones. Leading zeros must be stripped and a
1558 '0b' must be prepended.
1560 _rsft(obj,N,B) shift object in base B by N 'digits' right
1561 For unsupported bases B, return undef to signal failure
1562 _lsft(obj,N,B) shift object in base B by N 'digits' left
1563 For unsupported bases B, return undef to signal failure
1565 _xor(obj1,obj2) XOR (bit-wise) object 1 with object 2
1566 Note: XOR, AND and OR pad with zeros if size mismatches
1567 _and(obj1,obj2) AND (bit-wise) object 1 with object 2
1568 _or(obj1,obj2) OR (bit-wise) object 1 with object 2
1570 _mod(obj,obj) Return remainder of div of the 1st by the 2nd object
1571 _sqrt(obj) return the square root of object (truncate to int)
1572 _fac(obj) return factorial of object 1 (1*2*3*4..)
1573 _pow(obj,obj) return object 1 to the power of object 2
1574 _gcd(obj,obj) return Greatest Common Divisor of two objects
1576 _zeros(obj) return number of trailing decimal zeros
1578 Input strings come in as unsigned but with prefix (i.e. as '123', '0xabc'
1581 Testing of input parameter validity is done by the caller, so you need not
1582 worry about underflow (f.i. in C<_sub()>, C<_dec()>) nor about division by
1583 zero or similar cases.
1585 The first parameter can be modified, that includes the possibility that you
1586 return a reference to a completely different object instead. Although keeping
1587 the reference and just changing it's contents is prefered over creating and
1588 returning a different reference.
1590 Return values are always references to objects or strings. Exceptions are
1591 C<_lsft()> and C<_rsft()>, which return undef if they can not shift the
1592 argument. This is used to delegate shifting of bases different than the one
1593 you can support back to Math::BigInt, which will use some generic code to
1594 calculate the result.
1596 =head1 WRAP YOUR OWN
1598 If you want to port your own favourite c-lib for big numbers to the
1599 Math::BigInt interface, you can take any of the already existing modules as
1600 a rough guideline. You should really wrap up the latest BigInt and BigFloat
1601 testsuites with your module, and replace in them any of the following:
1607 use Math::BigInt lib => 'yourlib';
1609 This way you ensure that your library really works 100% within Math::BigInt.
1613 This program is free software; you may redistribute it and/or modify it under
1614 the same terms as Perl itself.
1618 Original math code by Mark Biggar, rewritten by Tels L<http://bloodgate.com/>
1620 Seperated from BigInt and shaped API with the help of John Peacock.
1624 L<Math::BigInt>, L<Math::BigFloat>, L<Math::BigInt::BitVect>,
1625 L<Math::BigInt::GMP>, L<Math::BigInt::Cached> and L<Math::BigInt::Pari>.