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Initial commit of OpenSPARC T2 design and verification files.
[OpenSPARC-T2-DV] / tools / perl-5.8.0 / lib / 5.8.0 / Math / BigInt / Calc.pm
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AT		 1package Math::BigInt::Calc;
2
3use 5.005;
4use strict;
5# use warnings; # dont use warnings for older Perls
6
7require Exporter;
8use vars qw/@ISA $VERSION/;
9@ISA = qw(Exporter);
10
11$VERSION = '0.30';
12
13# Package to store unsigned big integers in decimal and do math with them
14
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
18
19# todo:
20# - fully remove funky $# stuff (maybe)
21
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
26# correct result.
27
28##############################################################################
29# global constants, flags and accessory
30
31# constants for easier life
32my $nan = 'NaN';
33my ($MBASE,$BASE,$RBASE,$BASE_LEN,$MAX_VAL,$BASE_LEN2,$BASE_LEN_SMALL);
34my ($AND_BITS,$XOR_BITS,$OR_BITS);
35my ($AND_MASK,$XOR_MASK,$OR_MASK);
36my ($LEN_CONVERT);
37
38sub _base_len
39 {
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
42 shift;
43
44 my $b = shift;
45 if (defined $b)
46 {
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;
50 my $caught = 0;
51 while (--$BASE_LEN_SMALL > 5)
52 {
53 $MBASE = int("1e".$BASE_LEN_SMALL);
54 $RBASE = abs('1e-'.$BASE_LEN_SMALL); # see USE_MUL
55 $caught = 0;
56 $caught += 1 if (int($MBASE * $RBASE) != 1); # should be 1
57 $caught += 2 if (int($MBASE / $MBASE) != 1); # should be 1
58 last if $caught != 3;
59 }
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);
64
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
68 $MAX_VAL = $MBASE-1;
69 $LEN_CONVERT = 0;
70 $LEN_CONVERT = 1 if $BASE_LEN_SMALL != $BASE_LEN;
71
72 #print "BASE_LEN: $BASE_LEN MAX_VAL: $MAX_VAL BASE: $BASE RBASE: $RBASE ";
73 #print "BASE_LEN_SMALL: $BASE_LEN_SMALL MBASE: $MBASE\n";
74
75 undef &_mul;
76 undef &_div;
77
78 if ($caught & 1 != 0)
79 {
80 # must USE_MUL
81 *{_mul} = \&_mul_use_mul;
82 *{_div} = \&_div_use_mul;
83 }
84 else # $caught must be 2, since it can't be 1 nor 3
85 {
86 # can USE_DIV instead
87 *{_mul} = \&_mul_use_div;
88 *{_div} = \&_div_use_div;
89 }
90 }
91 return $BASE_LEN unless wantarray;
92 return ($BASE_LEN, $AND_BITS, $XOR_BITS, $OR_BITS, $BASE_LEN_SMALL, $MAX_VAL);
93 }
94
95BEGIN
96 {
97 # from Daniel Pfeiffer: determine largest group of digits that is precisely
98 # multipliable with itself plus carry
99 # Test now changed to expect the proper pattern, not a result off by 1 or 2
100 my ($e, $num) = 3; # lowest value we will use is 3+1-1 = 3
101 do
102 {
103 $num = ('9' x ++$e) + 0;
104 $num *= $num + 1.0;
105 } while ("$num" =~ /9{$e}0{$e}/); # must be a certain pattern
106 $e--; # last test failed, so retract one step
107 # the limits below brush the problems with the test above under the rug:
108 # the test should be able to find the proper $e automatically
109 $e = 5 if $^O =~ /^uts/; # UTS get's some special treatment
110 $e = 5 if $^O =~ /^unicos/; # unicos is also problematic (6 seems to work
111 # there, but we play safe)
112 $e = 5 if $] < 5.006; # cap, for older Perls
113 $e = 7 if $e > 7; # cap, for VMS, OS/390 and other 64 bit systems
114 # 8 fails inside random testsuite, so take 7
115
116 # determine how many digits fit into an integer and can be safely added
117 # together plus carry w/o causing an overflow
118
119 # this below detects 15 on a 64 bit system, because after that it becomes
120 # 1e16 and not 1000000 :/ I can make it detect 18, but then I get a lot of
121 # test failures. Ugh! (Tomake detect 18: uncomment lines marked with *)
122 use integer;
123 my $bi = 5; # approx. 16 bit
124 $num = int('9' x $bi);
125 # $num = 99999; # *
126 # while ( ($num+$num+1) eq '1' . '9' x $bi) # *
127 while ( int($num+$num+1) eq '1' . '9' x $bi)
128 {
129 $bi++; $num = int('9' x $bi);
130 # $bi++; $num *= 10; $num += 9; # *
131 }
132 $bi--; # back off one step
133 # by setting them equal, we ignore the findings and use the default
134 # one-size-fits-all approach from former versions
135 $bi = $e; # XXX, this should work always
136
137 __PACKAGE__->_base_len($e,$bi); # set and store
138
139 # find out how many bits _and, _or and _xor can take (old default = 16)
140 # I don't think anybody has yet 128 bit scalars, so let's play safe.
141 local $^W = 0; # don't warn about 'nonportable number'
142 $AND_BITS = 15; $XOR_BITS = 15; $OR_BITS = 15;
143
144 # find max bits, we will not go higher than numberofbits that fit into $BASE
145 # to make _and etc simpler (and faster for smaller, slower for large numbers)
146 my $max = 16;
147 while (2 ** $max < $BASE) { $max++; }
148 {
149 no integer;
150 $max = 16 if $] < 5.006; # older Perls might not take >16 too well
151 }
152 my ($x,$y,$z);
153 do {
154 $AND_BITS++;
155 $x = oct('0b' . '1' x $AND_BITS); $y = $x & $x;
156 $z = (2 ** $AND_BITS) - 1;
157 } while ($AND_BITS < $max && $x == $z && $y == $x);
158 $AND_BITS --; # retreat one step
159 do {
160 $XOR_BITS++;
161 $x = oct('0b' . '1' x $XOR_BITS); $y = $x ^ 0;
162 $z = (2 ** $XOR_BITS) - 1;
163 } while ($XOR_BITS < $max && $x == $z && $y == $x);
164 $XOR_BITS --; # retreat one step
165 do {
166 $OR_BITS++;
167 $x = oct('0b' . '1' x $OR_BITS); $y = $x | $x;
168 $z = (2 ** $OR_BITS) - 1;
169 } while ($OR_BITS < $max && $x == $z && $y == $x);
170 $OR_BITS --; # retreat one step
171
172 }
173
174##############################################################################
175# convert between the "small" and the "large" representation
176
177sub _to_large
178 {
179 # take an array in base $BASE_LEN_SMALL and convert it in-place to $BASE_LEN
180 my ($c,$x) = @_;
181
182# print "_to_large $BASE_LEN_SMALL => $BASE_LEN\n";
183
184 return $x if $LEN_CONVERT == 0 || # nothing to converconvertor
185 @$x == 1; # only one element => early out
186
187 # 12345 67890 12345 67890 contents
188 # to 3 2 1 0 index
189 # 123456 7890123 4567890 contents
190
191# # faster variant
192# my @d; my $str = '';
193# my $z = '0' x $BASE_LEN_SMALL;
194# foreach (@$x)
195# {
196# # ... . 04321 . 000321
197# $str = substr($z.$_,-$BASE_LEN_SMALL,$BASE_LEN_SMALL) . $str;
198# if (length($str) > $BASE_LEN)
199# {
200# push @d, substr($str,-$BASE_LEN,$BASE_LEN); # extract one piece
201# substr($str,-$BASE_LEN,$BASE_LEN) = ''; # remove it
202# }
203# }
204# push @d, $str if $str !~ /^0*$/; # extract last piece
205# @$x = @d;
206# $x->[-1] = int($x->[-1]); # strip leading zero
207# $x;
208
209 my $ret = "";
210 my $l = scalar @$x; # number of parts
211 $l --; $ret .= int($x->[$l]); $l--;
212 my $z = '0' x ($BASE_LEN_SMALL-1);
213 while ($l >= 0)
214 {
215 $ret .= substr($z.$x->[$l],-$BASE_LEN_SMALL);
216 $l--;
217 }
218 my $str = _new($c,\$ret); # make array
219 @$x = @$str; # clobber contents of $x
220 $x->[-1] = int($x->[-1]); # strip leading zero
221 }
222
223sub _to_small
224 {
225 # take an array in base $BASE_LEN and convert it in-place to $BASE_LEN_SMALL
226 my ($c,$x) = @_;
227
228 return $x if $LEN_CONVERT == 0; # nothing to do
229 return $x if @$x == 1 && length(int($x->[0])) <= $BASE_LEN_SMALL;
230
231 my $d = _str($c,$x);
232 my $il = length($$d)-1;
233 ## this leaves '00000' instead of int 0 and will be corrected after any op
234 # clobber contents of $x
235 @$x = reverse(unpack("a" . ($il % $BASE_LEN_SMALL+1)
236 . ("a$BASE_LEN_SMALL" x ($il / $BASE_LEN_SMALL)), $$d));
237
238 $x->[-1] = int($x->[-1]); # strip leading zero
239 }
240
241###############################################################################
242
243sub _new
244 {
245 # (ref to string) return ref to num_array
246 # Convert a number from string format (without sign) to internal base
247 # 1ex format. Assumes normalized value as input.
248 my $d = $_[1];
249 my $il = length($$d)-1;
250 # this leaves '00000' instead of int 0 and will be corrected after any op
251 [ reverse(unpack("a" . ($il % $BASE_LEN+1)
252 . ("a$BASE_LEN" x ($il / $BASE_LEN)), $$d)) ];
253 }
254
255BEGIN
256 {
257 $AND_MASK = __PACKAGE__->_new( \( 2 ** $AND_BITS ));
258 $XOR_MASK = __PACKAGE__->_new( \( 2 ** $XOR_BITS ));
259 $OR_MASK = __PACKAGE__->_new( \( 2 ** $OR_BITS ));
260 }
261
262sub _zero
263 {
264 # create a zero
265 [ 0 ];
266 }
267
268sub _one
269 {
270 # create a one
271 [ 1 ];
272 }
273
274sub _two
275 {
276 # create a two (used internally for shifting)
277 [ 2 ];
278 }
279
280sub _copy
281 {
282 [ @{$_[1]} ];
283 }
284
285# catch and throw away
286sub import { }
287
288##############################################################################
289# convert back to string and number
290
291sub _str
292 {
293 # (ref to BINT) return num_str
294 # Convert number from internal base 100000 format to string format.
295 # internal format is always normalized (no leading zeros, "-0" => "+0")
296 my $ar = $_[1];
297 my $ret = "";
298
299 my $l = scalar @$ar; # number of parts
300 return $nan if $l < 1; # should not happen
301
302 # handle first one different to strip leading zeros from it (there are no
303 # leading zero parts in internal representation)
304 $l --; $ret .= int($ar->[$l]); $l--;
305 # Interestingly, the pre-padd method uses more time
306 # the old grep variant takes longer (14 to 10 sec)
307 my $z = '0' x ($BASE_LEN-1);
308 while ($l >= 0)
309 {
310 $ret .= substr($z.$ar->[$l],-$BASE_LEN); # fastest way I could think of
311 $l--;
312 }
313 \$ret;
314 }
315
316sub _num
317 {
318 # Make a number (scalar int/float) from a BigInt object
319 my $x = $_[1];
320 return $x->[0] if scalar @$x == 1; # below $BASE
321 my $fac = 1;
322 my $num = 0;
323 foreach (@$x)
324 {
325 $num += $fac*$_; $fac *= $BASE;
326 }
327 $num;
328 }
329
330##############################################################################
331# actual math code
332
333sub _add
334 {
335 # (ref to int_num_array, ref to int_num_array)
336 # routine to add two base 1eX numbers
337 # stolen from Knuth Vol 2 Algorithm A pg 231
338 # there are separate routines to add and sub as per Knuth pg 233
339 # This routine clobbers up array x, but not y.
340
341 my ($c,$x,$y) = @_;
342
343 return $x if (@$y == 1) && $y->[0] == 0; # $x + 0 => $x
344 if ((@$x == 1) && $x->[0] == 0) # 0 + $y => $y->copy
345 {
346 # twice as slow as $x = [ @$y ], but necc. to retain $x as ref :(
347 @$x = @$y; return $x;
348 }
349
350 # for each in Y, add Y to X and carry. If after that, something is left in
351 # X, foreach in X add carry to X and then return X, carry
352 # Trades one "$j++" for having to shift arrays, $j could be made integer
353 # but this would impose a limit to number-length of 2**32.
354 my $i; my $car = 0; my $j = 0;
355 for $i (@$y)
356 {
357 $x->[$j] -= $BASE if $car = (($x->[$j] += $i + $car) >= $BASE) ? 1 : 0;
358 $j++;
359 }
360 while ($car != 0)
361 {
362 $x->[$j] -= $BASE if $car = (($x->[$j] += $car) >= $BASE) ? 1 : 0; $j++;
363 }
364 $x;
365 }
366
367sub _inc
368 {
369 # (ref to int_num_array, ref to int_num_array)
370 # routine to add 1 to a base 1eX numbers
371 # This routine clobbers up array x, but not y.
372 my ($c,$x) = @_;
373
374 for my $i (@$x)
375 {
376 return $x if (($i += 1) < $BASE); # early out
377 $i = 0; # overflow, next
378 }
379 push @$x,1 if ($x->[-1] == 0); # last overflowed, so extend
380 $x;
381 }
382
383sub _dec
384 {
385 # (ref to int_num_array, ref to int_num_array)
386 # routine to add 1 to a base 1eX numbers
387 # This routine clobbers up array x, but not y.
388 my ($c,$x) = @_;
389
390 my $MAX = $BASE-1; # since MAX_VAL based on MBASE
391 for my $i (@$x)
392 {
393 last if (($i -= 1) >= 0); # early out
394 $i = $MAX; # overflow, next
395 }
396 pop @$x if $x->[-1] == 0 && @$x > 1; # last overflowed (but leave 0)
397 $x;
398 }
399
400sub _sub
401 {
402 # (ref to int_num_array, ref to int_num_array, swap)
403 # subtract base 1eX numbers -- stolen from Knuth Vol 2 pg 232, $x > $y
404 # subtract Y from X by modifying x in place
405 my ($c,$sx,$sy,$s) = @_;
406
407 my $car = 0; my $i; my $j = 0;
408 if (!$s)
409 {
410 #print "case 2\n";
411 for $i (@$sx)
412 {
413 last unless defined $sy->[$j] || $car;
414 $i += $BASE if $car = (($i -= ($sy->[$j] || 0) + $car) < 0); $j++;
415 }
416 # might leave leading zeros, so fix that
417 return __strip_zeros($sx);
418 }
419 #print "case 1 (swap)\n";
420 for $i (@$sx)
421 {
422 # we can't do an early out if $x is < than $y, since we
423 # need to copy the high chunks from $y. Found by Bob Mathews.
424 #last unless defined $sy->[$j] || $car;
425 $sy->[$j] += $BASE
426 if $car = (($sy->[$j] = $i-($sy->[$j]||0) - $car) < 0);
427 $j++;
428 }
429 # might leave leading zeros, so fix that
430 __strip_zeros($sy);
431 }
432
433sub _square_use_mul
434 {
435 # compute $x ** 2 or $x * $x in-place and return $x
436 my ($c,$x) = @_;
437
438 # From: Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and
439 # S. Vanstone., Chapter 14
440
441 #14.16 Algorithm Multiple-precision squaring
442 #INPUT: positive integer x = (xt 1 xt 2 ... x1 x0)b.
443 #OUTPUT: x * x = x ** 2 in radix b representation.
444 #1. For i from 0 to (2t - 1) do: wi <- 0.
445 #2. For i from 0 to (t - 1) do the following:
446 # 2.1 (uv)b w2i + xi * xi, w2i v, c u.
447 # 2.2 For j from (i + 1)to (t - 1) do the following:
448 # (uv)b <- wi+j + 2*xj * xi + c, wi+j <- v, c <- u.
449 # 2.3 wi+t <- u.
450 #3. Return((w2t-1 w2t-2 ... w1 w0)b).
451
452# # Note: That description is crap. Half of the symbols are not explained or
453# # used with out beeing set.
454# my $t = scalar @$x; # count
455# my ($c,$i,$j);
456# for ($i = 0; $i < $t; $i++)
457# {
458# $x->[$i] = $x->[$i*2] + $x[$i]*$x[$i];
459# $x->[$i*2] = $x[$i]; $c = $x[$i];
460# for ($j = $i+1; $j < $t; $j++)
461# {
462# $x->[$i] = $x->[$i+$j] + 2 * $x->[$i] * $x->[$j];
463# $x->[$i+$j] = $x[$j]; $c = $x[$i];
464# }
465# $x->[$i+$t] = $x[$i];
466# }
467 $x;
468 }
469
470sub _mul_use_mul
471 {
472 # (ref to int_num_array, ref to int_num_array)
473 # multiply two numbers in internal representation
474 # modifies first arg, second need not be different from first
475 my ($c,$xv,$yv) = @_;
476
477 # shortcut for two very short numbers (improved by Nathan Zook)
478 # works also if xv and yv are the same reference
479 if ((@$xv == 1) && (@$yv == 1))
480 {
481 if (($xv->[0] *= $yv->[0]) >= $MBASE)
482 {
483 $xv->[0] = $xv->[0] - ($xv->[1] = int($xv->[0] * $RBASE)) * $MBASE;
484 };
485 return $xv;
486 }
487 # shortcut for result == 0
488 if ( ((@$xv == 1) && ($xv->[0] == 0)) ||
489 ((@$yv == 1) && ($yv->[0] == 0)) )
490 {
491 @$xv = (0);
492 return $xv;
493 }
494
495 # since multiplying $x with $x fails, make copy in this case
496 $yv = [@$xv] if $xv == $yv; # same references?
497# $yv = [@$xv] if "$xv" eq "$yv"; # same references?
498
499 # since multiplying $x with $x would fail here, use the faster squaring
500# return _square($c,$xv) if $xv == $yv; # same reference?
501
502 if ($LEN_CONVERT != 0)
503 {
504 $c->_to_small($xv); $c->_to_small($yv);
505 }
506
507 my @prod = (); my ($prod,$car,$cty,$xi,$yi);
508
509 for $xi (@$xv)
510 {
511 $car = 0; $cty = 0;
512
513 # slow variant
514# for $yi (@$yv)
515# {
516# $prod = $xi * $yi + ($prod[$cty] || 0) + $car;
517# $prod[$cty++] =
518# $prod - ($car = int($prod * RBASE)) * $MBASE; # see USE_MUL
519# }
520# $prod[$cty] += $car if $car; # need really to check for 0?
521# $xi = shift @prod;
522
523 # faster variant
524 # looping through this if $xi == 0 is silly - so optimize it away!
525 $xi = (shift @prod || 0), next if $xi == 0;
526 for $yi (@$yv)
527 {
528 $prod = $xi * $yi + ($prod[$cty] || 0) + $car;
529## this is actually a tad slower
530## $prod = $prod[$cty]; $prod += ($car + $xi * $yi); # no ||0 here
531 $prod[$cty++] =
532 $prod - ($car = int($prod * $RBASE)) * $MBASE; # see USE_MUL
533 }
534 $prod[$cty] += $car if $car; # need really to check for 0?
535 $xi = shift @prod || 0; # || 0 makes v5.005_3 happy
536 }
537 push @$xv, @prod;
538 if ($LEN_CONVERT != 0)
539 {
540 $c->_to_large($yv);
541 $c->_to_large($xv);
542 }
543 else
544 {
545 __strip_zeros($xv);
546 }
547 $xv;
548 }
549
550sub _mul_use_div
551 {
552 # (ref to int_num_array, ref to int_num_array)
553 # multiply two numbers in internal representation
554 # modifies first arg, second need not be different from first
555 my ($c,$xv,$yv) = @_;
556
557 # shortcut for two very short numbers (improved by Nathan Zook)
558 # works also if xv and yv are the same reference
559 if ((@$xv == 1) && (@$yv == 1))
560 {
561 if (($xv->[0] *= $yv->[0]) >= $MBASE)
562 {
563 $xv->[0] =
564 $xv->[0] - ($xv->[1] = int($xv->[0] / $MBASE)) * $MBASE;
565 };
566 return $xv;
567 }
568 # shortcut for result == 0
569 if ( ((@$xv == 1) && ($xv->[0] == 0)) ||
570 ((@$yv == 1) && ($yv->[0] == 0)) )
571 {
572 @$xv = (0);
573 return $xv;
574 }
575
576
577 # since multiplying $x with $x fails, make copy in this case
578 $yv = [@$xv] if $xv == $yv; # same references?
579# $yv = [@$xv] if "$xv" eq "$yv"; # same references?
580 # since multiplying $x with $x would fail here, use the faster squaring
581# return _square($c,$xv) if $xv == $yv; # same reference?
582
583 if ($LEN_CONVERT != 0)
584 {
585 $c->_to_small($xv); $c->_to_small($yv);
586 }
587
588 my @prod = (); my ($prod,$car,$cty,$xi,$yi);
589 for $xi (@$xv)
590 {
591 $car = 0; $cty = 0;
592 # looping through this if $xi == 0 is silly - so optimize it away!
593 $xi = (shift @prod || 0), next if $xi == 0;
594 for $yi (@$yv)
595 {
596 $prod = $xi * $yi + ($prod[$cty] || 0) + $car;
597 $prod[$cty++] =
598 $prod - ($car = int($prod / $MBASE)) * $MBASE;
599 }
600 $prod[$cty] += $car if $car; # need really to check for 0?
601 $xi = shift @prod || 0; # || 0 makes v5.005_3 happy
602 }
603 push @$xv, @prod;
604 if ($LEN_CONVERT != 0)
605 {
606 $c->_to_large($yv);
607 $c->_to_large($xv);
608 }
609 else
610 {
611 __strip_zeros($xv);
612 }
613 $xv;
614 }
615
616sub _div_use_mul
617 {
618 # ref to array, ref to array, modify first array and return remainder if
619 # in list context
620 my ($c,$x,$yorg) = @_;
621
622 if (@$x == 1 && @$yorg == 1)
623 {
624 # shortcut, $yorg and $x are two small numbers
625 if (wantarray)
626 {
627 my $r = [ $x->[0] % $yorg->[0] ];
628 $x->[0] = int($x->[0] / $yorg->[0]);
629 return ($x,$r);
630 }
631 else
632 {
633 $x->[0] = int($x->[0] / $yorg->[0]);
634 return $x;
635 }
636 }
637 if (@$yorg == 1)
638 {
639 my $rem;
640 $rem = _mod($c,[ @$x ],$yorg) if wantarray;
641
642 # shortcut, $y is < $BASE
643 my $j = scalar @$x; my $r = 0;
644 my $y = $yorg->[0]; my $b;
645 while ($j-- > 0)
646 {
647 $b = $r * $MBASE + $x->[$j];
648 $x->[$j] = int($b/$y);
649 $r = $b % $y;
650 }
651 pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero
652 return ($x,$rem) if wantarray;
653 return $x;
654 }
655
656 my $y = [ @$yorg ]; # always make copy to preserve
657 if ($LEN_CONVERT != 0)
658 {
659 $c->_to_small($x); $c->_to_small($y);
660 }
661
662 my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0);
663
664 $car = $bar = $prd = 0;
665 if (($dd = int($MBASE/($y->[-1]+1))) != 1)
666 {
667 for $xi (@$x)
668 {
669 $xi = $xi * $dd + $car;
670 $xi -= ($car = int($xi * $RBASE)) * $MBASE; # see USE_MUL
671 }
672 push(@$x, $car); $car = 0;
673 for $yi (@$y)
674 {
675 $yi = $yi * $dd + $car;
676 $yi -= ($car = int($yi * $RBASE)) * $MBASE; # see USE_MUL
677 }
678 }
679 else
680 {
681 push(@$x, 0);
682 }
683 @q = (); ($v2,$v1) = @$y[-2,-1];
684 $v2 = 0 unless $v2;
685 while ($#$x > $#$y)
686 {
687 ($u2,$u1,$u0) = @$x[-3..-1];
688 $u2 = 0 unless $u2;
689 #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n"
690 # if $v1 == 0;
691 $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$MBASE+$u1)/$v1));
692 --$q while ($v2*$q > ($u0*$MBASE+$u1-$q*$v1)*$MBASE+$u2);
693 if ($q)
694 {
695 ($car, $bar) = (0,0);
696 for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi)
697 {
698 $prd = $q * $y->[$yi] + $car;
699 $prd -= ($car = int($prd * $RBASE)) * $MBASE; # see USE_MUL
700 $x->[$xi] += $MBASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0));
701 }
702 if ($x->[-1] < $car + $bar)
703 {
704 $car = 0; --$q;
705 for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi)
706 {
707 $x->[$xi] -= $MBASE
708 if ($car = (($x->[$xi] += $y->[$yi] + $car) > $MBASE));
709 }
710 }
711 }
712 pop(@$x); unshift(@q, $q);
713 }
714 if (wantarray)
715 {
716 @d = ();
717 if ($dd != 1)
718 {
719 $car = 0;
720 for $xi (reverse @$x)
721 {
722 $prd = $car * $MBASE + $xi;
723 $car = $prd - ($tmp = int($prd / $dd)) * $dd; # see USE_MUL
724 unshift(@d, $tmp);
725 }
726 }
727 else
728 {
729 @d = @$x;
730 }
731 @$x = @q;
732 my $d = \@d;
733 if ($LEN_CONVERT != 0)
734 {
735 $c->_to_large($x); $c->_to_large($d);
736 }
737 else
738 {
739 __strip_zeros($x);
740 __strip_zeros($d);
741 }
742 return ($x,$d);
743 }
744 @$x = @q;
745 if ($LEN_CONVERT != 0)
746 {
747 $c->_to_large($x);
748 }
749 else
750 {
751 __strip_zeros($x);
752 }
753 $x;
754 }
755
756sub _div_use_div
757 {
758 # ref to array, ref to array, modify first array and return remainder if
759 # in list context
760 my ($c,$x,$yorg) = @_;
761
762 if (@$x == 1 && @$yorg == 1)
763 {
764 # shortcut, $yorg and $x are two small numbers
765 if (wantarray)
766 {
767 my $r = [ $x->[0] % $yorg->[0] ];
768 $x->[0] = int($x->[0] / $yorg->[0]);
769 return ($x,$r);
770 }
771 else
772 {
773 $x->[0] = int($x->[0] / $yorg->[0]);
774 return $x;
775 }
776 }
777 if (@$yorg == 1)
778 {
779 my $rem;
780 $rem = _mod($c,[ @$x ],$yorg) if wantarray;
781
782 # shortcut, $y is < $BASE
783 my $j = scalar @$x; my $r = 0;
784 my $y = $yorg->[0]; my $b;
785 while ($j-- > 0)
786 {
787 $b = $r * $MBASE + $x->[$j];
788 $x->[$j] = int($b/$y);
789 $r = $b % $y;
790 }
791 pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero
792 return ($x,$rem) if wantarray;
793 return $x;
794 }
795
796 my $y = [ @$yorg ]; # always make copy to preserve
797 if ($LEN_CONVERT != 0)
798 {
799 $c->_to_small($x); $c->_to_small($y);
800 }
801
802 my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0);
803
804 $car = $bar = $prd = 0;
805 if (($dd = int($MBASE/($y->[-1]+1))) != 1)
806 {
807 for $xi (@$x)
808 {
809 $xi = $xi * $dd + $car;
810 $xi -= ($car = int($xi / $MBASE)) * $MBASE;
811 }
812 push(@$x, $car); $car = 0;
813 for $yi (@$y)
814 {
815 $yi = $yi * $dd + $car;
816 $yi -= ($car = int($yi / $MBASE)) * $MBASE;
817 }
818 }
819 else
820 {
821 push(@$x, 0);
822 }
823 @q = (); ($v2,$v1) = @$y[-2,-1];
824 $v2 = 0 unless $v2;
825 while ($#$x > $#$y)
826 {
827 ($u2,$u1,$u0) = @$x[-3..-1];
828 $u2 = 0 unless $u2;
829 #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n"
830 # if $v1 == 0;
831 $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$MBASE+$u1)/$v1));
832 --$q while ($v2*$q > ($u0*$MBASE+$u1-$q*$v1)*$MBASE+$u2);
833 if ($q)
834 {
835 ($car, $bar) = (0,0);
836 for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi)
837 {
838 $prd = $q * $y->[$yi] + $car;
839 $prd -= ($car = int($prd / $MBASE)) * $MBASE;
840 $x->[$xi] += $MBASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0));
841 }
842 if ($x->[-1] < $car + $bar)
843 {
844 $car = 0; --$q;
845 for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi)
846 {
847 $x->[$xi] -= $MBASE
848 if ($car = (($x->[$xi] += $y->[$yi] + $car) > $MBASE));
849 }
850 }
851 }
852 pop(@$x); unshift(@q, $q);
853 }
854 if (wantarray)
855 {
856 @d = ();
857 if ($dd != 1)
858 {
859 $car = 0;
860 for $xi (reverse @$x)
861 {
862 $prd = $car * $MBASE + $xi;
863 $car = $prd - ($tmp = int($prd / $dd)) * $dd;
864 unshift(@d, $tmp);
865 }
866 }
867 else
868 {
869 @d = @$x;
870 }
871 @$x = @q;
872 my $d = \@d;
873 if ($LEN_CONVERT != 0)
874 {
875 $c->_to_large($x); $c->_to_large($d);
876 }
877 else
878 {
879 __strip_zeros($x);
880 __strip_zeros($d);
881 }
882 return ($x,$d);
883 }
884 @$x = @q;
885 if ($LEN_CONVERT != 0)
886 {
887 $c->_to_large($x);
888 }
889 else
890 {
891 __strip_zeros($x);
892 }
893 $x;
894 }
895
896##############################################################################
897# testing
898
899sub _acmp
900 {
901 # internal absolute post-normalized compare (ignore signs)
902 # ref to array, ref to array, return <0, 0, >0
903 # arrays must have at least one entry; this is not checked for
904
905 my ($c,$cx,$cy) = @_;
906
907 # fast comp based on number of array elements (aka pseudo-length)
908 my $lxy = scalar @$cx - scalar @$cy;
909 return -1 if $lxy < 0; # already differs, ret
910 return 1 if $lxy > 0; # ditto
911
912 # now calculate length based on digits, not parts
913 $lxy = _len($c,$cx) - _len($c,$cy); # difference
914 return -1 if $lxy < 0;
915 return 1 if $lxy > 0;
916
917 # hm, same lengths, but same contents?
918 my $i = 0; my $a;
919 # first way takes 5.49 sec instead of 4.87, but has the early out advantage
920 # so grep is slightly faster, but more inflexible. hm. $_ instead of $k
921 # yields 5.6 instead of 5.5 sec huh?
922 # manual way (abort if unequal, good for early ne)
923 my $j = scalar @$cx - 1;
924 while ($j >= 0)
925 {
926 last if ($a = $cx->[$j] - $cy->[$j]); $j--;
927 }
928# my $j = scalar @$cx;
929# while (--$j >= 0)
930# {
931# last if ($a = $cx->[$j] - $cy->[$j]);
932# }
933 return 1 if $a > 0;
934 return -1 if $a < 0;
935 0; # equal
936
937 # while it early aborts, it is even slower than the manual variant
938 #grep { return $a if ($a = $_ - $cy->[$i++]); } @$cx;
939 # grep way, go trough all (bad for early ne)
940 #grep { $a = $_ - $cy->[$i++]; } @$cx;
941 #return $a;
942 }
943
944sub _len
945 {
946 # compute number of digits in bigint, minus the sign
947
948 # int() because add/sub sometimes leaves strings (like '00005') instead of
949 # '5' in this place, thus causing length() to report wrong length
950 my $cx = $_[1];
951
952 return (@$cx-1)*$BASE_LEN+length(int($cx->[-1]));
953 }
954
955sub _digit
956 {
957 # return the nth digit, negative values count backward
958 # zero is rightmost, so _digit(123,0) will give 3
959 my ($c,$x,$n) = @_;
960
961 my $len = _len('',$x);
962
963 $n = $len+$n if $n < 0; # -1 last, -2 second-to-last
964 $n = abs($n); # if negative was too big
965 $len--; $n = $len if $n > $len; # n to big?
966
967 my $elem = int($n / $BASE_LEN); # which array element
968 my $digit = $n % $BASE_LEN; # which digit in this element
969 $elem = '0000'.@$x[$elem]; # get element padded with 0's
970 return substr($elem,-$digit-1,1);
971 }
972
973sub _zeros
974 {
975 # return amount of trailing zeros in decimal
976 # check each array elem in _m for having 0 at end as long as elem == 0
977 # Upon finding a elem != 0, stop
978 my $x = $_[1];
979 my $zeros = 0; my $elem;
980 foreach my $e (@$x)
981 {
982 if ($e != 0)
983 {
984 $elem = "$e"; # preserve x
985 $elem =~ s/.*?(0*$)/$1/; # strip anything not zero
986 $zeros *= $BASE_LEN; # elems * 5
987 $zeros += length($elem); # count trailing zeros
988 last; # early out
989 }
990 $zeros ++; # real else branch: 50% slower!
991 }
992 $zeros;
993 }
994
995##############################################################################
996# _is_* routines
997
998sub _is_zero
999 {
1000 # return true if arg (BINT or num_str) is zero (array '+', '0')
1001 my $x = $_[1];
1002
1003 (((scalar @$x == 1) && ($x->[0] == 0))) <=> 0;
1004 }
1005
1006sub _is_even
1007 {
1008 # return true if arg (BINT or num_str) is even
1009 my $x = $_[1];
1010 (!($x->[0] & 1)) <=> 0;
1011 }
1012
1013sub _is_odd
1014 {
1015 # return true if arg (BINT or num_str) is even
1016 my $x = $_[1];
1017
1018 (($x->[0] & 1)) <=> 0;
1019 }
1020
1021sub _is_one
1022 {
1023 # return true if arg (BINT or num_str) is one (array '+', '1')
1024 my $x = $_[1];
1025
1026 (scalar @$x == 1) && ($x->[0] == 1) <=> 0;
1027 }
1028
1029sub __strip_zeros
1030 {
1031 # internal normalization function that strips leading zeros from the array
1032 # args: ref to array
1033 my $s = shift;
1034
1035 my $cnt = scalar @$s; # get count of parts
1036 my $i = $cnt-1;
1037 push @$s,0 if $i < 0; # div might return empty results, so fix it
1038
1039 return $s if @$s == 1; # early out
1040
1041 #print "strip: cnt $cnt i $i\n";
1042 # '0', '3', '4', '0', '0',
1043 # 0 1 2 3 4
1044 # cnt = 5, i = 4
1045 # i = 4
1046 # i = 3
1047 # => fcnt = cnt - i (5-2 => 3, cnt => 5-1 = 4, throw away from 4th pos)
1048 # >= 1: skip first part (this can be zero)
1049 while ($i > 0) { last if $s->[$i] != 0; $i--; }
1050 $i++; splice @$s,$i if ($i < $cnt); # $i cant be 0
1051 $s;
1052 }
1053
1054###############################################################################
1055# check routine to test internal state of corruptions
1056
1057sub _check
1058 {
1059 # used by the test suite
1060 my $x = $_[1];
1061
1062 return "$x is not a reference" if !ref($x);
1063
1064 # are all parts are valid?
1065 my $i = 0; my $j = scalar @$x; my ($e,$try);
1066 while ($i < $j)
1067 {
1068 $e = $x->[$i]; $e = 'undef' unless defined $e;
1069 $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e)";
1070 last if $e !~ /^[+]?[0-9]+$/;
1071 $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (stringify)";
1072 last if "$e" !~ /^[+]?[0-9]+$/;
1073 $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (cat-stringify)";
1074 last if '' . "$e" !~ /^[+]?[0-9]+$/;
1075 $try = ' < 0 || >= $BASE; '."($x, $e)";
1076 last if $e <0 || $e >= $BASE;
1077 # this test is disabled, since new/bnorm and certain ops (like early out
1078 # in add/sub) are allowed/expected to leave '00000' in some elements
1079 #$try = '=~ /^00+/; '."($x, $e)";
1080 #last if $e =~ /^00+/;
1081 $i++;
1082 }
1083 return "Illegal part '$e' at pos $i (tested: $try)" if $i < $j;
1084 return 0;
1085 }
1086
1087
1088###############################################################################
1089###############################################################################
1090# some optional routines to make BigInt faster
1091
1092sub _mod
1093 {
1094 # if possible, use mod shortcut
1095 my ($c,$x,$yo) = @_;
1096
1097 # slow way since $y to big
1098 if (scalar @$yo > 1)
1099 {
1100 my ($xo,$rem) = _div($c,$x,$yo);
1101 return $rem;
1102 }
1103 my $y = $yo->[0];
1104 # both are single element arrays
1105 if (scalar @$x == 1)
1106 {
1107 $x->[0] %= $y;
1108 return $x;
1109 }
1110
1111 # @y is single element, but @x has more than one
1112 my $b = $BASE % $y;
1113 if ($b == 0)
1114 {
1115 # when BASE % Y == 0 then (B * BASE) % Y == 0
1116 # (B * BASE) % $y + A % Y => A % Y
1117 # so need to consider only last element: O(1)
1118 $x->[0] %= $y;
1119 }
1120 elsif ($b == 1)
1121 {
1122 # else need to go trough all elements: O(N), but loop is a bit simplified
1123 my $r = 0;
1124 foreach (@$x)
1125 {
1126 $r = ($r + $_) % $y; # not much faster, but heh...
1127 #$r += $_ % $y; $r %= $y;
1128 }
1129 $r = 0 if $r == $y;
1130 $x->[0] = $r;
1131 }
1132 else
1133 {
1134 # else need to go trough all elements: O(N)
1135 my $r = 0; my $bm = 1;
1136 foreach (@$x)
1137 {
1138 $r = ($_ * $bm + $r) % $y;
1139 $bm = ($bm * $b) % $y;
1140
1141 #$r += ($_ % $y) * $bm;
1142 #$bm *= $b;
1143 #$bm %= $y;
1144 #$r %= $y;
1145 }
1146 $r = 0 if $r == $y;
1147 $x->[0] = $r;
1148 }
1149 splice (@$x,1);
1150 $x;
1151 }
1152
1153##############################################################################
1154# shifts
1155
1156sub _rsft
1157 {
1158 my ($c,$x,$y,$n) = @_;
1159
1160 if ($n != 10)
1161 {
1162 $n = _new($c,\$n); return _div($c,$x, _pow($c,$n,$y));
1163 }
1164
1165 # shortcut (faster) for shifting by 10)
1166 # multiples of $BASE_LEN
1167 my $dst = 0; # destination
1168 my $src = _num($c,$y); # as normal int
1169 my $rem = $src % $BASE_LEN; # remainder to shift
1170 $src = int($src / $BASE_LEN); # source
1171 if ($rem == 0)
1172 {
1173 splice (@$x,0,$src); # even faster, 38.4 => 39.3
1174 }
1175 else
1176 {
1177 my $len = scalar @$x - $src; # elems to go
1178 my $vd; my $z = '0'x $BASE_LEN;
1179 $x->[scalar @$x] = 0; # avoid || 0 test inside loop
1180 while ($dst < $len)
1181 {
1182 $vd = $z.$x->[$src];
1183 $vd = substr($vd,-$BASE_LEN,$BASE_LEN-$rem);
1184 $src++;
1185 $vd = substr($z.$x->[$src],-$rem,$rem) . $vd;
1186 $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN;
1187 $x->[$dst] = int($vd);
1188 $dst++;
1189 }
1190 splice (@$x,$dst) if $dst > 0; # kill left-over array elems
1191 pop @$x if $x->[-1] == 0 && @$x > 1; # kill last element if 0
1192 } # else rem == 0
1193 $x;
1194 }
1195
1196sub _lsft
1197 {
1198 my ($c,$x,$y,$n) = @_;
1199
1200 if ($n != 10)
1201 {
1202 $n = _new($c,\$n); return _mul($c,$x, _pow($c,$n,$y));
1203 }
1204
1205 # shortcut (faster) for shifting by 10) since we are in base 10eX
1206 # multiples of $BASE_LEN:
1207 my $src = scalar @$x; # source
1208 my $len = _num($c,$y); # shift-len as normal int
1209 my $rem = $len % $BASE_LEN; # remainder to shift
1210 my $dst = $src + int($len/$BASE_LEN); # destination
1211 my $vd; # further speedup
1212 $x->[$src] = 0; # avoid first ||0 for speed
1213 my $z = '0' x $BASE_LEN;
1214 while ($src >= 0)
1215 {
1216 $vd = $x->[$src]; $vd = $z.$vd;
1217 $vd = substr($vd,-$BASE_LEN+$rem,$BASE_LEN-$rem);
1218 $vd .= $src > 0 ? substr($z.$x->[$src-1],-$BASE_LEN,$rem) : '0' x $rem;
1219 $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN;
1220 $x->[$dst] = int($vd);
1221 $dst--; $src--;
1222 }
1223 # set lowest parts to 0
1224 while ($dst >= 0) { $x->[$dst--] = 0; }
1225 # fix spurios last zero element
1226 splice @$x,-1 if $x->[-1] == 0;
1227 $x;
1228 }
1229
1230sub _pow
1231 {
1232 # power of $x to $y
1233 # ref to array, ref to array, return ref to array
1234 my ($c,$cx,$cy) = @_;
1235
1236 my $pow2 = _one();
1237
1238 my $y_bin = ${_as_bin($c,$cy)}; $y_bin =~ s/^0b//;
1239 my $len = length($y_bin);
1240 while (--$len > 0)
1241 {
1242 _mul($c,$pow2,$cx) if substr($y_bin,$len,1) eq '1'; # is odd?
1243 _mul($c,$cx,$cx);
1244 }
1245
1246 _mul($c,$cx,$pow2);
1247 $cx;
1248 }
1249
1250sub _fac
1251 {
1252 # factorial of $x
1253 # ref to array, return ref to array
1254 my ($c,$cx) = @_;
1255
1256 if ((@$cx == 1) && ($cx->[0] <= 2))
1257 {
1258 $cx->[0] = 1 * ($cx->[0]||1); # 0,1 => 1, 2 => 2
1259 return $cx;
1260 }
1261
1262 # go forward until $base is exceeded
1263 # limit is either $x or $base (x == 100 means as result too high)
1264 my $steps = 100; $steps = $cx->[0] if @$cx == 1;
1265 my $r = 2; my $cf = 3; my $step = 1; my $last = $r;
1266 while ($r < $BASE && $step < $steps)
1267 {
1268 $last = $r; $r *= $cf++; $step++;
1269 }
1270 if ((@$cx == 1) && ($step == $cx->[0]))
1271 {
1272 # completely done
1273 $cx = [$last];
1274 return $cx;
1275 }
1276 my $n = _copy($c,$cx);
1277 $cx = [$last];
1278
1279 #$cx = _one();
1280 while (!(@$n == 1 && $n->[0] == $step))
1281 {
1282 _mul($c,$cx,$n); _dec($c,$n);
1283 }
1284 $cx;
1285 }
1286
1287use constant DEBUG => 0;
1288
1289my $steps = 0;
1290
1291sub steps { $steps };
1292
1293sub _sqrt
1294 {
1295 # square-root of $x
1296 # ref to array, return ref to array
1297 my ($c,$x) = @_;
1298
1299 if (scalar @$x == 1)
1300 {
1301 # fit's into one Perl scalar
1302 $x->[0] = int(sqrt($x->[0]));
1303 return $x;
1304 }
1305 my $y = _copy($c,$x);
1306 # hopefully _len/2 is < $BASE, the -1 is to always undershot the guess
1307 # since our guess will "grow"
1308 my $l = int((_len($c,$x)-1) / 2);
1309
1310 my $lastelem = $x->[-1]; # for guess
1311 my $elems = scalar @$x - 1;
1312 # not enough digits, but could have more?
1313 if ((length($lastelem) <= 3) && ($elems > 1))
1314 {
1315 # right-align with zero pad
1316 my $len = length($lastelem) & 1;
1317 print "$lastelem => " if DEBUG;
1318 $lastelem .= substr($x->[-2] . '0' x $BASE_LEN,0,$BASE_LEN);
1319 # former odd => make odd again, or former even to even again
1320 $lastelem = $lastelem / 10 if (length($lastelem) & 1) != $len;
1321 print "$lastelem\n" if DEBUG;
1322 }
1323
1324 # construct $x (instead of _lsft($c,$x,$l,10)
1325 my $r = $l % $BASE_LEN; # 10000 00000 00000 00000 ($BASE_LEN=5)
1326 $l = int($l / $BASE_LEN);
1327 print "l = $l " if DEBUG;
1328
1329 splice @$x,$l; # keep ref($x), but modify it
1330
1331 # we make the first part of the guess not '1000...0' but int(sqrt($lastelem))
1332 # that gives us:
1333 # 14400 00000 => sqrt(14400) => 120
1334 # 144000 000000 => sqrt(144000) => 379
1335
1336 # $x->[$l--] = int('1' . '0' x $r); # old way of guessing
1337 print "$lastelem (elems $elems) => " if DEBUG;
1338 $lastelem = $lastelem / 10 if ($elems & 1 == 1); # odd or even?
1339 my $g = sqrt($lastelem); $g =~ s/\.//; # 2.345 => 2345
1340 $r -= 1 if $elems & 1 == 0; # 70 => 7
1341
1342 # padd with zeros if result is too short
1343 $x->[$l--] = int(substr($g . '0' x $r,0,$r+1));
1344 print "now ",$x->[-1] if DEBUG;
1345 print " would have been ", int('1' . '0' x $r),"\n" if DEBUG;
1346
1347 # If @$x > 1, we could compute the second elem of the guess, too, to create
1348 # an even better guess. Not implemented yet.
1349 $x->[$l--] = 0 while ($l >= 0); # all other digits of guess are zero
1350
1351 print "start x= ",${_str($c,$x)},"\n" if DEBUG;
1352 my $two = _two();
1353 my $last = _zero();
1354 my $lastlast = _zero();
1355 $steps = 0 if DEBUG;
1356 while (_acmp($c,$last,$x) != 0 && _acmp($c,$lastlast,$x) != 0)
1357 {
1358 $steps++ if DEBUG;
1359 $lastlast = _copy($c,$last);
1360 $last = _copy($c,$x);
1361 _add($c,$x, _div($c,_copy($c,$y),$x));
1362 _div($c,$x, $two );
1363 print " x= ",${_str($c,$x)},"\n" if DEBUG;
1364 }
1365 print "\nsteps in sqrt: $steps, " if DEBUG;
1366 _dec($c,$x) if _acmp($c,$y,_mul($c,_copy($c,$x),$x)) < 0; # overshot?
1367 print " final ",$x->[-1],"\n" if DEBUG;
1368 $x;
1369 }
1370
1371##############################################################################
1372# binary stuff
1373
1374sub _and
1375 {
1376 my ($c,$x,$y) = @_;
1377
1378 # the shortcut makes equal, large numbers _really_ fast, and makes only a
1379 # very small performance drop for small numbers (e.g. something with less
1380 # than 32 bit) Since we optimize for large numbers, this is enabled.
1381 return $x if _acmp($c,$x,$y) == 0; # shortcut
1382
1383 my $m = _one(); my ($xr,$yr);
1384 my $mask = $AND_MASK;
1385
1386 my $x1 = $x;
1387 my $y1 = _copy($c,$y); # make copy
1388 $x = _zero();
1389 my ($b,$xrr,$yrr);
1390 use integer;
1391 while (!_is_zero($c,$x1) && !_is_zero($c,$y1))
1392 {
1393 ($x1, $xr) = _div($c,$x1,$mask);
1394 ($y1, $yr) = _div($c,$y1,$mask);
1395
1396 # make ints() from $xr, $yr
1397 # this is when the AND_BITS are greater tahn $BASE and is slower for
1398 # small (<256 bits) numbers, but faster for large numbers. Disabled
1399 # due to KISS principle
1400
1401# $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; }
1402# $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; }
1403# _add($c,$x, _mul($c, _new( $c, \($xrr & $yrr) ), $m) );
1404
1405 # 0+ due to '&' doesn't work in strings
1406 _add($c,$x, _mul($c, [ 0+$xr->[0] & 0+$yr->[0] ], $m) );
1407 _mul($c,$m,$mask);
1408 }
1409 $x;
1410 }
1411
1412sub _xor
1413 {
1414 my ($c,$x,$y) = @_;
1415
1416 return _zero() if _acmp($c,$x,$y) == 0; # shortcut (see -and)
1417
1418 my $m = _one(); my ($xr,$yr);
1419 my $mask = $XOR_MASK;
1420
1421 my $x1 = $x;
1422 my $y1 = _copy($c,$y); # make copy
1423 $x = _zero();
1424 my ($b,$xrr,$yrr);
1425 use integer;
1426 while (!_is_zero($c,$x1) && !_is_zero($c,$y1))
1427 {
1428 ($x1, $xr) = _div($c,$x1,$mask);
1429 ($y1, $yr) = _div($c,$y1,$mask);
1430 # make ints() from $xr, $yr (see _and())
1431 #$b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; }
1432 #$b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; }
1433 #_add($c,$x, _mul($c, _new( $c, \($xrr ^ $yrr) ), $m) );
1434
1435 # 0+ due to '^' doesn't work in strings
1436 _add($c,$x, _mul($c, [ 0+$xr->[0] ^ 0+$yr->[0] ], $m) );
1437 _mul($c,$m,$mask);
1438 }
1439 # the loop stops when the shorter of the two numbers is exhausted
1440 # the remainder of the longer one will survive bit-by-bit, so we simple
1441 # multiply-add it in
1442 _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1);
1443 _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1);
1444
1445 $x;
1446 }
1447
1448sub _or
1449 {
1450 my ($c,$x,$y) = @_;
1451
1452 return $x if _acmp($c,$x,$y) == 0; # shortcut (see _and)
1453
1454 my $m = _one(); my ($xr,$yr);
1455 my $mask = $OR_MASK;
1456
1457 my $x1 = $x;
1458 my $y1 = _copy($c,$y); # make copy
1459 $x = _zero();
1460 my ($b,$xrr,$yrr);
1461 use integer;
1462 while (!_is_zero($c,$x1) && !_is_zero($c,$y1))
1463 {
1464 ($x1, $xr) = _div($c,$x1,$mask);
1465 ($y1, $yr) = _div($c,$y1,$mask);
1466 # make ints() from $xr, $yr (see _and())
1467# $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; }
1468# $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; }
1469# _add($c,$x, _mul($c, _new( $c, \($xrr | $yrr) ), $m) );
1470
1471 # 0+ due to '|' doesn't work in strings
1472 _add($c,$x, _mul($c, [ 0+$xr->[0] | 0+$yr->[0] ], $m) );
1473 _mul($c,$m,$mask);
1474 }
1475 # the loop stops when the shorter of the two numbers is exhausted
1476 # the remainder of the longer one will survive bit-by-bit, so we simple
1477 # multiply-add it in
1478 _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1);
1479 _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1);
1480
1481 $x;
1482 }
1483
1484sub _as_hex
1485 {
1486 # convert a decimal number to hex (ref to array, return ref to string)
1487 my ($c,$x) = @_;
1488
1489 my $x1 = _copy($c,$x);
1490
1491 my $es = '';
1492 my ($xr, $h, $x10000);
1493 if ($] >= 5.006)
1494 {
1495 $x10000 = [ 0x10000 ]; $h = 'h4';
1496 }
1497 else
1498 {
1499 $x10000 = [ 0x1000 ]; $h = 'h3';
1500 }
1501 while (! _is_zero($c,$x1))
1502 {
1503 ($x1, $xr) = _div($c,$x1,$x10000);
1504 $es .= unpack($h,pack('v',$xr->[0]));
1505 }
1506 $es = reverse $es;
1507 $es =~ s/^[0]+//; # strip leading zeros
1508 $es = '0x' . $es;
1509 \$es;
1510 }
1511
1512sub _as_bin
1513 {
1514 # convert a decimal number to bin (ref to array, return ref to string)
1515 my ($c,$x) = @_;
1516
1517 my $x1 = _copy($c,$x);
1518
1519 my $es = '';
1520 my ($xr, $b, $x10000);
1521 if ($] >= 5.006)
1522 {
1523 $x10000 = [ 0x10000 ]; $b = 'b16';
1524 }
1525 else
1526 {
1527 $x10000 = [ 0x1000 ]; $b = 'b12';
1528 }
1529 while (! _is_zero($c,$x1))
1530 {
1531 ($x1, $xr) = _div($c,$x1,$x10000);
1532 $es .= unpack($b,pack('v',$xr->[0]));
1533 }
1534 $es = reverse $es;
1535 $es =~ s/^[0]+//; # strip leading zeros
1536 $es = '0b' . $es;
1537 \$es;
1538 }
1539
1540sub _from_hex
1541 {
1542 # convert a hex number to decimal (ref to string, return ref to array)
1543 my ($c,$hs) = @_;
1544
1545 my $mul = _one();
1546 my $m = [ 0x10000 ]; # 16 bit at a time
1547 my $x = _zero();
1548
1549 my $len = length($$hs)-2;
1550 $len = int($len/4); # 4-digit parts, w/o '0x'
1551 my $val; my $i = -4;
1552 while ($len >= 0)
1553 {
1554 $val = substr($$hs,$i,4);
1555 $val =~ s/^[+-]?0x// if $len == 0; # for last part only because
1556 $val = hex($val); # hex does not like wrong chars
1557 $i -= 4; $len --;
1558 _add ($c, $x, _mul ($c, [ $val ], $mul ) ) if $val != 0;
1559 _mul ($c, $mul, $m ) if $len >= 0; # skip last mul
1560 }
1561 $x;
1562 }
1563
1564sub _from_bin
1565 {
1566 # convert a hex number to decimal (ref to string, return ref to array)
1567 my ($c,$bs) = @_;
1568
1569 # instead of converting 8 bit at a time, it is faster to convert the
1570 # number to hex, and then call _from_hex.
1571
1572 my $hs = $$bs;
1573 $hs =~ s/^[+-]?0b//; # remove sign and 0b
1574 my $l = length($hs); # bits
1575 $hs = '0' x (8-($l % 8)) . $hs if ($l % 8) != 0; # padd left side w/ 0
1576 my $h = unpack('H*', pack ('B*', $hs)); # repack as hex
1577 return $c->_from_hex(\('0x'.$h));
1578
1579 my $mul = _one();
1580 my $m = [ 0x100 ]; # 8 bit at a time
1581 my $x = _zero();
1582
1583 my $len = length($$bs)-2;
1584 $len = int($len/8); # 4-digit parts, w/o '0x'
1585 my $val; my $i = -8;
1586 while ($len >= 0)
1587 {
1588 $val = substr($$bs,$i,8);
1589 $val =~ s/^[+-]?0b// if $len == 0; # for last part only
1590
1591 $val = ord(pack('B8',substr('00000000'.$val,-8,8)));
1592
1593 $i -= 8; $len --;
1594 _add ($c, $x, _mul ($c, [ $val ], $mul ) ) if $val != 0;
1595 _mul ($c, $mul, $m ) if $len >= 0; # skip last mul
1596 }
1597 $x;
1598 }
1599
1600##############################################################################
1601# special modulus functions
1602
1603# not ready yet, since it would need to deal with unsigned numbers
1604sub _modinv1
1605 {
1606 # inverse modulus
1607 my ($c,$num,$mod) = @_;
1608
1609 my $u = _zero(); my $u1 = _one();
1610 my $a = _copy($c,$mod); my $b = _copy($c,$num);
1611
1612 # Euclid's Algorithm for bgcd(), only that we calc bgcd() ($a) and the
1613 # result ($u) at the same time
1614 while (!_is_zero($c,$b))
1615 {
1616# print ${_str($c,$a)}, " ", ${_str($c,$b)}, " ", ${_str($c,$u)}, " ",
1617# ${_str($c,$u1)}, "\n";
1618 ($a, my $q, $b) = ($b, _div($c,$a,$b));
1619# print ${_str($c,$a)}, " ", ${_str($c,$q)}, " ", ${_str($c,$b)}, "\n";
1620 # original: ($u,$u1) = ($u1, $u - $u1 * $q);
1621 my $t = _copy($c,$u);
1622 $u = _copy($c,$u1);
1623 _mul($c,$u1,$q);
1624 $u1 = _sub($t,$u1);
1625# print ${_str($c,$a)}, " ", ${_str($c,$b)}, " ", ${_str($c,$u)}, " ",
1626# ${_str($c,$u1)}, "\n";
1627 }
1628
1629 # if the gcd is not 1, then return NaN
1630 return undef unless _is_one($c,$a);
1631
1632 $num = _mod($c,$u,$mod);
1633# print ${_str($c,$num)},"\n";
1634 $num;
1635 }
1636
1637sub _modpow
1638 {
1639 # modulus of power ($x ** $y) % $z
1640 my ($c,$num,$exp,$mod) = @_;
1641
1642 # in the trivial case,
1643 if (_is_one($c,$mod))
1644 {
1645 splice @$num,0,1; $num->[0] = 0;
1646 return $num;
1647 }
1648 if ((scalar @$num == 1) && (($num->[0] == 0) || ($num->[0] == 1)))
1649 {
1650 $num->[0] = 1;
1651 return $num;
1652 }
1653
1654# $num = _mod($c,$num,$mod); # this does not make it faster
1655
1656 my $acc = _copy($c,$num); my $t = _one();
1657
1658 my $expbin = ${_as_bin($c,$exp)}; $expbin =~ s/^0b//;
1659 my $len = length($expbin);
1660 while (--$len >= 0)
1661 {
1662 if ( substr($expbin,$len,1) eq '1') # is_odd
1663 {
1664 _mul($c,$t,$acc);
1665 $t = _mod($c,$t,$mod);
1666 }
1667 _mul($c,$acc,$acc);
1668 $acc = _mod($c,$acc,$mod);
1669 }
1670 @$num = @$t;
1671 $num;
1672 }
1673
1674##############################################################################
1675##############################################################################
1676
16771;
1678__END__
1679
1680=head1 NAME
1681
1682Math::BigInt::Calc - Pure Perl module to support Math::BigInt
1683
1684=head1 SYNOPSIS
1685
1686Provides support for big integer calculations. Not intended to be used by other
1687modules (except Math::BigInt::Cached). Other modules which sport the same
1688functions can also be used to support Math::Bigint, like Math::BigInt::Pari.
1689
1690=head1 DESCRIPTION
1691
1692In order to allow for multiple big integer libraries, Math::BigInt was
1693rewritten to use library modules for core math routines. Any module which
1694follows the same API as this can be used instead by using the following:
1695
1696 use Math::BigInt lib => 'libname';
1697
1698'libname' is either the long name ('Math::BigInt::Pari'), or only the short
1699version like 'Pari'.
1700
1701=head1 EXPORT
1702
1703The following functions MUST be defined in order to support the use by
1704Math::BigInt:
1705
1706 _new(string) return ref to new object from ref to decimal string
1707 _zero() return a new object with value 0
1708 _one() return a new object with value 1
1709
1710 _str(obj) return ref to a string representing the object
1711 _num(obj) returns a Perl integer/floating point number
1712 NOTE: because of Perl numeric notation defaults,
1713 the _num'ified obj may lose accuracy due to
1714 machine-dependend floating point size limitations
1715
1716 _add(obj,obj) Simple addition of two objects
1717 _mul(obj,obj) Multiplication of two objects
1718 _div(obj,obj) Division of the 1st object by the 2nd
1719 In list context, returns (result,remainder).
1720 NOTE: this is integer math, so no
1721 fractional part will be returned.
1722 _sub(obj,obj) Simple subtraction of 1 object from another
1723 a third, optional parameter indicates that the params
1724 are swapped. In this case, the first param needs to
1725 be preserved, while you can destroy the second.
1726 sub (x,y,1) => return x - y and keep x intact!
1727 _dec(obj) decrement object by one (input is garant. to be > 0)
1728 _inc(obj) increment object by one
1729
1730
1731 _acmp(obj,obj) <=> operator for objects (return -1, 0 or 1)
1732
1733 _len(obj) returns count of the decimal digits of the object
1734 _digit(obj,n) returns the n'th decimal digit of object
1735
1736 _is_one(obj) return true if argument is +1
1737 _is_zero(obj) return true if argument is 0
1738 _is_even(obj) return true if argument is even (0,2,4,6..)
1739 _is_odd(obj) return true if argument is odd (1,3,5,7..)
1740
1741 _copy return a ref to a true copy of the object
1742
1743 _check(obj) check whether internal representation is still intact
1744 return 0 for ok, otherwise error message as string
1745
1746The following functions are optional, and can be defined if the underlying lib
1747has a fast way to do them. If undefined, Math::BigInt will use pure Perl (hence
1748slow) fallback routines to emulate these:
1749
1750 _from_hex(str) return ref to new object from ref to hexadecimal string
1751 _from_bin(str) return ref to new object from ref to binary string
1752
1753 _as_hex(str) return ref to scalar string containing the value as
1754 unsigned hex string, with the '0x' prepended.
1755 Leading zeros must be stripped.
1756 _as_bin(str) Like as_hex, only as binary string containing only
1757 zeros and ones. Leading zeros must be stripped and a
1758 '0b' must be prepended.
1759
1760 _rsft(obj,N,B) shift object in base B by N 'digits' right
1761 For unsupported bases B, return undef to signal failure
1762 _lsft(obj,N,B) shift object in base B by N 'digits' left
1763 For unsupported bases B, return undef to signal failure
1764
1765 _xor(obj1,obj2) XOR (bit-wise) object 1 with object 2
1766 Note: XOR, AND and OR pad with zeros if size mismatches
1767 _and(obj1,obj2) AND (bit-wise) object 1 with object 2
1768 _or(obj1,obj2) OR (bit-wise) object 1 with object 2
1769
1770 _mod(obj,obj) Return remainder of div of the 1st by the 2nd object
1771 _sqrt(obj) return the square root of object (truncate to int)
1772 _fac(obj) return factorial of object 1 (1*2*3*4..)
1773 _pow(obj,obj) return object 1 to the power of object 2
1774 _gcd(obj,obj) return Greatest Common Divisor of two objects
1775
1776 _zeros(obj) return number of trailing decimal zeros
1777 _modinv return inverse modulus
1778 _modpow return modulus of power ($x ** $y) % $z
1779
1780Input strings come in as unsigned but with prefix (i.e. as '123', '0xabc'
1781or '0b1101').
1782
1783Testing of input parameter validity is done by the caller, so you need not
1784worry about underflow (f.i. in C<_sub()>, C<_dec()>) nor about division by
1785zero or similar cases.
1786
1787The first parameter can be modified, that includes the possibility that you
1788return a reference to a completely different object instead. Although keeping
1789the reference and just changing it's contents is prefered over creating and
1790returning a different reference.
1791
1792Return values are always references to objects or strings. Exceptions are
1793C<_lsft()> and C<_rsft()>, which return undef if they can not shift the
1794argument. This is used to delegate shifting of bases different than the one
1795you can support back to Math::BigInt, which will use some generic code to
1796calculate the result.
1797
1798=head1 WRAP YOUR OWN
1799
1800If you want to port your own favourite c-lib for big numbers to the
1801Math::BigInt interface, you can take any of the already existing modules as
1802a rough guideline. You should really wrap up the latest BigInt and BigFloat
1803testsuites with your module, and replace in them any of the following:
1804
1805 use Math::BigInt;
1806
1807by this:
1808
1809 use Math::BigInt lib => 'yourlib';
1810
1811This way you ensure that your library really works 100% within Math::BigInt.
1812
1813=head1 LICENSE
1814
1815This program is free software; you may redistribute it and/or modify it under
1816the same terms as Perl itself.
1817
1818=head1 AUTHORS
1819
1820Original math code by Mark Biggar, rewritten by Tels L<http://bloodgate.com/>
1821in late 2000, 2001.
1822Seperated from BigInt and shaped API with the help of John Peacock.
1823
1824=head1 SEE ALSO
1825
1826L<Math::BigInt>, L<Math::BigFloat>, L<Math::BigInt::BitVect>,
1827L<Math::BigInt::GMP>, L<Math::BigInt::Cached> and L<Math::BigInt::Pari>.
1828
1829=cut
Full OpenSPARC design & verification tarball including full HDL.