Commit | Line | Data |
0716bf9b |
1 | package Math::BigInt::Calc; |
2 | |
3 | use 5.005; |
4 | use strict; |
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5 | # use warnings; # dont use warnings for older Perls |
0716bf9b |
6 | |
7 | require Exporter; |
bd05a461 |
8 | use vars qw/@ISA $VERSION/; |
0716bf9b |
9 | @ISA = qw(Exporter); |
10 | |
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11 | $VERSION = '0.32'; |
0716bf9b |
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 |
027dc388 |
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 |
0716bf9b |
18 | |
19 | # todo: |
20 | # - fully remove funky $# stuff (maybe) |
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21 | |
22 | # USE_MUL: due to problems on certain os (os390, posix-bc) "* 1e-5" is used |
ee15d750 |
23 | # instead of "/ 1e5" at some places, (marked with USE_MUL). Other platforms |
24 | # BS2000, some Crays need USE_DIV instead. |
bd05a461 |
25 | # The BEGIN block is used to determine which of the two variants gives the |
26 | # correct result. |
0716bf9b |
27 | |
28 | ############################################################################## |
29 | # global constants, flags and accessory |
30 | |
31 | # constants for easier life |
32 | my $nan = 'NaN'; |
61f5c3f5 |
33 | my ($MBASE,$BASE,$RBASE,$BASE_LEN,$MAX_VAL,$BASE_LEN2,$BASE_LEN_SMALL); |
394e6ffb |
34 | my ($AND_BITS,$XOR_BITS,$OR_BITS); |
35 | my ($AND_MASK,$XOR_MASK,$OR_MASK); |
61f5c3f5 |
36 | my ($LEN_CONVERT); |
ee15d750 |
37 | |
38 | sub _base_len |
39 | { |
dccbb853 |
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 |
394e6ffb |
42 | shift; |
43 | |
ee15d750 |
44 | my $b = shift; |
45 | if (defined $b) |
46 | { |
61f5c3f5 |
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) |
394e6ffb |
52 | { |
61f5c3f5 |
53 | $MBASE = int("1e".$BASE_LEN_SMALL); |
54 | $RBASE = abs('1e-'.$BASE_LEN_SMALL); # see USE_MUL |
394e6ffb |
55 | $caught = 0; |
61f5c3f5 |
56 | $caught += 1 if (int($MBASE * $RBASE) != 1); # should be 1 |
57 | $caught += 2 if (int($MBASE / $MBASE) != 1); # should be 1 |
394e6ffb |
58 | last if $caught != 3; |
59 | } |
61f5c3f5 |
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? |
ee15d750 |
63 | $BASE = int("1e".$BASE_LEN); |
61f5c3f5 |
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: $MBASE\n"; |
74 | |
b05afeb3 |
75 | undef &_mul; |
76 | undef &_div; |
1ddff52a |
77 | |
394e6ffb |
78 | if ($caught & 1 != 0) |
ee15d750 |
79 | { |
80 | # must USE_MUL |
ee15d750 |
81 | *{_mul} = \&_mul_use_mul; |
82 | *{_div} = \&_div_use_mul; |
83 | } |
394e6ffb |
84 | else # $caught must be 2, since it can't be 1 nor 3 |
ee15d750 |
85 | { |
ee15d750 |
86 | # can USE_DIV instead |
87 | *{_mul} = \&_mul_use_div; |
88 | *{_div} = \&_div_use_div; |
89 | } |
90 | } |
61f5c3f5 |
91 | return $BASE_LEN unless wantarray; |
92 | return ($BASE_LEN, $AND_BITS, $XOR_BITS, $OR_BITS, $BASE_LEN_SMALL, $MAX_VAL); |
ee15d750 |
93 | } |
574bacfe |
94 | |
95 | BEGIN |
96 | { |
bd05a461 |
97 | # from Daniel Pfeiffer: determine largest group of digits that is precisely |
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98 | # multipliable with itself plus carry |
dccbb853 |
99 | # Test now changed to expect the proper pattern, not a result off by 1 or 2 |
100 | my ($e, $num) = 3; # lowest value we will use is 3+1-1 = 3 |
bd05a461 |
101 | do |
102 | { |
103 | $num = ('9' x ++$e) + 0; |
394e6ffb |
104 | $num *= $num + 1.0; |
394e6ffb |
105 | } while ("$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) |
07d34614 |
112 | $e = 5 if $] < 5.006; # cap, for older Perls |
2e507a43 |
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 |
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115 | |
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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 |
394e6ffb |
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. |
394e6ffb |
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++; } |
1ddff52a |
148 | { |
149 | no integer; |
150 | $max = 16 if $] < 5.006; # older Perls might not take >16 too well |
151 | } |
394e6ffb |
152 | my ($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 | |
574bacfe |
172 | } |
173 | |
0716bf9b |
174 | ############################################################################## |
61f5c3f5 |
175 | # convert between the "small" and the "large" representation |
176 | |
177 | sub _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 | |
223 | sub _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 | ############################################################################### |
0716bf9b |
242 | |
243 | sub _new |
244 | { |
394e6ffb |
245 | # (ref to string) return ref to num_array |
9393ace2 |
246 | # Convert a number from string format (without sign) to internal base |
247 | # 1ex format. Assumes normalized value as input. |
574bacfe |
248 | my $d = $_[1]; |
61f5c3f5 |
249 | my $il = length($$d)-1; |
250 | # this leaves '00000' instead of int 0 and will be corrected after any op |
251 | [ reverse(unpack("a" . ($il % $BASE_LEN+1) |
574bacfe |
252 | . ("a$BASE_LEN" x ($il / $BASE_LEN)), $$d)) ]; |
0716bf9b |
253 | } |
394e6ffb |
254 | |
255 | BEGIN |
256 | { |
257 | $AND_MASK = __PACKAGE__->_new( \( 2 ** $AND_BITS )); |
258 | $XOR_MASK = __PACKAGE__->_new( \( 2 ** $XOR_BITS )); |
259 | $OR_MASK = __PACKAGE__->_new( \( 2 ** $OR_BITS )); |
260 | } |
0716bf9b |
261 | |
262 | sub _zero |
263 | { |
264 | # create a zero |
61f5c3f5 |
265 | [ 0 ]; |
0716bf9b |
266 | } |
267 | |
268 | sub _one |
269 | { |
270 | # create a one |
61f5c3f5 |
271 | [ 1 ]; |
0716bf9b |
272 | } |
273 | |
027dc388 |
274 | sub _two |
275 | { |
1ddff52a |
276 | # create a two (used internally for shifting) |
61f5c3f5 |
277 | [ 2 ]; |
027dc388 |
278 | } |
279 | |
0716bf9b |
280 | sub _copy |
281 | { |
61f5c3f5 |
282 | [ @{$_[1]} ]; |
0716bf9b |
283 | } |
284 | |
bd05a461 |
285 | # catch and throw away |
286 | sub import { } |
287 | |
0716bf9b |
288 | ############################################################################## |
289 | # convert back to string and number |
290 | |
291 | sub _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") |
574bacfe |
296 | my $ar = $_[1]; |
0716bf9b |
297 | my $ret = ""; |
61f5c3f5 |
298 | |
299 | my $l = scalar @$ar; # number of parts |
300 | return $nan if $l < 1; # should not happen |
301 | |
0716bf9b |
302 | # handle first one different to strip leading zeros from it (there are no |
303 | # leading zero parts in internal representation) |
61f5c3f5 |
304 | $l --; $ret .= int($ar->[$l]); $l--; |
0716bf9b |
305 | # Interestingly, the pre-padd method uses more time |
574bacfe |
306 | # the old grep variant takes longer (14 to 10 sec) |
307 | my $z = '0' x ($BASE_LEN-1); |
0716bf9b |
308 | while ($l >= 0) |
309 | { |
574bacfe |
310 | $ret .= substr($z.$ar->[$l],-$BASE_LEN); # fastest way I could think of |
0716bf9b |
311 | $l--; |
312 | } |
61f5c3f5 |
313 | \$ret; |
0716bf9b |
314 | } |
315 | |
316 | sub _num |
317 | { |
318 | # Make a number (scalar int/float) from a BigInt object |
574bacfe |
319 | my $x = $_[1]; |
0716bf9b |
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 | } |
61f5c3f5 |
327 | $num; |
0716bf9b |
328 | } |
329 | |
330 | ############################################################################## |
331 | # actual math code |
332 | |
333 | sub _add |
334 | { |
335 | # (ref to int_num_array, ref to int_num_array) |
574bacfe |
336 | # routine to add two base 1eX numbers |
0716bf9b |
337 | # stolen from Knuth Vol 2 Algorithm A pg 231 |
b22b3e31 |
338 | # there are separate routines to add and sub as per Knuth pg 233 |
0716bf9b |
339 | # This routine clobbers up array x, but not y. |
340 | |
574bacfe |
341 | my ($c,$x,$y) = @_; |
b3abae2a |
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 $x; |
348 | } |
0716bf9b |
349 | |
350 | # for each in Y, add Y to X and carry. If after that, something is left in |
351 | # X, foreach in X add carry to X and then return X, carry |
352 | # Trades one "$j++" for having to shift arrays, $j could be made integer |
b22b3e31 |
353 | # but this would impose a limit to number-length of 2**32. |
0716bf9b |
354 | my $i; my $car = 0; my $j = 0; |
355 | for $i (@$y) |
356 | { |
e745a66c |
357 | $x->[$j] -= $BASE if $car = (($x->[$j] += $i + $car) >= $BASE) ? 1 : 0; |
0716bf9b |
358 | $j++; |
359 | } |
360 | while ($car != 0) |
361 | { |
362 | $x->[$j] -= $BASE if $car = (($x->[$j] += $car) >= $BASE) ? 1 : 0; $j++; |
363 | } |
61f5c3f5 |
364 | $x; |
e745a66c |
365 | } |
366 | |
367 | sub _inc |
368 | { |
369 | # (ref to int_num_array, ref to int_num_array) |
370 | # routine to add 1 to a base 1eX numbers |
56d9de68 |
371 | # This routine modifies array x |
e745a66c |
372 | my ($c,$x) = @_; |
373 | |
374 | for my $i (@$x) |
375 | { |
376 | return $x if (($i += 1) < $BASE); # early out |
61f5c3f5 |
377 | $i = 0; # overflow, next |
e745a66c |
378 | } |
61f5c3f5 |
379 | push @$x,1 if ($x->[-1] == 0); # last overflowed, so extend |
380 | $x; |
e745a66c |
381 | } |
382 | |
383 | sub _dec |
384 | { |
385 | # (ref to int_num_array, ref to int_num_array) |
386 | # routine to add 1 to a base 1eX numbers |
56d9de68 |
387 | # This routine modifies array x |
e745a66c |
388 | my ($c,$x) = @_; |
389 | |
61f5c3f5 |
390 | my $MAX = $BASE-1; # since MAX_VAL based on MBASE |
e745a66c |
391 | for my $i (@$x) |
392 | { |
393 | last if (($i -= 1) >= 0); # early out |
61f5c3f5 |
394 | $i = $MAX; # overflow, next |
e745a66c |
395 | } |
396 | pop @$x if $x->[-1] == 0 && @$x > 1; # last overflowed (but leave 0) |
61f5c3f5 |
397 | $x; |
0716bf9b |
398 | } |
399 | |
400 | sub _sub |
401 | { |
9393ace2 |
402 | # (ref to int_num_array, ref to int_num_array, swap) |
574bacfe |
403 | # subtract base 1eX numbers -- stolen from Knuth Vol 2 pg 232, $x > $y |
56b9c951 |
404 | # subtract Y from X by modifying x in place |
574bacfe |
405 | my ($c,$sx,$sy,$s) = @_; |
0716bf9b |
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; |
0716bf9b |
414 | $i += $BASE if $car = (($i -= ($sy->[$j] || 0) + $car) < 0); $j++; |
0716bf9b |
415 | } |
416 | # might leave leading zeros, so fix that |
394e6ffb |
417 | return __strip_zeros($sx); |
0716bf9b |
418 | } |
394e6ffb |
419 | #print "case 1 (swap)\n"; |
420 | for $i (@$sx) |
0716bf9b |
421 | { |
07d34614 |
422 | # we can't do an early out if $x is < than $y, since we |
56b9c951 |
423 | # need to copy the high chunks from $y. Found by Bob Mathews. |
424 | #last unless defined $sy->[$j] || $car; |
394e6ffb |
425 | $sy->[$j] += $BASE |
426 | if $car = (($sy->[$j] = $i-($sy->[$j]||0) - $car) < 0); |
427 | $j++; |
0716bf9b |
428 | } |
394e6ffb |
429 | # might leave leading zeros, so fix that |
430 | __strip_zeros($sy); |
0716bf9b |
431 | } |
432 | |
ee15d750 |
433 | sub _mul_use_mul |
0716bf9b |
434 | { |
9393ace2 |
435 | # (ref to int_num_array, ref to int_num_array) |
0716bf9b |
436 | # multiply two numbers in internal representation |
b22b3e31 |
437 | # modifies first arg, second need not be different from first |
574bacfe |
438 | my ($c,$xv,$yv) = @_; |
dccbb853 |
439 | |
b3abae2a |
440 | # shortcut for two very short numbers (improved by Nathan Zook) |
61f5c3f5 |
441 | # works also if xv and yv are the same reference |
b3abae2a |
442 | if ((@$xv == 1) && (@$yv == 1)) |
443 | { |
444 | if (($xv->[0] *= $yv->[0]) >= $MBASE) |
445 | { |
446 | $xv->[0] = $xv->[0] - ($xv->[1] = int($xv->[0] * $RBASE)) * $MBASE; |
447 | }; |
448 | return $xv; |
449 | } |
450 | # shortcut for result == 0 |
451 | if ( ((@$xv == 1) && ($xv->[0] == 0)) || |
452 | ((@$yv == 1) && ($yv->[0] == 0)) ) |
453 | { |
454 | @$xv = (0); |
455 | return $xv; |
456 | } |
457 | |
0716bf9b |
458 | # since multiplying $x with $x fails, make copy in this case |
d614cd8b |
459 | $yv = [@$xv] if $xv == $yv; # same references? |
9393ace2 |
460 | |
61f5c3f5 |
461 | if ($LEN_CONVERT != 0) |
462 | { |
463 | $c->_to_small($xv); $c->_to_small($yv); |
464 | } |
465 | |
466 | my @prod = (); my ($prod,$car,$cty,$xi,$yi); |
467 | |
0716bf9b |
468 | for $xi (@$xv) |
469 | { |
470 | $car = 0; $cty = 0; |
574bacfe |
471 | |
472 | # slow variant |
473 | # for $yi (@$yv) |
474 | # { |
475 | # $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
476 | # $prod[$cty++] = |
61f5c3f5 |
477 | # $prod - ($car = int($prod * RBASE)) * $MBASE; # see USE_MUL |
574bacfe |
478 | # } |
479 | # $prod[$cty] += $car if $car; # need really to check for 0? |
480 | # $xi = shift @prod; |
481 | |
482 | # faster variant |
483 | # looping through this if $xi == 0 is silly - so optimize it away! |
484 | $xi = (shift @prod || 0), next if $xi == 0; |
0716bf9b |
485 | for $yi (@$yv) |
486 | { |
487 | $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
574bacfe |
488 | ## this is actually a tad slower |
489 | ## $prod = $prod[$cty]; $prod += ($car + $xi * $yi); # no ||0 here |
0716bf9b |
490 | $prod[$cty++] = |
61f5c3f5 |
491 | $prod - ($car = int($prod * $RBASE)) * $MBASE; # see USE_MUL |
0716bf9b |
492 | } |
493 | $prod[$cty] += $car if $car; # need really to check for 0? |
027dc388 |
494 | $xi = shift @prod || 0; # || 0 makes v5.005_3 happy |
0716bf9b |
495 | } |
0716bf9b |
496 | push @$xv, @prod; |
61f5c3f5 |
497 | if ($LEN_CONVERT != 0) |
498 | { |
499 | $c->_to_large($yv); |
500 | $c->_to_large($xv); |
501 | } |
502 | else |
503 | { |
504 | __strip_zeros($xv); |
505 | } |
506 | $xv; |
0716bf9b |
507 | } |
508 | |
ee15d750 |
509 | sub _mul_use_div |
510 | { |
9393ace2 |
511 | # (ref to int_num_array, ref to int_num_array) |
ee15d750 |
512 | # multiply two numbers in internal representation |
513 | # modifies first arg, second need not be different from first |
514 | my ($c,$xv,$yv) = @_; |
515 | |
b3abae2a |
516 | # shortcut for two very short numbers (improved by Nathan Zook) |
61f5c3f5 |
517 | # works also if xv and yv are the same reference |
b3abae2a |
518 | if ((@$xv == 1) && (@$yv == 1)) |
519 | { |
520 | if (($xv->[0] *= $yv->[0]) >= $MBASE) |
521 | { |
522 | $xv->[0] = |
523 | $xv->[0] - ($xv->[1] = int($xv->[0] / $MBASE)) * $MBASE; |
524 | }; |
525 | return $xv; |
526 | } |
527 | # shortcut for result == 0 |
528 | if ( ((@$xv == 1) && ($xv->[0] == 0)) || |
529 | ((@$yv == 1) && ($yv->[0] == 0)) ) |
530 | { |
531 | @$xv = (0); |
532 | return $xv; |
533 | } |
534 | |
61f5c3f5 |
535 | |
ee15d750 |
536 | # since multiplying $x with $x fails, make copy in this case |
d614cd8b |
537 | $yv = [@$xv] if $xv == $yv; # same references? |
9393ace2 |
538 | |
61f5c3f5 |
539 | if ($LEN_CONVERT != 0) |
540 | { |
541 | $c->_to_small($xv); $c->_to_small($yv); |
542 | } |
543 | |
544 | my @prod = (); my ($prod,$car,$cty,$xi,$yi); |
ee15d750 |
545 | for $xi (@$xv) |
546 | { |
547 | $car = 0; $cty = 0; |
548 | # looping through this if $xi == 0 is silly - so optimize it away! |
549 | $xi = (shift @prod || 0), next if $xi == 0; |
550 | for $yi (@$yv) |
551 | { |
552 | $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
553 | $prod[$cty++] = |
61f5c3f5 |
554 | $prod - ($car = int($prod / $MBASE)) * $MBASE; |
ee15d750 |
555 | } |
556 | $prod[$cty] += $car if $car; # need really to check for 0? |
027dc388 |
557 | $xi = shift @prod || 0; # || 0 makes v5.005_3 happy |
ee15d750 |
558 | } |
559 | push @$xv, @prod; |
61f5c3f5 |
560 | if ($LEN_CONVERT != 0) |
561 | { |
562 | $c->_to_large($yv); |
563 | $c->_to_large($xv); |
564 | } |
565 | else |
566 | { |
567 | __strip_zeros($xv); |
568 | } |
569 | $xv; |
ee15d750 |
570 | } |
571 | |
572 | sub _div_use_mul |
0716bf9b |
573 | { |
b22b3e31 |
574 | # ref to array, ref to array, modify first array and return remainder if |
0716bf9b |
575 | # in list context |
574bacfe |
576 | my ($c,$x,$yorg) = @_; |
0716bf9b |
577 | |
61f5c3f5 |
578 | if (@$x == 1 && @$yorg == 1) |
579 | { |
13a12e00 |
580 | # shortcut, $yorg and $x are two small numbers |
61f5c3f5 |
581 | if (wantarray) |
582 | { |
583 | my $r = [ $x->[0] % $yorg->[0] ]; |
584 | $x->[0] = int($x->[0] / $yorg->[0]); |
585 | return ($x,$r); |
586 | } |
587 | else |
588 | { |
589 | $x->[0] = int($x->[0] / $yorg->[0]); |
590 | return $x; |
591 | } |
592 | } |
28df3e88 |
593 | if (@$yorg == 1) |
594 | { |
595 | my $rem; |
596 | $rem = _mod($c,[ @$x ],$yorg) if wantarray; |
13a12e00 |
597 | |
28df3e88 |
598 | # shortcut, $y is < $BASE |
599 | my $j = scalar @$x; my $r = 0; |
600 | my $y = $yorg->[0]; my $b; |
601 | while ($j-- > 0) |
602 | { |
603 | $b = $r * $MBASE + $x->[$j]; |
604 | $x->[$j] = int($b/$y); |
605 | $r = $b % $y; |
606 | } |
607 | pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero |
608 | return ($x,$rem) if wantarray; |
609 | return $x; |
610 | } |
0716bf9b |
611 | |
d614cd8b |
612 | my $y = [ @$yorg ]; # always make copy to preserve |
61f5c3f5 |
613 | if ($LEN_CONVERT != 0) |
614 | { |
615 | $c->_to_small($x); $c->_to_small($y); |
616 | } |
617 | |
618 | my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0); |
619 | |
620 | $car = $bar = $prd = 0; |
621 | if (($dd = int($MBASE/($y->[-1]+1))) != 1) |
0716bf9b |
622 | { |
623 | for $xi (@$x) |
624 | { |
625 | $xi = $xi * $dd + $car; |
61f5c3f5 |
626 | $xi -= ($car = int($xi * $RBASE)) * $MBASE; # see USE_MUL |
0716bf9b |
627 | } |
628 | push(@$x, $car); $car = 0; |
629 | for $yi (@$y) |
630 | { |
631 | $yi = $yi * $dd + $car; |
61f5c3f5 |
632 | $yi -= ($car = int($yi * $RBASE)) * $MBASE; # see USE_MUL |
0716bf9b |
633 | } |
634 | } |
635 | else |
636 | { |
637 | push(@$x, 0); |
638 | } |
639 | @q = (); ($v2,$v1) = @$y[-2,-1]; |
640 | $v2 = 0 unless $v2; |
641 | while ($#$x > $#$y) |
642 | { |
643 | ($u2,$u1,$u0) = @$x[-3..-1]; |
644 | $u2 = 0 unless $u2; |
645 | #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" |
646 | # if $v1 == 0; |
61f5c3f5 |
647 | $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$MBASE+$u1)/$v1)); |
648 | --$q while ($v2*$q > ($u0*$MBASE+$u1-$q*$v1)*$MBASE+$u2); |
0716bf9b |
649 | if ($q) |
650 | { |
651 | ($car, $bar) = (0,0); |
652 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
653 | { |
654 | $prd = $q * $y->[$yi] + $car; |
61f5c3f5 |
655 | $prd -= ($car = int($prd * $RBASE)) * $MBASE; # see USE_MUL |
656 | $x->[$xi] += $MBASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0)); |
0716bf9b |
657 | } |
658 | if ($x->[-1] < $car + $bar) |
659 | { |
660 | $car = 0; --$q; |
661 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
662 | { |
61f5c3f5 |
663 | $x->[$xi] -= $MBASE |
664 | if ($car = (($x->[$xi] += $y->[$yi] + $car) > $MBASE)); |
0716bf9b |
665 | } |
666 | } |
667 | } |
668 | pop(@$x); unshift(@q, $q); |
669 | } |
670 | if (wantarray) |
671 | { |
672 | @d = (); |
673 | if ($dd != 1) |
674 | { |
675 | $car = 0; |
676 | for $xi (reverse @$x) |
677 | { |
61f5c3f5 |
678 | $prd = $car * $MBASE + $xi; |
0716bf9b |
679 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; # see USE_MUL |
680 | unshift(@d, $tmp); |
681 | } |
682 | } |
683 | else |
684 | { |
685 | @d = @$x; |
686 | } |
687 | @$x = @q; |
61f5c3f5 |
688 | my $d = \@d; |
689 | if ($LEN_CONVERT != 0) |
690 | { |
691 | $c->_to_large($x); $c->_to_large($d); |
692 | } |
693 | else |
694 | { |
695 | __strip_zeros($x); |
696 | __strip_zeros($d); |
697 | } |
698 | return ($x,$d); |
0716bf9b |
699 | } |
700 | @$x = @q; |
61f5c3f5 |
701 | if ($LEN_CONVERT != 0) |
702 | { |
703 | $c->_to_large($x); |
704 | } |
705 | else |
706 | { |
707 | __strip_zeros($x); |
708 | } |
709 | $x; |
0716bf9b |
710 | } |
711 | |
ee15d750 |
712 | sub _div_use_div |
713 | { |
714 | # ref to array, ref to array, modify first array and return remainder if |
715 | # in list context |
ee15d750 |
716 | my ($c,$x,$yorg) = @_; |
ee15d750 |
717 | |
61f5c3f5 |
718 | if (@$x == 1 && @$yorg == 1) |
719 | { |
13a12e00 |
720 | # shortcut, $yorg and $x are two small numbers |
61f5c3f5 |
721 | if (wantarray) |
722 | { |
723 | my $r = [ $x->[0] % $yorg->[0] ]; |
724 | $x->[0] = int($x->[0] / $yorg->[0]); |
725 | return ($x,$r); |
726 | } |
727 | else |
728 | { |
729 | $x->[0] = int($x->[0] / $yorg->[0]); |
730 | return $x; |
731 | } |
732 | } |
28df3e88 |
733 | if (@$yorg == 1) |
734 | { |
735 | my $rem; |
736 | $rem = _mod($c,[ @$x ],$yorg) if wantarray; |
737 | |
738 | # shortcut, $y is < $BASE |
739 | my $j = scalar @$x; my $r = 0; |
740 | my $y = $yorg->[0]; my $b; |
741 | while ($j-- > 0) |
742 | { |
743 | $b = $r * $MBASE + $x->[$j]; |
744 | $x->[$j] = int($b/$y); |
745 | $r = $b % $y; |
746 | } |
747 | pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero |
748 | return ($x,$rem) if wantarray; |
749 | return $x; |
750 | } |
ee15d750 |
751 | |
d614cd8b |
752 | my $y = [ @$yorg ]; # always make copy to preserve |
61f5c3f5 |
753 | if ($LEN_CONVERT != 0) |
754 | { |
755 | $c->_to_small($x); $c->_to_small($y); |
756 | } |
757 | |
758 | my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0); |
759 | |
760 | $car = $bar = $prd = 0; |
761 | if (($dd = int($MBASE/($y->[-1]+1))) != 1) |
ee15d750 |
762 | { |
763 | for $xi (@$x) |
764 | { |
765 | $xi = $xi * $dd + $car; |
61f5c3f5 |
766 | $xi -= ($car = int($xi / $MBASE)) * $MBASE; |
ee15d750 |
767 | } |
768 | push(@$x, $car); $car = 0; |
769 | for $yi (@$y) |
770 | { |
771 | $yi = $yi * $dd + $car; |
61f5c3f5 |
772 | $yi -= ($car = int($yi / $MBASE)) * $MBASE; |
ee15d750 |
773 | } |
774 | } |
775 | else |
776 | { |
777 | push(@$x, 0); |
778 | } |
779 | @q = (); ($v2,$v1) = @$y[-2,-1]; |
780 | $v2 = 0 unless $v2; |
781 | while ($#$x > $#$y) |
782 | { |
783 | ($u2,$u1,$u0) = @$x[-3..-1]; |
784 | $u2 = 0 unless $u2; |
785 | #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" |
786 | # if $v1 == 0; |
61f5c3f5 |
787 | $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$MBASE+$u1)/$v1)); |
788 | --$q while ($v2*$q > ($u0*$MBASE+$u1-$q*$v1)*$MBASE+$u2); |
ee15d750 |
789 | if ($q) |
790 | { |
791 | ($car, $bar) = (0,0); |
792 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
793 | { |
794 | $prd = $q * $y->[$yi] + $car; |
61f5c3f5 |
795 | $prd -= ($car = int($prd / $MBASE)) * $MBASE; |
796 | $x->[$xi] += $MBASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0)); |
ee15d750 |
797 | } |
798 | if ($x->[-1] < $car + $bar) |
799 | { |
800 | $car = 0; --$q; |
801 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
802 | { |
61f5c3f5 |
803 | $x->[$xi] -= $MBASE |
804 | if ($car = (($x->[$xi] += $y->[$yi] + $car) > $MBASE)); |
ee15d750 |
805 | } |
806 | } |
807 | } |
61f5c3f5 |
808 | pop(@$x); unshift(@q, $q); |
ee15d750 |
809 | } |
810 | if (wantarray) |
811 | { |
812 | @d = (); |
813 | if ($dd != 1) |
814 | { |
815 | $car = 0; |
816 | for $xi (reverse @$x) |
817 | { |
61f5c3f5 |
818 | $prd = $car * $MBASE + $xi; |
ee15d750 |
819 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; |
820 | unshift(@d, $tmp); |
821 | } |
822 | } |
823 | else |
824 | { |
825 | @d = @$x; |
826 | } |
827 | @$x = @q; |
61f5c3f5 |
828 | my $d = \@d; |
829 | if ($LEN_CONVERT != 0) |
830 | { |
831 | $c->_to_large($x); $c->_to_large($d); |
832 | } |
833 | else |
834 | { |
835 | __strip_zeros($x); |
836 | __strip_zeros($d); |
837 | } |
838 | return ($x,$d); |
ee15d750 |
839 | } |
840 | @$x = @q; |
61f5c3f5 |
841 | if ($LEN_CONVERT != 0) |
842 | { |
843 | $c->_to_large($x); |
844 | } |
845 | else |
846 | { |
847 | __strip_zeros($x); |
848 | } |
849 | $x; |
ee15d750 |
850 | } |
851 | |
394e6ffb |
852 | ############################################################################## |
853 | # testing |
854 | |
855 | sub _acmp |
856 | { |
857 | # internal absolute post-normalized compare (ignore signs) |
858 | # ref to array, ref to array, return <0, 0, >0 |
859 | # arrays must have at least one entry; this is not checked for |
860 | |
861 | my ($c,$cx,$cy) = @_; |
862 | |
f9a08e12 |
863 | # fast comp based on number of array elements (aka pseudo-length) |
394e6ffb |
864 | my $lxy = scalar @$cx - scalar @$cy; |
865 | return -1 if $lxy < 0; # already differs, ret |
866 | return 1 if $lxy > 0; # ditto |
56d9de68 |
867 | |
394e6ffb |
868 | # now calculate length based on digits, not parts |
56d9de68 |
869 | # we need only the length of the last element, since both array have the |
870 | # same number of parts |
871 | $lxy = length(int($cx->[-1])) - length(int($cy->[-1])); |
394e6ffb |
872 | return -1 if $lxy < 0; |
873 | return 1 if $lxy > 0; |
874 | |
56d9de68 |
875 | # hm, same lengths, but same contents? So we need to check all parts: |
876 | my $a; my $j = scalar @$cx - 1; |
394e6ffb |
877 | # manual way (abort if unequal, good for early ne) |
394e6ffb |
878 | while ($j >= 0) |
9393ace2 |
879 | { |
880 | last if ($a = $cx->[$j] - $cy->[$j]); $j--; |
881 | } |
394e6ffb |
882 | return 1 if $a > 0; |
883 | return -1 if $a < 0; |
56d9de68 |
884 | 0; # numbers are equal |
394e6ffb |
885 | } |
886 | |
887 | sub _len |
888 | { |
56d9de68 |
889 | # compute number of digits |
394e6ffb |
890 | |
891 | # int() because add/sub sometimes leaves strings (like '00005') instead of |
892 | # '5' in this place, thus causing length() to report wrong length |
893 | my $cx = $_[1]; |
894 | |
56d9de68 |
895 | (@$cx-1)*$BASE_LEN+length(int($cx->[-1])); |
394e6ffb |
896 | } |
897 | |
898 | sub _digit |
899 | { |
900 | # return the nth digit, negative values count backward |
901 | # zero is rightmost, so _digit(123,0) will give 3 |
902 | my ($c,$x,$n) = @_; |
903 | |
904 | my $len = _len('',$x); |
905 | |
906 | $n = $len+$n if $n < 0; # -1 last, -2 second-to-last |
907 | $n = abs($n); # if negative was too big |
908 | $len--; $n = $len if $n > $len; # n to big? |
909 | |
910 | my $elem = int($n / $BASE_LEN); # which array element |
911 | my $digit = $n % $BASE_LEN; # which digit in this element |
912 | $elem = '0000'.@$x[$elem]; # get element padded with 0's |
56d9de68 |
913 | substr($elem,-$digit-1,1); |
394e6ffb |
914 | } |
915 | |
916 | sub _zeros |
917 | { |
918 | # return amount of trailing zeros in decimal |
919 | # check each array elem in _m for having 0 at end as long as elem == 0 |
920 | # Upon finding a elem != 0, stop |
921 | my $x = $_[1]; |
922 | my $zeros = 0; my $elem; |
923 | foreach my $e (@$x) |
924 | { |
925 | if ($e != 0) |
926 | { |
927 | $elem = "$e"; # preserve x |
928 | $elem =~ s/.*?(0*$)/$1/; # strip anything not zero |
929 | $zeros *= $BASE_LEN; # elems * 5 |
61f5c3f5 |
930 | $zeros += length($elem); # count trailing zeros |
394e6ffb |
931 | last; # early out |
932 | } |
933 | $zeros ++; # real else branch: 50% slower! |
934 | } |
61f5c3f5 |
935 | $zeros; |
394e6ffb |
936 | } |
937 | |
938 | ############################################################################## |
939 | # _is_* routines |
940 | |
941 | sub _is_zero |
942 | { |
943 | # return true if arg (BINT or num_str) is zero (array '+', '0') |
944 | my $x = $_[1]; |
61f5c3f5 |
945 | |
946 | (((scalar @$x == 1) && ($x->[0] == 0))) <=> 0; |
394e6ffb |
947 | } |
948 | |
949 | sub _is_even |
950 | { |
951 | # return true if arg (BINT or num_str) is even |
952 | my $x = $_[1]; |
61f5c3f5 |
953 | (!($x->[0] & 1)) <=> 0; |
394e6ffb |
954 | } |
955 | |
956 | sub _is_odd |
957 | { |
958 | # return true if arg (BINT or num_str) is even |
959 | my $x = $_[1]; |
61f5c3f5 |
960 | |
961 | (($x->[0] & 1)) <=> 0; |
394e6ffb |
962 | } |
963 | |
964 | sub _is_one |
965 | { |
966 | # return true if arg (BINT or num_str) is one (array '+', '1') |
967 | my $x = $_[1]; |
61f5c3f5 |
968 | |
969 | (scalar @$x == 1) && ($x->[0] == 1) <=> 0; |
394e6ffb |
970 | } |
971 | |
972 | sub __strip_zeros |
973 | { |
974 | # internal normalization function that strips leading zeros from the array |
975 | # args: ref to array |
976 | my $s = shift; |
977 | |
978 | my $cnt = scalar @$s; # get count of parts |
979 | my $i = $cnt-1; |
980 | push @$s,0 if $i < 0; # div might return empty results, so fix it |
981 | |
61f5c3f5 |
982 | return $s if @$s == 1; # early out |
983 | |
394e6ffb |
984 | #print "strip: cnt $cnt i $i\n"; |
985 | # '0', '3', '4', '0', '0', |
986 | # 0 1 2 3 4 |
987 | # cnt = 5, i = 4 |
988 | # i = 4 |
989 | # i = 3 |
990 | # => fcnt = cnt - i (5-2 => 3, cnt => 5-1 = 4, throw away from 4th pos) |
991 | # >= 1: skip first part (this can be zero) |
992 | while ($i > 0) { last if $s->[$i] != 0; $i--; } |
993 | $i++; splice @$s,$i if ($i < $cnt); # $i cant be 0 |
994 | $s; |
995 | } |
996 | |
997 | ############################################################################### |
998 | # check routine to test internal state of corruptions |
999 | |
1000 | sub _check |
1001 | { |
1002 | # used by the test suite |
1003 | my $x = $_[1]; |
1004 | |
1005 | return "$x is not a reference" if !ref($x); |
1006 | |
1007 | # are all parts are valid? |
1008 | my $i = 0; my $j = scalar @$x; my ($e,$try); |
1009 | while ($i < $j) |
1010 | { |
1011 | $e = $x->[$i]; $e = 'undef' unless defined $e; |
1012 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e)"; |
1013 | last if $e !~ /^[+]?[0-9]+$/; |
1014 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (stringify)"; |
1015 | last if "$e" !~ /^[+]?[0-9]+$/; |
1016 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (cat-stringify)"; |
1017 | last if '' . "$e" !~ /^[+]?[0-9]+$/; |
1018 | $try = ' < 0 || >= $BASE; '."($x, $e)"; |
1019 | last if $e <0 || $e >= $BASE; |
1020 | # this test is disabled, since new/bnorm and certain ops (like early out |
1021 | # in add/sub) are allowed/expected to leave '00000' in some elements |
1022 | #$try = '=~ /^00+/; '."($x, $e)"; |
1023 | #last if $e =~ /^00+/; |
1024 | $i++; |
1025 | } |
1026 | return "Illegal part '$e' at pos $i (tested: $try)" if $i < $j; |
1027 | return 0; |
1028 | } |
1029 | |
1030 | |
1031 | ############################################################################### |
1032 | ############################################################################### |
1033 | # some optional routines to make BigInt faster |
1034 | |
dccbb853 |
1035 | sub _mod |
1036 | { |
1037 | # if possible, use mod shortcut |
1038 | my ($c,$x,$yo) = @_; |
1039 | |
1040 | # slow way since $y to big |
1041 | if (scalar @$yo > 1) |
1042 | { |
1043 | my ($xo,$rem) = _div($c,$x,$yo); |
1044 | return $rem; |
1045 | } |
1046 | my $y = $yo->[0]; |
027dc388 |
1047 | # both are single element arrays |
dccbb853 |
1048 | if (scalar @$x == 1) |
1049 | { |
1050 | $x->[0] %= $y; |
1051 | return $x; |
1052 | } |
1053 | |
61f5c3f5 |
1054 | # @y is single element, but @x has more than one |
dccbb853 |
1055 | my $b = $BASE % $y; |
1056 | if ($b == 0) |
1057 | { |
1058 | # when BASE % Y == 0 then (B * BASE) % Y == 0 |
1059 | # (B * BASE) % $y + A % Y => A % Y |
1060 | # so need to consider only last element: O(1) |
1061 | $x->[0] %= $y; |
1062 | } |
027dc388 |
1063 | elsif ($b == 1) |
1064 | { |
28df3e88 |
1065 | # else need to go trough all elements: O(N), but loop is a bit simplified |
027dc388 |
1066 | my $r = 0; |
1067 | foreach (@$x) |
1068 | { |
28df3e88 |
1069 | $r = ($r + $_) % $y; # not much faster, but heh... |
1070 | #$r += $_ % $y; $r %= $y; |
027dc388 |
1071 | } |
1072 | $r = 0 if $r == $y; |
1073 | $x->[0] = $r; |
1074 | } |
dccbb853 |
1075 | else |
1076 | { |
027dc388 |
1077 | # else need to go trough all elements: O(N) |
1078 | my $r = 0; my $bm = 1; |
1079 | foreach (@$x) |
1080 | { |
28df3e88 |
1081 | $r = ($_ * $bm + $r) % $y; |
1082 | $bm = ($bm * $b) % $y; |
1083 | |
1084 | #$r += ($_ % $y) * $bm; |
1085 | #$bm *= $b; |
1086 | #$bm %= $y; |
1087 | #$r %= $y; |
027dc388 |
1088 | } |
1089 | $r = 0 if $r == $y; |
1090 | $x->[0] = $r; |
dccbb853 |
1091 | } |
1092 | splice (@$x,1); |
61f5c3f5 |
1093 | $x; |
dccbb853 |
1094 | } |
1095 | |
0716bf9b |
1096 | ############################################################################## |
574bacfe |
1097 | # shifts |
1098 | |
1099 | sub _rsft |
1100 | { |
1101 | my ($c,$x,$y,$n) = @_; |
1102 | |
1103 | if ($n != 10) |
1104 | { |
61f5c3f5 |
1105 | $n = _new($c,\$n); return _div($c,$x, _pow($c,$n,$y)); |
1106 | } |
1107 | |
1108 | # shortcut (faster) for shifting by 10) |
1109 | # multiples of $BASE_LEN |
1110 | my $dst = 0; # destination |
1111 | my $src = _num($c,$y); # as normal int |
56d9de68 |
1112 | my $xlen = (@$x-1)*$BASE_LEN+length(int($x->[-1])); # len of x in digits |
1113 | if ($src > $xlen) |
1114 | { |
1115 | # 12345 67890 shifted right by more than 10 digits => 0 |
1116 | splice (@$x,1); # leave only one element |
1117 | $x->[0] = 0; # set to zero |
1118 | return $x; |
1119 | } |
61f5c3f5 |
1120 | my $rem = $src % $BASE_LEN; # remainder to shift |
1121 | $src = int($src / $BASE_LEN); # source |
1122 | if ($rem == 0) |
1123 | { |
1124 | splice (@$x,0,$src); # even faster, 38.4 => 39.3 |
574bacfe |
1125 | } |
1126 | else |
1127 | { |
61f5c3f5 |
1128 | my $len = scalar @$x - $src; # elems to go |
1129 | my $vd; my $z = '0'x $BASE_LEN; |
1130 | $x->[scalar @$x] = 0; # avoid || 0 test inside loop |
1131 | while ($dst < $len) |
574bacfe |
1132 | { |
61f5c3f5 |
1133 | $vd = $z.$x->[$src]; |
1134 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN-$rem); |
1135 | $src++; |
1136 | $vd = substr($z.$x->[$src],-$rem,$rem) . $vd; |
1137 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN; |
1138 | $x->[$dst] = int($vd); |
1139 | $dst++; |
574bacfe |
1140 | } |
61f5c3f5 |
1141 | splice (@$x,$dst) if $dst > 0; # kill left-over array elems |
56b9c951 |
1142 | pop @$x if $x->[-1] == 0 && @$x > 1; # kill last element if 0 |
61f5c3f5 |
1143 | } # else rem == 0 |
574bacfe |
1144 | $x; |
1145 | } |
1146 | |
1147 | sub _lsft |
1148 | { |
1149 | my ($c,$x,$y,$n) = @_; |
1150 | |
1151 | if ($n != 10) |
1152 | { |
61f5c3f5 |
1153 | $n = _new($c,\$n); return _mul($c,$x, _pow($c,$n,$y)); |
574bacfe |
1154 | } |
61f5c3f5 |
1155 | |
1156 | # shortcut (faster) for shifting by 10) since we are in base 10eX |
1157 | # multiples of $BASE_LEN: |
1158 | my $src = scalar @$x; # source |
1159 | my $len = _num($c,$y); # shift-len as normal int |
1160 | my $rem = $len % $BASE_LEN; # remainder to shift |
1161 | my $dst = $src + int($len/$BASE_LEN); # destination |
1162 | my $vd; # further speedup |
1163 | $x->[$src] = 0; # avoid first ||0 for speed |
1164 | my $z = '0' x $BASE_LEN; |
1165 | while ($src >= 0) |
574bacfe |
1166 | { |
61f5c3f5 |
1167 | $vd = $x->[$src]; $vd = $z.$vd; |
1168 | $vd = substr($vd,-$BASE_LEN+$rem,$BASE_LEN-$rem); |
1169 | $vd .= $src > 0 ? substr($z.$x->[$src-1],-$BASE_LEN,$rem) : '0' x $rem; |
1170 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN; |
1171 | $x->[$dst] = int($vd); |
1172 | $dst--; $src--; |
574bacfe |
1173 | } |
61f5c3f5 |
1174 | # set lowest parts to 0 |
1175 | while ($dst >= 0) { $x->[$dst--] = 0; } |
1176 | # fix spurios last zero element |
1177 | splice @$x,-1 if $x->[-1] == 0; |
574bacfe |
1178 | $x; |
1179 | } |
1180 | |
027dc388 |
1181 | sub _pow |
1182 | { |
1183 | # power of $x to $y |
1184 | # ref to array, ref to array, return ref to array |
1185 | my ($c,$cx,$cy) = @_; |
1186 | |
1187 | my $pow2 = _one(); |
1ddff52a |
1188 | |
1189 | my $y_bin = ${_as_bin($c,$cy)}; $y_bin =~ s/^0b//; |
1190 | my $len = length($y_bin); |
1191 | while (--$len > 0) |
027dc388 |
1192 | { |
1ddff52a |
1193 | _mul($c,$pow2,$cx) if substr($y_bin,$len,1) eq '1'; # is odd? |
027dc388 |
1194 | _mul($c,$cx,$cx); |
1195 | } |
1ddff52a |
1196 | |
1197 | _mul($c,$cx,$pow2); |
61f5c3f5 |
1198 | $cx; |
027dc388 |
1199 | } |
1200 | |
b3abae2a |
1201 | sub _fac |
1202 | { |
1203 | # factorial of $x |
1204 | # ref to array, return ref to array |
1205 | my ($c,$cx) = @_; |
1206 | |
1207 | if ((@$cx == 1) && ($cx->[0] <= 2)) |
1208 | { |
1209 | $cx->[0] = 1 * ($cx->[0]||1); # 0,1 => 1, 2 => 2 |
1210 | return $cx; |
1211 | } |
1212 | |
1213 | # go forward until $base is exceeded |
1214 | # limit is either $x or $base (x == 100 means as result too high) |
1215 | my $steps = 100; $steps = $cx->[0] if @$cx == 1; |
1216 | my $r = 2; my $cf = 3; my $step = 1; my $last = $r; |
1217 | while ($r < $BASE && $step < $steps) |
1218 | { |
1219 | $last = $r; $r *= $cf++; $step++; |
1220 | } |
1221 | if ((@$cx == 1) && ($step == $cx->[0])) |
1222 | { |
1223 | # completely done |
1224 | $cx = [$last]; |
1225 | return $cx; |
1226 | } |
1227 | my $n = _copy($c,$cx); |
1228 | $cx = [$last]; |
1229 | |
b3abae2a |
1230 | while (!(@$n == 1 && $n->[0] == $step)) |
1231 | { |
1232 | _mul($c,$cx,$n); _dec($c,$n); |
1233 | } |
1234 | $cx; |
1235 | } |
1236 | |
56d9de68 |
1237 | # for debugging: |
1238 | use constant DEBUG => 0; |
1239 | my $steps = 0; |
1240 | sub steps { $steps }; |
b3abae2a |
1241 | |
1242 | sub _sqrt |
0716bf9b |
1243 | { |
56d9de68 |
1244 | # square-root of $x in place |
1245 | # Compute a guess of the result (rule of thumb), then improve it via |
1246 | # Newton's method. |
394e6ffb |
1247 | my ($c,$x) = @_; |
0716bf9b |
1248 | |
394e6ffb |
1249 | if (scalar @$x == 1) |
1250 | { |
56d9de68 |
1251 | # fit's into one Perl scalar, so result can be computed directly |
394e6ffb |
1252 | $x->[0] = int(sqrt($x->[0])); |
1253 | return $x; |
1254 | } |
1255 | my $y = _copy($c,$x); |
b3abae2a |
1256 | # hopefully _len/2 is < $BASE, the -1 is to always undershot the guess |
1257 | # since our guess will "grow" |
1258 | my $l = int((_len($c,$x)-1) / 2); |
1259 | |
56d9de68 |
1260 | my $lastelem = $x->[-1]; # for guess |
b3abae2a |
1261 | my $elems = scalar @$x - 1; |
1262 | # not enough digits, but could have more? |
56d9de68 |
1263 | if ((length($lastelem) <= 3) && ($elems > 1)) |
b3abae2a |
1264 | { |
1265 | # right-align with zero pad |
1266 | my $len = length($lastelem) & 1; |
1267 | print "$lastelem => " if DEBUG; |
1268 | $lastelem .= substr($x->[-2] . '0' x $BASE_LEN,0,$BASE_LEN); |
1269 | # former odd => make odd again, or former even to even again |
56d9de68 |
1270 | $lastelem = $lastelem / 10 if (length($lastelem) & 1) != $len; |
b3abae2a |
1271 | print "$lastelem\n" if DEBUG; |
1272 | } |
0716bf9b |
1273 | |
61f5c3f5 |
1274 | # construct $x (instead of _lsft($c,$x,$l,10) |
1275 | my $r = $l % $BASE_LEN; # 10000 00000 00000 00000 ($BASE_LEN=5) |
1276 | $l = int($l / $BASE_LEN); |
b3abae2a |
1277 | print "l = $l " if DEBUG; |
56d9de68 |
1278 | |
1279 | splice @$x,$l; # keep ref($x), but modify it |
1280 | |
b3abae2a |
1281 | # we make the first part of the guess not '1000...0' but int(sqrt($lastelem)) |
1282 | # that gives us: |
56d9de68 |
1283 | # 14400 00000 => sqrt(14400) => guess first digits to be 120 |
1284 | # 144000 000000 => sqrt(144000) => guess 379 |
b3abae2a |
1285 | |
b3abae2a |
1286 | print "$lastelem (elems $elems) => " if DEBUG; |
1287 | $lastelem = $lastelem / 10 if ($elems & 1 == 1); # odd or even? |
1288 | my $g = sqrt($lastelem); $g =~ s/\.//; # 2.345 => 2345 |
1289 | $r -= 1 if $elems & 1 == 0; # 70 => 7 |
1290 | |
1291 | # padd with zeros if result is too short |
1292 | $x->[$l--] = int(substr($g . '0' x $r,0,$r+1)); |
1293 | print "now ",$x->[-1] if DEBUG; |
1294 | print " would have been ", int('1' . '0' x $r),"\n" if DEBUG; |
56d9de68 |
1295 | |
b3abae2a |
1296 | # If @$x > 1, we could compute the second elem of the guess, too, to create |
56d9de68 |
1297 | # an even better guess. Not implemented yet. Does it improve performance? |
b3abae2a |
1298 | $x->[$l--] = 0 while ($l >= 0); # all other digits of guess are zero |
56d9de68 |
1299 | |
b3abae2a |
1300 | print "start x= ",${_str($c,$x)},"\n" if DEBUG; |
394e6ffb |
1301 | my $two = _two(); |
1302 | my $last = _zero(); |
1303 | my $lastlast = _zero(); |
b3abae2a |
1304 | $steps = 0 if DEBUG; |
394e6ffb |
1305 | while (_acmp($c,$last,$x) != 0 && _acmp($c,$lastlast,$x) != 0) |
1306 | { |
b3abae2a |
1307 | $steps++ if DEBUG; |
394e6ffb |
1308 | $lastlast = _copy($c,$last); |
1309 | $last = _copy($c,$x); |
1310 | _add($c,$x, _div($c,_copy($c,$y),$x)); |
1311 | _div($c,$x, $two ); |
56d9de68 |
1312 | print " x= ",${_str($c,$x)},"\n" if DEBUG; |
394e6ffb |
1313 | } |
b3abae2a |
1314 | print "\nsteps in sqrt: $steps, " if DEBUG; |
394e6ffb |
1315 | _dec($c,$x) if _acmp($c,$y,_mul($c,_copy($c,$x),$x)) < 0; # overshot? |
b3abae2a |
1316 | print " final ",$x->[-1],"\n" if DEBUG; |
394e6ffb |
1317 | $x; |
0716bf9b |
1318 | } |
1319 | |
394e6ffb |
1320 | ############################################################################## |
1321 | # binary stuff |
0716bf9b |
1322 | |
394e6ffb |
1323 | sub _and |
1324 | { |
1325 | my ($c,$x,$y) = @_; |
0716bf9b |
1326 | |
394e6ffb |
1327 | # the shortcut makes equal, large numbers _really_ fast, and makes only a |
1328 | # very small performance drop for small numbers (e.g. something with less |
1329 | # than 32 bit) Since we optimize for large numbers, this is enabled. |
1330 | return $x if _acmp($c,$x,$y) == 0; # shortcut |
0716bf9b |
1331 | |
394e6ffb |
1332 | my $m = _one(); my ($xr,$yr); |
1333 | my $mask = $AND_MASK; |
1334 | |
1335 | my $x1 = $x; |
1336 | my $y1 = _copy($c,$y); # make copy |
1337 | $x = _zero(); |
1338 | my ($b,$xrr,$yrr); |
1339 | use integer; |
1340 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
1341 | { |
1342 | ($x1, $xr) = _div($c,$x1,$mask); |
1343 | ($y1, $yr) = _div($c,$y1,$mask); |
1344 | |
1345 | # make ints() from $xr, $yr |
1346 | # this is when the AND_BITS are greater tahn $BASE and is slower for |
1347 | # small (<256 bits) numbers, but faster for large numbers. Disabled |
1348 | # due to KISS principle |
1349 | |
1350 | # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
1351 | # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
1352 | # _add($c,$x, _mul($c, _new( $c, \($xrr & $yrr) ), $m) ); |
1353 | |
61f5c3f5 |
1354 | # 0+ due to '&' doesn't work in strings |
1355 | _add($c,$x, _mul($c, [ 0+$xr->[0] & 0+$yr->[0] ], $m) ); |
394e6ffb |
1356 | _mul($c,$m,$mask); |
1357 | } |
1358 | $x; |
0716bf9b |
1359 | } |
1360 | |
394e6ffb |
1361 | sub _xor |
0716bf9b |
1362 | { |
394e6ffb |
1363 | my ($c,$x,$y) = @_; |
1364 | |
1365 | return _zero() if _acmp($c,$x,$y) == 0; # shortcut (see -and) |
1366 | |
1367 | my $m = _one(); my ($xr,$yr); |
1368 | my $mask = $XOR_MASK; |
1369 | |
1370 | my $x1 = $x; |
1371 | my $y1 = _copy($c,$y); # make copy |
1372 | $x = _zero(); |
1373 | my ($b,$xrr,$yrr); |
1374 | use integer; |
1375 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
0716bf9b |
1376 | { |
394e6ffb |
1377 | ($x1, $xr) = _div($c,$x1,$mask); |
1378 | ($y1, $yr) = _div($c,$y1,$mask); |
1379 | # make ints() from $xr, $yr (see _and()) |
1380 | #$b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
1381 | #$b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
1382 | #_add($c,$x, _mul($c, _new( $c, \($xrr ^ $yrr) ), $m) ); |
61f5c3f5 |
1383 | |
1384 | # 0+ due to '^' doesn't work in strings |
1385 | _add($c,$x, _mul($c, [ 0+$xr->[0] ^ 0+$yr->[0] ], $m) ); |
394e6ffb |
1386 | _mul($c,$m,$mask); |
0716bf9b |
1387 | } |
394e6ffb |
1388 | # the loop stops when the shorter of the two numbers is exhausted |
1389 | # the remainder of the longer one will survive bit-by-bit, so we simple |
1390 | # multiply-add it in |
1391 | _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1); |
1392 | _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1); |
1393 | |
1394 | $x; |
0716bf9b |
1395 | } |
1396 | |
394e6ffb |
1397 | sub _or |
0716bf9b |
1398 | { |
394e6ffb |
1399 | my ($c,$x,$y) = @_; |
0716bf9b |
1400 | |
394e6ffb |
1401 | return $x if _acmp($c,$x,$y) == 0; # shortcut (see _and) |
0716bf9b |
1402 | |
394e6ffb |
1403 | my $m = _one(); my ($xr,$yr); |
1404 | my $mask = $OR_MASK; |
0716bf9b |
1405 | |
394e6ffb |
1406 | my $x1 = $x; |
1407 | my $y1 = _copy($c,$y); # make copy |
1408 | $x = _zero(); |
1409 | my ($b,$xrr,$yrr); |
1410 | use integer; |
1411 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
1412 | { |
1413 | ($x1, $xr) = _div($c,$x1,$mask); |
1414 | ($y1, $yr) = _div($c,$y1,$mask); |
1415 | # make ints() from $xr, $yr (see _and()) |
1416 | # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
1417 | # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
1418 | # _add($c,$x, _mul($c, _new( $c, \($xrr | $yrr) ), $m) ); |
1419 | |
61f5c3f5 |
1420 | # 0+ due to '|' doesn't work in strings |
1421 | _add($c,$x, _mul($c, [ 0+$xr->[0] | 0+$yr->[0] ], $m) ); |
394e6ffb |
1422 | _mul($c,$m,$mask); |
1423 | } |
1424 | # the loop stops when the shorter of the two numbers is exhausted |
1425 | # the remainder of the longer one will survive bit-by-bit, so we simple |
1426 | # multiply-add it in |
1427 | _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1); |
1428 | _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1); |
1429 | |
1430 | $x; |
0716bf9b |
1431 | } |
1432 | |
61f5c3f5 |
1433 | sub _as_hex |
1434 | { |
1435 | # convert a decimal number to hex (ref to array, return ref to string) |
1436 | my ($c,$x) = @_; |
1437 | |
1438 | my $x1 = _copy($c,$x); |
1439 | |
1440 | my $es = ''; |
1ddff52a |
1441 | my ($xr, $h, $x10000); |
1442 | if ($] >= 5.006) |
1443 | { |
1444 | $x10000 = [ 0x10000 ]; $h = 'h4'; |
1445 | } |
1446 | else |
1447 | { |
1448 | $x10000 = [ 0x1000 ]; $h = 'h3'; |
1449 | } |
61f5c3f5 |
1450 | while (! _is_zero($c,$x1)) |
1451 | { |
1452 | ($x1, $xr) = _div($c,$x1,$x10000); |
1ddff52a |
1453 | $es .= unpack($h,pack('v',$xr->[0])); |
61f5c3f5 |
1454 | } |
1455 | $es = reverse $es; |
1456 | $es =~ s/^[0]+//; # strip leading zeros |
1457 | $es = '0x' . $es; |
1458 | \$es; |
1459 | } |
1460 | |
1461 | sub _as_bin |
1462 | { |
1463 | # convert a decimal number to bin (ref to array, return ref to string) |
1464 | my ($c,$x) = @_; |
1465 | |
1466 | my $x1 = _copy($c,$x); |
1467 | |
1468 | my $es = ''; |
1ddff52a |
1469 | my ($xr, $b, $x10000); |
1470 | if ($] >= 5.006) |
1471 | { |
1472 | $x10000 = [ 0x10000 ]; $b = 'b16'; |
1473 | } |
1474 | else |
1475 | { |
1476 | $x10000 = [ 0x1000 ]; $b = 'b12'; |
1477 | } |
61f5c3f5 |
1478 | while (! _is_zero($c,$x1)) |
1479 | { |
1480 | ($x1, $xr) = _div($c,$x1,$x10000); |
1ddff52a |
1481 | $es .= unpack($b,pack('v',$xr->[0])); |
61f5c3f5 |
1482 | } |
1483 | $es = reverse $es; |
1484 | $es =~ s/^[0]+//; # strip leading zeros |
1485 | $es = '0b' . $es; |
1486 | \$es; |
1487 | } |
1488 | |
394e6ffb |
1489 | sub _from_hex |
0716bf9b |
1490 | { |
394e6ffb |
1491 | # convert a hex number to decimal (ref to string, return ref to array) |
1492 | my ($c,$hs) = @_; |
0716bf9b |
1493 | |
394e6ffb |
1494 | my $mul = _one(); |
1495 | my $m = [ 0x10000 ]; # 16 bit at a time |
1496 | my $x = _zero(); |
0716bf9b |
1497 | |
61f5c3f5 |
1498 | my $len = length($$hs)-2; |
394e6ffb |
1499 | $len = int($len/4); # 4-digit parts, w/o '0x' |
1500 | my $val; my $i = -4; |
1501 | while ($len >= 0) |
1502 | { |
1503 | $val = substr($$hs,$i,4); |
1504 | $val =~ s/^[+-]?0x// if $len == 0; # for last part only because |
1505 | $val = hex($val); # hex does not like wrong chars |
1506 | $i -= 4; $len --; |
1507 | _add ($c, $x, _mul ($c, [ $val ], $mul ) ) if $val != 0; |
1508 | _mul ($c, $mul, $m ) if $len >= 0; # skip last mul |
1509 | } |
1510 | $x; |
1511 | } |
1512 | |
1513 | sub _from_bin |
0716bf9b |
1514 | { |
394e6ffb |
1515 | # convert a hex number to decimal (ref to string, return ref to array) |
1516 | my ($c,$bs) = @_; |
0716bf9b |
1517 | |
13a12e00 |
1518 | # instead of converting 8 bit at a time, it is faster to convert the |
1519 | # number to hex, and then call _from_hex. |
1520 | |
1521 | my $hs = $$bs; |
1522 | $hs =~ s/^[+-]?0b//; # remove sign and 0b |
1523 | my $l = length($hs); # bits |
1524 | $hs = '0' x (8-($l % 8)) . $hs if ($l % 8) != 0; # padd left side w/ 0 |
1525 | my $h = unpack('H*', pack ('B*', $hs)); # repack as hex |
1526 | return $c->_from_hex(\('0x'.$h)); |
1527 | |
394e6ffb |
1528 | my $mul = _one(); |
1529 | my $m = [ 0x100 ]; # 8 bit at a time |
1530 | my $x = _zero(); |
0716bf9b |
1531 | |
61f5c3f5 |
1532 | my $len = length($$bs)-2; |
394e6ffb |
1533 | $len = int($len/8); # 4-digit parts, w/o '0x' |
1534 | my $val; my $i = -8; |
1535 | while ($len >= 0) |
0716bf9b |
1536 | { |
394e6ffb |
1537 | $val = substr($$bs,$i,8); |
1538 | $val =~ s/^[+-]?0b// if $len == 0; # for last part only |
1539 | |
394e6ffb |
1540 | $val = ord(pack('B8',substr('00000000'.$val,-8,8))); |
1541 | |
1542 | $i -= 8; $len --; |
1543 | _add ($c, $x, _mul ($c, [ $val ], $mul ) ) if $val != 0; |
1544 | _mul ($c, $mul, $m ) if $len >= 0; # skip last mul |
0716bf9b |
1545 | } |
394e6ffb |
1546 | $x; |
0716bf9b |
1547 | } |
1548 | |
07d34614 |
1549 | ############################################################################## |
1550 | # special modulus functions |
1551 | |
56d9de68 |
1552 | sub _modinv |
d614cd8b |
1553 | { |
56d9de68 |
1554 | # modular inverse |
1555 | my ($c,$x,$y) = @_; |
1ddff52a |
1556 | |
56d9de68 |
1557 | my $u = _zero($c); my $u1 = _one($c); |
1558 | my $a = _copy($c,$y); my $b = _copy($c,$x); |
1ddff52a |
1559 | |
1560 | # Euclid's Algorithm for bgcd(), only that we calc bgcd() ($a) and the |
56d9de68 |
1561 | # result ($u) at the same time. See comments in BigInt for why this works. |
1562 | my $q; |
1563 | ($a, $q, $b) = ($b, _div($c,$a,$b)); # step 1 |
1564 | my $sign = 1; |
1ddff52a |
1565 | while (!_is_zero($c,$b)) |
1566 | { |
56d9de68 |
1567 | my $t = _add($c, # step 2: |
1568 | _mul($c,_copy($c,$u1), $q) , # t = u1 * q |
1569 | $u ); # + u |
1570 | $u = $u1; # u = u1, u1 = t |
1571 | $u1 = $t; |
1572 | $sign = -$sign; |
1573 | ($a, $q, $b) = ($b, _div($c,$a,$b)); # step 1 |
1ddff52a |
1574 | } |
1575 | |
1576 | # if the gcd is not 1, then return NaN |
56d9de68 |
1577 | return (undef,undef) unless _is_one($c,$a); |
1578 | |
1579 | $sign = $sign == 1 ? '+' : '-'; |
1580 | ($u1,$sign); |
d614cd8b |
1581 | } |
1582 | |
1583 | sub _modpow |
1584 | { |
1585 | # modulus of power ($x ** $y) % $z |
07d34614 |
1586 | my ($c,$num,$exp,$mod) = @_; |
1587 | |
1588 | # in the trivial case, |
1589 | if (_is_one($c,$mod)) |
1590 | { |
1591 | splice @$num,0,1; $num->[0] = 0; |
1592 | return $num; |
1593 | } |
1594 | if ((scalar @$num == 1) && (($num->[0] == 0) || ($num->[0] == 1))) |
1595 | { |
1596 | $num->[0] = 1; |
1597 | return $num; |
1598 | } |
1ddff52a |
1599 | |
1600 | # $num = _mod($c,$num,$mod); # this does not make it faster |
07d34614 |
1601 | |
1602 | my $acc = _copy($c,$num); my $t = _one(); |
1603 | |
1ddff52a |
1604 | my $expbin = ${_as_bin($c,$exp)}; $expbin =~ s/^0b//; |
1605 | my $len = length($expbin); |
1606 | while (--$len >= 0) |
07d34614 |
1607 | { |
1ddff52a |
1608 | if ( substr($expbin,$len,1) eq '1') # is_odd |
07d34614 |
1609 | { |
1610 | _mul($c,$t,$acc); |
1611 | $t = _mod($c,$t,$mod); |
1612 | } |
1613 | _mul($c,$acc,$acc); |
1614 | $acc = _mod($c,$acc,$mod); |
07d34614 |
1615 | } |
1616 | @$num = @$t; |
1617 | $num; |
d614cd8b |
1618 | } |
1619 | |
394e6ffb |
1620 | ############################################################################## |
1621 | ############################################################################## |
1622 | |
0716bf9b |
1623 | 1; |
1624 | __END__ |
1625 | |
1626 | =head1 NAME |
1627 | |
1628 | Math::BigInt::Calc - Pure Perl module to support Math::BigInt |
1629 | |
1630 | =head1 SYNOPSIS |
1631 | |
ee15d750 |
1632 | Provides support for big integer calculations. Not intended to be used by other |
1633 | modules (except Math::BigInt::Cached). Other modules which sport the same |
1634 | functions can also be used to support Math::Bigint, like Math::BigInt::Pari. |
0716bf9b |
1635 | |
1636 | =head1 DESCRIPTION |
1637 | |
027dc388 |
1638 | In order to allow for multiple big integer libraries, Math::BigInt was |
1639 | rewritten to use library modules for core math routines. Any module which |
1640 | follows the same API as this can be used instead by using the following: |
0716bf9b |
1641 | |
ee15d750 |
1642 | use Math::BigInt lib => 'libname'; |
0716bf9b |
1643 | |
027dc388 |
1644 | 'libname' is either the long name ('Math::BigInt::Pari'), or only the short |
1645 | version like 'Pari'. |
1646 | |
0716bf9b |
1647 | =head1 EXPORT |
1648 | |
027dc388 |
1649 | The following functions MUST be defined in order to support the use by |
1650 | Math::BigInt: |
0716bf9b |
1651 | |
1652 | _new(string) return ref to new object from ref to decimal string |
1653 | _zero() return a new object with value 0 |
1654 | _one() return a new object with value 1 |
1655 | |
1656 | _str(obj) return ref to a string representing the object |
1657 | _num(obj) returns a Perl integer/floating point number |
1658 | NOTE: because of Perl numeric notation defaults, |
1659 | the _num'ified obj may lose accuracy due to |
1660 | machine-dependend floating point size limitations |
1661 | |
1662 | _add(obj,obj) Simple addition of two objects |
1663 | _mul(obj,obj) Multiplication of two objects |
1664 | _div(obj,obj) Division of the 1st object by the 2nd |
b22b3e31 |
1665 | In list context, returns (result,remainder). |
1666 | NOTE: this is integer math, so no |
1667 | fractional part will be returned. |
1668 | _sub(obj,obj) Simple subtraction of 1 object from another |
0716bf9b |
1669 | a third, optional parameter indicates that the params |
1670 | are swapped. In this case, the first param needs to |
1671 | be preserved, while you can destroy the second. |
1672 | sub (x,y,1) => return x - y and keep x intact! |
e745a66c |
1673 | _dec(obj) decrement object by one (input is garant. to be > 0) |
1674 | _inc(obj) increment object by one |
1675 | |
0716bf9b |
1676 | |
1677 | _acmp(obj,obj) <=> operator for objects (return -1, 0 or 1) |
1678 | |
1679 | _len(obj) returns count of the decimal digits of the object |
1680 | _digit(obj,n) returns the n'th decimal digit of object |
1681 | |
1682 | _is_one(obj) return true if argument is +1 |
1683 | _is_zero(obj) return true if argument is 0 |
1684 | _is_even(obj) return true if argument is even (0,2,4,6..) |
1685 | _is_odd(obj) return true if argument is odd (1,3,5,7..) |
1686 | |
1687 | _copy return a ref to a true copy of the object |
1688 | |
1689 | _check(obj) check whether internal representation is still intact |
1690 | return 0 for ok, otherwise error message as string |
1691 | |
bd05a461 |
1692 | The following functions are optional, and can be defined if the underlying lib |
027dc388 |
1693 | has a fast way to do them. If undefined, Math::BigInt will use pure Perl (hence |
1694 | slow) fallback routines to emulate these: |
0716bf9b |
1695 | |
1696 | _from_hex(str) return ref to new object from ref to hexadecimal string |
1697 | _from_bin(str) return ref to new object from ref to binary string |
1698 | |
ee15d750 |
1699 | _as_hex(str) return ref to scalar string containing the value as |
1700 | unsigned hex string, with the '0x' prepended. |
1701 | Leading zeros must be stripped. |
1702 | _as_bin(str) Like as_hex, only as binary string containing only |
1703 | zeros and ones. Leading zeros must be stripped and a |
1704 | '0b' must be prepended. |
1705 | |
0716bf9b |
1706 | _rsft(obj,N,B) shift object in base B by N 'digits' right |
dccbb853 |
1707 | For unsupported bases B, return undef to signal failure |
0716bf9b |
1708 | _lsft(obj,N,B) shift object in base B by N 'digits' left |
dccbb853 |
1709 | For unsupported bases B, return undef to signal failure |
0716bf9b |
1710 | |
1711 | _xor(obj1,obj2) XOR (bit-wise) object 1 with object 2 |
dccbb853 |
1712 | Note: XOR, AND and OR pad with zeros if size mismatches |
0716bf9b |
1713 | _and(obj1,obj2) AND (bit-wise) object 1 with object 2 |
1714 | _or(obj1,obj2) OR (bit-wise) object 1 with object 2 |
1715 | |
dccbb853 |
1716 | _mod(obj,obj) Return remainder of div of the 1st by the 2nd object |
394e6ffb |
1717 | _sqrt(obj) return the square root of object (truncate to int) |
b3abae2a |
1718 | _fac(obj) return factorial of object 1 (1*2*3*4..) |
0716bf9b |
1719 | _pow(obj,obj) return object 1 to the power of object 2 |
1720 | _gcd(obj,obj) return Greatest Common Divisor of two objects |
1721 | |
b22b3e31 |
1722 | _zeros(obj) return number of trailing decimal zeros |
d614cd8b |
1723 | _modinv return inverse modulus |
1724 | _modpow return modulus of power ($x ** $y) % $z |
0716bf9b |
1725 | |
b22b3e31 |
1726 | Input strings come in as unsigned but with prefix (i.e. as '123', '0xabc' |
0716bf9b |
1727 | or '0b1101'). |
1728 | |
b22b3e31 |
1729 | Testing of input parameter validity is done by the caller, so you need not |
574bacfe |
1730 | worry about underflow (f.i. in C<_sub()>, C<_dec()>) nor about division by |
1731 | zero or similar cases. |
1732 | |
1733 | The first parameter can be modified, that includes the possibility that you |
1734 | return a reference to a completely different object instead. Although keeping |
dccbb853 |
1735 | the reference and just changing it's contents is prefered over creating and |
1736 | returning a different reference. |
574bacfe |
1737 | |
1738 | Return values are always references to objects or strings. Exceptions are |
1739 | C<_lsft()> and C<_rsft()>, which return undef if they can not shift the |
027dc388 |
1740 | argument. This is used to delegate shifting of bases different than the one |
1741 | you can support back to Math::BigInt, which will use some generic code to |
1742 | calculate the result. |
574bacfe |
1743 | |
1744 | =head1 WRAP YOUR OWN |
1745 | |
1746 | If you want to port your own favourite c-lib for big numbers to the |
1747 | Math::BigInt interface, you can take any of the already existing modules as |
1748 | a rough guideline. You should really wrap up the latest BigInt and BigFloat |
bd05a461 |
1749 | testsuites with your module, and replace in them any of the following: |
574bacfe |
1750 | |
1751 | use Math::BigInt; |
1752 | |
bd05a461 |
1753 | by this: |
574bacfe |
1754 | |
1755 | use Math::BigInt lib => 'yourlib'; |
1756 | |
1757 | This way you ensure that your library really works 100% within Math::BigInt. |
0716bf9b |
1758 | |
1759 | =head1 LICENSE |
1760 | |
1761 | This program is free software; you may redistribute it and/or modify it under |
1762 | the same terms as Perl itself. |
1763 | |
1764 | =head1 AUTHORS |
1765 | |
1766 | Original math code by Mark Biggar, rewritten by Tels L<http://bloodgate.com/> |
1767 | in late 2000, 2001. |
1768 | Seperated from BigInt and shaped API with the help of John Peacock. |
1769 | |
1770 | =head1 SEE ALSO |
1771 | |
ee15d750 |
1772 | L<Math::BigInt>, L<Math::BigFloat>, L<Math::BigInt::BitVect>, |
1773 | L<Math::BigInt::GMP>, L<Math::BigInt::Cached> and L<Math::BigInt::Pari>. |
0716bf9b |
1774 | |
1775 | =cut |