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 |
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6 | |
7 | require Exporter; |
bd05a461 |
8 | use vars qw/@ISA $VERSION/; |
0716bf9b |
9 | @ISA = qw(Exporter); |
10 | |
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11 | $VERSION = '0.29'; |
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: $MBASE\n"; |
74 | |
b05afeb3 |
75 | undef &_mul; |
76 | undef &_div; |
394e6ffb |
77 | if ($caught & 1 != 0) |
ee15d750 |
78 | { |
79 | # must USE_MUL |
ee15d750 |
80 | *{_mul} = \&_mul_use_mul; |
81 | *{_div} = \&_div_use_mul; |
82 | } |
394e6ffb |
83 | else # $caught must be 2, since it can't be 1 nor 3 |
ee15d750 |
84 | { |
ee15d750 |
85 | # can USE_DIV instead |
86 | *{_mul} = \&_mul_use_div; |
87 | *{_div} = \&_div_use_div; |
88 | } |
89 | } |
61f5c3f5 |
90 | return $BASE_LEN unless wantarray; |
91 | return ($BASE_LEN, $AND_BITS, $XOR_BITS, $OR_BITS, $BASE_LEN_SMALL, $MAX_VAL); |
ee15d750 |
92 | } |
574bacfe |
93 | |
94 | BEGIN |
95 | { |
bd05a461 |
96 | # from Daniel Pfeiffer: determine largest group of digits that is precisely |
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97 | # multipliable with itself plus carry |
dccbb853 |
98 | # Test now changed to expect the proper pattern, not a result off by 1 or 2 |
99 | my ($e, $num) = 3; # lowest value we will use is 3+1-1 = 3 |
bd05a461 |
100 | do |
101 | { |
102 | $num = ('9' x ++$e) + 0; |
394e6ffb |
103 | $num *= $num + 1.0; |
394e6ffb |
104 | } while ("$num" =~ /9{$e}0{$e}/); # must be a certain pattern |
105 | $e--; # last test failed, so retract one step |
106 | # the limits below brush the problems with the test above under the rug: |
107 | # the test should be able to find the proper $e automatically |
108 | $e = 5 if $^O =~ /^uts/; # UTS get's some special treatment |
109 | $e = 5 if $^O =~ /^unicos/; # unicos is also problematic (6 seems to work |
110 | # there, but we play safe) |
07d34614 |
111 | $e = 5 if $] < 5.006; # cap, for older Perls |
2e507a43 |
112 | $e = 7 if $e > 7; # cap, for VMS, OS/390 and other 64 bit systems |
113 | # 8 fails inside random testsuite, so take 7 |
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114 | |
61f5c3f5 |
115 | # determine how many digits fit into an integer and can be safely added |
116 | # together plus carry w/o causing an overflow |
117 | |
118 | # this below detects 15 on a 64 bit system, because after that it becomes |
119 | # 1e16 and not 1000000 :/ I can make it detect 18, but then I get a lot of |
120 | # test failures. Ugh! (Tomake detect 18: uncomment lines marked with *) |
121 | use integer; |
122 | my $bi = 5; # approx. 16 bit |
123 | $num = int('9' x $bi); |
124 | # $num = 99999; # * |
125 | # while ( ($num+$num+1) eq '1' . '9' x $bi) # * |
126 | while ( int($num+$num+1) eq '1' . '9' x $bi) |
127 | { |
128 | $bi++; $num = int('9' x $bi); |
129 | # $bi++; $num *= 10; $num += 9; # * |
130 | } |
131 | $bi--; # back off one step |
132 | # by setting them equal, we ignore the findings and use the default |
133 | # one-size-fits-all approach from former versions |
134 | $bi = $e; # XXX, this should work always |
135 | |
136 | __PACKAGE__->_base_len($e,$bi); # set and store |
394e6ffb |
137 | |
138 | # find out how many bits _and, _or and _xor can take (old default = 16) |
139 | # I don't think anybody has yet 128 bit scalars, so let's play safe. |
394e6ffb |
140 | local $^W = 0; # don't warn about 'nonportable number' |
141 | $AND_BITS = 15; $XOR_BITS = 15; $OR_BITS = 15; |
142 | |
143 | # find max bits, we will not go higher than numberofbits that fit into $BASE |
144 | # to make _and etc simpler (and faster for smaller, slower for large numbers) |
145 | my $max = 16; |
146 | while (2 ** $max < $BASE) { $max++; } |
147 | my ($x,$y,$z); |
148 | do { |
149 | $AND_BITS++; |
150 | $x = oct('0b' . '1' x $AND_BITS); $y = $x & $x; |
151 | $z = (2 ** $AND_BITS) - 1; |
152 | } while ($AND_BITS < $max && $x == $z && $y == $x); |
153 | $AND_BITS --; # retreat one step |
154 | do { |
155 | $XOR_BITS++; |
156 | $x = oct('0b' . '1' x $XOR_BITS); $y = $x ^ 0; |
157 | $z = (2 ** $XOR_BITS) - 1; |
158 | } while ($XOR_BITS < $max && $x == $z && $y == $x); |
159 | $XOR_BITS --; # retreat one step |
160 | do { |
161 | $OR_BITS++; |
162 | $x = oct('0b' . '1' x $OR_BITS); $y = $x | $x; |
163 | $z = (2 ** $OR_BITS) - 1; |
164 | } while ($OR_BITS < $max && $x == $z && $y == $x); |
165 | $OR_BITS --; # retreat one step |
166 | |
574bacfe |
167 | } |
168 | |
0716bf9b |
169 | ############################################################################## |
61f5c3f5 |
170 | # convert between the "small" and the "large" representation |
171 | |
172 | sub _to_large |
173 | { |
174 | # take an array in base $BASE_LEN_SMALL and convert it in-place to $BASE_LEN |
175 | my ($c,$x) = @_; |
176 | |
177 | # print "_to_large $BASE_LEN_SMALL => $BASE_LEN\n"; |
178 | |
179 | return $x if $LEN_CONVERT == 0 || # nothing to converconvertor |
180 | @$x == 1; # only one element => early out |
181 | |
182 | # 12345 67890 12345 67890 contents |
183 | # to 3 2 1 0 index |
184 | # 123456 7890123 4567890 contents |
185 | |
186 | # # faster variant |
187 | # my @d; my $str = ''; |
188 | # my $z = '0' x $BASE_LEN_SMALL; |
189 | # foreach (@$x) |
190 | # { |
191 | # # ... . 04321 . 000321 |
192 | # $str = substr($z.$_,-$BASE_LEN_SMALL,$BASE_LEN_SMALL) . $str; |
193 | # if (length($str) > $BASE_LEN) |
194 | # { |
195 | # push @d, substr($str,-$BASE_LEN,$BASE_LEN); # extract one piece |
196 | # substr($str,-$BASE_LEN,$BASE_LEN) = ''; # remove it |
197 | # } |
198 | # } |
199 | # push @d, $str if $str !~ /^0*$/; # extract last piece |
200 | # @$x = @d; |
201 | # $x->[-1] = int($x->[-1]); # strip leading zero |
202 | # $x; |
203 | |
204 | my $ret = ""; |
205 | my $l = scalar @$x; # number of parts |
206 | $l --; $ret .= int($x->[$l]); $l--; |
207 | my $z = '0' x ($BASE_LEN_SMALL-1); |
208 | while ($l >= 0) |
209 | { |
210 | $ret .= substr($z.$x->[$l],-$BASE_LEN_SMALL); |
211 | $l--; |
212 | } |
213 | my $str = _new($c,\$ret); # make array |
214 | @$x = @$str; # clobber contents of $x |
215 | $x->[-1] = int($x->[-1]); # strip leading zero |
216 | } |
217 | |
218 | sub _to_small |
219 | { |
220 | # take an array in base $BASE_LEN and convert it in-place to $BASE_LEN_SMALL |
221 | my ($c,$x) = @_; |
222 | |
223 | return $x if $LEN_CONVERT == 0; # nothing to do |
224 | return $x if @$x == 1 && length(int($x->[0])) <= $BASE_LEN_SMALL; |
225 | |
226 | my $d = _str($c,$x); |
227 | my $il = length($$d)-1; |
228 | ## this leaves '00000' instead of int 0 and will be corrected after any op |
229 | # clobber contents of $x |
230 | @$x = reverse(unpack("a" . ($il % $BASE_LEN_SMALL+1) |
231 | . ("a$BASE_LEN_SMALL" x ($il / $BASE_LEN_SMALL)), $$d)); |
232 | |
233 | $x->[-1] = int($x->[-1]); # strip leading zero |
234 | } |
235 | |
236 | ############################################################################### |
0716bf9b |
237 | |
238 | sub _new |
239 | { |
394e6ffb |
240 | # (ref to string) return ref to num_array |
9393ace2 |
241 | # Convert a number from string format (without sign) to internal base |
242 | # 1ex format. Assumes normalized value as input. |
574bacfe |
243 | my $d = $_[1]; |
61f5c3f5 |
244 | my $il = length($$d)-1; |
245 | # this leaves '00000' instead of int 0 and will be corrected after any op |
246 | [ reverse(unpack("a" . ($il % $BASE_LEN+1) |
574bacfe |
247 | . ("a$BASE_LEN" x ($il / $BASE_LEN)), $$d)) ]; |
0716bf9b |
248 | } |
394e6ffb |
249 | |
250 | BEGIN |
251 | { |
252 | $AND_MASK = __PACKAGE__->_new( \( 2 ** $AND_BITS )); |
253 | $XOR_MASK = __PACKAGE__->_new( \( 2 ** $XOR_BITS )); |
254 | $OR_MASK = __PACKAGE__->_new( \( 2 ** $OR_BITS )); |
255 | } |
0716bf9b |
256 | |
257 | sub _zero |
258 | { |
259 | # create a zero |
61f5c3f5 |
260 | [ 0 ]; |
0716bf9b |
261 | } |
262 | |
263 | sub _one |
264 | { |
265 | # create a one |
61f5c3f5 |
266 | [ 1 ]; |
0716bf9b |
267 | } |
268 | |
027dc388 |
269 | sub _two |
270 | { |
271 | # create a two (for _pow) |
61f5c3f5 |
272 | [ 2 ]; |
027dc388 |
273 | } |
274 | |
0716bf9b |
275 | sub _copy |
276 | { |
61f5c3f5 |
277 | [ @{$_[1]} ]; |
0716bf9b |
278 | } |
279 | |
bd05a461 |
280 | # catch and throw away |
281 | sub import { } |
282 | |
0716bf9b |
283 | ############################################################################## |
284 | # convert back to string and number |
285 | |
286 | sub _str |
287 | { |
288 | # (ref to BINT) return num_str |
289 | # Convert number from internal base 100000 format to string format. |
290 | # internal format is always normalized (no leading zeros, "-0" => "+0") |
574bacfe |
291 | my $ar = $_[1]; |
0716bf9b |
292 | my $ret = ""; |
61f5c3f5 |
293 | |
294 | my $l = scalar @$ar; # number of parts |
295 | return $nan if $l < 1; # should not happen |
296 | |
0716bf9b |
297 | # handle first one different to strip leading zeros from it (there are no |
298 | # leading zero parts in internal representation) |
61f5c3f5 |
299 | $l --; $ret .= int($ar->[$l]); $l--; |
0716bf9b |
300 | # Interestingly, the pre-padd method uses more time |
574bacfe |
301 | # the old grep variant takes longer (14 to 10 sec) |
302 | my $z = '0' x ($BASE_LEN-1); |
0716bf9b |
303 | while ($l >= 0) |
304 | { |
574bacfe |
305 | $ret .= substr($z.$ar->[$l],-$BASE_LEN); # fastest way I could think of |
0716bf9b |
306 | $l--; |
307 | } |
61f5c3f5 |
308 | \$ret; |
0716bf9b |
309 | } |
310 | |
311 | sub _num |
312 | { |
313 | # Make a number (scalar int/float) from a BigInt object |
574bacfe |
314 | my $x = $_[1]; |
0716bf9b |
315 | return $x->[0] if scalar @$x == 1; # below $BASE |
316 | my $fac = 1; |
317 | my $num = 0; |
318 | foreach (@$x) |
319 | { |
320 | $num += $fac*$_; $fac *= $BASE; |
321 | } |
61f5c3f5 |
322 | $num; |
0716bf9b |
323 | } |
324 | |
325 | ############################################################################## |
326 | # actual math code |
327 | |
328 | sub _add |
329 | { |
330 | # (ref to int_num_array, ref to int_num_array) |
574bacfe |
331 | # routine to add two base 1eX numbers |
0716bf9b |
332 | # stolen from Knuth Vol 2 Algorithm A pg 231 |
b22b3e31 |
333 | # there are separate routines to add and sub as per Knuth pg 233 |
0716bf9b |
334 | # This routine clobbers up array x, but not y. |
335 | |
574bacfe |
336 | my ($c,$x,$y) = @_; |
b3abae2a |
337 | |
338 | return $x if (@$y == 1) && $y->[0] == 0; # $x + 0 => $x |
339 | if ((@$x == 1) && $x->[0] == 0) # 0 + $y => $y->copy |
340 | { |
341 | # twice as slow as $x = [ @$y ], but necc. to retain $x as ref :( |
342 | @$x = @$y; return $x; |
343 | } |
0716bf9b |
344 | |
345 | # for each in Y, add Y to X and carry. If after that, something is left in |
346 | # X, foreach in X add carry to X and then return X, carry |
347 | # Trades one "$j++" for having to shift arrays, $j could be made integer |
b22b3e31 |
348 | # but this would impose a limit to number-length of 2**32. |
0716bf9b |
349 | my $i; my $car = 0; my $j = 0; |
350 | for $i (@$y) |
351 | { |
e745a66c |
352 | $x->[$j] -= $BASE if $car = (($x->[$j] += $i + $car) >= $BASE) ? 1 : 0; |
0716bf9b |
353 | $j++; |
354 | } |
355 | while ($car != 0) |
356 | { |
357 | $x->[$j] -= $BASE if $car = (($x->[$j] += $car) >= $BASE) ? 1 : 0; $j++; |
358 | } |
61f5c3f5 |
359 | $x; |
e745a66c |
360 | } |
361 | |
362 | sub _inc |
363 | { |
364 | # (ref to int_num_array, ref to int_num_array) |
365 | # routine to add 1 to a base 1eX numbers |
366 | # This routine clobbers up array x, but not y. |
367 | my ($c,$x) = @_; |
368 | |
369 | for my $i (@$x) |
370 | { |
371 | return $x if (($i += 1) < $BASE); # early out |
61f5c3f5 |
372 | $i = 0; # overflow, next |
e745a66c |
373 | } |
61f5c3f5 |
374 | push @$x,1 if ($x->[-1] == 0); # last overflowed, so extend |
375 | $x; |
e745a66c |
376 | } |
377 | |
378 | sub _dec |
379 | { |
380 | # (ref to int_num_array, ref to int_num_array) |
381 | # routine to add 1 to a base 1eX numbers |
382 | # This routine clobbers up array x, but not y. |
383 | my ($c,$x) = @_; |
384 | |
61f5c3f5 |
385 | my $MAX = $BASE-1; # since MAX_VAL based on MBASE |
e745a66c |
386 | for my $i (@$x) |
387 | { |
388 | last if (($i -= 1) >= 0); # early out |
61f5c3f5 |
389 | $i = $MAX; # overflow, next |
e745a66c |
390 | } |
391 | pop @$x if $x->[-1] == 0 && @$x > 1; # last overflowed (but leave 0) |
61f5c3f5 |
392 | $x; |
0716bf9b |
393 | } |
394 | |
395 | sub _sub |
396 | { |
9393ace2 |
397 | # (ref to int_num_array, ref to int_num_array, swap) |
574bacfe |
398 | # subtract base 1eX numbers -- stolen from Knuth Vol 2 pg 232, $x > $y |
56b9c951 |
399 | # subtract Y from X by modifying x in place |
574bacfe |
400 | my ($c,$sx,$sy,$s) = @_; |
0716bf9b |
401 | |
402 | my $car = 0; my $i; my $j = 0; |
403 | if (!$s) |
404 | { |
405 | #print "case 2\n"; |
406 | for $i (@$sx) |
407 | { |
408 | last unless defined $sy->[$j] || $car; |
0716bf9b |
409 | $i += $BASE if $car = (($i -= ($sy->[$j] || 0) + $car) < 0); $j++; |
0716bf9b |
410 | } |
411 | # might leave leading zeros, so fix that |
394e6ffb |
412 | return __strip_zeros($sx); |
0716bf9b |
413 | } |
394e6ffb |
414 | #print "case 1 (swap)\n"; |
415 | for $i (@$sx) |
0716bf9b |
416 | { |
07d34614 |
417 | # we can't do an early out if $x is < than $y, since we |
56b9c951 |
418 | # need to copy the high chunks from $y. Found by Bob Mathews. |
419 | #last unless defined $sy->[$j] || $car; |
394e6ffb |
420 | $sy->[$j] += $BASE |
421 | if $car = (($sy->[$j] = $i-($sy->[$j]||0) - $car) < 0); |
422 | $j++; |
0716bf9b |
423 | } |
394e6ffb |
424 | # might leave leading zeros, so fix that |
425 | __strip_zeros($sy); |
0716bf9b |
426 | } |
427 | |
9393ace2 |
428 | sub _square_use_mul |
429 | { |
430 | # compute $x ** 2 or $x * $x in-place and return $x |
431 | my ($c,$x) = @_; |
432 | |
433 | # From: Handbook of Applied Cryptography by A. Menezes, P. van Oorschot and |
434 | # S. Vanstone., Chapter 14 |
435 | |
436 | #14.16 Algorithm Multiple-precision squaring |
437 | #INPUT: positive integer x = (xt 1 xt 2 ... x1 x0)b. |
438 | #OUTPUT: x * x = x ** 2 in radix b representation. |
439 | #1. For i from 0 to (2t - 1) do: wi <- 0. |
440 | #2. For i from 0 to (t - 1) do the following: |
441 | # 2.1 (uv)b w2i + xi * xi, w2i v, c u. |
442 | # 2.2 For j from (i + 1)to (t - 1) do the following: |
443 | # (uv)b <- wi+j + 2*xj * xi + c, wi+j <- v, c <- u. |
444 | # 2.3 wi+t <- u. |
445 | #3. Return((w2t-1 w2t-2 ... w1 w0)b). |
446 | |
447 | # # Note: That description is crap. Half of the symbols are not explained or |
448 | # # used with out beeing set. |
449 | # my $t = scalar @$x; # count |
450 | # my ($c,$i,$j); |
451 | # for ($i = 0; $i < $t; $i++) |
452 | # { |
453 | # $x->[$i] = $x->[$i*2] + $x[$i]*$x[$i]; |
454 | # $x->[$i*2] = $x[$i]; $c = $x[$i]; |
455 | # for ($j = $i+1; $j < $t; $j++) |
456 | # { |
457 | # $x->[$i] = $x->[$i+$j] + 2 * $x->[$i] * $x->[$j]; |
458 | # $x->[$i+$j] = $x[$j]; $c = $x[$i]; |
459 | # } |
460 | # $x->[$i+$t] = $x[$i]; |
461 | # } |
462 | $x; |
463 | } |
464 | |
ee15d750 |
465 | sub _mul_use_mul |
0716bf9b |
466 | { |
9393ace2 |
467 | # (ref to int_num_array, ref to int_num_array) |
0716bf9b |
468 | # multiply two numbers in internal representation |
b22b3e31 |
469 | # modifies first arg, second need not be different from first |
574bacfe |
470 | my ($c,$xv,$yv) = @_; |
dccbb853 |
471 | |
b3abae2a |
472 | # shortcut for two very short numbers (improved by Nathan Zook) |
61f5c3f5 |
473 | # works also if xv and yv are the same reference |
b3abae2a |
474 | if ((@$xv == 1) && (@$yv == 1)) |
475 | { |
476 | if (($xv->[0] *= $yv->[0]) >= $MBASE) |
477 | { |
478 | $xv->[0] = $xv->[0] - ($xv->[1] = int($xv->[0] * $RBASE)) * $MBASE; |
479 | }; |
480 | return $xv; |
481 | } |
482 | # shortcut for result == 0 |
483 | if ( ((@$xv == 1) && ($xv->[0] == 0)) || |
484 | ((@$yv == 1) && ($yv->[0] == 0)) ) |
485 | { |
486 | @$xv = (0); |
487 | return $xv; |
488 | } |
489 | |
0716bf9b |
490 | # since multiplying $x with $x fails, make copy in this case |
d614cd8b |
491 | $yv = [@$xv] if $xv == $yv; # same references? |
492 | # $yv = [@$xv] if "$xv" eq "$yv"; # same references? |
493 | |
9393ace2 |
494 | # since multiplying $x with $x would fail here, use the faster squaring |
d614cd8b |
495 | # return _square($c,$xv) if $xv == $yv; # same reference? |
9393ace2 |
496 | |
61f5c3f5 |
497 | if ($LEN_CONVERT != 0) |
498 | { |
499 | $c->_to_small($xv); $c->_to_small($yv); |
500 | } |
501 | |
502 | my @prod = (); my ($prod,$car,$cty,$xi,$yi); |
503 | |
0716bf9b |
504 | for $xi (@$xv) |
505 | { |
506 | $car = 0; $cty = 0; |
574bacfe |
507 | |
508 | # slow variant |
509 | # for $yi (@$yv) |
510 | # { |
511 | # $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
512 | # $prod[$cty++] = |
61f5c3f5 |
513 | # $prod - ($car = int($prod * RBASE)) * $MBASE; # see USE_MUL |
574bacfe |
514 | # } |
515 | # $prod[$cty] += $car if $car; # need really to check for 0? |
516 | # $xi = shift @prod; |
517 | |
518 | # faster variant |
519 | # looping through this if $xi == 0 is silly - so optimize it away! |
520 | $xi = (shift @prod || 0), next if $xi == 0; |
0716bf9b |
521 | for $yi (@$yv) |
522 | { |
523 | $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
574bacfe |
524 | ## this is actually a tad slower |
525 | ## $prod = $prod[$cty]; $prod += ($car + $xi * $yi); # no ||0 here |
0716bf9b |
526 | $prod[$cty++] = |
61f5c3f5 |
527 | $prod - ($car = int($prod * $RBASE)) * $MBASE; # see USE_MUL |
0716bf9b |
528 | } |
529 | $prod[$cty] += $car if $car; # need really to check for 0? |
027dc388 |
530 | $xi = shift @prod || 0; # || 0 makes v5.005_3 happy |
0716bf9b |
531 | } |
0716bf9b |
532 | push @$xv, @prod; |
61f5c3f5 |
533 | if ($LEN_CONVERT != 0) |
534 | { |
535 | $c->_to_large($yv); |
536 | $c->_to_large($xv); |
537 | } |
538 | else |
539 | { |
540 | __strip_zeros($xv); |
541 | } |
542 | $xv; |
0716bf9b |
543 | } |
544 | |
ee15d750 |
545 | sub _mul_use_div |
546 | { |
9393ace2 |
547 | # (ref to int_num_array, ref to int_num_array) |
ee15d750 |
548 | # multiply two numbers in internal representation |
549 | # modifies first arg, second need not be different from first |
550 | my ($c,$xv,$yv) = @_; |
551 | |
b3abae2a |
552 | # shortcut for two very short numbers (improved by Nathan Zook) |
61f5c3f5 |
553 | # works also if xv and yv are the same reference |
b3abae2a |
554 | if ((@$xv == 1) && (@$yv == 1)) |
555 | { |
556 | if (($xv->[0] *= $yv->[0]) >= $MBASE) |
557 | { |
558 | $xv->[0] = |
559 | $xv->[0] - ($xv->[1] = int($xv->[0] / $MBASE)) * $MBASE; |
560 | }; |
561 | return $xv; |
562 | } |
563 | # shortcut for result == 0 |
564 | if ( ((@$xv == 1) && ($xv->[0] == 0)) || |
565 | ((@$yv == 1) && ($yv->[0] == 0)) ) |
566 | { |
567 | @$xv = (0); |
568 | return $xv; |
569 | } |
570 | |
61f5c3f5 |
571 | |
ee15d750 |
572 | # since multiplying $x with $x fails, make copy in this case |
d614cd8b |
573 | $yv = [@$xv] if $xv == $yv; # same references? |
574 | # $yv = [@$xv] if "$xv" eq "$yv"; # same references? |
9393ace2 |
575 | # since multiplying $x with $x would fail here, use the faster squaring |
d614cd8b |
576 | # return _square($c,$xv) if $xv == $yv; # same reference? |
9393ace2 |
577 | |
61f5c3f5 |
578 | if ($LEN_CONVERT != 0) |
579 | { |
580 | $c->_to_small($xv); $c->_to_small($yv); |
581 | } |
582 | |
583 | my @prod = (); my ($prod,$car,$cty,$xi,$yi); |
ee15d750 |
584 | for $xi (@$xv) |
585 | { |
586 | $car = 0; $cty = 0; |
587 | # looping through this if $xi == 0 is silly - so optimize it away! |
588 | $xi = (shift @prod || 0), next if $xi == 0; |
589 | for $yi (@$yv) |
590 | { |
591 | $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
592 | $prod[$cty++] = |
61f5c3f5 |
593 | $prod - ($car = int($prod / $MBASE)) * $MBASE; |
ee15d750 |
594 | } |
595 | $prod[$cty] += $car if $car; # need really to check for 0? |
027dc388 |
596 | $xi = shift @prod || 0; # || 0 makes v5.005_3 happy |
ee15d750 |
597 | } |
598 | push @$xv, @prod; |
61f5c3f5 |
599 | if ($LEN_CONVERT != 0) |
600 | { |
601 | $c->_to_large($yv); |
602 | $c->_to_large($xv); |
603 | } |
604 | else |
605 | { |
606 | __strip_zeros($xv); |
607 | } |
608 | $xv; |
ee15d750 |
609 | } |
610 | |
611 | sub _div_use_mul |
0716bf9b |
612 | { |
b22b3e31 |
613 | # ref to array, ref to array, modify first array and return remainder if |
0716bf9b |
614 | # in list context |
574bacfe |
615 | my ($c,$x,$yorg) = @_; |
0716bf9b |
616 | |
61f5c3f5 |
617 | if (@$x == 1 && @$yorg == 1) |
618 | { |
13a12e00 |
619 | # shortcut, $yorg and $x are two small numbers |
61f5c3f5 |
620 | if (wantarray) |
621 | { |
622 | my $r = [ $x->[0] % $yorg->[0] ]; |
623 | $x->[0] = int($x->[0] / $yorg->[0]); |
624 | return ($x,$r); |
625 | } |
626 | else |
627 | { |
628 | $x->[0] = int($x->[0] / $yorg->[0]); |
629 | return $x; |
630 | } |
631 | } |
28df3e88 |
632 | if (@$yorg == 1) |
633 | { |
634 | my $rem; |
635 | $rem = _mod($c,[ @$x ],$yorg) if wantarray; |
13a12e00 |
636 | |
28df3e88 |
637 | # shortcut, $y is < $BASE |
638 | my $j = scalar @$x; my $r = 0; |
639 | my $y = $yorg->[0]; my $b; |
640 | while ($j-- > 0) |
641 | { |
642 | $b = $r * $MBASE + $x->[$j]; |
643 | $x->[$j] = int($b/$y); |
644 | $r = $b % $y; |
645 | } |
646 | pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero |
647 | return ($x,$rem) if wantarray; |
648 | return $x; |
649 | } |
0716bf9b |
650 | |
d614cd8b |
651 | my $y = [ @$yorg ]; # always make copy to preserve |
61f5c3f5 |
652 | if ($LEN_CONVERT != 0) |
653 | { |
654 | $c->_to_small($x); $c->_to_small($y); |
655 | } |
656 | |
657 | my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0); |
658 | |
659 | $car = $bar = $prd = 0; |
660 | if (($dd = int($MBASE/($y->[-1]+1))) != 1) |
0716bf9b |
661 | { |
662 | for $xi (@$x) |
663 | { |
664 | $xi = $xi * $dd + $car; |
61f5c3f5 |
665 | $xi -= ($car = int($xi * $RBASE)) * $MBASE; # see USE_MUL |
0716bf9b |
666 | } |
667 | push(@$x, $car); $car = 0; |
668 | for $yi (@$y) |
669 | { |
670 | $yi = $yi * $dd + $car; |
61f5c3f5 |
671 | $yi -= ($car = int($yi * $RBASE)) * $MBASE; # see USE_MUL |
0716bf9b |
672 | } |
673 | } |
674 | else |
675 | { |
676 | push(@$x, 0); |
677 | } |
678 | @q = (); ($v2,$v1) = @$y[-2,-1]; |
679 | $v2 = 0 unless $v2; |
680 | while ($#$x > $#$y) |
681 | { |
682 | ($u2,$u1,$u0) = @$x[-3..-1]; |
683 | $u2 = 0 unless $u2; |
684 | #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" |
685 | # if $v1 == 0; |
61f5c3f5 |
686 | $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$MBASE+$u1)/$v1)); |
687 | --$q while ($v2*$q > ($u0*$MBASE+$u1-$q*$v1)*$MBASE+$u2); |
0716bf9b |
688 | if ($q) |
689 | { |
690 | ($car, $bar) = (0,0); |
691 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
692 | { |
693 | $prd = $q * $y->[$yi] + $car; |
61f5c3f5 |
694 | $prd -= ($car = int($prd * $RBASE)) * $MBASE; # see USE_MUL |
695 | $x->[$xi] += $MBASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0)); |
0716bf9b |
696 | } |
697 | if ($x->[-1] < $car + $bar) |
698 | { |
699 | $car = 0; --$q; |
700 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
701 | { |
61f5c3f5 |
702 | $x->[$xi] -= $MBASE |
703 | if ($car = (($x->[$xi] += $y->[$yi] + $car) > $MBASE)); |
0716bf9b |
704 | } |
705 | } |
706 | } |
707 | pop(@$x); unshift(@q, $q); |
708 | } |
709 | if (wantarray) |
710 | { |
711 | @d = (); |
712 | if ($dd != 1) |
713 | { |
714 | $car = 0; |
715 | for $xi (reverse @$x) |
716 | { |
61f5c3f5 |
717 | $prd = $car * $MBASE + $xi; |
0716bf9b |
718 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; # see USE_MUL |
719 | unshift(@d, $tmp); |
720 | } |
721 | } |
722 | else |
723 | { |
724 | @d = @$x; |
725 | } |
726 | @$x = @q; |
61f5c3f5 |
727 | my $d = \@d; |
728 | if ($LEN_CONVERT != 0) |
729 | { |
730 | $c->_to_large($x); $c->_to_large($d); |
731 | } |
732 | else |
733 | { |
734 | __strip_zeros($x); |
735 | __strip_zeros($d); |
736 | } |
737 | return ($x,$d); |
0716bf9b |
738 | } |
739 | @$x = @q; |
61f5c3f5 |
740 | if ($LEN_CONVERT != 0) |
741 | { |
742 | $c->_to_large($x); |
743 | } |
744 | else |
745 | { |
746 | __strip_zeros($x); |
747 | } |
748 | $x; |
0716bf9b |
749 | } |
750 | |
ee15d750 |
751 | sub _div_use_div |
752 | { |
753 | # ref to array, ref to array, modify first array and return remainder if |
754 | # in list context |
ee15d750 |
755 | my ($c,$x,$yorg) = @_; |
ee15d750 |
756 | |
61f5c3f5 |
757 | if (@$x == 1 && @$yorg == 1) |
758 | { |
13a12e00 |
759 | # shortcut, $yorg and $x are two small numbers |
61f5c3f5 |
760 | if (wantarray) |
761 | { |
762 | my $r = [ $x->[0] % $yorg->[0] ]; |
763 | $x->[0] = int($x->[0] / $yorg->[0]); |
764 | return ($x,$r); |
765 | } |
766 | else |
767 | { |
768 | $x->[0] = int($x->[0] / $yorg->[0]); |
769 | return $x; |
770 | } |
771 | } |
28df3e88 |
772 | if (@$yorg == 1) |
773 | { |
774 | my $rem; |
775 | $rem = _mod($c,[ @$x ],$yorg) if wantarray; |
776 | |
777 | # shortcut, $y is < $BASE |
778 | my $j = scalar @$x; my $r = 0; |
779 | my $y = $yorg->[0]; my $b; |
780 | while ($j-- > 0) |
781 | { |
782 | $b = $r * $MBASE + $x->[$j]; |
783 | $x->[$j] = int($b/$y); |
784 | $r = $b % $y; |
785 | } |
786 | pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero |
787 | return ($x,$rem) if wantarray; |
788 | return $x; |
789 | } |
ee15d750 |
790 | |
d614cd8b |
791 | my $y = [ @$yorg ]; # always make copy to preserve |
61f5c3f5 |
792 | if ($LEN_CONVERT != 0) |
793 | { |
794 | $c->_to_small($x); $c->_to_small($y); |
795 | } |
796 | |
797 | my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0); |
798 | |
799 | $car = $bar = $prd = 0; |
800 | if (($dd = int($MBASE/($y->[-1]+1))) != 1) |
ee15d750 |
801 | { |
802 | for $xi (@$x) |
803 | { |
804 | $xi = $xi * $dd + $car; |
61f5c3f5 |
805 | $xi -= ($car = int($xi / $MBASE)) * $MBASE; |
ee15d750 |
806 | } |
807 | push(@$x, $car); $car = 0; |
808 | for $yi (@$y) |
809 | { |
810 | $yi = $yi * $dd + $car; |
61f5c3f5 |
811 | $yi -= ($car = int($yi / $MBASE)) * $MBASE; |
ee15d750 |
812 | } |
813 | } |
814 | else |
815 | { |
816 | push(@$x, 0); |
817 | } |
818 | @q = (); ($v2,$v1) = @$y[-2,-1]; |
819 | $v2 = 0 unless $v2; |
820 | while ($#$x > $#$y) |
821 | { |
822 | ($u2,$u1,$u0) = @$x[-3..-1]; |
823 | $u2 = 0 unless $u2; |
824 | #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" |
825 | # if $v1 == 0; |
61f5c3f5 |
826 | $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$MBASE+$u1)/$v1)); |
827 | --$q while ($v2*$q > ($u0*$MBASE+$u1-$q*$v1)*$MBASE+$u2); |
ee15d750 |
828 | if ($q) |
829 | { |
830 | ($car, $bar) = (0,0); |
831 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
832 | { |
833 | $prd = $q * $y->[$yi] + $car; |
61f5c3f5 |
834 | $prd -= ($car = int($prd / $MBASE)) * $MBASE; |
835 | $x->[$xi] += $MBASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0)); |
ee15d750 |
836 | } |
837 | if ($x->[-1] < $car + $bar) |
838 | { |
839 | $car = 0; --$q; |
840 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
841 | { |
61f5c3f5 |
842 | $x->[$xi] -= $MBASE |
843 | if ($car = (($x->[$xi] += $y->[$yi] + $car) > $MBASE)); |
ee15d750 |
844 | } |
845 | } |
846 | } |
61f5c3f5 |
847 | pop(@$x); unshift(@q, $q); |
ee15d750 |
848 | } |
849 | if (wantarray) |
850 | { |
851 | @d = (); |
852 | if ($dd != 1) |
853 | { |
854 | $car = 0; |
855 | for $xi (reverse @$x) |
856 | { |
61f5c3f5 |
857 | $prd = $car * $MBASE + $xi; |
ee15d750 |
858 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; |
859 | unshift(@d, $tmp); |
860 | } |
861 | } |
862 | else |
863 | { |
864 | @d = @$x; |
865 | } |
866 | @$x = @q; |
61f5c3f5 |
867 | my $d = \@d; |
868 | if ($LEN_CONVERT != 0) |
869 | { |
870 | $c->_to_large($x); $c->_to_large($d); |
871 | } |
872 | else |
873 | { |
874 | __strip_zeros($x); |
875 | __strip_zeros($d); |
876 | } |
877 | return ($x,$d); |
ee15d750 |
878 | } |
879 | @$x = @q; |
61f5c3f5 |
880 | if ($LEN_CONVERT != 0) |
881 | { |
882 | $c->_to_large($x); |
883 | } |
884 | else |
885 | { |
886 | __strip_zeros($x); |
887 | } |
888 | $x; |
ee15d750 |
889 | } |
890 | |
394e6ffb |
891 | ############################################################################## |
892 | # testing |
893 | |
894 | sub _acmp |
895 | { |
896 | # internal absolute post-normalized compare (ignore signs) |
897 | # ref to array, ref to array, return <0, 0, >0 |
898 | # arrays must have at least one entry; this is not checked for |
899 | |
900 | my ($c,$cx,$cy) = @_; |
901 | |
61f5c3f5 |
902 | # fast comp based on array elements |
394e6ffb |
903 | my $lxy = scalar @$cx - scalar @$cy; |
904 | return -1 if $lxy < 0; # already differs, ret |
905 | return 1 if $lxy > 0; # ditto |
906 | |
907 | # now calculate length based on digits, not parts |
908 | $lxy = _len($c,$cx) - _len($c,$cy); # difference |
909 | return -1 if $lxy < 0; |
910 | return 1 if $lxy > 0; |
911 | |
912 | # hm, same lengths, but same contents? |
913 | my $i = 0; my $a; |
914 | # first way takes 5.49 sec instead of 4.87, but has the early out advantage |
915 | # so grep is slightly faster, but more inflexible. hm. $_ instead of $k |
916 | # yields 5.6 instead of 5.5 sec huh? |
917 | # manual way (abort if unequal, good for early ne) |
918 | my $j = scalar @$cx - 1; |
919 | while ($j >= 0) |
9393ace2 |
920 | { |
921 | last if ($a = $cx->[$j] - $cy->[$j]); $j--; |
922 | } |
923 | # my $j = scalar @$cx; |
924 | # while (--$j >= 0) |
925 | # { |
926 | # last if ($a = $cx->[$j] - $cy->[$j]); |
927 | # } |
394e6ffb |
928 | return 1 if $a > 0; |
929 | return -1 if $a < 0; |
61f5c3f5 |
930 | 0; # equal |
931 | |
394e6ffb |
932 | # while it early aborts, it is even slower than the manual variant |
933 | #grep { return $a if ($a = $_ - $cy->[$i++]); } @$cx; |
934 | # grep way, go trough all (bad for early ne) |
935 | #grep { $a = $_ - $cy->[$i++]; } @$cx; |
936 | #return $a; |
937 | } |
938 | |
939 | sub _len |
940 | { |
941 | # compute number of digits in bigint, minus the sign |
942 | |
943 | # int() because add/sub sometimes leaves strings (like '00005') instead of |
944 | # '5' in this place, thus causing length() to report wrong length |
945 | my $cx = $_[1]; |
946 | |
947 | return (@$cx-1)*$BASE_LEN+length(int($cx->[-1])); |
948 | } |
949 | |
950 | sub _digit |
951 | { |
952 | # return the nth digit, negative values count backward |
953 | # zero is rightmost, so _digit(123,0) will give 3 |
954 | my ($c,$x,$n) = @_; |
955 | |
956 | my $len = _len('',$x); |
957 | |
958 | $n = $len+$n if $n < 0; # -1 last, -2 second-to-last |
959 | $n = abs($n); # if negative was too big |
960 | $len--; $n = $len if $n > $len; # n to big? |
961 | |
962 | my $elem = int($n / $BASE_LEN); # which array element |
963 | my $digit = $n % $BASE_LEN; # which digit in this element |
964 | $elem = '0000'.@$x[$elem]; # get element padded with 0's |
965 | return substr($elem,-$digit-1,1); |
966 | } |
967 | |
968 | sub _zeros |
969 | { |
970 | # return amount of trailing zeros in decimal |
971 | # check each array elem in _m for having 0 at end as long as elem == 0 |
972 | # Upon finding a elem != 0, stop |
973 | my $x = $_[1]; |
974 | my $zeros = 0; my $elem; |
975 | foreach my $e (@$x) |
976 | { |
977 | if ($e != 0) |
978 | { |
979 | $elem = "$e"; # preserve x |
980 | $elem =~ s/.*?(0*$)/$1/; # strip anything not zero |
981 | $zeros *= $BASE_LEN; # elems * 5 |
61f5c3f5 |
982 | $zeros += length($elem); # count trailing zeros |
394e6ffb |
983 | last; # early out |
984 | } |
985 | $zeros ++; # real else branch: 50% slower! |
986 | } |
61f5c3f5 |
987 | $zeros; |
394e6ffb |
988 | } |
989 | |
990 | ############################################################################## |
991 | # _is_* routines |
992 | |
993 | sub _is_zero |
994 | { |
995 | # return true if arg (BINT or num_str) is zero (array '+', '0') |
996 | my $x = $_[1]; |
61f5c3f5 |
997 | |
998 | (((scalar @$x == 1) && ($x->[0] == 0))) <=> 0; |
394e6ffb |
999 | } |
1000 | |
1001 | sub _is_even |
1002 | { |
1003 | # return true if arg (BINT or num_str) is even |
1004 | my $x = $_[1]; |
61f5c3f5 |
1005 | (!($x->[0] & 1)) <=> 0; |
394e6ffb |
1006 | } |
1007 | |
1008 | sub _is_odd |
1009 | { |
1010 | # return true if arg (BINT or num_str) is even |
1011 | my $x = $_[1]; |
61f5c3f5 |
1012 | |
1013 | (($x->[0] & 1)) <=> 0; |
394e6ffb |
1014 | } |
1015 | |
1016 | sub _is_one |
1017 | { |
1018 | # return true if arg (BINT or num_str) is one (array '+', '1') |
1019 | my $x = $_[1]; |
61f5c3f5 |
1020 | |
1021 | (scalar @$x == 1) && ($x->[0] == 1) <=> 0; |
394e6ffb |
1022 | } |
1023 | |
1024 | sub __strip_zeros |
1025 | { |
1026 | # internal normalization function that strips leading zeros from the array |
1027 | # args: ref to array |
1028 | my $s = shift; |
1029 | |
1030 | my $cnt = scalar @$s; # get count of parts |
1031 | my $i = $cnt-1; |
1032 | push @$s,0 if $i < 0; # div might return empty results, so fix it |
1033 | |
61f5c3f5 |
1034 | return $s if @$s == 1; # early out |
1035 | |
394e6ffb |
1036 | #print "strip: cnt $cnt i $i\n"; |
1037 | # '0', '3', '4', '0', '0', |
1038 | # 0 1 2 3 4 |
1039 | # cnt = 5, i = 4 |
1040 | # i = 4 |
1041 | # i = 3 |
1042 | # => fcnt = cnt - i (5-2 => 3, cnt => 5-1 = 4, throw away from 4th pos) |
1043 | # >= 1: skip first part (this can be zero) |
1044 | while ($i > 0) { last if $s->[$i] != 0; $i--; } |
1045 | $i++; splice @$s,$i if ($i < $cnt); # $i cant be 0 |
1046 | $s; |
1047 | } |
1048 | |
1049 | ############################################################################### |
1050 | # check routine to test internal state of corruptions |
1051 | |
1052 | sub _check |
1053 | { |
1054 | # used by the test suite |
1055 | my $x = $_[1]; |
1056 | |
1057 | return "$x is not a reference" if !ref($x); |
1058 | |
1059 | # are all parts are valid? |
1060 | my $i = 0; my $j = scalar @$x; my ($e,$try); |
1061 | while ($i < $j) |
1062 | { |
1063 | $e = $x->[$i]; $e = 'undef' unless defined $e; |
1064 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e)"; |
1065 | last if $e !~ /^[+]?[0-9]+$/; |
1066 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (stringify)"; |
1067 | last if "$e" !~ /^[+]?[0-9]+$/; |
1068 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (cat-stringify)"; |
1069 | last if '' . "$e" !~ /^[+]?[0-9]+$/; |
1070 | $try = ' < 0 || >= $BASE; '."($x, $e)"; |
1071 | last if $e <0 || $e >= $BASE; |
1072 | # this test is disabled, since new/bnorm and certain ops (like early out |
1073 | # in add/sub) are allowed/expected to leave '00000' in some elements |
1074 | #$try = '=~ /^00+/; '."($x, $e)"; |
1075 | #last if $e =~ /^00+/; |
1076 | $i++; |
1077 | } |
1078 | return "Illegal part '$e' at pos $i (tested: $try)" if $i < $j; |
1079 | return 0; |
1080 | } |
1081 | |
1082 | |
1083 | ############################################################################### |
1084 | ############################################################################### |
1085 | # some optional routines to make BigInt faster |
1086 | |
dccbb853 |
1087 | sub _mod |
1088 | { |
1089 | # if possible, use mod shortcut |
1090 | my ($c,$x,$yo) = @_; |
1091 | |
1092 | # slow way since $y to big |
1093 | if (scalar @$yo > 1) |
1094 | { |
1095 | my ($xo,$rem) = _div($c,$x,$yo); |
1096 | return $rem; |
1097 | } |
1098 | my $y = $yo->[0]; |
027dc388 |
1099 | # both are single element arrays |
dccbb853 |
1100 | if (scalar @$x == 1) |
1101 | { |
1102 | $x->[0] %= $y; |
1103 | return $x; |
1104 | } |
1105 | |
61f5c3f5 |
1106 | # @y is single element, but @x has more than one |
dccbb853 |
1107 | my $b = $BASE % $y; |
1108 | if ($b == 0) |
1109 | { |
1110 | # when BASE % Y == 0 then (B * BASE) % Y == 0 |
1111 | # (B * BASE) % $y + A % Y => A % Y |
1112 | # so need to consider only last element: O(1) |
1113 | $x->[0] %= $y; |
1114 | } |
027dc388 |
1115 | elsif ($b == 1) |
1116 | { |
28df3e88 |
1117 | # else need to go trough all elements: O(N), but loop is a bit simplified |
027dc388 |
1118 | my $r = 0; |
1119 | foreach (@$x) |
1120 | { |
28df3e88 |
1121 | $r = ($r + $_) % $y; # not much faster, but heh... |
1122 | #$r += $_ % $y; $r %= $y; |
027dc388 |
1123 | } |
1124 | $r = 0 if $r == $y; |
1125 | $x->[0] = $r; |
1126 | } |
dccbb853 |
1127 | else |
1128 | { |
027dc388 |
1129 | # else need to go trough all elements: O(N) |
1130 | my $r = 0; my $bm = 1; |
1131 | foreach (@$x) |
1132 | { |
28df3e88 |
1133 | $r = ($_ * $bm + $r) % $y; |
1134 | $bm = ($bm * $b) % $y; |
1135 | |
1136 | #$r += ($_ % $y) * $bm; |
1137 | #$bm *= $b; |
1138 | #$bm %= $y; |
1139 | #$r %= $y; |
027dc388 |
1140 | } |
1141 | $r = 0 if $r == $y; |
1142 | $x->[0] = $r; |
dccbb853 |
1143 | } |
1144 | splice (@$x,1); |
61f5c3f5 |
1145 | $x; |
dccbb853 |
1146 | } |
1147 | |
0716bf9b |
1148 | ############################################################################## |
574bacfe |
1149 | # shifts |
1150 | |
1151 | sub _rsft |
1152 | { |
1153 | my ($c,$x,$y,$n) = @_; |
1154 | |
1155 | if ($n != 10) |
1156 | { |
61f5c3f5 |
1157 | $n = _new($c,\$n); return _div($c,$x, _pow($c,$n,$y)); |
1158 | } |
1159 | |
1160 | # shortcut (faster) for shifting by 10) |
1161 | # multiples of $BASE_LEN |
1162 | my $dst = 0; # destination |
1163 | my $src = _num($c,$y); # as normal int |
1164 | my $rem = $src % $BASE_LEN; # remainder to shift |
1165 | $src = int($src / $BASE_LEN); # source |
1166 | if ($rem == 0) |
1167 | { |
1168 | splice (@$x,0,$src); # even faster, 38.4 => 39.3 |
574bacfe |
1169 | } |
1170 | else |
1171 | { |
61f5c3f5 |
1172 | my $len = scalar @$x - $src; # elems to go |
1173 | my $vd; my $z = '0'x $BASE_LEN; |
1174 | $x->[scalar @$x] = 0; # avoid || 0 test inside loop |
1175 | while ($dst < $len) |
574bacfe |
1176 | { |
61f5c3f5 |
1177 | $vd = $z.$x->[$src]; |
1178 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN-$rem); |
1179 | $src++; |
1180 | $vd = substr($z.$x->[$src],-$rem,$rem) . $vd; |
1181 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN; |
1182 | $x->[$dst] = int($vd); |
1183 | $dst++; |
574bacfe |
1184 | } |
61f5c3f5 |
1185 | splice (@$x,$dst) if $dst > 0; # kill left-over array elems |
56b9c951 |
1186 | pop @$x if $x->[-1] == 0 && @$x > 1; # kill last element if 0 |
61f5c3f5 |
1187 | } # else rem == 0 |
574bacfe |
1188 | $x; |
1189 | } |
1190 | |
1191 | sub _lsft |
1192 | { |
1193 | my ($c,$x,$y,$n) = @_; |
1194 | |
1195 | if ($n != 10) |
1196 | { |
61f5c3f5 |
1197 | $n = _new($c,\$n); return _mul($c,$x, _pow($c,$n,$y)); |
574bacfe |
1198 | } |
61f5c3f5 |
1199 | |
1200 | # shortcut (faster) for shifting by 10) since we are in base 10eX |
1201 | # multiples of $BASE_LEN: |
1202 | my $src = scalar @$x; # source |
1203 | my $len = _num($c,$y); # shift-len as normal int |
1204 | my $rem = $len % $BASE_LEN; # remainder to shift |
1205 | my $dst = $src + int($len/$BASE_LEN); # destination |
1206 | my $vd; # further speedup |
1207 | $x->[$src] = 0; # avoid first ||0 for speed |
1208 | my $z = '0' x $BASE_LEN; |
1209 | while ($src >= 0) |
574bacfe |
1210 | { |
61f5c3f5 |
1211 | $vd = $x->[$src]; $vd = $z.$vd; |
1212 | $vd = substr($vd,-$BASE_LEN+$rem,$BASE_LEN-$rem); |
1213 | $vd .= $src > 0 ? substr($z.$x->[$src-1],-$BASE_LEN,$rem) : '0' x $rem; |
1214 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN; |
1215 | $x->[$dst] = int($vd); |
1216 | $dst--; $src--; |
574bacfe |
1217 | } |
61f5c3f5 |
1218 | # set lowest parts to 0 |
1219 | while ($dst >= 0) { $x->[$dst--] = 0; } |
1220 | # fix spurios last zero element |
1221 | splice @$x,-1 if $x->[-1] == 0; |
574bacfe |
1222 | $x; |
1223 | } |
1224 | |
027dc388 |
1225 | sub _pow |
1226 | { |
1227 | # power of $x to $y |
1228 | # ref to array, ref to array, return ref to array |
1229 | my ($c,$cx,$cy) = @_; |
1230 | |
1231 | my $pow2 = _one(); |
1232 | my $two = _two(); |
1233 | my $y1 = _copy($c,$cy); |
1234 | while (!_is_one($c,$y1)) |
1235 | { |
1236 | _mul($c,$pow2,$cx) if _is_odd($c,$y1); |
1237 | _div($c,$y1,$two); |
1238 | _mul($c,$cx,$cx); |
1239 | } |
1240 | _mul($c,$cx,$pow2) unless _is_one($c,$pow2); |
61f5c3f5 |
1241 | $cx; |
027dc388 |
1242 | } |
1243 | |
b3abae2a |
1244 | sub _fac |
1245 | { |
1246 | # factorial of $x |
1247 | # ref to array, return ref to array |
1248 | my ($c,$cx) = @_; |
1249 | |
1250 | if ((@$cx == 1) && ($cx->[0] <= 2)) |
1251 | { |
1252 | $cx->[0] = 1 * ($cx->[0]||1); # 0,1 => 1, 2 => 2 |
1253 | return $cx; |
1254 | } |
1255 | |
1256 | # go forward until $base is exceeded |
1257 | # limit is either $x or $base (x == 100 means as result too high) |
1258 | my $steps = 100; $steps = $cx->[0] if @$cx == 1; |
1259 | my $r = 2; my $cf = 3; my $step = 1; my $last = $r; |
1260 | while ($r < $BASE && $step < $steps) |
1261 | { |
1262 | $last = $r; $r *= $cf++; $step++; |
1263 | } |
1264 | if ((@$cx == 1) && ($step == $cx->[0])) |
1265 | { |
1266 | # completely done |
1267 | $cx = [$last]; |
1268 | return $cx; |
1269 | } |
1270 | my $n = _copy($c,$cx); |
1271 | $cx = [$last]; |
1272 | |
1273 | #$cx = _one(); |
1274 | while (!(@$n == 1 && $n->[0] == $step)) |
1275 | { |
1276 | _mul($c,$cx,$n); _dec($c,$n); |
1277 | } |
1278 | $cx; |
1279 | } |
1280 | |
1281 | use constant DEBUG => 0; |
1282 | |
1283 | my $steps = 0; |
1284 | |
1285 | sub steps { $steps }; |
1286 | |
1287 | sub _sqrt |
0716bf9b |
1288 | { |
394e6ffb |
1289 | # square-root of $x |
1290 | # ref to array, return ref to array |
1291 | my ($c,$x) = @_; |
0716bf9b |
1292 | |
394e6ffb |
1293 | if (scalar @$x == 1) |
1294 | { |
1295 | # fit's into one Perl scalar |
1296 | $x->[0] = int(sqrt($x->[0])); |
1297 | return $x; |
1298 | } |
1299 | my $y = _copy($c,$x); |
b3abae2a |
1300 | # hopefully _len/2 is < $BASE, the -1 is to always undershot the guess |
1301 | # since our guess will "grow" |
1302 | my $l = int((_len($c,$x)-1) / 2); |
1303 | |
1304 | my $lastelem = $x->[-1]; # for guess |
1305 | my $elems = scalar @$x - 1; |
1306 | # not enough digits, but could have more? |
1307 | if ((length($lastelem) <= 3) && ($elems > 1)) |
1308 | { |
1309 | # right-align with zero pad |
1310 | my $len = length($lastelem) & 1; |
1311 | print "$lastelem => " if DEBUG; |
1312 | $lastelem .= substr($x->[-2] . '0' x $BASE_LEN,0,$BASE_LEN); |
1313 | # former odd => make odd again, or former even to even again |
1314 | $lastelem = $lastelem / 10 if (length($lastelem) & 1) != $len; |
1315 | print "$lastelem\n" if DEBUG; |
1316 | } |
0716bf9b |
1317 | |
61f5c3f5 |
1318 | # construct $x (instead of _lsft($c,$x,$l,10) |
1319 | my $r = $l % $BASE_LEN; # 10000 00000 00000 00000 ($BASE_LEN=5) |
1320 | $l = int($l / $BASE_LEN); |
b3abae2a |
1321 | print "l = $l " if DEBUG; |
1322 | |
1323 | splice @$x,$l; # keep ref($x), but modify it |
1324 | |
1325 | # we make the first part of the guess not '1000...0' but int(sqrt($lastelem)) |
1326 | # that gives us: |
1327 | # 14400 00000 => sqrt(14400) => 120 |
1328 | # 144000 000000 => sqrt(144000) => 379 |
1329 | |
1330 | # $x->[$l--] = int('1' . '0' x $r); # old way of guessing |
1331 | print "$lastelem (elems $elems) => " if DEBUG; |
1332 | $lastelem = $lastelem / 10 if ($elems & 1 == 1); # odd or even? |
1333 | my $g = sqrt($lastelem); $g =~ s/\.//; # 2.345 => 2345 |
1334 | $r -= 1 if $elems & 1 == 0; # 70 => 7 |
1335 | |
1336 | # padd with zeros if result is too short |
1337 | $x->[$l--] = int(substr($g . '0' x $r,0,$r+1)); |
1338 | print "now ",$x->[-1] if DEBUG; |
1339 | print " would have been ", int('1' . '0' x $r),"\n" if DEBUG; |
1340 | |
1341 | # If @$x > 1, we could compute the second elem of the guess, too, to create |
1342 | # an even better guess. Not implemented yet. |
1343 | $x->[$l--] = 0 while ($l >= 0); # all other digits of guess are zero |
61f5c3f5 |
1344 | |
b3abae2a |
1345 | print "start x= ",${_str($c,$x)},"\n" if DEBUG; |
394e6ffb |
1346 | my $two = _two(); |
1347 | my $last = _zero(); |
1348 | my $lastlast = _zero(); |
b3abae2a |
1349 | $steps = 0 if DEBUG; |
394e6ffb |
1350 | while (_acmp($c,$last,$x) != 0 && _acmp($c,$lastlast,$x) != 0) |
1351 | { |
b3abae2a |
1352 | $steps++ if DEBUG; |
394e6ffb |
1353 | $lastlast = _copy($c,$last); |
1354 | $last = _copy($c,$x); |
1355 | _add($c,$x, _div($c,_copy($c,$y),$x)); |
1356 | _div($c,$x, $two ); |
b3abae2a |
1357 | print " x= ",${_str($c,$x)},"\n" if DEBUG; |
394e6ffb |
1358 | } |
b3abae2a |
1359 | print "\nsteps in sqrt: $steps, " if DEBUG; |
394e6ffb |
1360 | _dec($c,$x) if _acmp($c,$y,_mul($c,_copy($c,$x),$x)) < 0; # overshot? |
b3abae2a |
1361 | print " final ",$x->[-1],"\n" if DEBUG; |
394e6ffb |
1362 | $x; |
0716bf9b |
1363 | } |
1364 | |
394e6ffb |
1365 | ############################################################################## |
1366 | # binary stuff |
0716bf9b |
1367 | |
394e6ffb |
1368 | sub _and |
1369 | { |
1370 | my ($c,$x,$y) = @_; |
0716bf9b |
1371 | |
394e6ffb |
1372 | # the shortcut makes equal, large numbers _really_ fast, and makes only a |
1373 | # very small performance drop for small numbers (e.g. something with less |
1374 | # than 32 bit) Since we optimize for large numbers, this is enabled. |
1375 | return $x if _acmp($c,$x,$y) == 0; # shortcut |
0716bf9b |
1376 | |
394e6ffb |
1377 | my $m = _one(); my ($xr,$yr); |
1378 | my $mask = $AND_MASK; |
1379 | |
1380 | my $x1 = $x; |
1381 | my $y1 = _copy($c,$y); # make copy |
1382 | $x = _zero(); |
1383 | my ($b,$xrr,$yrr); |
1384 | use integer; |
1385 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
1386 | { |
1387 | ($x1, $xr) = _div($c,$x1,$mask); |
1388 | ($y1, $yr) = _div($c,$y1,$mask); |
1389 | |
1390 | # make ints() from $xr, $yr |
1391 | # this is when the AND_BITS are greater tahn $BASE and is slower for |
1392 | # small (<256 bits) numbers, but faster for large numbers. Disabled |
1393 | # due to KISS principle |
1394 | |
1395 | # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
1396 | # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
1397 | # _add($c,$x, _mul($c, _new( $c, \($xrr & $yrr) ), $m) ); |
1398 | |
61f5c3f5 |
1399 | # 0+ due to '&' doesn't work in strings |
1400 | _add($c,$x, _mul($c, [ 0+$xr->[0] & 0+$yr->[0] ], $m) ); |
394e6ffb |
1401 | _mul($c,$m,$mask); |
1402 | } |
1403 | $x; |
0716bf9b |
1404 | } |
1405 | |
394e6ffb |
1406 | sub _xor |
0716bf9b |
1407 | { |
394e6ffb |
1408 | my ($c,$x,$y) = @_; |
1409 | |
1410 | return _zero() if _acmp($c,$x,$y) == 0; # shortcut (see -and) |
1411 | |
1412 | my $m = _one(); my ($xr,$yr); |
1413 | my $mask = $XOR_MASK; |
1414 | |
1415 | my $x1 = $x; |
1416 | my $y1 = _copy($c,$y); # make copy |
1417 | $x = _zero(); |
1418 | my ($b,$xrr,$yrr); |
1419 | use integer; |
1420 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
0716bf9b |
1421 | { |
394e6ffb |
1422 | ($x1, $xr) = _div($c,$x1,$mask); |
1423 | ($y1, $yr) = _div($c,$y1,$mask); |
1424 | # make ints() from $xr, $yr (see _and()) |
1425 | #$b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
1426 | #$b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
1427 | #_add($c,$x, _mul($c, _new( $c, \($xrr ^ $yrr) ), $m) ); |
61f5c3f5 |
1428 | |
1429 | # 0+ due to '^' doesn't work in strings |
1430 | _add($c,$x, _mul($c, [ 0+$xr->[0] ^ 0+$yr->[0] ], $m) ); |
394e6ffb |
1431 | _mul($c,$m,$mask); |
0716bf9b |
1432 | } |
394e6ffb |
1433 | # the loop stops when the shorter of the two numbers is exhausted |
1434 | # the remainder of the longer one will survive bit-by-bit, so we simple |
1435 | # multiply-add it in |
1436 | _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1); |
1437 | _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1); |
1438 | |
1439 | $x; |
0716bf9b |
1440 | } |
1441 | |
394e6ffb |
1442 | sub _or |
0716bf9b |
1443 | { |
394e6ffb |
1444 | my ($c,$x,$y) = @_; |
0716bf9b |
1445 | |
394e6ffb |
1446 | return $x if _acmp($c,$x,$y) == 0; # shortcut (see _and) |
0716bf9b |
1447 | |
394e6ffb |
1448 | my $m = _one(); my ($xr,$yr); |
1449 | my $mask = $OR_MASK; |
0716bf9b |
1450 | |
394e6ffb |
1451 | my $x1 = $x; |
1452 | my $y1 = _copy($c,$y); # make copy |
1453 | $x = _zero(); |
1454 | my ($b,$xrr,$yrr); |
1455 | use integer; |
1456 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
1457 | { |
1458 | ($x1, $xr) = _div($c,$x1,$mask); |
1459 | ($y1, $yr) = _div($c,$y1,$mask); |
1460 | # make ints() from $xr, $yr (see _and()) |
1461 | # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
1462 | # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
1463 | # _add($c,$x, _mul($c, _new( $c, \($xrr | $yrr) ), $m) ); |
1464 | |
61f5c3f5 |
1465 | # 0+ due to '|' doesn't work in strings |
1466 | _add($c,$x, _mul($c, [ 0+$xr->[0] | 0+$yr->[0] ], $m) ); |
394e6ffb |
1467 | _mul($c,$m,$mask); |
1468 | } |
1469 | # the loop stops when the shorter of the two numbers is exhausted |
1470 | # the remainder of the longer one will survive bit-by-bit, so we simple |
1471 | # multiply-add it in |
1472 | _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1); |
1473 | _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1); |
1474 | |
1475 | $x; |
0716bf9b |
1476 | } |
1477 | |
61f5c3f5 |
1478 | sub _as_hex |
1479 | { |
1480 | # convert a decimal number to hex (ref to array, return ref to string) |
1481 | my ($c,$x) = @_; |
1482 | |
1483 | my $x1 = _copy($c,$x); |
1484 | |
1485 | my $es = ''; |
1486 | my $xr; |
1487 | my $x10000 = [ 0x10000 ]; |
1488 | while (! _is_zero($c,$x1)) |
1489 | { |
1490 | ($x1, $xr) = _div($c,$x1,$x10000); |
1491 | $es .= unpack('h4',pack('v',$xr->[0])); |
1492 | } |
1493 | $es = reverse $es; |
1494 | $es =~ s/^[0]+//; # strip leading zeros |
1495 | $es = '0x' . $es; |
1496 | \$es; |
1497 | } |
1498 | |
1499 | sub _as_bin |
1500 | { |
1501 | # convert a decimal number to bin (ref to array, return ref to string) |
1502 | my ($c,$x) = @_; |
1503 | |
1504 | my $x1 = _copy($c,$x); |
1505 | |
1506 | my $es = ''; |
1507 | my $xr; |
1508 | my $x10000 = [ 0x10000 ]; |
1509 | while (! _is_zero($c,$x1)) |
1510 | { |
1511 | ($x1, $xr) = _div($c,$x1,$x10000); |
1512 | $es .= unpack('b16',pack('v',$xr->[0])); |
1513 | } |
1514 | $es = reverse $es; |
1515 | $es =~ s/^[0]+//; # strip leading zeros |
1516 | $es = '0b' . $es; |
1517 | \$es; |
1518 | } |
1519 | |
394e6ffb |
1520 | sub _from_hex |
0716bf9b |
1521 | { |
394e6ffb |
1522 | # convert a hex number to decimal (ref to string, return ref to array) |
1523 | my ($c,$hs) = @_; |
0716bf9b |
1524 | |
394e6ffb |
1525 | my $mul = _one(); |
1526 | my $m = [ 0x10000 ]; # 16 bit at a time |
1527 | my $x = _zero(); |
0716bf9b |
1528 | |
61f5c3f5 |
1529 | my $len = length($$hs)-2; |
394e6ffb |
1530 | $len = int($len/4); # 4-digit parts, w/o '0x' |
1531 | my $val; my $i = -4; |
1532 | while ($len >= 0) |
1533 | { |
1534 | $val = substr($$hs,$i,4); |
1535 | $val =~ s/^[+-]?0x// if $len == 0; # for last part only because |
1536 | $val = hex($val); # hex does not like wrong chars |
1537 | $i -= 4; $len --; |
1538 | _add ($c, $x, _mul ($c, [ $val ], $mul ) ) if $val != 0; |
1539 | _mul ($c, $mul, $m ) if $len >= 0; # skip last mul |
1540 | } |
1541 | $x; |
1542 | } |
1543 | |
1544 | sub _from_bin |
0716bf9b |
1545 | { |
394e6ffb |
1546 | # convert a hex number to decimal (ref to string, return ref to array) |
1547 | my ($c,$bs) = @_; |
0716bf9b |
1548 | |
13a12e00 |
1549 | # instead of converting 8 bit at a time, it is faster to convert the |
1550 | # number to hex, and then call _from_hex. |
1551 | |
1552 | my $hs = $$bs; |
1553 | $hs =~ s/^[+-]?0b//; # remove sign and 0b |
1554 | my $l = length($hs); # bits |
1555 | $hs = '0' x (8-($l % 8)) . $hs if ($l % 8) != 0; # padd left side w/ 0 |
1556 | my $h = unpack('H*', pack ('B*', $hs)); # repack as hex |
1557 | return $c->_from_hex(\('0x'.$h)); |
1558 | |
394e6ffb |
1559 | my $mul = _one(); |
1560 | my $m = [ 0x100 ]; # 8 bit at a time |
1561 | my $x = _zero(); |
0716bf9b |
1562 | |
61f5c3f5 |
1563 | my $len = length($$bs)-2; |
394e6ffb |
1564 | $len = int($len/8); # 4-digit parts, w/o '0x' |
1565 | my $val; my $i = -8; |
1566 | while ($len >= 0) |
0716bf9b |
1567 | { |
394e6ffb |
1568 | $val = substr($$bs,$i,8); |
1569 | $val =~ s/^[+-]?0b// if $len == 0; # for last part only |
1570 | |
394e6ffb |
1571 | $val = ord(pack('B8',substr('00000000'.$val,-8,8))); |
1572 | |
1573 | $i -= 8; $len --; |
1574 | _add ($c, $x, _mul ($c, [ $val ], $mul ) ) if $val != 0; |
1575 | _mul ($c, $mul, $m ) if $len >= 0; # skip last mul |
0716bf9b |
1576 | } |
394e6ffb |
1577 | $x; |
0716bf9b |
1578 | } |
1579 | |
07d34614 |
1580 | ############################################################################## |
1581 | # special modulus functions |
1582 | |
1583 | sub _modinv1 |
d614cd8b |
1584 | { |
1585 | # inverse modulus |
1586 | } |
1587 | |
1588 | sub _modpow |
1589 | { |
1590 | # modulus of power ($x ** $y) % $z |
07d34614 |
1591 | my ($c,$num,$exp,$mod) = @_; |
1592 | |
1593 | # in the trivial case, |
1594 | if (_is_one($c,$mod)) |
1595 | { |
1596 | splice @$num,0,1; $num->[0] = 0; |
1597 | return $num; |
1598 | } |
1599 | if ((scalar @$num == 1) && (($num->[0] == 0) || ($num->[0] == 1))) |
1600 | { |
1601 | $num->[0] = 1; |
1602 | return $num; |
1603 | } |
1604 | |
1605 | # $num = _mod($c,$num,$mod); |
1606 | |
1607 | my $acc = _copy($c,$num); my $t = _one(); |
1608 | |
1609 | my $two = _two(); |
1610 | my $exp1 = _copy($c,$exp); # keep arguments |
1611 | while (!_is_zero($c,$exp1)) |
1612 | { |
1613 | if (_is_odd($c,$exp1)) |
1614 | { |
1615 | _mul($c,$t,$acc); |
1616 | $t = _mod($c,$t,$mod); |
1617 | } |
1618 | _mul($c,$acc,$acc); |
1619 | $acc = _mod($c,$acc,$mod); |
1620 | _div($c,$exp1,$two); |
1621 | # print "exp ",${_str($c,$exp1)},"\n"; |
1622 | # print "acc ",${_str($c,$acc)},"\n"; |
1623 | # print "num ",${_str($c,$num)},"\n"; |
1624 | # print "mod ",${_str($c,$mod)},"\n"; |
1625 | } |
1626 | @$num = @$t; |
1627 | $num; |
d614cd8b |
1628 | } |
1629 | |
394e6ffb |
1630 | ############################################################################## |
1631 | ############################################################################## |
1632 | |
0716bf9b |
1633 | 1; |
1634 | __END__ |
1635 | |
1636 | =head1 NAME |
1637 | |
1638 | Math::BigInt::Calc - Pure Perl module to support Math::BigInt |
1639 | |
1640 | =head1 SYNOPSIS |
1641 | |
ee15d750 |
1642 | Provides support for big integer calculations. Not intended to be used by other |
1643 | modules (except Math::BigInt::Cached). Other modules which sport the same |
1644 | functions can also be used to support Math::Bigint, like Math::BigInt::Pari. |
0716bf9b |
1645 | |
1646 | =head1 DESCRIPTION |
1647 | |
027dc388 |
1648 | In order to allow for multiple big integer libraries, Math::BigInt was |
1649 | rewritten to use library modules for core math routines. Any module which |
1650 | follows the same API as this can be used instead by using the following: |
0716bf9b |
1651 | |
ee15d750 |
1652 | use Math::BigInt lib => 'libname'; |
0716bf9b |
1653 | |
027dc388 |
1654 | 'libname' is either the long name ('Math::BigInt::Pari'), or only the short |
1655 | version like 'Pari'. |
1656 | |
0716bf9b |
1657 | =head1 EXPORT |
1658 | |
027dc388 |
1659 | The following functions MUST be defined in order to support the use by |
1660 | Math::BigInt: |
0716bf9b |
1661 | |
1662 | _new(string) return ref to new object from ref to decimal string |
1663 | _zero() return a new object with value 0 |
1664 | _one() return a new object with value 1 |
1665 | |
1666 | _str(obj) return ref to a string representing the object |
1667 | _num(obj) returns a Perl integer/floating point number |
1668 | NOTE: because of Perl numeric notation defaults, |
1669 | the _num'ified obj may lose accuracy due to |
1670 | machine-dependend floating point size limitations |
1671 | |
1672 | _add(obj,obj) Simple addition of two objects |
1673 | _mul(obj,obj) Multiplication of two objects |
1674 | _div(obj,obj) Division of the 1st object by the 2nd |
b22b3e31 |
1675 | In list context, returns (result,remainder). |
1676 | NOTE: this is integer math, so no |
1677 | fractional part will be returned. |
1678 | _sub(obj,obj) Simple subtraction of 1 object from another |
0716bf9b |
1679 | a third, optional parameter indicates that the params |
1680 | are swapped. In this case, the first param needs to |
1681 | be preserved, while you can destroy the second. |
1682 | sub (x,y,1) => return x - y and keep x intact! |
e745a66c |
1683 | _dec(obj) decrement object by one (input is garant. to be > 0) |
1684 | _inc(obj) increment object by one |
1685 | |
0716bf9b |
1686 | |
1687 | _acmp(obj,obj) <=> operator for objects (return -1, 0 or 1) |
1688 | |
1689 | _len(obj) returns count of the decimal digits of the object |
1690 | _digit(obj,n) returns the n'th decimal digit of object |
1691 | |
1692 | _is_one(obj) return true if argument is +1 |
1693 | _is_zero(obj) return true if argument is 0 |
1694 | _is_even(obj) return true if argument is even (0,2,4,6..) |
1695 | _is_odd(obj) return true if argument is odd (1,3,5,7..) |
1696 | |
1697 | _copy return a ref to a true copy of the object |
1698 | |
1699 | _check(obj) check whether internal representation is still intact |
1700 | return 0 for ok, otherwise error message as string |
1701 | |
bd05a461 |
1702 | The following functions are optional, and can be defined if the underlying lib |
027dc388 |
1703 | has a fast way to do them. If undefined, Math::BigInt will use pure Perl (hence |
1704 | slow) fallback routines to emulate these: |
0716bf9b |
1705 | |
1706 | _from_hex(str) return ref to new object from ref to hexadecimal string |
1707 | _from_bin(str) return ref to new object from ref to binary string |
1708 | |
ee15d750 |
1709 | _as_hex(str) return ref to scalar string containing the value as |
1710 | unsigned hex string, with the '0x' prepended. |
1711 | Leading zeros must be stripped. |
1712 | _as_bin(str) Like as_hex, only as binary string containing only |
1713 | zeros and ones. Leading zeros must be stripped and a |
1714 | '0b' must be prepended. |
1715 | |
0716bf9b |
1716 | _rsft(obj,N,B) shift object in base B by N 'digits' right |
dccbb853 |
1717 | For unsupported bases B, return undef to signal failure |
0716bf9b |
1718 | _lsft(obj,N,B) shift object in base B by N 'digits' left |
dccbb853 |
1719 | For unsupported bases B, return undef to signal failure |
0716bf9b |
1720 | |
1721 | _xor(obj1,obj2) XOR (bit-wise) object 1 with object 2 |
dccbb853 |
1722 | Note: XOR, AND and OR pad with zeros if size mismatches |
0716bf9b |
1723 | _and(obj1,obj2) AND (bit-wise) object 1 with object 2 |
1724 | _or(obj1,obj2) OR (bit-wise) object 1 with object 2 |
1725 | |
dccbb853 |
1726 | _mod(obj,obj) Return remainder of div of the 1st by the 2nd object |
394e6ffb |
1727 | _sqrt(obj) return the square root of object (truncate to int) |
b3abae2a |
1728 | _fac(obj) return factorial of object 1 (1*2*3*4..) |
0716bf9b |
1729 | _pow(obj,obj) return object 1 to the power of object 2 |
1730 | _gcd(obj,obj) return Greatest Common Divisor of two objects |
1731 | |
b22b3e31 |
1732 | _zeros(obj) return number of trailing decimal zeros |
d614cd8b |
1733 | _modinv return inverse modulus |
1734 | _modpow return modulus of power ($x ** $y) % $z |
0716bf9b |
1735 | |
b22b3e31 |
1736 | Input strings come in as unsigned but with prefix (i.e. as '123', '0xabc' |
0716bf9b |
1737 | or '0b1101'). |
1738 | |
b22b3e31 |
1739 | Testing of input parameter validity is done by the caller, so you need not |
574bacfe |
1740 | worry about underflow (f.i. in C<_sub()>, C<_dec()>) nor about division by |
1741 | zero or similar cases. |
1742 | |
1743 | The first parameter can be modified, that includes the possibility that you |
1744 | return a reference to a completely different object instead. Although keeping |
dccbb853 |
1745 | the reference and just changing it's contents is prefered over creating and |
1746 | returning a different reference. |
574bacfe |
1747 | |
1748 | Return values are always references to objects or strings. Exceptions are |
1749 | C<_lsft()> and C<_rsft()>, which return undef if they can not shift the |
027dc388 |
1750 | argument. This is used to delegate shifting of bases different than the one |
1751 | you can support back to Math::BigInt, which will use some generic code to |
1752 | calculate the result. |
574bacfe |
1753 | |
1754 | =head1 WRAP YOUR OWN |
1755 | |
1756 | If you want to port your own favourite c-lib for big numbers to the |
1757 | Math::BigInt interface, you can take any of the already existing modules as |
1758 | a rough guideline. You should really wrap up the latest BigInt and BigFloat |
bd05a461 |
1759 | testsuites with your module, and replace in them any of the following: |
574bacfe |
1760 | |
1761 | use Math::BigInt; |
1762 | |
bd05a461 |
1763 | by this: |
574bacfe |
1764 | |
1765 | use Math::BigInt lib => 'yourlib'; |
1766 | |
1767 | This way you ensure that your library really works 100% within Math::BigInt. |
0716bf9b |
1768 | |
1769 | =head1 LICENSE |
1770 | |
1771 | This program is free software; you may redistribute it and/or modify it under |
1772 | the same terms as Perl itself. |
1773 | |
1774 | =head1 AUTHORS |
1775 | |
1776 | Original math code by Mark Biggar, rewritten by Tels L<http://bloodgate.com/> |
1777 | in late 2000, 2001. |
1778 | Seperated from BigInt and shaped API with the help of John Peacock. |
1779 | |
1780 | =head1 SEE ALSO |
1781 | |
ee15d750 |
1782 | L<Math::BigInt>, L<Math::BigFloat>, L<Math::BigInt::BitVect>, |
1783 | L<Math::BigInt::GMP>, L<Math::BigInt::Cached> and L<Math::BigInt::Pari>. |
0716bf9b |
1784 | |
1785 | =cut |