Commit | Line | Data |
0716bf9b |
1 | package Math::BigInt::Calc; |
2 | |
08a3f4a9 |
3 | use 5.006; |
0716bf9b |
4 | use strict; |
574bacfe |
5 | # use warnings; # dont use warnings for older Perls |
0716bf9b |
6 | |
08a3f4a9 |
7 | our $VERSION = '0.52'; |
0716bf9b |
8 | |
9 | # Package to store unsigned big integers in decimal and do math with them |
10 | |
11 | # Internally the numbers are stored in an array with at least 1 element, no |
027dc388 |
12 | # leading zero parts (except the first) and in base 1eX where X is determined |
13 | # automatically at loading time to be the maximum possible value |
0716bf9b |
14 | |
15 | # todo: |
a87115f0 |
16 | # - fully remove funky $# stuff in div() (maybe - that code scares me...) |
0716bf9b |
17 | |
18 | # USE_MUL: due to problems on certain os (os390, posix-bc) "* 1e-5" is used |
ee15d750 |
19 | # instead of "/ 1e5" at some places, (marked with USE_MUL). Other platforms |
20 | # BS2000, some Crays need USE_DIV instead. |
bd05a461 |
21 | # The BEGIN block is used to determine which of the two variants gives the |
22 | # correct result. |
0716bf9b |
23 | |
990fb837 |
24 | # Beware of things like: |
2ebb273f |
25 | # $i = $i * $y + $car; $car = int($i / $BASE); $i = $i % $BASE; |
93c87d9d |
26 | # This works on x86, but fails on ARM (SA1100, iPAQ) due to whoknows what |
990fb837 |
27 | # reasons. So, use this instead (slower, but correct): |
2ebb273f |
28 | # $i = $i * $y + $car; $car = int($i / $BASE); $i -= $BASE * $car; |
990fb837 |
29 | |
0716bf9b |
30 | ############################################################################## |
31 | # global constants, flags and accessory |
9b924220 |
32 | |
2ebb273f |
33 | # announce that we are compatible with MBI v1.83 and up |
50109ad0 |
34 | sub api_version () { 2; } |
0716bf9b |
35 | |
36 | # constants for easier life |
2ebb273f |
37 | my ($BASE,$BASE_LEN,$RBASE,$MAX_VAL); |
394e6ffb |
38 | my ($AND_BITS,$XOR_BITS,$OR_BITS); |
39 | my ($AND_MASK,$XOR_MASK,$OR_MASK); |
ee15d750 |
40 | |
41 | sub _base_len |
42 | { |
50109ad0 |
43 | # Set/get the BASE_LEN and assorted other, connected values. |
44 | # Used only by the testsuite, the set variant is used only by the BEGIN |
45 | # block below: |
394e6ffb |
46 | shift; |
47 | |
8d8376c6 |
48 | my ($b, $int) = @_; |
ee15d750 |
49 | if (defined $b) |
50 | { |
2ebb273f |
51 | # avoid redefinitions |
52 | undef &_mul; |
53 | undef &_div; |
54 | |
08a3f4a9 |
55 | if ($] >= 5.008 && $int && $b > 7) |
2ebb273f |
56 | { |
57 | $BASE_LEN = $b; |
58 | *_mul = \&_mul_use_div_64; |
59 | *_div = \&_div_use_div_64; |
60 | $BASE = int("1e".$BASE_LEN); |
61 | $MAX_VAL = $BASE-1; |
62 | return $BASE_LEN unless wantarray; |
63 | return ($BASE_LEN, $AND_BITS, $XOR_BITS, $OR_BITS, $BASE_LEN, $MAX_VAL, $BASE); |
64 | } |
65 | |
66 | # find whether we can use mul or div in mul()/div() |
67 | $BASE_LEN = $b+1; |
61f5c3f5 |
68 | my $caught = 0; |
2ebb273f |
69 | while (--$BASE_LEN > 5) |
394e6ffb |
70 | { |
2ebb273f |
71 | $BASE = int("1e".$BASE_LEN); |
72 | $RBASE = abs('1e-'.$BASE_LEN); # see USE_MUL |
394e6ffb |
73 | $caught = 0; |
2ebb273f |
74 | $caught += 1 if (int($BASE * $RBASE) != 1); # should be 1 |
75 | $caught += 2 if (int($BASE / $BASE) != 1); # should be 1 |
394e6ffb |
76 | last if $caught != 3; |
77 | } |
ee15d750 |
78 | $BASE = int("1e".$BASE_LEN); |
2ebb273f |
79 | $RBASE = abs('1e-'.$BASE_LEN); # see USE_MUL |
80 | $MAX_VAL = $BASE-1; |
b68b7ab1 |
81 | |
50109ad0 |
82 | # ($caught & 1) != 0 => cannot use MUL |
83 | # ($caught & 2) != 0 => cannot use DIV |
990fb837 |
84 | if ($caught == 2) # 2 |
ee15d750 |
85 | { |
990fb837 |
86 | # must USE_MUL since we cannot use DIV |
2ebb273f |
87 | *_mul = \&_mul_use_mul; |
88 | *_div = \&_div_use_mul; |
ee15d750 |
89 | } |
990fb837 |
90 | else # 0 or 1 |
ee15d750 |
91 | { |
ee15d750 |
92 | # can USE_DIV instead |
2ebb273f |
93 | *_mul = \&_mul_use_div; |
94 | *_div = \&_div_use_div; |
ee15d750 |
95 | } |
96 | } |
61f5c3f5 |
97 | return $BASE_LEN unless wantarray; |
2ebb273f |
98 | return ($BASE_LEN, $AND_BITS, $XOR_BITS, $OR_BITS, $BASE_LEN, $MAX_VAL, $BASE); |
ee15d750 |
99 | } |
574bacfe |
100 | |
03874afe |
101 | sub _new |
102 | { |
103 | # (ref to string) return ref to num_array |
104 | # Convert a number from string format (without sign) to internal base |
105 | # 1ex format. Assumes normalized value as input. |
106 | my $il = length($_[1])-1; |
107 | |
108 | # < BASE_LEN due len-1 above |
109 | return [ int($_[1]) ] if $il < $BASE_LEN; # shortcut for short numbers |
110 | |
111 | # this leaves '00000' instead of int 0 and will be corrected after any op |
112 | [ reverse(unpack("a" . ($il % $BASE_LEN+1) |
113 | . ("a$BASE_LEN" x ($il / $BASE_LEN)), $_[1])) ]; |
114 | } |
115 | |
574bacfe |
116 | BEGIN |
117 | { |
bd05a461 |
118 | # from Daniel Pfeiffer: determine largest group of digits that is precisely |
574bacfe |
119 | # multipliable with itself plus carry |
dccbb853 |
120 | # Test now changed to expect the proper pattern, not a result off by 1 or 2 |
121 | my ($e, $num) = 3; # lowest value we will use is 3+1-1 = 3 |
bd05a461 |
122 | do |
123 | { |
124 | $num = ('9' x ++$e) + 0; |
394e6ffb |
125 | $num *= $num + 1.0; |
394e6ffb |
126 | } while ("$num" =~ /9{$e}0{$e}/); # must be a certain pattern |
127 | $e--; # last test failed, so retract one step |
128 | # the limits below brush the problems with the test above under the rug: |
129 | # the test should be able to find the proper $e automatically |
130 | $e = 5 if $^O =~ /^uts/; # UTS get's some special treatment |
131 | $e = 5 if $^O =~ /^unicos/; # unicos is also problematic (6 seems to work |
132 | # there, but we play safe) |
2ebb273f |
133 | |
8d8376c6 |
134 | my $int = 0; |
135 | if ($e > 7) |
136 | { |
137 | use integer; |
138 | my $e1 = 7; |
139 | $num = 7; |
140 | do |
141 | { |
142 | $num = ('9' x ++$e1) + 0; |
143 | $num *= $num + 1; |
144 | } while ("$num" =~ /9{$e1}0{$e1}/); # must be a certain pattern |
145 | $e1--; # last test failed, so retract one step |
146 | if ($e1 > 7) |
147 | { |
148 | $int = 1; $e = $e1; |
149 | } |
150 | } |
151 | |
152 | __PACKAGE__->_base_len($e,$int); # set and store |
394e6ffb |
153 | |
b68b7ab1 |
154 | use integer; |
394e6ffb |
155 | # find out how many bits _and, _or and _xor can take (old default = 16) |
156 | # I don't think anybody has yet 128 bit scalars, so let's play safe. |
394e6ffb |
157 | local $^W = 0; # don't warn about 'nonportable number' |
c38b2de2 |
158 | $AND_BITS = 15; $XOR_BITS = 15; $OR_BITS = 15; |
394e6ffb |
159 | |
160 | # find max bits, we will not go higher than numberofbits that fit into $BASE |
161 | # to make _and etc simpler (and faster for smaller, slower for large numbers) |
162 | my $max = 16; |
163 | while (2 ** $max < $BASE) { $max++; } |
1ddff52a |
164 | { |
165 | no integer; |
166 | $max = 16 if $] < 5.006; # older Perls might not take >16 too well |
167 | } |
394e6ffb |
168 | my ($x,$y,$z); |
169 | do { |
170 | $AND_BITS++; |
0617f807 |
171 | $x = CORE::oct('0b' . '1' x $AND_BITS); $y = $x & $x; |
394e6ffb |
172 | $z = (2 ** $AND_BITS) - 1; |
173 | } while ($AND_BITS < $max && $x == $z && $y == $x); |
174 | $AND_BITS --; # retreat one step |
175 | do { |
176 | $XOR_BITS++; |
0617f807 |
177 | $x = CORE::oct('0b' . '1' x $XOR_BITS); $y = $x ^ 0; |
394e6ffb |
178 | $z = (2 ** $XOR_BITS) - 1; |
179 | } while ($XOR_BITS < $max && $x == $z && $y == $x); |
180 | $XOR_BITS --; # retreat one step |
181 | do { |
182 | $OR_BITS++; |
0617f807 |
183 | $x = CORE::oct('0b' . '1' x $OR_BITS); $y = $x | $x; |
394e6ffb |
184 | $z = (2 ** $OR_BITS) - 1; |
185 | } while ($OR_BITS < $max && $x == $z && $y == $x); |
186 | $OR_BITS --; # retreat one step |
187 | |
9b924220 |
188 | $AND_MASK = __PACKAGE__->_new( ( 2 ** $AND_BITS )); |
189 | $XOR_MASK = __PACKAGE__->_new( ( 2 ** $XOR_BITS )); |
190 | $OR_MASK = __PACKAGE__->_new( ( 2 ** $OR_BITS )); |
50109ad0 |
191 | |
192 | # We can compute the approximate lenght no faster than the real length: |
193 | *_alen = \&_len; |
394e6ffb |
194 | } |
0716bf9b |
195 | |
03874afe |
196 | ############################################################################### |
197 | |
0716bf9b |
198 | sub _zero |
199 | { |
200 | # create a zero |
61f5c3f5 |
201 | [ 0 ]; |
0716bf9b |
202 | } |
203 | |
204 | sub _one |
205 | { |
206 | # create a one |
61f5c3f5 |
207 | [ 1 ]; |
0716bf9b |
208 | } |
209 | |
027dc388 |
210 | sub _two |
211 | { |
1ddff52a |
212 | # create a two (used internally for shifting) |
61f5c3f5 |
213 | [ 2 ]; |
027dc388 |
214 | } |
215 | |
9b924220 |
216 | sub _ten |
217 | { |
218 | # create a 10 (used internally for shifting) |
219 | [ 10 ]; |
220 | } |
221 | |
50109ad0 |
222 | sub _1ex |
223 | { |
224 | # create a 1Ex |
225 | my $rem = $_[1] % $BASE_LEN; # remainder |
226 | my $parts = $_[1] / $BASE_LEN; # parts |
227 | |
228 | # 000000, 000000, 100 |
229 | [ (0) x $parts, '1' . ('0' x $rem) ]; |
230 | } |
231 | |
0716bf9b |
232 | sub _copy |
233 | { |
091c87b1 |
234 | # make a true copy |
61f5c3f5 |
235 | [ @{$_[1]} ]; |
0716bf9b |
236 | } |
237 | |
bd05a461 |
238 | # catch and throw away |
239 | sub import { } |
240 | |
0716bf9b |
241 | ############################################################################## |
242 | # convert back to string and number |
243 | |
244 | sub _str |
245 | { |
246 | # (ref to BINT) return num_str |
247 | # Convert number from internal base 100000 format to string format. |
248 | # internal format is always normalized (no leading zeros, "-0" => "+0") |
574bacfe |
249 | my $ar = $_[1]; |
61f5c3f5 |
250 | |
b68b7ab1 |
251 | my $l = scalar @$ar; # number of parts |
252 | if ($l < 1) # should not happen |
253 | { |
254 | require Carp; |
255 | Carp::croak("$_[1] has no elements"); |
256 | } |
61f5c3f5 |
257 | |
b68b7ab1 |
258 | my $ret = ""; |
0716bf9b |
259 | # handle first one different to strip leading zeros from it (there are no |
260 | # leading zero parts in internal representation) |
61f5c3f5 |
261 | $l --; $ret .= int($ar->[$l]); $l--; |
0716bf9b |
262 | # Interestingly, the pre-padd method uses more time |
091c87b1 |
263 | # the old grep variant takes longer (14 vs. 10 sec) |
574bacfe |
264 | my $z = '0' x ($BASE_LEN-1); |
0716bf9b |
265 | while ($l >= 0) |
266 | { |
574bacfe |
267 | $ret .= substr($z.$ar->[$l],-$BASE_LEN); # fastest way I could think of |
0716bf9b |
268 | $l--; |
269 | } |
9b924220 |
270 | $ret; |
0716bf9b |
271 | } |
272 | |
273 | sub _num |
274 | { |
9b924220 |
275 | # Make a number (scalar int/float) from a BigInt object |
574bacfe |
276 | my $x = $_[1]; |
9b924220 |
277 | |
278 | return 0+$x->[0] if scalar @$x == 1; # below $BASE |
0716bf9b |
279 | my $fac = 1; |
280 | my $num = 0; |
281 | foreach (@$x) |
282 | { |
283 | $num += $fac*$_; $fac *= $BASE; |
284 | } |
61f5c3f5 |
285 | $num; |
0716bf9b |
286 | } |
287 | |
288 | ############################################################################## |
289 | # actual math code |
290 | |
291 | sub _add |
292 | { |
293 | # (ref to int_num_array, ref to int_num_array) |
574bacfe |
294 | # routine to add two base 1eX numbers |
0716bf9b |
295 | # stolen from Knuth Vol 2 Algorithm A pg 231 |
b22b3e31 |
296 | # there are separate routines to add and sub as per Knuth pg 233 |
0716bf9b |
297 | # This routine clobbers up array x, but not y. |
298 | |
574bacfe |
299 | my ($c,$x,$y) = @_; |
b3abae2a |
300 | |
301 | return $x if (@$y == 1) && $y->[0] == 0; # $x + 0 => $x |
302 | if ((@$x == 1) && $x->[0] == 0) # 0 + $y => $y->copy |
303 | { |
7b29e1e6 |
304 | # twice as slow as $x = [ @$y ], but nec. to retain $x as ref :( |
b3abae2a |
305 | @$x = @$y; return $x; |
306 | } |
0716bf9b |
307 | |
308 | # for each in Y, add Y to X and carry. If after that, something is left in |
309 | # X, foreach in X add carry to X and then return X, carry |
091c87b1 |
310 | # Trades one "$j++" for having to shift arrays |
0716bf9b |
311 | my $i; my $car = 0; my $j = 0; |
312 | for $i (@$y) |
313 | { |
e745a66c |
314 | $x->[$j] -= $BASE if $car = (($x->[$j] += $i + $car) >= $BASE) ? 1 : 0; |
0716bf9b |
315 | $j++; |
316 | } |
317 | while ($car != 0) |
318 | { |
319 | $x->[$j] -= $BASE if $car = (($x->[$j] += $car) >= $BASE) ? 1 : 0; $j++; |
320 | } |
61f5c3f5 |
321 | $x; |
e745a66c |
322 | } |
323 | |
324 | sub _inc |
325 | { |
326 | # (ref to int_num_array, ref to int_num_array) |
091c87b1 |
327 | # Add 1 to $x, modify $x in place |
e745a66c |
328 | my ($c,$x) = @_; |
329 | |
330 | for my $i (@$x) |
331 | { |
332 | return $x if (($i += 1) < $BASE); # early out |
61f5c3f5 |
333 | $i = 0; # overflow, next |
e745a66c |
334 | } |
ae161977 |
335 | push @$x,1 if (($x->[-1] || 0) == 0); # last overflowed, so extend |
61f5c3f5 |
336 | $x; |
e745a66c |
337 | } |
338 | |
339 | sub _dec |
340 | { |
341 | # (ref to int_num_array, ref to int_num_array) |
091c87b1 |
342 | # Sub 1 from $x, modify $x in place |
e745a66c |
343 | my ($c,$x) = @_; |
344 | |
2ebb273f |
345 | my $MAX = $BASE-1; # since MAX_VAL based on BASE |
e745a66c |
346 | for my $i (@$x) |
347 | { |
348 | last if (($i -= 1) >= 0); # early out |
091c87b1 |
349 | $i = $MAX; # underflow, next |
e745a66c |
350 | } |
091c87b1 |
351 | pop @$x if $x->[-1] == 0 && @$x > 1; # last underflowed (but leave 0) |
61f5c3f5 |
352 | $x; |
0716bf9b |
353 | } |
354 | |
355 | sub _sub |
356 | { |
9393ace2 |
357 | # (ref to int_num_array, ref to int_num_array, swap) |
574bacfe |
358 | # subtract base 1eX numbers -- stolen from Knuth Vol 2 pg 232, $x > $y |
56b9c951 |
359 | # subtract Y from X by modifying x in place |
574bacfe |
360 | my ($c,$sx,$sy,$s) = @_; |
0716bf9b |
361 | |
362 | my $car = 0; my $i; my $j = 0; |
363 | if (!$s) |
364 | { |
0716bf9b |
365 | for $i (@$sx) |
366 | { |
367 | last unless defined $sy->[$j] || $car; |
0716bf9b |
368 | $i += $BASE if $car = (($i -= ($sy->[$j] || 0) + $car) < 0); $j++; |
0716bf9b |
369 | } |
370 | # might leave leading zeros, so fix that |
394e6ffb |
371 | return __strip_zeros($sx); |
0716bf9b |
372 | } |
394e6ffb |
373 | for $i (@$sx) |
0716bf9b |
374 | { |
07d34614 |
375 | # we can't do an early out if $x is < than $y, since we |
56b9c951 |
376 | # need to copy the high chunks from $y. Found by Bob Mathews. |
377 | #last unless defined $sy->[$j] || $car; |
394e6ffb |
378 | $sy->[$j] += $BASE |
379 | if $car = (($sy->[$j] = $i-($sy->[$j]||0) - $car) < 0); |
380 | $j++; |
0716bf9b |
381 | } |
394e6ffb |
382 | # might leave leading zeros, so fix that |
383 | __strip_zeros($sy); |
0716bf9b |
384 | } |
385 | |
ee15d750 |
386 | sub _mul_use_mul |
0716bf9b |
387 | { |
9393ace2 |
388 | # (ref to int_num_array, ref to int_num_array) |
0716bf9b |
389 | # multiply two numbers in internal representation |
b22b3e31 |
390 | # modifies first arg, second need not be different from first |
574bacfe |
391 | my ($c,$xv,$yv) = @_; |
dccbb853 |
392 | |
990fb837 |
393 | if (@$yv == 1) |
b3abae2a |
394 | { |
990fb837 |
395 | # shortcut for two very short numbers (improved by Nathan Zook) |
396 | # works also if xv and yv are the same reference, and handles also $x == 0 |
397 | if (@$xv == 1) |
398 | { |
2ebb273f |
399 | if (($xv->[0] *= $yv->[0]) >= $BASE) |
990fb837 |
400 | { |
2ebb273f |
401 | $xv->[0] = $xv->[0] - ($xv->[1] = int($xv->[0] * $RBASE)) * $BASE; |
990fb837 |
402 | }; |
403 | return $xv; |
404 | } |
405 | # $x * 0 => 0 |
406 | if ($yv->[0] == 0) |
407 | { |
408 | @$xv = (0); |
409 | return $xv; |
410 | } |
411 | # multiply a large number a by a single element one, so speed up |
412 | my $y = $yv->[0]; my $car = 0; |
413 | foreach my $i (@$xv) |
414 | { |
2ebb273f |
415 | $i = $i * $y + $car; $car = int($i * $RBASE); $i -= $car * $BASE; |
990fb837 |
416 | } |
417 | push @$xv, $car if $car != 0; |
b3abae2a |
418 | return $xv; |
419 | } |
990fb837 |
420 | # shortcut for result $x == 0 => result = 0 |
421 | return $xv if ( ((@$xv == 1) && ($xv->[0] == 0)) ); |
b3abae2a |
422 | |
0716bf9b |
423 | # since multiplying $x with $x fails, make copy in this case |
d614cd8b |
424 | $yv = [@$xv] if $xv == $yv; # same references? |
9393ace2 |
425 | |
61f5c3f5 |
426 | my @prod = (); my ($prod,$car,$cty,$xi,$yi); |
427 | |
0716bf9b |
428 | for $xi (@$xv) |
429 | { |
430 | $car = 0; $cty = 0; |
574bacfe |
431 | |
432 | # slow variant |
433 | # for $yi (@$yv) |
434 | # { |
435 | # $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
436 | # $prod[$cty++] = |
2ebb273f |
437 | # $prod - ($car = int($prod * RBASE)) * $BASE; # see USE_MUL |
574bacfe |
438 | # } |
439 | # $prod[$cty] += $car if $car; # need really to check for 0? |
440 | # $xi = shift @prod; |
441 | |
442 | # faster variant |
443 | # looping through this if $xi == 0 is silly - so optimize it away! |
444 | $xi = (shift @prod || 0), next if $xi == 0; |
0716bf9b |
445 | for $yi (@$yv) |
446 | { |
447 | $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
574bacfe |
448 | ## this is actually a tad slower |
449 | ## $prod = $prod[$cty]; $prod += ($car + $xi * $yi); # no ||0 here |
0716bf9b |
450 | $prod[$cty++] = |
2ebb273f |
451 | $prod - ($car = int($prod * $RBASE)) * $BASE; # see USE_MUL |
0716bf9b |
452 | } |
453 | $prod[$cty] += $car if $car; # need really to check for 0? |
027dc388 |
454 | $xi = shift @prod || 0; # || 0 makes v5.005_3 happy |
0716bf9b |
455 | } |
0716bf9b |
456 | push @$xv, @prod; |
50109ad0 |
457 | # can't have leading zeros |
458 | # __strip_zeros($xv); |
61f5c3f5 |
459 | $xv; |
0716bf9b |
460 | } |
461 | |
2ebb273f |
462 | sub _mul_use_div_64 |
463 | { |
464 | # (ref to int_num_array, ref to int_num_array) |
465 | # multiply two numbers in internal representation |
466 | # modifies first arg, second need not be different from first |
467 | # works for 64 bit integer with "use integer" |
468 | my ($c,$xv,$yv) = @_; |
469 | |
470 | use integer; |
471 | if (@$yv == 1) |
472 | { |
473 | # shortcut for two small numbers, also handles $x == 0 |
474 | if (@$xv == 1) |
475 | { |
476 | # shortcut for two very short numbers (improved by Nathan Zook) |
477 | # works also if xv and yv are the same reference, and handles also $x == 0 |
478 | if (($xv->[0] *= $yv->[0]) >= $BASE) |
479 | { |
480 | $xv->[0] = |
481 | $xv->[0] - ($xv->[1] = $xv->[0] / $BASE) * $BASE; |
482 | }; |
483 | return $xv; |
484 | } |
485 | # $x * 0 => 0 |
486 | if ($yv->[0] == 0) |
487 | { |
488 | @$xv = (0); |
489 | return $xv; |
490 | } |
491 | # multiply a large number a by a single element one, so speed up |
492 | my $y = $yv->[0]; my $car = 0; |
493 | foreach my $i (@$xv) |
494 | { |
495 | #$i = $i * $y + $car; $car = $i / $BASE; $i -= $car * $BASE; |
496 | $i = $i * $y + $car; $i -= ($car = $i / $BASE) * $BASE; |
497 | } |
498 | push @$xv, $car if $car != 0; |
499 | return $xv; |
500 | } |
501 | # shortcut for result $x == 0 => result = 0 |
502 | return $xv if ( ((@$xv == 1) && ($xv->[0] == 0)) ); |
503 | |
504 | # since multiplying $x with $x fails, make copy in this case |
505 | $yv = [@$xv] if $xv == $yv; # same references? |
506 | |
507 | my @prod = (); my ($prod,$car,$cty,$xi,$yi); |
508 | for $xi (@$xv) |
509 | { |
510 | $car = 0; $cty = 0; |
511 | # looping through this if $xi == 0 is silly - so optimize it away! |
512 | $xi = (shift @prod || 0), next if $xi == 0; |
513 | for $yi (@$yv) |
514 | { |
515 | $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
516 | $prod[$cty++] = $prod - ($car = $prod / $BASE) * $BASE; |
517 | } |
518 | $prod[$cty] += $car if $car; # need really to check for 0? |
519 | $xi = shift @prod || 0; # || 0 makes v5.005_3 happy |
520 | } |
521 | push @$xv, @prod; |
522 | $xv; |
523 | } |
524 | |
ee15d750 |
525 | sub _mul_use_div |
526 | { |
9393ace2 |
527 | # (ref to int_num_array, ref to int_num_array) |
ee15d750 |
528 | # multiply two numbers in internal representation |
529 | # modifies first arg, second need not be different from first |
530 | my ($c,$xv,$yv) = @_; |
50109ad0 |
531 | |
990fb837 |
532 | if (@$yv == 1) |
b3abae2a |
533 | { |
990fb837 |
534 | # shortcut for two small numbers, also handles $x == 0 |
535 | if (@$xv == 1) |
536 | { |
537 | # shortcut for two very short numbers (improved by Nathan Zook) |
538 | # works also if xv and yv are the same reference, and handles also $x == 0 |
2ebb273f |
539 | if (($xv->[0] *= $yv->[0]) >= $BASE) |
990fb837 |
540 | { |
541 | $xv->[0] = |
2ebb273f |
542 | $xv->[0] - ($xv->[1] = int($xv->[0] / $BASE)) * $BASE; |
990fb837 |
543 | }; |
544 | return $xv; |
545 | } |
546 | # $x * 0 => 0 |
547 | if ($yv->[0] == 0) |
548 | { |
549 | @$xv = (0); |
550 | return $xv; |
551 | } |
552 | # multiply a large number a by a single element one, so speed up |
553 | my $y = $yv->[0]; my $car = 0; |
554 | foreach my $i (@$xv) |
555 | { |
2ebb273f |
556 | $i = $i * $y + $car; $car = int($i / $BASE); $i -= $car * $BASE; |
557 | # This (together with use integer;) does not work on 32-bit Perls |
558 | #$i = $i * $y + $car; $i -= ($car = $i / $BASE) * $BASE; |
990fb837 |
559 | } |
560 | push @$xv, $car if $car != 0; |
b3abae2a |
561 | return $xv; |
562 | } |
990fb837 |
563 | # shortcut for result $x == 0 => result = 0 |
564 | return $xv if ( ((@$xv == 1) && ($xv->[0] == 0)) ); |
b3abae2a |
565 | |
ee15d750 |
566 | # since multiplying $x with $x fails, make copy in this case |
d614cd8b |
567 | $yv = [@$xv] if $xv == $yv; # same references? |
9393ace2 |
568 | |
61f5c3f5 |
569 | my @prod = (); my ($prod,$car,$cty,$xi,$yi); |
ee15d750 |
570 | for $xi (@$xv) |
571 | { |
572 | $car = 0; $cty = 0; |
573 | # looping through this if $xi == 0 is silly - so optimize it away! |
574 | $xi = (shift @prod || 0), next if $xi == 0; |
575 | for $yi (@$yv) |
576 | { |
577 | $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
2ebb273f |
578 | $prod[$cty++] = $prod - ($car = int($prod / $BASE)) * $BASE; |
ee15d750 |
579 | } |
580 | $prod[$cty] += $car if $car; # need really to check for 0? |
027dc388 |
581 | $xi = shift @prod || 0; # || 0 makes v5.005_3 happy |
ee15d750 |
582 | } |
583 | push @$xv, @prod; |
50109ad0 |
584 | # can't have leading zeros |
585 | # __strip_zeros($xv); |
61f5c3f5 |
586 | $xv; |
ee15d750 |
587 | } |
588 | |
589 | sub _div_use_mul |
0716bf9b |
590 | { |
b22b3e31 |
591 | # ref to array, ref to array, modify first array and return remainder if |
0716bf9b |
592 | # in list context |
aef458a0 |
593 | |
594 | # see comments in _div_use_div() for more explanations |
595 | |
574bacfe |
596 | my ($c,$x,$yorg) = @_; |
aef458a0 |
597 | |
598 | # the general div algorithmn here is about O(N*N) and thus quite slow, so |
599 | # we first check for some special cases and use shortcuts to handle them. |
0716bf9b |
600 | |
aef458a0 |
601 | # This works, because we store the numbers in a chunked format where each |
602 | # element contains 5..7 digits (depending on system). |
603 | |
604 | # if both numbers have only one element: |
61f5c3f5 |
605 | if (@$x == 1 && @$yorg == 1) |
606 | { |
13a12e00 |
607 | # shortcut, $yorg and $x are two small numbers |
61f5c3f5 |
608 | if (wantarray) |
609 | { |
610 | my $r = [ $x->[0] % $yorg->[0] ]; |
611 | $x->[0] = int($x->[0] / $yorg->[0]); |
612 | return ($x,$r); |
613 | } |
614 | else |
615 | { |
616 | $x->[0] = int($x->[0] / $yorg->[0]); |
617 | return $x; |
618 | } |
619 | } |
aef458a0 |
620 | |
621 | # if x has more than one, but y has only one element: |
28df3e88 |
622 | if (@$yorg == 1) |
623 | { |
624 | my $rem; |
625 | $rem = _mod($c,[ @$x ],$yorg) if wantarray; |
13a12e00 |
626 | |
28df3e88 |
627 | # shortcut, $y is < $BASE |
628 | my $j = scalar @$x; my $r = 0; |
629 | my $y = $yorg->[0]; my $b; |
630 | while ($j-- > 0) |
631 | { |
2ebb273f |
632 | $b = $r * $BASE + $x->[$j]; |
28df3e88 |
633 | $x->[$j] = int($b/$y); |
634 | $r = $b % $y; |
635 | } |
636 | pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero |
637 | return ($x,$rem) if wantarray; |
638 | return $x; |
639 | } |
0716bf9b |
640 | |
aef458a0 |
641 | # now x and y have more than one element |
642 | |
643 | # check whether y has more elements than x, if yet, the result will be 0 |
644 | if (@$yorg > @$x) |
645 | { |
646 | my $rem; |
647 | $rem = [@$x] if wantarray; # make copy |
648 | splice (@$x,1); # keep ref to original array |
649 | $x->[0] = 0; # set to 0 |
650 | return ($x,$rem) if wantarray; # including remainder? |
651 | return $x; # only x, which is [0] now |
652 | } |
653 | # check whether the numbers have the same number of elements, in that case |
654 | # the result will fit into one element and can be computed efficiently |
655 | if (@$yorg == @$x) |
656 | { |
657 | my $rem; |
658 | # if $yorg has more digits than $x (it's leading element is longer than |
659 | # the one from $x), the result will also be 0: |
660 | if (length(int($yorg->[-1])) > length(int($x->[-1]))) |
661 | { |
662 | $rem = [@$x] if wantarray; # make copy |
663 | splice (@$x,1); # keep ref to org array |
664 | $x->[0] = 0; # set to 0 |
665 | return ($x,$rem) if wantarray; # including remainder? |
666 | return $x; |
667 | } |
668 | # now calculate $x / $yorg |
669 | if (length(int($yorg->[-1])) == length(int($x->[-1]))) |
670 | { |
b68b7ab1 |
671 | # same length, so make full compare |
672 | |
aef458a0 |
673 | my $a = 0; my $j = scalar @$x - 1; |
674 | # manual way (abort if unequal, good for early ne) |
675 | while ($j >= 0) |
676 | { |
677 | last if ($a = $x->[$j] - $yorg->[$j]); $j--; |
678 | } |
679 | # $a contains the result of the compare between X and Y |
b68b7ab1 |
680 | # a < 0: x < y, a == 0: x == y, a > 0: x > y |
aef458a0 |
681 | if ($a <= 0) |
682 | { |
b68b7ab1 |
683 | $rem = [ 0 ]; # a = 0 => x == y => rem 0 |
684 | $rem = [@$x] if $a != 0; # a < 0 => x < y => rem = x |
685 | splice(@$x,1); # keep single element |
686 | $x->[0] = 0; # if $a < 0 |
687 | $x->[0] = 1 if $a == 0; # $x == $y |
7596a890 |
688 | return ($x,$rem) if wantarray; |
689 | return $x; |
aef458a0 |
690 | } |
b68b7ab1 |
691 | # $x >= $y, so proceed normally |
aef458a0 |
692 | } |
693 | } |
694 | |
695 | # all other cases: |
696 | |
d614cd8b |
697 | my $y = [ @$yorg ]; # always make copy to preserve |
61f5c3f5 |
698 | |
699 | my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0); |
700 | |
701 | $car = $bar = $prd = 0; |
2ebb273f |
702 | if (($dd = int($BASE/($y->[-1]+1))) != 1) |
0716bf9b |
703 | { |
704 | for $xi (@$x) |
705 | { |
706 | $xi = $xi * $dd + $car; |
2ebb273f |
707 | $xi -= ($car = int($xi * $RBASE)) * $BASE; # see USE_MUL |
0716bf9b |
708 | } |
709 | push(@$x, $car); $car = 0; |
710 | for $yi (@$y) |
711 | { |
712 | $yi = $yi * $dd + $car; |
2ebb273f |
713 | $yi -= ($car = int($yi * $RBASE)) * $BASE; # see USE_MUL |
0716bf9b |
714 | } |
715 | } |
716 | else |
717 | { |
718 | push(@$x, 0); |
719 | } |
720 | @q = (); ($v2,$v1) = @$y[-2,-1]; |
721 | $v2 = 0 unless $v2; |
722 | while ($#$x > $#$y) |
723 | { |
724 | ($u2,$u1,$u0) = @$x[-3..-1]; |
725 | $u2 = 0 unless $u2; |
726 | #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" |
727 | # if $v1 == 0; |
2ebb273f |
728 | $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$BASE+$u1)/$v1)); |
729 | --$q while ($v2*$q > ($u0*$BASE+$u1-$q*$v1)*$BASE+$u2); |
0716bf9b |
730 | if ($q) |
731 | { |
732 | ($car, $bar) = (0,0); |
733 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
734 | { |
735 | $prd = $q * $y->[$yi] + $car; |
2ebb273f |
736 | $prd -= ($car = int($prd * $RBASE)) * $BASE; # see USE_MUL |
737 | $x->[$xi] += $BASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0)); |
0716bf9b |
738 | } |
739 | if ($x->[-1] < $car + $bar) |
740 | { |
741 | $car = 0; --$q; |
742 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
743 | { |
2ebb273f |
744 | $x->[$xi] -= $BASE |
745 | if ($car = (($x->[$xi] += $y->[$yi] + $car) >= $BASE)); |
0716bf9b |
746 | } |
747 | } |
748 | } |
aef458a0 |
749 | pop(@$x); |
750 | unshift(@q, $q); |
0716bf9b |
751 | } |
752 | if (wantarray) |
753 | { |
754 | @d = (); |
755 | if ($dd != 1) |
756 | { |
757 | $car = 0; |
758 | for $xi (reverse @$x) |
759 | { |
2ebb273f |
760 | $prd = $car * $BASE + $xi; |
0716bf9b |
761 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; # see USE_MUL |
762 | unshift(@d, $tmp); |
763 | } |
764 | } |
765 | else |
766 | { |
767 | @d = @$x; |
768 | } |
769 | @$x = @q; |
61f5c3f5 |
770 | my $d = \@d; |
990fb837 |
771 | __strip_zeros($x); |
772 | __strip_zeros($d); |
61f5c3f5 |
773 | return ($x,$d); |
0716bf9b |
774 | } |
775 | @$x = @q; |
990fb837 |
776 | __strip_zeros($x); |
61f5c3f5 |
777 | $x; |
0716bf9b |
778 | } |
779 | |
2ebb273f |
780 | sub _div_use_div_64 |
781 | { |
782 | # ref to array, ref to array, modify first array and return remainder if |
783 | # in list context |
784 | # This version works on 64 bit integers |
785 | my ($c,$x,$yorg) = @_; |
786 | |
787 | use integer; |
788 | # the general div algorithmn here is about O(N*N) and thus quite slow, so |
789 | # we first check for some special cases and use shortcuts to handle them. |
790 | |
791 | # This works, because we store the numbers in a chunked format where each |
792 | # element contains 5..7 digits (depending on system). |
793 | |
794 | # if both numbers have only one element: |
795 | if (@$x == 1 && @$yorg == 1) |
796 | { |
797 | # shortcut, $yorg and $x are two small numbers |
798 | if (wantarray) |
799 | { |
800 | my $r = [ $x->[0] % $yorg->[0] ]; |
801 | $x->[0] = int($x->[0] / $yorg->[0]); |
802 | return ($x,$r); |
803 | } |
804 | else |
805 | { |
806 | $x->[0] = int($x->[0] / $yorg->[0]); |
807 | return $x; |
808 | } |
809 | } |
810 | # if x has more than one, but y has only one element: |
811 | if (@$yorg == 1) |
812 | { |
813 | my $rem; |
814 | $rem = _mod($c,[ @$x ],$yorg) if wantarray; |
815 | |
816 | # shortcut, $y is < $BASE |
817 | my $j = scalar @$x; my $r = 0; |
818 | my $y = $yorg->[0]; my $b; |
819 | while ($j-- > 0) |
820 | { |
821 | $b = $r * $BASE + $x->[$j]; |
822 | $x->[$j] = int($b/$y); |
823 | $r = $b % $y; |
824 | } |
825 | pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero |
826 | return ($x,$rem) if wantarray; |
827 | return $x; |
828 | } |
829 | # now x and y have more than one element |
830 | |
831 | # check whether y has more elements than x, if yet, the result will be 0 |
832 | if (@$yorg > @$x) |
833 | { |
834 | my $rem; |
835 | $rem = [@$x] if wantarray; # make copy |
836 | splice (@$x,1); # keep ref to original array |
837 | $x->[0] = 0; # set to 0 |
838 | return ($x,$rem) if wantarray; # including remainder? |
839 | return $x; # only x, which is [0] now |
840 | } |
841 | # check whether the numbers have the same number of elements, in that case |
842 | # the result will fit into one element and can be computed efficiently |
843 | if (@$yorg == @$x) |
844 | { |
845 | my $rem; |
846 | # if $yorg has more digits than $x (it's leading element is longer than |
847 | # the one from $x), the result will also be 0: |
848 | if (length(int($yorg->[-1])) > length(int($x->[-1]))) |
849 | { |
850 | $rem = [@$x] if wantarray; # make copy |
851 | splice (@$x,1); # keep ref to org array |
852 | $x->[0] = 0; # set to 0 |
853 | return ($x,$rem) if wantarray; # including remainder? |
854 | return $x; |
855 | } |
856 | # now calculate $x / $yorg |
857 | |
858 | if (length(int($yorg->[-1])) == length(int($x->[-1]))) |
859 | { |
860 | # same length, so make full compare |
861 | |
862 | my $a = 0; my $j = scalar @$x - 1; |
863 | # manual way (abort if unequal, good for early ne) |
864 | while ($j >= 0) |
865 | { |
866 | last if ($a = $x->[$j] - $yorg->[$j]); $j--; |
867 | } |
868 | # $a contains the result of the compare between X and Y |
869 | # a < 0: x < y, a == 0: x == y, a > 0: x > y |
870 | if ($a <= 0) |
871 | { |
872 | $rem = [ 0 ]; # a = 0 => x == y => rem 0 |
873 | $rem = [@$x] if $a != 0; # a < 0 => x < y => rem = x |
874 | splice(@$x,1); # keep single element |
875 | $x->[0] = 0; # if $a < 0 |
876 | $x->[0] = 1 if $a == 0; # $x == $y |
877 | return ($x,$rem) if wantarray; # including remainder? |
878 | return $x; |
879 | } |
880 | # $x >= $y, so proceed normally |
881 | |
882 | } |
883 | } |
884 | |
885 | # all other cases: |
886 | |
887 | my $y = [ @$yorg ]; # always make copy to preserve |
888 | |
889 | my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0); |
890 | |
891 | $car = $bar = $prd = 0; |
892 | if (($dd = int($BASE/($y->[-1]+1))) != 1) |
893 | { |
894 | for $xi (@$x) |
895 | { |
896 | $xi = $xi * $dd + $car; |
897 | $xi -= ($car = int($xi / $BASE)) * $BASE; |
898 | } |
899 | push(@$x, $car); $car = 0; |
900 | for $yi (@$y) |
901 | { |
902 | $yi = $yi * $dd + $car; |
903 | $yi -= ($car = int($yi / $BASE)) * $BASE; |
904 | } |
905 | } |
906 | else |
907 | { |
908 | push(@$x, 0); |
909 | } |
910 | |
911 | # @q will accumulate the final result, $q contains the current computed |
912 | # part of the final result |
913 | |
914 | @q = (); ($v2,$v1) = @$y[-2,-1]; |
915 | $v2 = 0 unless $v2; |
916 | while ($#$x > $#$y) |
917 | { |
918 | ($u2,$u1,$u0) = @$x[-3..-1]; |
919 | $u2 = 0 unless $u2; |
920 | #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" |
921 | # if $v1 == 0; |
922 | $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$BASE+$u1)/$v1)); |
923 | --$q while ($v2*$q > ($u0*$BASE+$u1-$q*$v1)*$BASE+$u2); |
924 | if ($q) |
925 | { |
926 | ($car, $bar) = (0,0); |
927 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
928 | { |
929 | $prd = $q * $y->[$yi] + $car; |
930 | $prd -= ($car = int($prd / $BASE)) * $BASE; |
931 | $x->[$xi] += $BASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0)); |
932 | } |
933 | if ($x->[-1] < $car + $bar) |
934 | { |
935 | $car = 0; --$q; |
936 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
937 | { |
938 | $x->[$xi] -= $BASE |
939 | if ($car = (($x->[$xi] += $y->[$yi] + $car) >= $BASE)); |
940 | } |
941 | } |
942 | } |
943 | pop(@$x); unshift(@q, $q); |
944 | } |
945 | if (wantarray) |
946 | { |
947 | @d = (); |
948 | if ($dd != 1) |
949 | { |
950 | $car = 0; |
951 | for $xi (reverse @$x) |
952 | { |
953 | $prd = $car * $BASE + $xi; |
954 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; |
955 | unshift(@d, $tmp); |
956 | } |
957 | } |
958 | else |
959 | { |
960 | @d = @$x; |
961 | } |
962 | @$x = @q; |
963 | my $d = \@d; |
964 | __strip_zeros($x); |
965 | __strip_zeros($d); |
966 | return ($x,$d); |
967 | } |
968 | @$x = @q; |
969 | __strip_zeros($x); |
970 | $x; |
971 | } |
972 | |
ee15d750 |
973 | sub _div_use_div |
974 | { |
975 | # ref to array, ref to array, modify first array and return remainder if |
976 | # in list context |
ee15d750 |
977 | my ($c,$x,$yorg) = @_; |
ee15d750 |
978 | |
990fb837 |
979 | # the general div algorithmn here is about O(N*N) and thus quite slow, so |
980 | # we first check for some special cases and use shortcuts to handle them. |
981 | |
982 | # This works, because we store the numbers in a chunked format where each |
983 | # element contains 5..7 digits (depending on system). |
984 | |
985 | # if both numbers have only one element: |
61f5c3f5 |
986 | if (@$x == 1 && @$yorg == 1) |
987 | { |
13a12e00 |
988 | # shortcut, $yorg and $x are two small numbers |
61f5c3f5 |
989 | if (wantarray) |
990 | { |
991 | my $r = [ $x->[0] % $yorg->[0] ]; |
992 | $x->[0] = int($x->[0] / $yorg->[0]); |
993 | return ($x,$r); |
994 | } |
995 | else |
996 | { |
997 | $x->[0] = int($x->[0] / $yorg->[0]); |
998 | return $x; |
999 | } |
1000 | } |
990fb837 |
1001 | # if x has more than one, but y has only one element: |
28df3e88 |
1002 | if (@$yorg == 1) |
1003 | { |
1004 | my $rem; |
1005 | $rem = _mod($c,[ @$x ],$yorg) if wantarray; |
1006 | |
1007 | # shortcut, $y is < $BASE |
1008 | my $j = scalar @$x; my $r = 0; |
1009 | my $y = $yorg->[0]; my $b; |
1010 | while ($j-- > 0) |
1011 | { |
2ebb273f |
1012 | $b = $r * $BASE + $x->[$j]; |
28df3e88 |
1013 | $x->[$j] = int($b/$y); |
1014 | $r = $b % $y; |
1015 | } |
1016 | pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero |
1017 | return ($x,$rem) if wantarray; |
1018 | return $x; |
1019 | } |
990fb837 |
1020 | # now x and y have more than one element |
ee15d750 |
1021 | |
990fb837 |
1022 | # check whether y has more elements than x, if yet, the result will be 0 |
1023 | if (@$yorg > @$x) |
61f5c3f5 |
1024 | { |
990fb837 |
1025 | my $rem; |
1026 | $rem = [@$x] if wantarray; # make copy |
1027 | splice (@$x,1); # keep ref to original array |
1028 | $x->[0] = 0; # set to 0 |
1029 | return ($x,$rem) if wantarray; # including remainder? |
aef458a0 |
1030 | return $x; # only x, which is [0] now |
61f5c3f5 |
1031 | } |
990fb837 |
1032 | # check whether the numbers have the same number of elements, in that case |
1033 | # the result will fit into one element and can be computed efficiently |
1034 | if (@$yorg == @$x) |
1035 | { |
1036 | my $rem; |
1037 | # if $yorg has more digits than $x (it's leading element is longer than |
1038 | # the one from $x), the result will also be 0: |
1039 | if (length(int($yorg->[-1])) > length(int($x->[-1]))) |
1040 | { |
1041 | $rem = [@$x] if wantarray; # make copy |
1042 | splice (@$x,1); # keep ref to org array |
1043 | $x->[0] = 0; # set to 0 |
1044 | return ($x,$rem) if wantarray; # including remainder? |
1045 | return $x; |
1046 | } |
1047 | # now calculate $x / $yorg |
091c87b1 |
1048 | |
990fb837 |
1049 | if (length(int($yorg->[-1])) == length(int($x->[-1]))) |
1050 | { |
b68b7ab1 |
1051 | # same length, so make full compare |
1052 | |
990fb837 |
1053 | my $a = 0; my $j = scalar @$x - 1; |
1054 | # manual way (abort if unequal, good for early ne) |
1055 | while ($j >= 0) |
1056 | { |
1057 | last if ($a = $x->[$j] - $yorg->[$j]); $j--; |
1058 | } |
aef458a0 |
1059 | # $a contains the result of the compare between X and Y |
b68b7ab1 |
1060 | # a < 0: x < y, a == 0: x == y, a > 0: x > y |
990fb837 |
1061 | if ($a <= 0) |
1062 | { |
b68b7ab1 |
1063 | $rem = [ 0 ]; # a = 0 => x == y => rem 0 |
1064 | $rem = [@$x] if $a != 0; # a < 0 => x < y => rem = x |
aef458a0 |
1065 | splice(@$x,1); # keep single element |
990fb837 |
1066 | $x->[0] = 0; # if $a < 0 |
b68b7ab1 |
1067 | $x->[0] = 1 if $a == 0; # $x == $y |
7596a890 |
1068 | return ($x,$rem) if wantarray; # including remainder? |
1069 | return $x; |
990fb837 |
1070 | } |
aef458a0 |
1071 | # $x >= $y, so proceed normally |
b68b7ab1 |
1072 | |
990fb837 |
1073 | } |
990fb837 |
1074 | } |
1075 | |
1076 | # all other cases: |
1077 | |
1078 | my $y = [ @$yorg ]; # always make copy to preserve |
61f5c3f5 |
1079 | |
1080 | my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0); |
1081 | |
1082 | $car = $bar = $prd = 0; |
2ebb273f |
1083 | if (($dd = int($BASE/($y->[-1]+1))) != 1) |
ee15d750 |
1084 | { |
1085 | for $xi (@$x) |
1086 | { |
1087 | $xi = $xi * $dd + $car; |
2ebb273f |
1088 | $xi -= ($car = int($xi / $BASE)) * $BASE; |
ee15d750 |
1089 | } |
1090 | push(@$x, $car); $car = 0; |
1091 | for $yi (@$y) |
1092 | { |
1093 | $yi = $yi * $dd + $car; |
2ebb273f |
1094 | $yi -= ($car = int($yi / $BASE)) * $BASE; |
ee15d750 |
1095 | } |
1096 | } |
1097 | else |
1098 | { |
1099 | push(@$x, 0); |
1100 | } |
aef458a0 |
1101 | |
1102 | # @q will accumulate the final result, $q contains the current computed |
1103 | # part of the final result |
1104 | |
ee15d750 |
1105 | @q = (); ($v2,$v1) = @$y[-2,-1]; |
1106 | $v2 = 0 unless $v2; |
1107 | while ($#$x > $#$y) |
1108 | { |
1109 | ($u2,$u1,$u0) = @$x[-3..-1]; |
1110 | $u2 = 0 unless $u2; |
1111 | #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" |
1112 | # if $v1 == 0; |
2ebb273f |
1113 | $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$BASE+$u1)/$v1)); |
1114 | --$q while ($v2*$q > ($u0*$BASE+$u1-$q*$v1)*$BASE+$u2); |
ee15d750 |
1115 | if ($q) |
1116 | { |
1117 | ($car, $bar) = (0,0); |
1118 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
1119 | { |
1120 | $prd = $q * $y->[$yi] + $car; |
2ebb273f |
1121 | $prd -= ($car = int($prd / $BASE)) * $BASE; |
1122 | $x->[$xi] += $BASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0)); |
ee15d750 |
1123 | } |
1124 | if ($x->[-1] < $car + $bar) |
1125 | { |
1126 | $car = 0; --$q; |
1127 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
1128 | { |
2ebb273f |
1129 | $x->[$xi] -= $BASE |
1130 | if ($car = (($x->[$xi] += $y->[$yi] + $car) >= $BASE)); |
ee15d750 |
1131 | } |
1132 | } |
1133 | } |
61f5c3f5 |
1134 | pop(@$x); unshift(@q, $q); |
ee15d750 |
1135 | } |
1136 | if (wantarray) |
1137 | { |
1138 | @d = (); |
1139 | if ($dd != 1) |
1140 | { |
1141 | $car = 0; |
1142 | for $xi (reverse @$x) |
1143 | { |
2ebb273f |
1144 | $prd = $car * $BASE + $xi; |
ee15d750 |
1145 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; |
1146 | unshift(@d, $tmp); |
1147 | } |
1148 | } |
1149 | else |
1150 | { |
1151 | @d = @$x; |
1152 | } |
1153 | @$x = @q; |
61f5c3f5 |
1154 | my $d = \@d; |
990fb837 |
1155 | __strip_zeros($x); |
1156 | __strip_zeros($d); |
61f5c3f5 |
1157 | return ($x,$d); |
ee15d750 |
1158 | } |
1159 | @$x = @q; |
990fb837 |
1160 | __strip_zeros($x); |
61f5c3f5 |
1161 | $x; |
ee15d750 |
1162 | } |
1163 | |
394e6ffb |
1164 | ############################################################################## |
1165 | # testing |
1166 | |
1167 | sub _acmp |
1168 | { |
1169 | # internal absolute post-normalized compare (ignore signs) |
1170 | # ref to array, ref to array, return <0, 0, >0 |
1171 | # arrays must have at least one entry; this is not checked for |
394e6ffb |
1172 | my ($c,$cx,$cy) = @_; |
091c87b1 |
1173 | |
1174 | # shortcut for short numbers |
1175 | return (($cx->[0] <=> $cy->[0]) <=> 0) |
1176 | if scalar @$cx == scalar @$cy && scalar @$cx == 1; |
394e6ffb |
1177 | |
f9a08e12 |
1178 | # fast comp based on number of array elements (aka pseudo-length) |
091c87b1 |
1179 | my $lxy = (scalar @$cx - scalar @$cy) |
1180 | # or length of first element if same number of elements (aka difference 0) |
1181 | || |
1182 | # need int() here because sometimes the last element is '00018' vs '18' |
1183 | (length(int($cx->[-1])) - length(int($cy->[-1]))); |
394e6ffb |
1184 | return -1 if $lxy < 0; # already differs, ret |
1185 | return 1 if $lxy > 0; # ditto |
56d9de68 |
1186 | |
394e6ffb |
1187 | # manual way (abort if unequal, good for early ne) |
091c87b1 |
1188 | my $a; my $j = scalar @$cx; |
1189 | while (--$j >= 0) |
9393ace2 |
1190 | { |
091c87b1 |
1191 | last if ($a = $cx->[$j] - $cy->[$j]); |
9393ace2 |
1192 | } |
091c87b1 |
1193 | $a <=> 0; |
394e6ffb |
1194 | } |
1195 | |
1196 | sub _len |
1197 | { |
50109ad0 |
1198 | # compute number of digits in base 10 |
394e6ffb |
1199 | |
1200 | # int() because add/sub sometimes leaves strings (like '00005') instead of |
1201 | # '5' in this place, thus causing length() to report wrong length |
1202 | my $cx = $_[1]; |
1203 | |
56d9de68 |
1204 | (@$cx-1)*$BASE_LEN+length(int($cx->[-1])); |
394e6ffb |
1205 | } |
1206 | |
1207 | sub _digit |
1208 | { |
1209 | # return the nth digit, negative values count backward |
1210 | # zero is rightmost, so _digit(123,0) will give 3 |
1211 | my ($c,$x,$n) = @_; |
1212 | |
1213 | my $len = _len('',$x); |
1214 | |
1215 | $n = $len+$n if $n < 0; # -1 last, -2 second-to-last |
1216 | $n = abs($n); # if negative was too big |
1217 | $len--; $n = $len if $n > $len; # n to big? |
1218 | |
1219 | my $elem = int($n / $BASE_LEN); # which array element |
1220 | my $digit = $n % $BASE_LEN; # which digit in this element |
52edfb59 |
1221 | $elem = '0' x $BASE_LEN . @$x[$elem]; # get element padded with 0's |
56d9de68 |
1222 | substr($elem,-$digit-1,1); |
394e6ffb |
1223 | } |
1224 | |
1225 | sub _zeros |
1226 | { |
1227 | # return amount of trailing zeros in decimal |
1228 | # check each array elem in _m for having 0 at end as long as elem == 0 |
1229 | # Upon finding a elem != 0, stop |
1230 | my $x = $_[1]; |
9b924220 |
1231 | |
1232 | return 0 if scalar @$x == 1 && $x->[0] == 0; |
1233 | |
394e6ffb |
1234 | my $zeros = 0; my $elem; |
1235 | foreach my $e (@$x) |
1236 | { |
1237 | if ($e != 0) |
1238 | { |
1239 | $elem = "$e"; # preserve x |
1240 | $elem =~ s/.*?(0*$)/$1/; # strip anything not zero |
1241 | $zeros *= $BASE_LEN; # elems * 5 |
61f5c3f5 |
1242 | $zeros += length($elem); # count trailing zeros |
394e6ffb |
1243 | last; # early out |
1244 | } |
1245 | $zeros ++; # real else branch: 50% slower! |
1246 | } |
61f5c3f5 |
1247 | $zeros; |
394e6ffb |
1248 | } |
1249 | |
1250 | ############################################################################## |
1251 | # _is_* routines |
1252 | |
1253 | sub _is_zero |
1254 | { |
9b924220 |
1255 | # return true if arg is zero |
1256 | (((scalar @{$_[1]} == 1) && ($_[1]->[0] == 0))) <=> 0; |
394e6ffb |
1257 | } |
1258 | |
1259 | sub _is_even |
1260 | { |
9b924220 |
1261 | # return true if arg is even |
1262 | (!($_[1]->[0] & 1)) <=> 0; |
394e6ffb |
1263 | } |
1264 | |
1265 | sub _is_odd |
1266 | { |
9b924220 |
1267 | # return true if arg is even |
1268 | (($_[1]->[0] & 1)) <=> 0; |
394e6ffb |
1269 | } |
1270 | |
1271 | sub _is_one |
1272 | { |
9b924220 |
1273 | # return true if arg is one |
1274 | (scalar @{$_[1]} == 1) && ($_[1]->[0] == 1) <=> 0; |
1275 | } |
1276 | |
1277 | sub _is_two |
1278 | { |
1279 | # return true if arg is two |
1280 | (scalar @{$_[1]} == 1) && ($_[1]->[0] == 2) <=> 0; |
1281 | } |
61f5c3f5 |
1282 | |
9b924220 |
1283 | sub _is_ten |
1284 | { |
1285 | # return true if arg is ten |
1286 | (scalar @{$_[1]} == 1) && ($_[1]->[0] == 10) <=> 0; |
394e6ffb |
1287 | } |
1288 | |
1289 | sub __strip_zeros |
1290 | { |
1291 | # internal normalization function that strips leading zeros from the array |
1292 | # args: ref to array |
1293 | my $s = shift; |
1294 | |
1295 | my $cnt = scalar @$s; # get count of parts |
1296 | my $i = $cnt-1; |
1297 | push @$s,0 if $i < 0; # div might return empty results, so fix it |
1298 | |
61f5c3f5 |
1299 | return $s if @$s == 1; # early out |
1300 | |
394e6ffb |
1301 | #print "strip: cnt $cnt i $i\n"; |
1302 | # '0', '3', '4', '0', '0', |
1303 | # 0 1 2 3 4 |
1304 | # cnt = 5, i = 4 |
1305 | # i = 4 |
1306 | # i = 3 |
1307 | # => fcnt = cnt - i (5-2 => 3, cnt => 5-1 = 4, throw away from 4th pos) |
1308 | # >= 1: skip first part (this can be zero) |
1309 | while ($i > 0) { last if $s->[$i] != 0; $i--; } |
1310 | $i++; splice @$s,$i if ($i < $cnt); # $i cant be 0 |
1311 | $s; |
1312 | } |
1313 | |
1314 | ############################################################################### |
3a427a11 |
1315 | # check routine to test internal state for corruptions |
394e6ffb |
1316 | |
1317 | sub _check |
1318 | { |
1319 | # used by the test suite |
1320 | my $x = $_[1]; |
1321 | |
1322 | return "$x is not a reference" if !ref($x); |
1323 | |
1324 | # are all parts are valid? |
1325 | my $i = 0; my $j = scalar @$x; my ($e,$try); |
1326 | while ($i < $j) |
1327 | { |
1328 | $e = $x->[$i]; $e = 'undef' unless defined $e; |
1329 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e)"; |
1330 | last if $e !~ /^[+]?[0-9]+$/; |
1331 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (stringify)"; |
1332 | last if "$e" !~ /^[+]?[0-9]+$/; |
1333 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (cat-stringify)"; |
1334 | last if '' . "$e" !~ /^[+]?[0-9]+$/; |
1335 | $try = ' < 0 || >= $BASE; '."($x, $e)"; |
1336 | last if $e <0 || $e >= $BASE; |
1337 | # this test is disabled, since new/bnorm and certain ops (like early out |
1338 | # in add/sub) are allowed/expected to leave '00000' in some elements |
1339 | #$try = '=~ /^00+/; '."($x, $e)"; |
1340 | #last if $e =~ /^00+/; |
1341 | $i++; |
1342 | } |
1343 | return "Illegal part '$e' at pos $i (tested: $try)" if $i < $j; |
3a427a11 |
1344 | 0; |
394e6ffb |
1345 | } |
1346 | |
1347 | |
1348 | ############################################################################### |
394e6ffb |
1349 | |
dccbb853 |
1350 | sub _mod |
1351 | { |
1352 | # if possible, use mod shortcut |
1353 | my ($c,$x,$yo) = @_; |
1354 | |
1355 | # slow way since $y to big |
1356 | if (scalar @$yo > 1) |
1357 | { |
1358 | my ($xo,$rem) = _div($c,$x,$yo); |
1359 | return $rem; |
1360 | } |
aef458a0 |
1361 | |
dccbb853 |
1362 | my $y = $yo->[0]; |
027dc388 |
1363 | # both are single element arrays |
dccbb853 |
1364 | if (scalar @$x == 1) |
1365 | { |
1366 | $x->[0] %= $y; |
1367 | return $x; |
1368 | } |
1369 | |
aef458a0 |
1370 | # @y is a single element, but @x has more than one element |
dccbb853 |
1371 | my $b = $BASE % $y; |
1372 | if ($b == 0) |
1373 | { |
1374 | # when BASE % Y == 0 then (B * BASE) % Y == 0 |
1375 | # (B * BASE) % $y + A % Y => A % Y |
1376 | # so need to consider only last element: O(1) |
1377 | $x->[0] %= $y; |
1378 | } |
027dc388 |
1379 | elsif ($b == 1) |
1380 | { |
3a427a11 |
1381 | # else need to go through all elements: O(N), but loop is a bit simplified |
027dc388 |
1382 | my $r = 0; |
1383 | foreach (@$x) |
1384 | { |
28df3e88 |
1385 | $r = ($r + $_) % $y; # not much faster, but heh... |
1386 | #$r += $_ % $y; $r %= $y; |
027dc388 |
1387 | } |
1388 | $r = 0 if $r == $y; |
1389 | $x->[0] = $r; |
1390 | } |
dccbb853 |
1391 | else |
1392 | { |
3a427a11 |
1393 | # else need to go through all elements: O(N) |
027dc388 |
1394 | my $r = 0; my $bm = 1; |
1395 | foreach (@$x) |
1396 | { |
28df3e88 |
1397 | $r = ($_ * $bm + $r) % $y; |
1398 | $bm = ($bm * $b) % $y; |
1399 | |
1400 | #$r += ($_ % $y) * $bm; |
1401 | #$bm *= $b; |
1402 | #$bm %= $y; |
1403 | #$r %= $y; |
027dc388 |
1404 | } |
1405 | $r = 0 if $r == $y; |
1406 | $x->[0] = $r; |
dccbb853 |
1407 | } |
091c87b1 |
1408 | splice (@$x,1); # keep one element of $x |
61f5c3f5 |
1409 | $x; |
dccbb853 |
1410 | } |
1411 | |
0716bf9b |
1412 | ############################################################################## |
574bacfe |
1413 | # shifts |
1414 | |
1415 | sub _rsft |
1416 | { |
1417 | my ($c,$x,$y,$n) = @_; |
1418 | |
1419 | if ($n != 10) |
1420 | { |
9b924220 |
1421 | $n = _new($c,$n); return _div($c,$x, _pow($c,$n,$y)); |
61f5c3f5 |
1422 | } |
1423 | |
1424 | # shortcut (faster) for shifting by 10) |
1425 | # multiples of $BASE_LEN |
1426 | my $dst = 0; # destination |
1427 | my $src = _num($c,$y); # as normal int |
56d9de68 |
1428 | my $xlen = (@$x-1)*$BASE_LEN+length(int($x->[-1])); # len of x in digits |
90d1b129 |
1429 | if ($src >= $xlen or ($src == $xlen and ! defined $x->[1])) |
56d9de68 |
1430 | { |
1431 | # 12345 67890 shifted right by more than 10 digits => 0 |
1432 | splice (@$x,1); # leave only one element |
1433 | $x->[0] = 0; # set to zero |
1434 | return $x; |
1435 | } |
61f5c3f5 |
1436 | my $rem = $src % $BASE_LEN; # remainder to shift |
1437 | $src = int($src / $BASE_LEN); # source |
1438 | if ($rem == 0) |
1439 | { |
1440 | splice (@$x,0,$src); # even faster, 38.4 => 39.3 |
574bacfe |
1441 | } |
1442 | else |
1443 | { |
61f5c3f5 |
1444 | my $len = scalar @$x - $src; # elems to go |
1445 | my $vd; my $z = '0'x $BASE_LEN; |
1446 | $x->[scalar @$x] = 0; # avoid || 0 test inside loop |
1447 | while ($dst < $len) |
574bacfe |
1448 | { |
61f5c3f5 |
1449 | $vd = $z.$x->[$src]; |
1450 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN-$rem); |
1451 | $src++; |
1452 | $vd = substr($z.$x->[$src],-$rem,$rem) . $vd; |
1453 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN; |
1454 | $x->[$dst] = int($vd); |
1455 | $dst++; |
574bacfe |
1456 | } |
61f5c3f5 |
1457 | splice (@$x,$dst) if $dst > 0; # kill left-over array elems |
56b9c951 |
1458 | pop @$x if $x->[-1] == 0 && @$x > 1; # kill last element if 0 |
61f5c3f5 |
1459 | } # else rem == 0 |
574bacfe |
1460 | $x; |
1461 | } |
1462 | |
1463 | sub _lsft |
1464 | { |
1465 | my ($c,$x,$y,$n) = @_; |
1466 | |
1467 | if ($n != 10) |
1468 | { |
9b924220 |
1469 | $n = _new($c,$n); return _mul($c,$x, _pow($c,$n,$y)); |
574bacfe |
1470 | } |
61f5c3f5 |
1471 | |
1472 | # shortcut (faster) for shifting by 10) since we are in base 10eX |
1473 | # multiples of $BASE_LEN: |
1474 | my $src = scalar @$x; # source |
1475 | my $len = _num($c,$y); # shift-len as normal int |
1476 | my $rem = $len % $BASE_LEN; # remainder to shift |
1477 | my $dst = $src + int($len/$BASE_LEN); # destination |
1478 | my $vd; # further speedup |
1479 | $x->[$src] = 0; # avoid first ||0 for speed |
1480 | my $z = '0' x $BASE_LEN; |
1481 | while ($src >= 0) |
574bacfe |
1482 | { |
61f5c3f5 |
1483 | $vd = $x->[$src]; $vd = $z.$vd; |
1484 | $vd = substr($vd,-$BASE_LEN+$rem,$BASE_LEN-$rem); |
1485 | $vd .= $src > 0 ? substr($z.$x->[$src-1],-$BASE_LEN,$rem) : '0' x $rem; |
1486 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN; |
1487 | $x->[$dst] = int($vd); |
1488 | $dst--; $src--; |
574bacfe |
1489 | } |
61f5c3f5 |
1490 | # set lowest parts to 0 |
1491 | while ($dst >= 0) { $x->[$dst--] = 0; } |
1492 | # fix spurios last zero element |
1493 | splice @$x,-1 if $x->[-1] == 0; |
574bacfe |
1494 | $x; |
1495 | } |
1496 | |
027dc388 |
1497 | sub _pow |
1498 | { |
1499 | # power of $x to $y |
1500 | # ref to array, ref to array, return ref to array |
1501 | my ($c,$cx,$cy) = @_; |
1502 | |
b282a552 |
1503 | if (scalar @$cy == 1 && $cy->[0] == 0) |
1504 | { |
1505 | splice (@$cx,1); $cx->[0] = 1; # y == 0 => x => 1 |
1506 | return $cx; |
1507 | } |
1508 | if ((scalar @$cx == 1 && $cx->[0] == 1) || # x == 1 |
1509 | (scalar @$cy == 1 && $cy->[0] == 1)) # or y == 1 |
1510 | { |
1511 | return $cx; |
1512 | } |
1513 | if (scalar @$cx == 1 && $cx->[0] == 0) |
1514 | { |
1515 | splice (@$cx,1); $cx->[0] = 0; # 0 ** y => 0 (if not y <= 0) |
1516 | return $cx; |
1517 | } |
1518 | |
027dc388 |
1519 | my $pow2 = _one(); |
1ddff52a |
1520 | |
9b924220 |
1521 | my $y_bin = _as_bin($c,$cy); $y_bin =~ s/^0b//; |
1ddff52a |
1522 | my $len = length($y_bin); |
1523 | while (--$len > 0) |
027dc388 |
1524 | { |
1ddff52a |
1525 | _mul($c,$pow2,$cx) if substr($y_bin,$len,1) eq '1'; # is odd? |
027dc388 |
1526 | _mul($c,$cx,$cx); |
1527 | } |
1ddff52a |
1528 | |
1529 | _mul($c,$cx,$pow2); |
61f5c3f5 |
1530 | $cx; |
027dc388 |
1531 | } |
1532 | |
50109ad0 |
1533 | sub _nok |
1534 | { |
1535 | # n over k |
1536 | # ref to array, return ref to array |
1537 | my ($c,$n,$k) = @_; |
1538 | |
1539 | # ( 7 ) 7! 7*6*5 * 4*3*2*1 7 * 6 * 5 |
1540 | # ( - ) = --------- = --------------- = --------- |
1541 | # ( 3 ) 3! (7-3)! 3*2*1 * 4*3*2*1 3 * 2 * 1 |
1542 | |
1543 | # compute n - k + 2 (so we start with 5 in the example above) |
1544 | my $x = _copy($c,$n); |
1545 | |
1546 | _sub($c,$n,$k); |
1547 | if (!_is_one($c,$n)) |
1548 | { |
1549 | _inc($c,$n); |
1550 | my $f = _copy($c,$n); _inc($c,$f); # n = 5, f = 6, d = 2 |
1551 | my $d = _two($c); |
1552 | while (_acmp($c,$f,$x) <= 0) # f < n ? |
1553 | { |
1554 | # n = (n * f / d) == 5 * 6 / 2 => n == 3 |
1555 | $n = _mul($c,$n,$f); $n = _div($c,$n,$d); |
1556 | # f = 7, d = 3 |
1557 | _inc($c,$f); _inc($c,$d); |
1558 | } |
1559 | } |
1560 | else |
1561 | { |
1562 | # keep ref to $n and set it to 1 |
1563 | splice (@$n,1); $n->[0] = 1; |
1564 | } |
1565 | $n; |
1566 | } |
1567 | |
1568 | my @factorials = ( |
1569 | 1, |
1570 | 1, |
1571 | 2, |
1572 | 2*3, |
1573 | 2*3*4, |
1574 | 2*3*4*5, |
1575 | 2*3*4*5*6, |
1576 | 2*3*4*5*6*7, |
1577 | ); |
1578 | |
b3abae2a |
1579 | sub _fac |
1580 | { |
1581 | # factorial of $x |
1582 | # ref to array, return ref to array |
1583 | my ($c,$cx) = @_; |
1584 | |
50109ad0 |
1585 | if ((@$cx == 1) && ($cx->[0] <= 7)) |
b3abae2a |
1586 | { |
50109ad0 |
1587 | $cx->[0] = $factorials[$cx->[0]]; # 0 => 1, 1 => 1, 2 => 2 etc. |
b3abae2a |
1588 | return $cx; |
1589 | } |
1590 | |
50109ad0 |
1591 | if ((@$cx == 1) && # we do this only if $x >= 12 and $x <= 7000 |
1592 | ($cx->[0] >= 12 && $cx->[0] < 7000)) |
1593 | { |
1594 | |
1595 | # Calculate (k-j) * (k-j+1) ... k .. (k+j-1) * (k + j) |
1596 | # See http://blogten.blogspot.com/2007/01/calculating-n.html |
1597 | # The above series can be expressed as factors: |
1598 | # k * k - (j - i) * 2 |
1599 | # We cache k*k, and calculate (j * j) as the sum of the first j odd integers |
1600 | |
1601 | # This will not work when N exceeds the storage of a Perl scalar, however, |
1602 | # in this case the algorithm would be way to slow to terminate, anyway. |
1603 | |
1604 | # As soon as the last element of $cx is 0, we split it up and remember |
1605 | # how many zeors we got so far. The reason is that n! will accumulate |
1606 | # zeros at the end rather fast. |
1607 | my $zero_elements = 0; |
1608 | |
1609 | # If n is even, set n = n -1 |
1610 | my $k = _num($c,$cx); my $even = 1; |
1611 | if (($k & 1) == 0) |
1612 | { |
1613 | $even = $k; $k --; |
1614 | } |
1615 | # set k to the center point |
1616 | $k = ($k + 1) / 2; |
1617 | # print "k $k even: $even\n"; |
1618 | # now calculate k * k |
1619 | my $k2 = $k * $k; |
1620 | my $odd = 1; my $sum = 1; |
1621 | my $i = $k - 1; |
1622 | # keep reference to x |
1623 | my $new_x = _new($c, $k * $even); |
1624 | @$cx = @$new_x; |
1625 | if ($cx->[0] == 0) |
1626 | { |
1627 | $zero_elements ++; shift @$cx; |
1628 | } |
1629 | # print STDERR "x = ", _str($c,$cx),"\n"; |
1630 | my $BASE2 = int(sqrt($BASE))-1; |
1631 | my $j = 1; |
1632 | while ($j <= $i) |
1633 | { |
1634 | my $m = ($k2 - $sum); $odd += 2; $sum += $odd; $j++; |
1635 | while ($j <= $i && ($m < $BASE2) && (($k2 - $sum) < $BASE2)) |
1636 | { |
1637 | $m *= ($k2 - $sum); |
1638 | $odd += 2; $sum += $odd; $j++; |
1639 | # print STDERR "\n k2 $k2 m $m sum $sum odd $odd\n"; sleep(1); |
1640 | } |
1641 | if ($m < $BASE) |
1642 | { |
1643 | _mul($c,$cx,[$m]); |
1644 | } |
1645 | else |
1646 | { |
1647 | _mul($c,$cx,$c->_new($m)); |
1648 | } |
1649 | if ($cx->[0] == 0) |
1650 | { |
1651 | $zero_elements ++; shift @$cx; |
1652 | } |
1653 | # print STDERR "Calculate $k2 - $sum = $m (x = ", _str($c,$cx),")\n"; |
1654 | } |
1655 | # multiply in the zeros again |
1656 | unshift @$cx, (0) x $zero_elements; |
1657 | return $cx; |
1658 | } |
1659 | |
b3abae2a |
1660 | # go forward until $base is exceeded |
091c87b1 |
1661 | # limit is either $x steps (steps == 100 means a result always too high) or |
1662 | # $base. |
b3abae2a |
1663 | my $steps = 100; $steps = $cx->[0] if @$cx == 1; |
091c87b1 |
1664 | my $r = 2; my $cf = 3; my $step = 2; my $last = $r; |
1665 | while ($r*$cf < $BASE && $step < $steps) |
b3abae2a |
1666 | { |
1667 | $last = $r; $r *= $cf++; $step++; |
1668 | } |
091c87b1 |
1669 | if ((@$cx == 1) && $step == $cx->[0]) |
b3abae2a |
1670 | { |
091c87b1 |
1671 | # completely done, so keep reference to $x and return |
1672 | $cx->[0] = $r; |
b3abae2a |
1673 | return $cx; |
1674 | } |
091c87b1 |
1675 | |
990fb837 |
1676 | # now we must do the left over steps |
091c87b1 |
1677 | my $n; # steps still to do |
1678 | if (scalar @$cx == 1) |
1679 | { |
1680 | $n = $cx->[0]; |
1681 | } |
1682 | else |
1683 | { |
1684 | $n = _copy($c,$cx); |
1685 | } |
b3abae2a |
1686 | |
50109ad0 |
1687 | # Set $cx to the last result below $BASE (but keep ref to $x) |
1688 | $cx->[0] = $last; splice (@$cx,1); |
1689 | # As soon as the last element of $cx is 0, we split it up and remember |
1690 | # how many zeors we got so far. The reason is that n! will accumulate |
1691 | # zeros at the end rather fast. |
990fb837 |
1692 | my $zero_elements = 0; |
091c87b1 |
1693 | |
1694 | # do left-over steps fit into a scalar? |
1695 | if (ref $n eq 'ARRAY') |
b3abae2a |
1696 | { |
091c87b1 |
1697 | # No, so use slower inc() & cmp() |
50109ad0 |
1698 | # ($n is at least $BASE here) |
1699 | my $base_2 = int(sqrt($BASE)) - 1; |
1700 | #print STDERR "base_2: $base_2\n"; |
1701 | while ($step < $base_2) |
1702 | { |
1703 | if ($cx->[0] == 0) |
1704 | { |
1705 | $zero_elements ++; shift @$cx; |
1706 | } |
1707 | my $b = $step * ($step + 1); $step += 2; |
1708 | _mul($c,$cx,[$b]); |
1709 | } |
091c87b1 |
1710 | $step = [$step]; |
50109ad0 |
1711 | while (_acmp($c,$step,$n) <= 0) |
990fb837 |
1712 | { |
1713 | if ($cx->[0] == 0) |
1714 | { |
1715 | $zero_elements ++; shift @$cx; |
1716 | } |
091c87b1 |
1717 | _mul($c,$cx,$step); _inc($c,$step); |
990fb837 |
1718 | } |
990fb837 |
1719 | } |
091c87b1 |
1720 | else |
990fb837 |
1721 | { |
091c87b1 |
1722 | # Yes, so we can speed it up slightly |
50109ad0 |
1723 | |
1724 | # print "# left over steps $n\n"; |
1725 | |
1726 | my $base_4 = int(sqrt(sqrt($BASE))) - 2; |
1727 | #print STDERR "base_4: $base_4\n"; |
1728 | my $n4 = $n - 4; |
1729 | while ($step < $n4 && $step < $base_4) |
990fb837 |
1730 | { |
091c87b1 |
1731 | if ($cx->[0] == 0) |
1732 | { |
1733 | $zero_elements ++; shift @$cx; |
1734 | } |
50109ad0 |
1735 | my $b = $step * ($step + 1); $step += 2; $b *= $step * ($step + 1); $step += 2; |
1736 | _mul($c,$cx,[$b]); |
1737 | } |
1738 | my $base_2 = int(sqrt($BASE)) - 1; |
1739 | my $n2 = $n - 2; |
1740 | #print STDERR "base_2: $base_2\n"; |
1741 | while ($step < $n2 && $step < $base_2) |
1742 | { |
1743 | if ($cx->[0] == 0) |
1744 | { |
1745 | $zero_elements ++; shift @$cx; |
1746 | } |
1747 | my $b = $step * ($step + 1); $step += 2; |
1748 | _mul($c,$cx,[$b]); |
1749 | } |
1750 | # do what's left over |
1751 | while ($step <= $n) |
1752 | { |
091c87b1 |
1753 | _mul($c,$cx,[$step]); $step++; |
50109ad0 |
1754 | if ($cx->[0] == 0) |
1755 | { |
1756 | $zero_elements ++; shift @$cx; |
1757 | } |
990fb837 |
1758 | } |
990fb837 |
1759 | } |
1760 | # multiply in the zeros again |
50109ad0 |
1761 | unshift @$cx, (0) x $zero_elements; |
091c87b1 |
1762 | $cx; # return result |
1763 | } |
1764 | |
9b924220 |
1765 | ############################################################################# |
1766 | |
091c87b1 |
1767 | sub _log_int |
1768 | { |
1769 | # calculate integer log of $x to base $base |
1770 | # ref to array, ref to array - return ref to array |
1771 | my ($c,$x,$base) = @_; |
1772 | |
1773 | # X == 0 => NaN |
1774 | return if (scalar @$x == 1 && $x->[0] == 0); |
1775 | # BASE 0 or 1 => NaN |
1776 | return if (scalar @$base == 1 && $base->[0] < 2); |
b282a552 |
1777 | my $cmp = _acmp($c,$x,$base); # X == BASE => 1 |
091c87b1 |
1778 | if ($cmp == 0) |
1779 | { |
1780 | splice (@$x,1); $x->[0] = 1; |
8df1e0a2 |
1781 | return ($x,1) |
091c87b1 |
1782 | } |
1783 | # X < BASE |
1784 | if ($cmp < 0) |
1785 | { |
1786 | splice (@$x,1); $x->[0] = 0; |
8df1e0a2 |
1787 | return ($x,undef); |
091c87b1 |
1788 | } |
1789 | |
091c87b1 |
1790 | my $x_org = _copy($c,$x); # preserve x |
8df1e0a2 |
1791 | splice(@$x,1); $x->[0] = 1; # keep ref to $x |
1792 | |
86b76201 |
1793 | # Compute a guess for the result based on: |
1794 | # $guess = int ( length_in_base_10(X) / ( log(base) / log(10) ) ) |
2ebb273f |
1795 | my $len = _len($c,$x_org); |
1796 | my $log = log($base->[-1]) / log(10); |
b282a552 |
1797 | |
2ebb273f |
1798 | # for each additional element in $base, we add $BASE_LEN to the result, |
1799 | # based on the observation that log($BASE,10) is BASE_LEN and |
1800 | # log(x*y) == log(x) + log(y): |
1801 | $log += ((scalar @$base)-1) * $BASE_LEN; |
b282a552 |
1802 | |
2ebb273f |
1803 | # calculate now a guess based on the values obtained above: |
1804 | my $res = int($len / $log); |
1805 | |
1806 | $x->[0] = $res; |
1807 | my $trial = _pow ($c, _copy($c, $base), $x); |
1808 | my $a = _acmp($c,$trial,$x_org); |
1809 | |
1810 | # print STDERR "# trial ", _str($c,$x)," was: $a (0 = exact, -1 too small, +1 too big)\n"; |
1811 | |
1812 | # found an exact result? |
1813 | return ($x,1) if $a == 0; |
1814 | |
1815 | if ($a > 0) |
1816 | { |
1817 | # or too big |
1818 | _div($c,$trial,$base); _dec($c, $x); |
1819 | while (($a = _acmp($c,$trial,$x_org)) > 0) |
b282a552 |
1820 | { |
2ebb273f |
1821 | # print STDERR "# big _log_int at ", _str($c,$x), "\n"; |
1822 | _div($c,$trial,$base); _dec($c, $x); |
b282a552 |
1823 | } |
2ebb273f |
1824 | # result is now exact (a == 0), or too small (a < 0) |
1825 | return ($x, $a == 0 ? 1 : 0); |
1826 | } |
1827 | |
1828 | # else: result was to small |
1829 | _mul($c,$trial,$base); |
1830 | |
1831 | # did we now get the right result? |
1832 | $a = _acmp($c,$trial,$x_org); |
1833 | |
1834 | if ($a == 0) # yes, exactly |
1835 | { |
1836 | _inc($c, $x); |
1837 | return ($x,1); |
1838 | } |
1839 | return ($x,0) if $a > 0; |
1840 | |
1841 | # Result still too small (we should come here only if the estimate above |
1842 | # was very off base): |
1843 | |
1844 | # Now let the normal trial run obtain the real result |
1845 | # Simple loop that increments $x by 2 in each step, possible overstepping |
1846 | # the real result |
091c87b1 |
1847 | |
2ebb273f |
1848 | my $base_mul = _mul($c, _copy($c,$base), $base); # $base * $base |
8df1e0a2 |
1849 | |
9b924220 |
1850 | while (($a = _acmp($c,$trial,$x_org)) < 0) |
091c87b1 |
1851 | { |
2ebb273f |
1852 | # print STDERR "# small _log_int at ", _str($c,$x), "\n"; |
8df1e0a2 |
1853 | _mul($c,$trial,$base_mul); _add($c, $x, [2]); |
091c87b1 |
1854 | } |
8df1e0a2 |
1855 | |
1856 | my $exact = 1; |
1857 | if ($a > 0) |
091c87b1 |
1858 | { |
8df1e0a2 |
1859 | # overstepped the result |
1860 | _dec($c, $x); |
1861 | _div($c,$trial,$base); |
9b924220 |
1862 | $a = _acmp($c,$trial,$x_org); |
8df1e0a2 |
1863 | if ($a > 0) |
091c87b1 |
1864 | { |
8df1e0a2 |
1865 | _dec($c, $x); |
091c87b1 |
1866 | } |
2ebb273f |
1867 | $exact = 0 if $a != 0; # a = -1 => not exact result, a = 0 => exact |
091c87b1 |
1868 | } |
1869 | |
8df1e0a2 |
1870 | ($x,$exact); # return result |
b3abae2a |
1871 | } |
1872 | |
56d9de68 |
1873 | # for debugging: |
1874 | use constant DEBUG => 0; |
1875 | my $steps = 0; |
1876 | sub steps { $steps }; |
b3abae2a |
1877 | |
1878 | sub _sqrt |
0716bf9b |
1879 | { |
56d9de68 |
1880 | # square-root of $x in place |
990fb837 |
1881 | # Compute a guess of the result (by rule of thumb), then improve it via |
56d9de68 |
1882 | # Newton's method. |
394e6ffb |
1883 | my ($c,$x) = @_; |
0716bf9b |
1884 | |
394e6ffb |
1885 | if (scalar @$x == 1) |
1886 | { |
50109ad0 |
1887 | # fits into one Perl scalar, so result can be computed directly |
394e6ffb |
1888 | $x->[0] = int(sqrt($x->[0])); |
1889 | return $x; |
1890 | } |
1891 | my $y = _copy($c,$x); |
b3abae2a |
1892 | # hopefully _len/2 is < $BASE, the -1 is to always undershot the guess |
1893 | # since our guess will "grow" |
1894 | my $l = int((_len($c,$x)-1) / 2); |
1895 | |
56d9de68 |
1896 | my $lastelem = $x->[-1]; # for guess |
b3abae2a |
1897 | my $elems = scalar @$x - 1; |
1898 | # not enough digits, but could have more? |
56d9de68 |
1899 | if ((length($lastelem) <= 3) && ($elems > 1)) |
b3abae2a |
1900 | { |
1901 | # right-align with zero pad |
1902 | my $len = length($lastelem) & 1; |
1903 | print "$lastelem => " if DEBUG; |
1904 | $lastelem .= substr($x->[-2] . '0' x $BASE_LEN,0,$BASE_LEN); |
1905 | # former odd => make odd again, or former even to even again |
56d9de68 |
1906 | $lastelem = $lastelem / 10 if (length($lastelem) & 1) != $len; |
b3abae2a |
1907 | print "$lastelem\n" if DEBUG; |
1908 | } |
0716bf9b |
1909 | |
61f5c3f5 |
1910 | # construct $x (instead of _lsft($c,$x,$l,10) |
1911 | my $r = $l % $BASE_LEN; # 10000 00000 00000 00000 ($BASE_LEN=5) |
1912 | $l = int($l / $BASE_LEN); |
b3abae2a |
1913 | print "l = $l " if DEBUG; |
56d9de68 |
1914 | |
1915 | splice @$x,$l; # keep ref($x), but modify it |
1916 | |
b3abae2a |
1917 | # we make the first part of the guess not '1000...0' but int(sqrt($lastelem)) |
1918 | # that gives us: |
56d9de68 |
1919 | # 14400 00000 => sqrt(14400) => guess first digits to be 120 |
1920 | # 144000 000000 => sqrt(144000) => guess 379 |
b3abae2a |
1921 | |
b3abae2a |
1922 | print "$lastelem (elems $elems) => " if DEBUG; |
1923 | $lastelem = $lastelem / 10 if ($elems & 1 == 1); # odd or even? |
1924 | my $g = sqrt($lastelem); $g =~ s/\.//; # 2.345 => 2345 |
1925 | $r -= 1 if $elems & 1 == 0; # 70 => 7 |
1926 | |
1927 | # padd with zeros if result is too short |
1928 | $x->[$l--] = int(substr($g . '0' x $r,0,$r+1)); |
1929 | print "now ",$x->[-1] if DEBUG; |
1930 | print " would have been ", int('1' . '0' x $r),"\n" if DEBUG; |
56d9de68 |
1931 | |
b3abae2a |
1932 | # If @$x > 1, we could compute the second elem of the guess, too, to create |
56d9de68 |
1933 | # an even better guess. Not implemented yet. Does it improve performance? |
b3abae2a |
1934 | $x->[$l--] = 0 while ($l >= 0); # all other digits of guess are zero |
56d9de68 |
1935 | |
9b924220 |
1936 | print "start x= ",_str($c,$x),"\n" if DEBUG; |
394e6ffb |
1937 | my $two = _two(); |
1938 | my $last = _zero(); |
1939 | my $lastlast = _zero(); |
b3abae2a |
1940 | $steps = 0 if DEBUG; |
394e6ffb |
1941 | while (_acmp($c,$last,$x) != 0 && _acmp($c,$lastlast,$x) != 0) |
1942 | { |
b3abae2a |
1943 | $steps++ if DEBUG; |
394e6ffb |
1944 | $lastlast = _copy($c,$last); |
1945 | $last = _copy($c,$x); |
1946 | _add($c,$x, _div($c,_copy($c,$y),$x)); |
1947 | _div($c,$x, $two ); |
9b924220 |
1948 | print " x= ",_str($c,$x),"\n" if DEBUG; |
394e6ffb |
1949 | } |
b3abae2a |
1950 | print "\nsteps in sqrt: $steps, " if DEBUG; |
394e6ffb |
1951 | _dec($c,$x) if _acmp($c,$y,_mul($c,_copy($c,$x),$x)) < 0; # overshot? |
b3abae2a |
1952 | print " final ",$x->[-1],"\n" if DEBUG; |
394e6ffb |
1953 | $x; |
0716bf9b |
1954 | } |
1955 | |
990fb837 |
1956 | sub _root |
1957 | { |
1958 | # take n'th root of $x in place (n >= 3) |
990fb837 |
1959 | my ($c,$x,$n) = @_; |
1960 | |
1961 | if (scalar @$x == 1) |
1962 | { |
1963 | if (scalar @$n > 1) |
1964 | { |
1965 | # result will always be smaller than 2 so trunc to 1 at once |
1966 | $x->[0] = 1; |
1967 | } |
1968 | else |
1969 | { |
50109ad0 |
1970 | # fits into one Perl scalar, so result can be computed directly |
091c87b1 |
1971 | # cannot use int() here, because it rounds wrongly (try |
1972 | # (81 ** 3) ** (1/3) to see what I mean) |
1973 | #$x->[0] = int( $x->[0] ** (1 / $n->[0]) ); |
1974 | # round to 8 digits, then truncate result to integer |
1975 | $x->[0] = int ( sprintf ("%.8f", $x->[0] ** (1 / $n->[0]) ) ); |
990fb837 |
1976 | } |
1977 | return $x; |
1978 | } |
1979 | |
3a427a11 |
1980 | # we know now that X is more than one element long |
1981 | |
c38b2de2 |
1982 | # if $n is a power of two, we can repeatedly take sqrt($X) and find the |
1983 | # proper result, because sqrt(sqrt($x)) == root($x,4) |
1984 | my $b = _as_bin($c,$n); |
9b924220 |
1985 | if ($b =~ /0b1(0+)$/) |
c38b2de2 |
1986 | { |
1987 | my $count = CORE::length($1); # 0b100 => len('00') => 2 |
1988 | my $cnt = $count; # counter for loop |
1989 | unshift (@$x, 0); # add one element, together with one |
1990 | # more below in the loop this makes 2 |
1991 | while ($cnt-- > 0) |
1992 | { |
1993 | # 'inflate' $X by adding one element, basically computing |
1994 | # $x * $BASE * $BASE. This gives us more $BASE_LEN digits for result |
1995 | # since len(sqrt($X)) approx == len($x) / 2. |
1996 | unshift (@$x, 0); |
1997 | # calculate sqrt($x), $x is now one element to big, again. In the next |
1998 | # round we make that two, again. |
1999 | _sqrt($c,$x); |
2000 | } |
2001 | # $x is now one element to big, so truncate result by removing it |
2002 | splice (@$x,0,1); |
2003 | } |
2004 | else |
2005 | { |
091c87b1 |
2006 | # trial computation by starting with 2,4,8,16 etc until we overstep |
3a427a11 |
2007 | my $step; |
091c87b1 |
2008 | my $trial = _two(); |
2009 | |
3a427a11 |
2010 | # while still to do more than X steps |
2011 | do |
091c87b1 |
2012 | { |
3a427a11 |
2013 | $step = _two(); |
2014 | while (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) < 0) |
2015 | { |
2016 | _mul ($c, $step, [2]); |
2017 | _add ($c, $trial, $step); |
2018 | } |
2019 | |
2020 | # hit exactly? |
2021 | if (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) == 0) |
2022 | { |
2023 | @$x = @$trial; # make copy while preserving ref to $x |
2024 | return $x; |
2025 | } |
2026 | # overstepped, so go back on step |
2027 | _sub($c, $trial, $step); |
2028 | } while (scalar @$step > 1 || $step->[0] > 128); |
091c87b1 |
2029 | |
3a427a11 |
2030 | # reset step to 2 |
2031 | $step = _two(); |
091c87b1 |
2032 | # add two, because $trial cannot be exactly the result (otherwise we would |
2033 | # alrady have found it) |
2034 | _add($c, $trial, $step); |
2035 | |
3a427a11 |
2036 | # and now add more and more (2,4,6,8,10 etc) |
2037 | while (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) < 0) |
2038 | { |
2039 | _add ($c, $trial, $step); |
2040 | } |
091c87b1 |
2041 | |
2042 | # hit not exactly? (overstepped) |
091c87b1 |
2043 | if (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) > 0) |
2044 | { |
2045 | _dec($c,$trial); |
2046 | } |
3a427a11 |
2047 | |
2048 | # hit not exactly? (overstepped) |
2049 | # 80 too small, 81 slightly too big, 82 too big |
091c87b1 |
2050 | if (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) > 0) |
2051 | { |
3a427a11 |
2052 | _dec ($c, $trial); |
091c87b1 |
2053 | } |
3a427a11 |
2054 | |
091c87b1 |
2055 | @$x = @$trial; # make copy while preserving ref to $x |
2056 | return $x; |
c38b2de2 |
2057 | } |
990fb837 |
2058 | $x; |
2059 | } |
2060 | |
394e6ffb |
2061 | ############################################################################## |
2062 | # binary stuff |
0716bf9b |
2063 | |
394e6ffb |
2064 | sub _and |
2065 | { |
2066 | my ($c,$x,$y) = @_; |
0716bf9b |
2067 | |
394e6ffb |
2068 | # the shortcut makes equal, large numbers _really_ fast, and makes only a |
2069 | # very small performance drop for small numbers (e.g. something with less |
2070 | # than 32 bit) Since we optimize for large numbers, this is enabled. |
2071 | return $x if _acmp($c,$x,$y) == 0; # shortcut |
0716bf9b |
2072 | |
394e6ffb |
2073 | my $m = _one(); my ($xr,$yr); |
2074 | my $mask = $AND_MASK; |
2075 | |
2076 | my $x1 = $x; |
2077 | my $y1 = _copy($c,$y); # make copy |
2078 | $x = _zero(); |
2079 | my ($b,$xrr,$yrr); |
2080 | use integer; |
2081 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
2082 | { |
2083 | ($x1, $xr) = _div($c,$x1,$mask); |
2084 | ($y1, $yr) = _div($c,$y1,$mask); |
2085 | |
2086 | # make ints() from $xr, $yr |
9b924220 |
2087 | # this is when the AND_BITS are greater than $BASE and is slower for |
394e6ffb |
2088 | # small (<256 bits) numbers, but faster for large numbers. Disabled |
2089 | # due to KISS principle |
2090 | |
2091 | # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
2092 | # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
9b924220 |
2093 | # _add($c,$x, _mul($c, _new( $c, ($xrr & $yrr) ), $m) ); |
394e6ffb |
2094 | |
61f5c3f5 |
2095 | # 0+ due to '&' doesn't work in strings |
2096 | _add($c,$x, _mul($c, [ 0+$xr->[0] & 0+$yr->[0] ], $m) ); |
394e6ffb |
2097 | _mul($c,$m,$mask); |
2098 | } |
2099 | $x; |
0716bf9b |
2100 | } |
2101 | |
394e6ffb |
2102 | sub _xor |
0716bf9b |
2103 | { |
394e6ffb |
2104 | my ($c,$x,$y) = @_; |
2105 | |
2106 | return _zero() if _acmp($c,$x,$y) == 0; # shortcut (see -and) |
2107 | |
2108 | my $m = _one(); my ($xr,$yr); |
2109 | my $mask = $XOR_MASK; |
2110 | |
2111 | my $x1 = $x; |
2112 | my $y1 = _copy($c,$y); # make copy |
2113 | $x = _zero(); |
2114 | my ($b,$xrr,$yrr); |
2115 | use integer; |
2116 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
0716bf9b |
2117 | { |
394e6ffb |
2118 | ($x1, $xr) = _div($c,$x1,$mask); |
2119 | ($y1, $yr) = _div($c,$y1,$mask); |
2120 | # make ints() from $xr, $yr (see _and()) |
2121 | #$b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
2122 | #$b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
9b924220 |
2123 | #_add($c,$x, _mul($c, _new( $c, ($xrr ^ $yrr) ), $m) ); |
61f5c3f5 |
2124 | |
2125 | # 0+ due to '^' doesn't work in strings |
2126 | _add($c,$x, _mul($c, [ 0+$xr->[0] ^ 0+$yr->[0] ], $m) ); |
394e6ffb |
2127 | _mul($c,$m,$mask); |
0716bf9b |
2128 | } |
394e6ffb |
2129 | # the loop stops when the shorter of the two numbers is exhausted |
2130 | # the remainder of the longer one will survive bit-by-bit, so we simple |
2131 | # multiply-add it in |
2132 | _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1); |
2133 | _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1); |
2134 | |
2135 | $x; |
0716bf9b |
2136 | } |
2137 | |
394e6ffb |
2138 | sub _or |
0716bf9b |
2139 | { |
394e6ffb |
2140 | my ($c,$x,$y) = @_; |
0716bf9b |
2141 | |
394e6ffb |
2142 | return $x if _acmp($c,$x,$y) == 0; # shortcut (see _and) |
0716bf9b |
2143 | |
394e6ffb |
2144 | my $m = _one(); my ($xr,$yr); |
2145 | my $mask = $OR_MASK; |
0716bf9b |
2146 | |
394e6ffb |
2147 | my $x1 = $x; |
2148 | my $y1 = _copy($c,$y); # make copy |
2149 | $x = _zero(); |
2150 | my ($b,$xrr,$yrr); |
2151 | use integer; |
2152 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
2153 | { |
2154 | ($x1, $xr) = _div($c,$x1,$mask); |
2155 | ($y1, $yr) = _div($c,$y1,$mask); |
2156 | # make ints() from $xr, $yr (see _and()) |
2157 | # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
2158 | # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
9b924220 |
2159 | # _add($c,$x, _mul($c, _new( $c, ($xrr | $yrr) ), $m) ); |
394e6ffb |
2160 | |
61f5c3f5 |
2161 | # 0+ due to '|' doesn't work in strings |
2162 | _add($c,$x, _mul($c, [ 0+$xr->[0] | 0+$yr->[0] ], $m) ); |
394e6ffb |
2163 | _mul($c,$m,$mask); |
2164 | } |
2165 | # the loop stops when the shorter of the two numbers is exhausted |
2166 | # the remainder of the longer one will survive bit-by-bit, so we simple |
2167 | # multiply-add it in |
2168 | _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1); |
2169 | _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1); |
2170 | |
2171 | $x; |
0716bf9b |
2172 | } |
2173 | |
61f5c3f5 |
2174 | sub _as_hex |
2175 | { |
2176 | # convert a decimal number to hex (ref to array, return ref to string) |
2177 | my ($c,$x) = @_; |
2178 | |
50109ad0 |
2179 | # fits into one element (handle also 0x0 case) |
03874afe |
2180 | return sprintf("0x%x",$x->[0]) if @$x == 1; |
990fb837 |
2181 | |
61f5c3f5 |
2182 | my $x1 = _copy($c,$x); |
2183 | |
2184 | my $es = ''; |
1ddff52a |
2185 | my ($xr, $h, $x10000); |
2186 | if ($] >= 5.006) |
2187 | { |
2188 | $x10000 = [ 0x10000 ]; $h = 'h4'; |
2189 | } |
2190 | else |
2191 | { |
2192 | $x10000 = [ 0x1000 ]; $h = 'h3'; |
2193 | } |
091c87b1 |
2194 | while (@$x1 != 1 || $x1->[0] != 0) # _is_zero() |
61f5c3f5 |
2195 | { |
2196 | ($x1, $xr) = _div($c,$x1,$x10000); |
7b29e1e6 |
2197 | $es .= unpack($h,pack('V',$xr->[0])); |
61f5c3f5 |
2198 | } |
2199 | $es = reverse $es; |
2200 | $es =~ s/^[0]+//; # strip leading zeros |
03874afe |
2201 | '0x' . $es; # return result prepended with 0x |
61f5c3f5 |
2202 | } |
2203 | |
2204 | sub _as_bin |
2205 | { |
2206 | # convert a decimal number to bin (ref to array, return ref to string) |
2207 | my ($c,$x) = @_; |
2208 | |
50109ad0 |
2209 | # fits into one element (and Perl recent enough), handle also 0b0 case |
091c87b1 |
2210 | # handle zero case for older Perls |
2211 | if ($] <= 5.005 && @$x == 1 && $x->[0] == 0) |
2212 | { |
9b924220 |
2213 | my $t = '0b0'; return $t; |
091c87b1 |
2214 | } |
2215 | if (@$x == 1 && $] >= 5.006) |
990fb837 |
2216 | { |
091c87b1 |
2217 | my $t = sprintf("0b%b",$x->[0]); |
9b924220 |
2218 | return $t; |
990fb837 |
2219 | } |
61f5c3f5 |
2220 | my $x1 = _copy($c,$x); |
2221 | |
2222 | my $es = ''; |
1ddff52a |
2223 | my ($xr, $b, $x10000); |
2224 | if ($] >= 5.006) |
2225 | { |
2226 | $x10000 = [ 0x10000 ]; $b = 'b16'; |
2227 | } |
2228 | else |
2229 | { |
2230 | $x10000 = [ 0x1000 ]; $b = 'b12'; |
2231 | } |
091c87b1 |
2232 | while (!(@$x1 == 1 && $x1->[0] == 0)) # _is_zero() |
61f5c3f5 |
2233 | { |
2234 | ($x1, $xr) = _div($c,$x1,$x10000); |
7b29e1e6 |
2235 | $es .= unpack($b,pack('v',$xr->[0])); |
61f5c3f5 |
2236 | } |
2237 | $es = reverse $es; |
2238 | $es =~ s/^[0]+//; # strip leading zeros |
03874afe |
2239 | '0b' . $es; # return result prepended with 0b |
61f5c3f5 |
2240 | } |
2241 | |
7b29e1e6 |
2242 | sub _as_oct |
2243 | { |
2244 | # convert a decimal number to octal (ref to array, return ref to string) |
2245 | my ($c,$x) = @_; |
2246 | |
50109ad0 |
2247 | # fits into one element (handle also 0 case) |
7b29e1e6 |
2248 | return sprintf("0%o",$x->[0]) if @$x == 1; |
2249 | |
2250 | my $x1 = _copy($c,$x); |
2251 | |
2252 | my $es = ''; |
2253 | my $xr; |
2254 | my $x1000 = [ 0100000 ]; |
2255 | while (@$x1 != 1 || $x1->[0] != 0) # _is_zero() |
2256 | { |
2257 | ($x1, $xr) = _div($c,$x1,$x1000); |
2258 | $es .= reverse sprintf("%05o", $xr->[0]); |
2259 | } |
2260 | $es = reverse $es; |
2261 | $es =~ s/^[0]+//; # strip leading zeros |
2262 | '0' . $es; # return result prepended with 0 |
2263 | } |
2264 | |
2265 | sub _from_oct |
2266 | { |
50109ad0 |
2267 | # convert a octal number to decimal (string, return ref to array) |
7b29e1e6 |
2268 | my ($c,$os) = @_; |
2269 | |
2270 | # for older Perls, play safe |
2271 | my $m = [ 0100000 ]; |
2272 | my $d = 5; # 5 digits at a time |
2273 | |
2274 | my $mul = _one(); |
2275 | my $x = _zero(); |
2276 | |
2277 | my $len = int( (length($os)-1)/$d ); # $d digit parts, w/o the '0' |
2278 | my $val; my $i = -$d; |
2279 | while ($len >= 0) |
2280 | { |
2281 | $val = substr($os,$i,$d); # get oct digits |
0617f807 |
2282 | $val = CORE::oct($val); |
7b29e1e6 |
2283 | $i -= $d; $len --; |
2284 | my $adder = [ $val ]; |
2285 | _add ($c, $x, _mul ($c, $adder, $mul ) ) if $val != 0; |
2286 | _mul ($c, $mul, $m ) if $len >= 0; # skip last mul |
2287 | } |
2288 | $x; |
2289 | } |
2290 | |
394e6ffb |
2291 | sub _from_hex |
0716bf9b |
2292 | { |
50109ad0 |
2293 | # convert a hex number to decimal (string, return ref to array) |
394e6ffb |
2294 | my ($c,$hs) = @_; |
0716bf9b |
2295 | |
2d2b2744 |
2296 | my $m = _new($c, 0x10000000); # 28 bit at a time (<32 bit!) |
03874afe |
2297 | my $d = 7; # 7 digits at a time |
2298 | if ($] <= 5.006) |
2299 | { |
2300 | # for older Perls, play safe |
2301 | $m = [ 0x10000 ]; # 16 bit at a time (<32 bit!) |
2302 | $d = 4; # 4 digits at a time |
2303 | } |
2304 | |
394e6ffb |
2305 | my $mul = _one(); |
394e6ffb |
2306 | my $x = _zero(); |
0716bf9b |
2307 | |
03874afe |
2308 | my $len = int( (length($hs)-2)/$d ); # $d digit parts, w/o the '0x' |
2309 | my $val; my $i = -$d; |
394e6ffb |
2310 | while ($len >= 0) |
2311 | { |
03874afe |
2312 | $val = substr($hs,$i,$d); # get hex digits |
7b29e1e6 |
2313 | $val =~ s/^0x// if $len == 0; # for last part only because |
0617f807 |
2314 | $val = CORE::hex($val); # hex does not like wrong chars |
03874afe |
2315 | $i -= $d; $len --; |
2d2b2744 |
2316 | my $adder = [ $val ]; |
2317 | # if the resulting number was to big to fit into one element, create a |
2318 | # two-element version (bug found by Mark Lakata - Thanx!) |
2319 | if (CORE::length($val) > $BASE_LEN) |
2320 | { |
2321 | $adder = _new($c,$val); |
2322 | } |
2323 | _add ($c, $x, _mul ($c, $adder, $mul ) ) if $val != 0; |
394e6ffb |
2324 | _mul ($c, $mul, $m ) if $len >= 0; # skip last mul |
2325 | } |
2326 | $x; |
2327 | } |
2328 | |
2329 | sub _from_bin |
0716bf9b |
2330 | { |
50109ad0 |
2331 | # convert a hex number to decimal (string, return ref to array) |
394e6ffb |
2332 | my ($c,$bs) = @_; |
0716bf9b |
2333 | |
091c87b1 |
2334 | # instead of converting X (8) bit at a time, it is faster to "convert" the |
13a12e00 |
2335 | # number to hex, and then call _from_hex. |
2336 | |
9b924220 |
2337 | my $hs = $bs; |
13a12e00 |
2338 | $hs =~ s/^[+-]?0b//; # remove sign and 0b |
2339 | my $l = length($hs); # bits |
2340 | $hs = '0' x (8-($l % 8)) . $hs if ($l % 8) != 0; # padd left side w/ 0 |
03874afe |
2341 | my $h = '0x' . unpack('H*', pack ('B*', $hs)); # repack as hex |
091c87b1 |
2342 | |
03874afe |
2343 | $c->_from_hex($h); |
0716bf9b |
2344 | } |
2345 | |
07d34614 |
2346 | ############################################################################## |
2347 | # special modulus functions |
2348 | |
56d9de68 |
2349 | sub _modinv |
d614cd8b |
2350 | { |
56d9de68 |
2351 | # modular inverse |
2352 | my ($c,$x,$y) = @_; |
1ddff52a |
2353 | |
56d9de68 |
2354 | my $u = _zero($c); my $u1 = _one($c); |
2355 | my $a = _copy($c,$y); my $b = _copy($c,$x); |
1ddff52a |
2356 | |
2357 | # Euclid's Algorithm for bgcd(), only that we calc bgcd() ($a) and the |
56d9de68 |
2358 | # result ($u) at the same time. See comments in BigInt for why this works. |
2359 | my $q; |
2360 | ($a, $q, $b) = ($b, _div($c,$a,$b)); # step 1 |
2361 | my $sign = 1; |
1ddff52a |
2362 | while (!_is_zero($c,$b)) |
2363 | { |
56d9de68 |
2364 | my $t = _add($c, # step 2: |
2365 | _mul($c,_copy($c,$u1), $q) , # t = u1 * q |
2366 | $u ); # + u |
2367 | $u = $u1; # u = u1, u1 = t |
2368 | $u1 = $t; |
2369 | $sign = -$sign; |
2370 | ($a, $q, $b) = ($b, _div($c,$a,$b)); # step 1 |
1ddff52a |
2371 | } |
2372 | |
2373 | # if the gcd is not 1, then return NaN |
56d9de68 |
2374 | return (undef,undef) unless _is_one($c,$a); |
2375 | |
03874afe |
2376 | ($u1, $sign == 1 ? '+' : '-'); |
d614cd8b |
2377 | } |
2378 | |
2379 | sub _modpow |
2380 | { |
2381 | # modulus of power ($x ** $y) % $z |
07d34614 |
2382 | my ($c,$num,$exp,$mod) = @_; |
2383 | |
2384 | # in the trivial case, |
2385 | if (_is_one($c,$mod)) |
2386 | { |
2387 | splice @$num,0,1; $num->[0] = 0; |
2388 | return $num; |
2389 | } |
2390 | if ((scalar @$num == 1) && (($num->[0] == 0) || ($num->[0] == 1))) |
2391 | { |
2392 | $num->[0] = 1; |
2393 | return $num; |
2394 | } |
1ddff52a |
2395 | |
2396 | # $num = _mod($c,$num,$mod); # this does not make it faster |
07d34614 |
2397 | |
2398 | my $acc = _copy($c,$num); my $t = _one(); |
2399 | |
9b924220 |
2400 | my $expbin = _as_bin($c,$exp); $expbin =~ s/^0b//; |
1ddff52a |
2401 | my $len = length($expbin); |
2402 | while (--$len >= 0) |
07d34614 |
2403 | { |
1ddff52a |
2404 | if ( substr($expbin,$len,1) eq '1') # is_odd |
07d34614 |
2405 | { |
2406 | _mul($c,$t,$acc); |
2407 | $t = _mod($c,$t,$mod); |
2408 | } |
2409 | _mul($c,$acc,$acc); |
2410 | $acc = _mod($c,$acc,$mod); |
07d34614 |
2411 | } |
2412 | @$num = @$t; |
2413 | $num; |
d614cd8b |
2414 | } |
2415 | |
9b924220 |
2416 | sub _gcd |
2417 | { |
2418 | # greatest common divisor |
2419 | my ($c,$x,$y) = @_; |
2420 | |
b68b7ab1 |
2421 | while ( (scalar @$y != 1) || ($y->[0] != 0) ) # while ($y != 0) |
9b924220 |
2422 | { |
2423 | my $t = _copy($c,$y); |
2424 | $y = _mod($c, $x, $y); |
2425 | $x = $t; |
2426 | } |
2427 | $x; |
2428 | } |
2429 | |
394e6ffb |
2430 | ############################################################################## |
2431 | ############################################################################## |
2432 | |
0716bf9b |
2433 | 1; |
2434 | __END__ |
2435 | |
2436 | =head1 NAME |
2437 | |
2438 | Math::BigInt::Calc - Pure Perl module to support Math::BigInt |
2439 | |
2440 | =head1 SYNOPSIS |
2441 | |
ee15d750 |
2442 | Provides support for big integer calculations. Not intended to be used by other |
091c87b1 |
2443 | modules. Other modules which sport the same functions can also be used to support |
2444 | Math::BigInt, like Math::BigInt::GMP or Math::BigInt::Pari. |
0716bf9b |
2445 | |
2446 | =head1 DESCRIPTION |
2447 | |
027dc388 |
2448 | In order to allow for multiple big integer libraries, Math::BigInt was |
2449 | rewritten to use library modules for core math routines. Any module which |
2450 | follows the same API as this can be used instead by using the following: |
0716bf9b |
2451 | |
ee15d750 |
2452 | use Math::BigInt lib => 'libname'; |
0716bf9b |
2453 | |
027dc388 |
2454 | 'libname' is either the long name ('Math::BigInt::Pari'), or only the short |
2455 | version like 'Pari'. |
2456 | |
990fb837 |
2457 | =head1 STORAGE |
2458 | |
2459 | =head1 METHODS |
0716bf9b |
2460 | |
027dc388 |
2461 | The following functions MUST be defined in order to support the use by |
9b924220 |
2462 | Math::BigInt v1.70 or later: |
0716bf9b |
2463 | |
50109ad0 |
2464 | api_version() return API version, 1 for v1.70, 2 for v1.83 |
0716bf9b |
2465 | _new(string) return ref to new object from ref to decimal string |
2466 | _zero() return a new object with value 0 |
2467 | _one() return a new object with value 1 |
9b924220 |
2468 | _two() return a new object with value 2 |
2469 | _ten() return a new object with value 10 |
0716bf9b |
2470 | |
2471 | _str(obj) return ref to a string representing the object |
2472 | _num(obj) returns a Perl integer/floating point number |
2473 | NOTE: because of Perl numeric notation defaults, |
2474 | the _num'ified obj may lose accuracy due to |
3c4b39be |
2475 | machine-dependent floating point size limitations |
0716bf9b |
2476 | |
2477 | _add(obj,obj) Simple addition of two objects |
2478 | _mul(obj,obj) Multiplication of two objects |
2479 | _div(obj,obj) Division of the 1st object by the 2nd |
b22b3e31 |
2480 | In list context, returns (result,remainder). |
2481 | NOTE: this is integer math, so no |
2482 | fractional part will be returned. |
990fb837 |
2483 | The second operand will be not be 0, so no need to |
2484 | check for that. |
b22b3e31 |
2485 | _sub(obj,obj) Simple subtraction of 1 object from another |
0716bf9b |
2486 | a third, optional parameter indicates that the params |
2487 | are swapped. In this case, the first param needs to |
2488 | be preserved, while you can destroy the second. |
2489 | sub (x,y,1) => return x - y and keep x intact! |
3c4b39be |
2490 | _dec(obj) decrement object by one (input is guaranteed to be > 0) |
e745a66c |
2491 | _inc(obj) increment object by one |
2492 | |
0716bf9b |
2493 | |
2494 | _acmp(obj,obj) <=> operator for objects (return -1, 0 or 1) |
2495 | |
2496 | _len(obj) returns count of the decimal digits of the object |
2497 | _digit(obj,n) returns the n'th decimal digit of object |
2498 | |
9b924220 |
2499 | _is_one(obj) return true if argument is 1 |
2500 | _is_two(obj) return true if argument is 2 |
2501 | _is_ten(obj) return true if argument is 10 |
0716bf9b |
2502 | _is_zero(obj) return true if argument is 0 |
2503 | _is_even(obj) return true if argument is even (0,2,4,6..) |
2504 | _is_odd(obj) return true if argument is odd (1,3,5,7..) |
2505 | |
2506 | _copy return a ref to a true copy of the object |
2507 | |
2508 | _check(obj) check whether internal representation is still intact |
2509 | return 0 for ok, otherwise error message as string |
2510 | |
50109ad0 |
2511 | _from_hex(str) return new object from a hexadecimal string |
2512 | _from_bin(str) return new object from a binary string |
2513 | _from_oct(str) return new object from an octal string |
0716bf9b |
2514 | |
9b924220 |
2515 | _as_hex(str) return string containing the value as |
ee15d750 |
2516 | unsigned hex string, with the '0x' prepended. |
2517 | Leading zeros must be stripped. |
2518 | _as_bin(str) Like as_hex, only as binary string containing only |
2519 | zeros and ones. Leading zeros must be stripped and a |
2520 | '0b' must be prepended. |
2521 | |
0716bf9b |
2522 | _rsft(obj,N,B) shift object in base B by N 'digits' right |
2523 | _lsft(obj,N,B) shift object in base B by N 'digits' left |
2524 | |
2525 | _xor(obj1,obj2) XOR (bit-wise) object 1 with object 2 |
dccbb853 |
2526 | Note: XOR, AND and OR pad with zeros if size mismatches |
0716bf9b |
2527 | _and(obj1,obj2) AND (bit-wise) object 1 with object 2 |
2528 | _or(obj1,obj2) OR (bit-wise) object 1 with object 2 |
2529 | |
50109ad0 |
2530 | _mod(obj1,obj2) Return remainder of div of the 1st by the 2nd object |
990fb837 |
2531 | _sqrt(obj) return the square root of object (truncated to int) |
2532 | _root(obj) return the n'th (n >= 3) root of obj (truncated to int) |
b3abae2a |
2533 | _fac(obj) return factorial of object 1 (1*2*3*4..) |
50109ad0 |
2534 | _pow(obj1,obj2) return object 1 to the power of object 2 |
b282a552 |
2535 | return undef for NaN |
b22b3e31 |
2536 | _zeros(obj) return number of trailing decimal zeros |
d614cd8b |
2537 | _modinv return inverse modulus |
2538 | _modpow return modulus of power ($x ** $y) % $z |
091c87b1 |
2539 | _log_int(X,N) calculate integer log() of X in base N |
2540 | X >= 0, N >= 0 (return undef for NaN) |
8df1e0a2 |
2541 | returns (RESULT, EXACT) where EXACT is: |
2542 | 1 : result is exactly RESULT |
2543 | 0 : result was truncated to RESULT |
2544 | undef : unknown whether result is exactly RESULT |
9b924220 |
2545 | _gcd(obj,obj) return Greatest Common Divisor of two objects |
2546 | |
50109ad0 |
2547 | The following functions are REQUIRED for an api_version of 2 or greater: |
2548 | |
2549 | _1ex($x) create the number 1Ex where x >= 0 |
2550 | _alen(obj) returns approximate count of the decimal digits of the |
2551 | object. This estimate MUST always be greater or equal |
2552 | to what _len() returns. |
2553 | _nok(n,k) calculate n over k (binomial coefficient) |
2554 | |
9b924220 |
2555 | The following functions are optional, and can be defined if the underlying lib |
2556 | has a fast way to do them. If undefined, Math::BigInt will use pure Perl (hence |
2557 | slow) fallback routines to emulate these: |
2558 | |
2559 | _signed_or |
2560 | _signed_and |
2561 | _signed_xor |
2562 | |
b22b3e31 |
2563 | Input strings come in as unsigned but with prefix (i.e. as '123', '0xabc' |
0716bf9b |
2564 | or '0b1101'). |
2565 | |
990fb837 |
2566 | So the library needs only to deal with unsigned big integers. Testing of input |
2567 | parameter validity is done by the caller, so you need not worry about |
2568 | underflow (f.i. in C<_sub()>, C<_dec()>) nor about division by zero or similar |
2569 | cases. |
574bacfe |
2570 | |
2571 | The first parameter can be modified, that includes the possibility that you |
2572 | return a reference to a completely different object instead. Although keeping |
50109ad0 |
2573 | the reference and just changing its contents is preferred over creating and |
dccbb853 |
2574 | returning a different reference. |
574bacfe |
2575 | |
990fb837 |
2576 | Return values are always references to objects, strings, or true/false for |
3c4b39be |
2577 | comparison routines. |
990fb837 |
2578 | |
574bacfe |
2579 | =head1 WRAP YOUR OWN |
2580 | |
2581 | If you want to port your own favourite c-lib for big numbers to the |
2582 | Math::BigInt interface, you can take any of the already existing modules as |
2583 | a rough guideline. You should really wrap up the latest BigInt and BigFloat |
bd05a461 |
2584 | testsuites with your module, and replace in them any of the following: |
574bacfe |
2585 | |
2586 | use Math::BigInt; |
2587 | |
bd05a461 |
2588 | by this: |
574bacfe |
2589 | |
2590 | use Math::BigInt lib => 'yourlib'; |
2591 | |
2592 | This way you ensure that your library really works 100% within Math::BigInt. |
0716bf9b |
2593 | |
2594 | =head1 LICENSE |
2595 | |
2596 | This program is free software; you may redistribute it and/or modify it under |
2597 | the same terms as Perl itself. |
2598 | |
2599 | =head1 AUTHORS |
2600 | |
2601 | Original math code by Mark Biggar, rewritten by Tels L<http://bloodgate.com/> |
990fb837 |
2602 | in late 2000. |
0716bf9b |
2603 | Seperated from BigInt and shaped API with the help of John Peacock. |
b68b7ab1 |
2604 | |
7b29e1e6 |
2605 | Fixed, speed-up, streamlined and enhanced by Tels 2001 - 2007. |
0716bf9b |
2606 | |
2607 | =head1 SEE ALSO |
2608 | |
50109ad0 |
2609 | L<Math::BigInt>, L<Math::BigFloat>, |
990fb837 |
2610 | L<Math::BigInt::GMP>, L<Math::BigInt::FastCalc> and L<Math::BigInt::Pari>. |
0716bf9b |
2611 | |
2612 | =cut |