Commit | Line | Data |
0716bf9b |
1 | package Math::BigInt::Calc; |
2 | |
a0ac753d |
3 | use 5.006002; |
0716bf9b |
4 | use strict; |
574bacfe |
5 | # use warnings; # dont use warnings for older Perls |
0716bf9b |
6 | |
2ebb273f |
7 | our $VERSION = '0.51'; |
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 $b) |
50 | { |
2ebb273f |
51 | # avoid redefinitions |
52 | undef &_mul; |
53 | undef &_div; |
54 | |
8d8376c6 |
55 | if ($] > 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 | |
134 | # $e = 5 if $] < 5.006; # cap, for older Perls |
135 | # $e = 7 if $e > 7; # cap, for VMS, OS/390 and other 64 bit systems |
136 | # # 8 fails inside random testsuite, so take 7 |
8d8376c6 |
137 | my $int = 0; |
138 | if ($e > 7) |
139 | { |
140 | use integer; |
141 | my $e1 = 7; |
142 | $num = 7; |
143 | do |
144 | { |
145 | $num = ('9' x ++$e1) + 0; |
146 | $num *= $num + 1; |
147 | } while ("$num" =~ /9{$e1}0{$e1}/); # must be a certain pattern |
148 | $e1--; # last test failed, so retract one step |
149 | if ($e1 > 7) |
150 | { |
151 | $int = 1; $e = $e1; |
152 | } |
153 | } |
154 | |
155 | __PACKAGE__->_base_len($e,$int); # set and store |
394e6ffb |
156 | |
b68b7ab1 |
157 | use integer; |
394e6ffb |
158 | # find out how many bits _and, _or and _xor can take (old default = 16) |
159 | # I don't think anybody has yet 128 bit scalars, so let's play safe. |
394e6ffb |
160 | local $^W = 0; # don't warn about 'nonportable number' |
c38b2de2 |
161 | $AND_BITS = 15; $XOR_BITS = 15; $OR_BITS = 15; |
394e6ffb |
162 | |
163 | # find max bits, we will not go higher than numberofbits that fit into $BASE |
164 | # to make _and etc simpler (and faster for smaller, slower for large numbers) |
165 | my $max = 16; |
166 | while (2 ** $max < $BASE) { $max++; } |
1ddff52a |
167 | { |
168 | no integer; |
169 | $max = 16 if $] < 5.006; # older Perls might not take >16 too well |
170 | } |
394e6ffb |
171 | my ($x,$y,$z); |
172 | do { |
173 | $AND_BITS++; |
174 | $x = oct('0b' . '1' x $AND_BITS); $y = $x & $x; |
175 | $z = (2 ** $AND_BITS) - 1; |
176 | } while ($AND_BITS < $max && $x == $z && $y == $x); |
177 | $AND_BITS --; # retreat one step |
178 | do { |
179 | $XOR_BITS++; |
180 | $x = oct('0b' . '1' x $XOR_BITS); $y = $x ^ 0; |
181 | $z = (2 ** $XOR_BITS) - 1; |
182 | } while ($XOR_BITS < $max && $x == $z && $y == $x); |
183 | $XOR_BITS --; # retreat one step |
184 | do { |
185 | $OR_BITS++; |
186 | $x = oct('0b' . '1' x $OR_BITS); $y = $x | $x; |
187 | $z = (2 ** $OR_BITS) - 1; |
188 | } while ($OR_BITS < $max && $x == $z && $y == $x); |
189 | $OR_BITS --; # retreat one step |
190 | |
9b924220 |
191 | $AND_MASK = __PACKAGE__->_new( ( 2 ** $AND_BITS )); |
192 | $XOR_MASK = __PACKAGE__->_new( ( 2 ** $XOR_BITS )); |
193 | $OR_MASK = __PACKAGE__->_new( ( 2 ** $OR_BITS )); |
50109ad0 |
194 | |
195 | # We can compute the approximate lenght no faster than the real length: |
196 | *_alen = \&_len; |
394e6ffb |
197 | } |
0716bf9b |
198 | |
03874afe |
199 | ############################################################################### |
200 | |
0716bf9b |
201 | sub _zero |
202 | { |
203 | # create a zero |
61f5c3f5 |
204 | [ 0 ]; |
0716bf9b |
205 | } |
206 | |
207 | sub _one |
208 | { |
209 | # create a one |
61f5c3f5 |
210 | [ 1 ]; |
0716bf9b |
211 | } |
212 | |
027dc388 |
213 | sub _two |
214 | { |
1ddff52a |
215 | # create a two (used internally for shifting) |
61f5c3f5 |
216 | [ 2 ]; |
027dc388 |
217 | } |
218 | |
9b924220 |
219 | sub _ten |
220 | { |
221 | # create a 10 (used internally for shifting) |
222 | [ 10 ]; |
223 | } |
224 | |
50109ad0 |
225 | sub _1ex |
226 | { |
227 | # create a 1Ex |
228 | my $rem = $_[1] % $BASE_LEN; # remainder |
229 | my $parts = $_[1] / $BASE_LEN; # parts |
230 | |
231 | # 000000, 000000, 100 |
232 | [ (0) x $parts, '1' . ('0' x $rem) ]; |
233 | } |
234 | |
0716bf9b |
235 | sub _copy |
236 | { |
091c87b1 |
237 | # make a true copy |
61f5c3f5 |
238 | [ @{$_[1]} ]; |
0716bf9b |
239 | } |
240 | |
bd05a461 |
241 | # catch and throw away |
242 | sub import { } |
243 | |
0716bf9b |
244 | ############################################################################## |
245 | # convert back to string and number |
246 | |
247 | sub _str |
248 | { |
249 | # (ref to BINT) return num_str |
250 | # Convert number from internal base 100000 format to string format. |
251 | # internal format is always normalized (no leading zeros, "-0" => "+0") |
574bacfe |
252 | my $ar = $_[1]; |
61f5c3f5 |
253 | |
b68b7ab1 |
254 | my $l = scalar @$ar; # number of parts |
255 | if ($l < 1) # should not happen |
256 | { |
257 | require Carp; |
258 | Carp::croak("$_[1] has no elements"); |
259 | } |
61f5c3f5 |
260 | |
b68b7ab1 |
261 | my $ret = ""; |
0716bf9b |
262 | # handle first one different to strip leading zeros from it (there are no |
263 | # leading zero parts in internal representation) |
61f5c3f5 |
264 | $l --; $ret .= int($ar->[$l]); $l--; |
0716bf9b |
265 | # Interestingly, the pre-padd method uses more time |
091c87b1 |
266 | # the old grep variant takes longer (14 vs. 10 sec) |
574bacfe |
267 | my $z = '0' x ($BASE_LEN-1); |
0716bf9b |
268 | while ($l >= 0) |
269 | { |
574bacfe |
270 | $ret .= substr($z.$ar->[$l],-$BASE_LEN); # fastest way I could think of |
0716bf9b |
271 | $l--; |
272 | } |
9b924220 |
273 | $ret; |
0716bf9b |
274 | } |
275 | |
276 | sub _num |
277 | { |
9b924220 |
278 | # Make a number (scalar int/float) from a BigInt object |
574bacfe |
279 | my $x = $_[1]; |
9b924220 |
280 | |
281 | return 0+$x->[0] if scalar @$x == 1; # below $BASE |
0716bf9b |
282 | my $fac = 1; |
283 | my $num = 0; |
284 | foreach (@$x) |
285 | { |
286 | $num += $fac*$_; $fac *= $BASE; |
287 | } |
61f5c3f5 |
288 | $num; |
0716bf9b |
289 | } |
290 | |
291 | ############################################################################## |
292 | # actual math code |
293 | |
294 | sub _add |
295 | { |
296 | # (ref to int_num_array, ref to int_num_array) |
574bacfe |
297 | # routine to add two base 1eX numbers |
0716bf9b |
298 | # stolen from Knuth Vol 2 Algorithm A pg 231 |
b22b3e31 |
299 | # there are separate routines to add and sub as per Knuth pg 233 |
0716bf9b |
300 | # This routine clobbers up array x, but not y. |
301 | |
574bacfe |
302 | my ($c,$x,$y) = @_; |
b3abae2a |
303 | |
304 | return $x if (@$y == 1) && $y->[0] == 0; # $x + 0 => $x |
305 | if ((@$x == 1) && $x->[0] == 0) # 0 + $y => $y->copy |
306 | { |
7b29e1e6 |
307 | # twice as slow as $x = [ @$y ], but nec. to retain $x as ref :( |
b3abae2a |
308 | @$x = @$y; return $x; |
309 | } |
0716bf9b |
310 | |
311 | # for each in Y, add Y to X and carry. If after that, something is left in |
312 | # X, foreach in X add carry to X and then return X, carry |
091c87b1 |
313 | # Trades one "$j++" for having to shift arrays |
0716bf9b |
314 | my $i; my $car = 0; my $j = 0; |
315 | for $i (@$y) |
316 | { |
e745a66c |
317 | $x->[$j] -= $BASE if $car = (($x->[$j] += $i + $car) >= $BASE) ? 1 : 0; |
0716bf9b |
318 | $j++; |
319 | } |
320 | while ($car != 0) |
321 | { |
322 | $x->[$j] -= $BASE if $car = (($x->[$j] += $car) >= $BASE) ? 1 : 0; $j++; |
323 | } |
61f5c3f5 |
324 | $x; |
e745a66c |
325 | } |
326 | |
327 | sub _inc |
328 | { |
329 | # (ref to int_num_array, ref to int_num_array) |
091c87b1 |
330 | # Add 1 to $x, modify $x in place |
e745a66c |
331 | my ($c,$x) = @_; |
332 | |
333 | for my $i (@$x) |
334 | { |
335 | return $x if (($i += 1) < $BASE); # early out |
61f5c3f5 |
336 | $i = 0; # overflow, next |
e745a66c |
337 | } |
ae161977 |
338 | push @$x,1 if (($x->[-1] || 0) == 0); # last overflowed, so extend |
61f5c3f5 |
339 | $x; |
e745a66c |
340 | } |
341 | |
342 | sub _dec |
343 | { |
344 | # (ref to int_num_array, ref to int_num_array) |
091c87b1 |
345 | # Sub 1 from $x, modify $x in place |
e745a66c |
346 | my ($c,$x) = @_; |
347 | |
2ebb273f |
348 | my $MAX = $BASE-1; # since MAX_VAL based on BASE |
e745a66c |
349 | for my $i (@$x) |
350 | { |
351 | last if (($i -= 1) >= 0); # early out |
091c87b1 |
352 | $i = $MAX; # underflow, next |
e745a66c |
353 | } |
091c87b1 |
354 | pop @$x if $x->[-1] == 0 && @$x > 1; # last underflowed (but leave 0) |
61f5c3f5 |
355 | $x; |
0716bf9b |
356 | } |
357 | |
358 | sub _sub |
359 | { |
9393ace2 |
360 | # (ref to int_num_array, ref to int_num_array, swap) |
574bacfe |
361 | # subtract base 1eX numbers -- stolen from Knuth Vol 2 pg 232, $x > $y |
56b9c951 |
362 | # subtract Y from X by modifying x in place |
574bacfe |
363 | my ($c,$sx,$sy,$s) = @_; |
0716bf9b |
364 | |
365 | my $car = 0; my $i; my $j = 0; |
366 | if (!$s) |
367 | { |
0716bf9b |
368 | for $i (@$sx) |
369 | { |
370 | last unless defined $sy->[$j] || $car; |
0716bf9b |
371 | $i += $BASE if $car = (($i -= ($sy->[$j] || 0) + $car) < 0); $j++; |
0716bf9b |
372 | } |
373 | # might leave leading zeros, so fix that |
394e6ffb |
374 | return __strip_zeros($sx); |
0716bf9b |
375 | } |
394e6ffb |
376 | for $i (@$sx) |
0716bf9b |
377 | { |
07d34614 |
378 | # we can't do an early out if $x is < than $y, since we |
56b9c951 |
379 | # need to copy the high chunks from $y. Found by Bob Mathews. |
380 | #last unless defined $sy->[$j] || $car; |
394e6ffb |
381 | $sy->[$j] += $BASE |
382 | if $car = (($sy->[$j] = $i-($sy->[$j]||0) - $car) < 0); |
383 | $j++; |
0716bf9b |
384 | } |
394e6ffb |
385 | # might leave leading zeros, so fix that |
386 | __strip_zeros($sy); |
0716bf9b |
387 | } |
388 | |
ee15d750 |
389 | sub _mul_use_mul |
0716bf9b |
390 | { |
9393ace2 |
391 | # (ref to int_num_array, ref to int_num_array) |
0716bf9b |
392 | # multiply two numbers in internal representation |
b22b3e31 |
393 | # modifies first arg, second need not be different from first |
574bacfe |
394 | my ($c,$xv,$yv) = @_; |
dccbb853 |
395 | |
990fb837 |
396 | if (@$yv == 1) |
b3abae2a |
397 | { |
990fb837 |
398 | # shortcut for two very short numbers (improved by Nathan Zook) |
399 | # works also if xv and yv are the same reference, and handles also $x == 0 |
400 | if (@$xv == 1) |
401 | { |
2ebb273f |
402 | if (($xv->[0] *= $yv->[0]) >= $BASE) |
990fb837 |
403 | { |
2ebb273f |
404 | $xv->[0] = $xv->[0] - ($xv->[1] = int($xv->[0] * $RBASE)) * $BASE; |
990fb837 |
405 | }; |
406 | return $xv; |
407 | } |
408 | # $x * 0 => 0 |
409 | if ($yv->[0] == 0) |
410 | { |
411 | @$xv = (0); |
412 | return $xv; |
413 | } |
414 | # multiply a large number a by a single element one, so speed up |
415 | my $y = $yv->[0]; my $car = 0; |
416 | foreach my $i (@$xv) |
417 | { |
2ebb273f |
418 | $i = $i * $y + $car; $car = int($i * $RBASE); $i -= $car * $BASE; |
990fb837 |
419 | } |
420 | push @$xv, $car if $car != 0; |
b3abae2a |
421 | return $xv; |
422 | } |
990fb837 |
423 | # shortcut for result $x == 0 => result = 0 |
424 | return $xv if ( ((@$xv == 1) && ($xv->[0] == 0)) ); |
b3abae2a |
425 | |
0716bf9b |
426 | # since multiplying $x with $x fails, make copy in this case |
d614cd8b |
427 | $yv = [@$xv] if $xv == $yv; # same references? |
9393ace2 |
428 | |
61f5c3f5 |
429 | my @prod = (); my ($prod,$car,$cty,$xi,$yi); |
430 | |
0716bf9b |
431 | for $xi (@$xv) |
432 | { |
433 | $car = 0; $cty = 0; |
574bacfe |
434 | |
435 | # slow variant |
436 | # for $yi (@$yv) |
437 | # { |
438 | # $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
439 | # $prod[$cty++] = |
2ebb273f |
440 | # $prod - ($car = int($prod * RBASE)) * $BASE; # see USE_MUL |
574bacfe |
441 | # } |
442 | # $prod[$cty] += $car if $car; # need really to check for 0? |
443 | # $xi = shift @prod; |
444 | |
445 | # faster variant |
446 | # looping through this if $xi == 0 is silly - so optimize it away! |
447 | $xi = (shift @prod || 0), next if $xi == 0; |
0716bf9b |
448 | for $yi (@$yv) |
449 | { |
450 | $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
574bacfe |
451 | ## this is actually a tad slower |
452 | ## $prod = $prod[$cty]; $prod += ($car + $xi * $yi); # no ||0 here |
0716bf9b |
453 | $prod[$cty++] = |
2ebb273f |
454 | $prod - ($car = int($prod * $RBASE)) * $BASE; # see USE_MUL |
0716bf9b |
455 | } |
456 | $prod[$cty] += $car if $car; # need really to check for 0? |
027dc388 |
457 | $xi = shift @prod || 0; # || 0 makes v5.005_3 happy |
0716bf9b |
458 | } |
0716bf9b |
459 | push @$xv, @prod; |
50109ad0 |
460 | # can't have leading zeros |
461 | # __strip_zeros($xv); |
61f5c3f5 |
462 | $xv; |
0716bf9b |
463 | } |
464 | |
2ebb273f |
465 | sub _mul_use_div_64 |
466 | { |
467 | # (ref to int_num_array, ref to int_num_array) |
468 | # multiply two numbers in internal representation |
469 | # modifies first arg, second need not be different from first |
470 | # works for 64 bit integer with "use integer" |
471 | my ($c,$xv,$yv) = @_; |
472 | |
473 | use integer; |
474 | if (@$yv == 1) |
475 | { |
476 | # shortcut for two small numbers, also handles $x == 0 |
477 | if (@$xv == 1) |
478 | { |
479 | # shortcut for two very short numbers (improved by Nathan Zook) |
480 | # works also if xv and yv are the same reference, and handles also $x == 0 |
481 | if (($xv->[0] *= $yv->[0]) >= $BASE) |
482 | { |
483 | $xv->[0] = |
484 | $xv->[0] - ($xv->[1] = $xv->[0] / $BASE) * $BASE; |
485 | }; |
486 | return $xv; |
487 | } |
488 | # $x * 0 => 0 |
489 | if ($yv->[0] == 0) |
490 | { |
491 | @$xv = (0); |
492 | return $xv; |
493 | } |
494 | # multiply a large number a by a single element one, so speed up |
495 | my $y = $yv->[0]; my $car = 0; |
496 | foreach my $i (@$xv) |
497 | { |
498 | #$i = $i * $y + $car; $car = $i / $BASE; $i -= $car * $BASE; |
499 | $i = $i * $y + $car; $i -= ($car = $i / $BASE) * $BASE; |
500 | } |
501 | push @$xv, $car if $car != 0; |
502 | return $xv; |
503 | } |
504 | # shortcut for result $x == 0 => result = 0 |
505 | return $xv if ( ((@$xv == 1) && ($xv->[0] == 0)) ); |
506 | |
507 | # since multiplying $x with $x fails, make copy in this case |
508 | $yv = [@$xv] if $xv == $yv; # same references? |
509 | |
510 | my @prod = (); my ($prod,$car,$cty,$xi,$yi); |
511 | for $xi (@$xv) |
512 | { |
513 | $car = 0; $cty = 0; |
514 | # looping through this if $xi == 0 is silly - so optimize it away! |
515 | $xi = (shift @prod || 0), next if $xi == 0; |
516 | for $yi (@$yv) |
517 | { |
518 | $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
519 | $prod[$cty++] = $prod - ($car = $prod / $BASE) * $BASE; |
520 | } |
521 | $prod[$cty] += $car if $car; # need really to check for 0? |
522 | $xi = shift @prod || 0; # || 0 makes v5.005_3 happy |
523 | } |
524 | push @$xv, @prod; |
525 | $xv; |
526 | } |
527 | |
ee15d750 |
528 | sub _mul_use_div |
529 | { |
9393ace2 |
530 | # (ref to int_num_array, ref to int_num_array) |
ee15d750 |
531 | # multiply two numbers in internal representation |
532 | # modifies first arg, second need not be different from first |
533 | my ($c,$xv,$yv) = @_; |
50109ad0 |
534 | |
990fb837 |
535 | if (@$yv == 1) |
b3abae2a |
536 | { |
990fb837 |
537 | # shortcut for two small numbers, also handles $x == 0 |
538 | if (@$xv == 1) |
539 | { |
540 | # shortcut for two very short numbers (improved by Nathan Zook) |
541 | # works also if xv and yv are the same reference, and handles also $x == 0 |
2ebb273f |
542 | if (($xv->[0] *= $yv->[0]) >= $BASE) |
990fb837 |
543 | { |
544 | $xv->[0] = |
2ebb273f |
545 | $xv->[0] - ($xv->[1] = int($xv->[0] / $BASE)) * $BASE; |
990fb837 |
546 | }; |
547 | return $xv; |
548 | } |
549 | # $x * 0 => 0 |
550 | if ($yv->[0] == 0) |
551 | { |
552 | @$xv = (0); |
553 | return $xv; |
554 | } |
555 | # multiply a large number a by a single element one, so speed up |
556 | my $y = $yv->[0]; my $car = 0; |
557 | foreach my $i (@$xv) |
558 | { |
2ebb273f |
559 | $i = $i * $y + $car; $car = int($i / $BASE); $i -= $car * $BASE; |
560 | # This (together with use integer;) does not work on 32-bit Perls |
561 | #$i = $i * $y + $car; $i -= ($car = $i / $BASE) * $BASE; |
990fb837 |
562 | } |
563 | push @$xv, $car if $car != 0; |
b3abae2a |
564 | return $xv; |
565 | } |
990fb837 |
566 | # shortcut for result $x == 0 => result = 0 |
567 | return $xv if ( ((@$xv == 1) && ($xv->[0] == 0)) ); |
b3abae2a |
568 | |
ee15d750 |
569 | # since multiplying $x with $x fails, make copy in this case |
d614cd8b |
570 | $yv = [@$xv] if $xv == $yv; # same references? |
9393ace2 |
571 | |
61f5c3f5 |
572 | my @prod = (); my ($prod,$car,$cty,$xi,$yi); |
ee15d750 |
573 | for $xi (@$xv) |
574 | { |
575 | $car = 0; $cty = 0; |
576 | # looping through this if $xi == 0 is silly - so optimize it away! |
577 | $xi = (shift @prod || 0), next if $xi == 0; |
578 | for $yi (@$yv) |
579 | { |
580 | $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
2ebb273f |
581 | $prod[$cty++] = $prod - ($car = int($prod / $BASE)) * $BASE; |
ee15d750 |
582 | } |
583 | $prod[$cty] += $car if $car; # need really to check for 0? |
027dc388 |
584 | $xi = shift @prod || 0; # || 0 makes v5.005_3 happy |
ee15d750 |
585 | } |
586 | push @$xv, @prod; |
50109ad0 |
587 | # can't have leading zeros |
588 | # __strip_zeros($xv); |
61f5c3f5 |
589 | $xv; |
ee15d750 |
590 | } |
591 | |
592 | sub _div_use_mul |
0716bf9b |
593 | { |
b22b3e31 |
594 | # ref to array, ref to array, modify first array and return remainder if |
0716bf9b |
595 | # in list context |
aef458a0 |
596 | |
597 | # see comments in _div_use_div() for more explanations |
598 | |
574bacfe |
599 | my ($c,$x,$yorg) = @_; |
aef458a0 |
600 | |
601 | # the general div algorithmn here is about O(N*N) and thus quite slow, so |
602 | # we first check for some special cases and use shortcuts to handle them. |
0716bf9b |
603 | |
aef458a0 |
604 | # This works, because we store the numbers in a chunked format where each |
605 | # element contains 5..7 digits (depending on system). |
606 | |
607 | # if both numbers have only one element: |
61f5c3f5 |
608 | if (@$x == 1 && @$yorg == 1) |
609 | { |
13a12e00 |
610 | # shortcut, $yorg and $x are two small numbers |
61f5c3f5 |
611 | if (wantarray) |
612 | { |
613 | my $r = [ $x->[0] % $yorg->[0] ]; |
614 | $x->[0] = int($x->[0] / $yorg->[0]); |
615 | return ($x,$r); |
616 | } |
617 | else |
618 | { |
619 | $x->[0] = int($x->[0] / $yorg->[0]); |
620 | return $x; |
621 | } |
622 | } |
aef458a0 |
623 | |
624 | # if x has more than one, but y has only one element: |
28df3e88 |
625 | if (@$yorg == 1) |
626 | { |
627 | my $rem; |
628 | $rem = _mod($c,[ @$x ],$yorg) if wantarray; |
13a12e00 |
629 | |
28df3e88 |
630 | # shortcut, $y is < $BASE |
631 | my $j = scalar @$x; my $r = 0; |
632 | my $y = $yorg->[0]; my $b; |
633 | while ($j-- > 0) |
634 | { |
2ebb273f |
635 | $b = $r * $BASE + $x->[$j]; |
28df3e88 |
636 | $x->[$j] = int($b/$y); |
637 | $r = $b % $y; |
638 | } |
639 | pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero |
640 | return ($x,$rem) if wantarray; |
641 | return $x; |
642 | } |
0716bf9b |
643 | |
aef458a0 |
644 | # now x and y have more than one element |
645 | |
646 | # check whether y has more elements than x, if yet, the result will be 0 |
647 | if (@$yorg > @$x) |
648 | { |
649 | my $rem; |
650 | $rem = [@$x] if wantarray; # make copy |
651 | splice (@$x,1); # keep ref to original array |
652 | $x->[0] = 0; # set to 0 |
653 | return ($x,$rem) if wantarray; # including remainder? |
654 | return $x; # only x, which is [0] now |
655 | } |
656 | # check whether the numbers have the same number of elements, in that case |
657 | # the result will fit into one element and can be computed efficiently |
658 | if (@$yorg == @$x) |
659 | { |
660 | my $rem; |
661 | # if $yorg has more digits than $x (it's leading element is longer than |
662 | # the one from $x), the result will also be 0: |
663 | if (length(int($yorg->[-1])) > length(int($x->[-1]))) |
664 | { |
665 | $rem = [@$x] if wantarray; # make copy |
666 | splice (@$x,1); # keep ref to org array |
667 | $x->[0] = 0; # set to 0 |
668 | return ($x,$rem) if wantarray; # including remainder? |
669 | return $x; |
670 | } |
671 | # now calculate $x / $yorg |
672 | if (length(int($yorg->[-1])) == length(int($x->[-1]))) |
673 | { |
b68b7ab1 |
674 | # same length, so make full compare |
675 | |
aef458a0 |
676 | my $a = 0; my $j = scalar @$x - 1; |
677 | # manual way (abort if unequal, good for early ne) |
678 | while ($j >= 0) |
679 | { |
680 | last if ($a = $x->[$j] - $yorg->[$j]); $j--; |
681 | } |
682 | # $a contains the result of the compare between X and Y |
b68b7ab1 |
683 | # a < 0: x < y, a == 0: x == y, a > 0: x > y |
aef458a0 |
684 | if ($a <= 0) |
685 | { |
b68b7ab1 |
686 | $rem = [ 0 ]; # a = 0 => x == y => rem 0 |
687 | $rem = [@$x] if $a != 0; # a < 0 => x < y => rem = x |
688 | splice(@$x,1); # keep single element |
689 | $x->[0] = 0; # if $a < 0 |
690 | $x->[0] = 1 if $a == 0; # $x == $y |
7596a890 |
691 | return ($x,$rem) if wantarray; |
692 | return $x; |
aef458a0 |
693 | } |
b68b7ab1 |
694 | # $x >= $y, so proceed normally |
aef458a0 |
695 | } |
696 | } |
697 | |
698 | # all other cases: |
699 | |
d614cd8b |
700 | my $y = [ @$yorg ]; # always make copy to preserve |
61f5c3f5 |
701 | |
702 | my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0); |
703 | |
704 | $car = $bar = $prd = 0; |
2ebb273f |
705 | if (($dd = int($BASE/($y->[-1]+1))) != 1) |
0716bf9b |
706 | { |
707 | for $xi (@$x) |
708 | { |
709 | $xi = $xi * $dd + $car; |
2ebb273f |
710 | $xi -= ($car = int($xi * $RBASE)) * $BASE; # see USE_MUL |
0716bf9b |
711 | } |
712 | push(@$x, $car); $car = 0; |
713 | for $yi (@$y) |
714 | { |
715 | $yi = $yi * $dd + $car; |
2ebb273f |
716 | $yi -= ($car = int($yi * $RBASE)) * $BASE; # see USE_MUL |
0716bf9b |
717 | } |
718 | } |
719 | else |
720 | { |
721 | push(@$x, 0); |
722 | } |
723 | @q = (); ($v2,$v1) = @$y[-2,-1]; |
724 | $v2 = 0 unless $v2; |
725 | while ($#$x > $#$y) |
726 | { |
727 | ($u2,$u1,$u0) = @$x[-3..-1]; |
728 | $u2 = 0 unless $u2; |
729 | #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" |
730 | # if $v1 == 0; |
2ebb273f |
731 | $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$BASE+$u1)/$v1)); |
732 | --$q while ($v2*$q > ($u0*$BASE+$u1-$q*$v1)*$BASE+$u2); |
0716bf9b |
733 | if ($q) |
734 | { |
735 | ($car, $bar) = (0,0); |
736 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
737 | { |
738 | $prd = $q * $y->[$yi] + $car; |
2ebb273f |
739 | $prd -= ($car = int($prd * $RBASE)) * $BASE; # see USE_MUL |
740 | $x->[$xi] += $BASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0)); |
0716bf9b |
741 | } |
742 | if ($x->[-1] < $car + $bar) |
743 | { |
744 | $car = 0; --$q; |
745 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
746 | { |
2ebb273f |
747 | $x->[$xi] -= $BASE |
748 | if ($car = (($x->[$xi] += $y->[$yi] + $car) >= $BASE)); |
0716bf9b |
749 | } |
750 | } |
751 | } |
aef458a0 |
752 | pop(@$x); |
753 | unshift(@q, $q); |
0716bf9b |
754 | } |
755 | if (wantarray) |
756 | { |
757 | @d = (); |
758 | if ($dd != 1) |
759 | { |
760 | $car = 0; |
761 | for $xi (reverse @$x) |
762 | { |
2ebb273f |
763 | $prd = $car * $BASE + $xi; |
0716bf9b |
764 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; # see USE_MUL |
765 | unshift(@d, $tmp); |
766 | } |
767 | } |
768 | else |
769 | { |
770 | @d = @$x; |
771 | } |
772 | @$x = @q; |
61f5c3f5 |
773 | my $d = \@d; |
990fb837 |
774 | __strip_zeros($x); |
775 | __strip_zeros($d); |
61f5c3f5 |
776 | return ($x,$d); |
0716bf9b |
777 | } |
778 | @$x = @q; |
990fb837 |
779 | __strip_zeros($x); |
61f5c3f5 |
780 | $x; |
0716bf9b |
781 | } |
782 | |
2ebb273f |
783 | sub _div_use_div_64 |
784 | { |
785 | # ref to array, ref to array, modify first array and return remainder if |
786 | # in list context |
787 | # This version works on 64 bit integers |
788 | my ($c,$x,$yorg) = @_; |
789 | |
790 | use integer; |
791 | # the general div algorithmn here is about O(N*N) and thus quite slow, so |
792 | # we first check for some special cases and use shortcuts to handle them. |
793 | |
794 | # This works, because we store the numbers in a chunked format where each |
795 | # element contains 5..7 digits (depending on system). |
796 | |
797 | # if both numbers have only one element: |
798 | if (@$x == 1 && @$yorg == 1) |
799 | { |
800 | # shortcut, $yorg and $x are two small numbers |
801 | if (wantarray) |
802 | { |
803 | my $r = [ $x->[0] % $yorg->[0] ]; |
804 | $x->[0] = int($x->[0] / $yorg->[0]); |
805 | return ($x,$r); |
806 | } |
807 | else |
808 | { |
809 | $x->[0] = int($x->[0] / $yorg->[0]); |
810 | return $x; |
811 | } |
812 | } |
813 | # if x has more than one, but y has only one element: |
814 | if (@$yorg == 1) |
815 | { |
816 | my $rem; |
817 | $rem = _mod($c,[ @$x ],$yorg) if wantarray; |
818 | |
819 | # shortcut, $y is < $BASE |
820 | my $j = scalar @$x; my $r = 0; |
821 | my $y = $yorg->[0]; my $b; |
822 | while ($j-- > 0) |
823 | { |
824 | $b = $r * $BASE + $x->[$j]; |
825 | $x->[$j] = int($b/$y); |
826 | $r = $b % $y; |
827 | } |
828 | pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero |
829 | return ($x,$rem) if wantarray; |
830 | return $x; |
831 | } |
832 | # now x and y have more than one element |
833 | |
834 | # check whether y has more elements than x, if yet, the result will be 0 |
835 | if (@$yorg > @$x) |
836 | { |
837 | my $rem; |
838 | $rem = [@$x] if wantarray; # make copy |
839 | splice (@$x,1); # keep ref to original array |
840 | $x->[0] = 0; # set to 0 |
841 | return ($x,$rem) if wantarray; # including remainder? |
842 | return $x; # only x, which is [0] now |
843 | } |
844 | # check whether the numbers have the same number of elements, in that case |
845 | # the result will fit into one element and can be computed efficiently |
846 | if (@$yorg == @$x) |
847 | { |
848 | my $rem; |
849 | # if $yorg has more digits than $x (it's leading element is longer than |
850 | # the one from $x), the result will also be 0: |
851 | if (length(int($yorg->[-1])) > length(int($x->[-1]))) |
852 | { |
853 | $rem = [@$x] if wantarray; # make copy |
854 | splice (@$x,1); # keep ref to org array |
855 | $x->[0] = 0; # set to 0 |
856 | return ($x,$rem) if wantarray; # including remainder? |
857 | return $x; |
858 | } |
859 | # now calculate $x / $yorg |
860 | |
861 | if (length(int($yorg->[-1])) == length(int($x->[-1]))) |
862 | { |
863 | # same length, so make full compare |
864 | |
865 | my $a = 0; my $j = scalar @$x - 1; |
866 | # manual way (abort if unequal, good for early ne) |
867 | while ($j >= 0) |
868 | { |
869 | last if ($a = $x->[$j] - $yorg->[$j]); $j--; |
870 | } |
871 | # $a contains the result of the compare between X and Y |
872 | # a < 0: x < y, a == 0: x == y, a > 0: x > y |
873 | if ($a <= 0) |
874 | { |
875 | $rem = [ 0 ]; # a = 0 => x == y => rem 0 |
876 | $rem = [@$x] if $a != 0; # a < 0 => x < y => rem = x |
877 | splice(@$x,1); # keep single element |
878 | $x->[0] = 0; # if $a < 0 |
879 | $x->[0] = 1 if $a == 0; # $x == $y |
880 | return ($x,$rem) if wantarray; # including remainder? |
881 | return $x; |
882 | } |
883 | # $x >= $y, so proceed normally |
884 | |
885 | } |
886 | } |
887 | |
888 | # all other cases: |
889 | |
890 | my $y = [ @$yorg ]; # always make copy to preserve |
891 | |
892 | my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0); |
893 | |
894 | $car = $bar = $prd = 0; |
895 | if (($dd = int($BASE/($y->[-1]+1))) != 1) |
896 | { |
897 | for $xi (@$x) |
898 | { |
899 | $xi = $xi * $dd + $car; |
900 | $xi -= ($car = int($xi / $BASE)) * $BASE; |
901 | } |
902 | push(@$x, $car); $car = 0; |
903 | for $yi (@$y) |
904 | { |
905 | $yi = $yi * $dd + $car; |
906 | $yi -= ($car = int($yi / $BASE)) * $BASE; |
907 | } |
908 | } |
909 | else |
910 | { |
911 | push(@$x, 0); |
912 | } |
913 | |
914 | # @q will accumulate the final result, $q contains the current computed |
915 | # part of the final result |
916 | |
917 | @q = (); ($v2,$v1) = @$y[-2,-1]; |
918 | $v2 = 0 unless $v2; |
919 | while ($#$x > $#$y) |
920 | { |
921 | ($u2,$u1,$u0) = @$x[-3..-1]; |
922 | $u2 = 0 unless $u2; |
923 | #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" |
924 | # if $v1 == 0; |
925 | $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$BASE+$u1)/$v1)); |
926 | --$q while ($v2*$q > ($u0*$BASE+$u1-$q*$v1)*$BASE+$u2); |
927 | if ($q) |
928 | { |
929 | ($car, $bar) = (0,0); |
930 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
931 | { |
932 | $prd = $q * $y->[$yi] + $car; |
933 | $prd -= ($car = int($prd / $BASE)) * $BASE; |
934 | $x->[$xi] += $BASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0)); |
935 | } |
936 | if ($x->[-1] < $car + $bar) |
937 | { |
938 | $car = 0; --$q; |
939 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
940 | { |
941 | $x->[$xi] -= $BASE |
942 | if ($car = (($x->[$xi] += $y->[$yi] + $car) >= $BASE)); |
943 | } |
944 | } |
945 | } |
946 | pop(@$x); unshift(@q, $q); |
947 | } |
948 | if (wantarray) |
949 | { |
950 | @d = (); |
951 | if ($dd != 1) |
952 | { |
953 | $car = 0; |
954 | for $xi (reverse @$x) |
955 | { |
956 | $prd = $car * $BASE + $xi; |
957 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; |
958 | unshift(@d, $tmp); |
959 | } |
960 | } |
961 | else |
962 | { |
963 | @d = @$x; |
964 | } |
965 | @$x = @q; |
966 | my $d = \@d; |
967 | __strip_zeros($x); |
968 | __strip_zeros($d); |
969 | return ($x,$d); |
970 | } |
971 | @$x = @q; |
972 | __strip_zeros($x); |
973 | $x; |
974 | } |
975 | |
ee15d750 |
976 | sub _div_use_div |
977 | { |
978 | # ref to array, ref to array, modify first array and return remainder if |
979 | # in list context |
ee15d750 |
980 | my ($c,$x,$yorg) = @_; |
ee15d750 |
981 | |
990fb837 |
982 | # the general div algorithmn here is about O(N*N) and thus quite slow, so |
983 | # we first check for some special cases and use shortcuts to handle them. |
984 | |
985 | # This works, because we store the numbers in a chunked format where each |
986 | # element contains 5..7 digits (depending on system). |
987 | |
988 | # if both numbers have only one element: |
61f5c3f5 |
989 | if (@$x == 1 && @$yorg == 1) |
990 | { |
13a12e00 |
991 | # shortcut, $yorg and $x are two small numbers |
61f5c3f5 |
992 | if (wantarray) |
993 | { |
994 | my $r = [ $x->[0] % $yorg->[0] ]; |
995 | $x->[0] = int($x->[0] / $yorg->[0]); |
996 | return ($x,$r); |
997 | } |
998 | else |
999 | { |
1000 | $x->[0] = int($x->[0] / $yorg->[0]); |
1001 | return $x; |
1002 | } |
1003 | } |
990fb837 |
1004 | # if x has more than one, but y has only one element: |
28df3e88 |
1005 | if (@$yorg == 1) |
1006 | { |
1007 | my $rem; |
1008 | $rem = _mod($c,[ @$x ],$yorg) if wantarray; |
1009 | |
1010 | # shortcut, $y is < $BASE |
1011 | my $j = scalar @$x; my $r = 0; |
1012 | my $y = $yorg->[0]; my $b; |
1013 | while ($j-- > 0) |
1014 | { |
2ebb273f |
1015 | $b = $r * $BASE + $x->[$j]; |
28df3e88 |
1016 | $x->[$j] = int($b/$y); |
1017 | $r = $b % $y; |
1018 | } |
1019 | pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero |
1020 | return ($x,$rem) if wantarray; |
1021 | return $x; |
1022 | } |
990fb837 |
1023 | # now x and y have more than one element |
ee15d750 |
1024 | |
990fb837 |
1025 | # check whether y has more elements than x, if yet, the result will be 0 |
1026 | if (@$yorg > @$x) |
61f5c3f5 |
1027 | { |
990fb837 |
1028 | my $rem; |
1029 | $rem = [@$x] if wantarray; # make copy |
1030 | splice (@$x,1); # keep ref to original array |
1031 | $x->[0] = 0; # set to 0 |
1032 | return ($x,$rem) if wantarray; # including remainder? |
aef458a0 |
1033 | return $x; # only x, which is [0] now |
61f5c3f5 |
1034 | } |
990fb837 |
1035 | # check whether the numbers have the same number of elements, in that case |
1036 | # the result will fit into one element and can be computed efficiently |
1037 | if (@$yorg == @$x) |
1038 | { |
1039 | my $rem; |
1040 | # if $yorg has more digits than $x (it's leading element is longer than |
1041 | # the one from $x), the result will also be 0: |
1042 | if (length(int($yorg->[-1])) > length(int($x->[-1]))) |
1043 | { |
1044 | $rem = [@$x] if wantarray; # make copy |
1045 | splice (@$x,1); # keep ref to org array |
1046 | $x->[0] = 0; # set to 0 |
1047 | return ($x,$rem) if wantarray; # including remainder? |
1048 | return $x; |
1049 | } |
1050 | # now calculate $x / $yorg |
091c87b1 |
1051 | |
990fb837 |
1052 | if (length(int($yorg->[-1])) == length(int($x->[-1]))) |
1053 | { |
b68b7ab1 |
1054 | # same length, so make full compare |
1055 | |
990fb837 |
1056 | my $a = 0; my $j = scalar @$x - 1; |
1057 | # manual way (abort if unequal, good for early ne) |
1058 | while ($j >= 0) |
1059 | { |
1060 | last if ($a = $x->[$j] - $yorg->[$j]); $j--; |
1061 | } |
aef458a0 |
1062 | # $a contains the result of the compare between X and Y |
b68b7ab1 |
1063 | # a < 0: x < y, a == 0: x == y, a > 0: x > y |
990fb837 |
1064 | if ($a <= 0) |
1065 | { |
b68b7ab1 |
1066 | $rem = [ 0 ]; # a = 0 => x == y => rem 0 |
1067 | $rem = [@$x] if $a != 0; # a < 0 => x < y => rem = x |
aef458a0 |
1068 | splice(@$x,1); # keep single element |
990fb837 |
1069 | $x->[0] = 0; # if $a < 0 |
b68b7ab1 |
1070 | $x->[0] = 1 if $a == 0; # $x == $y |
7596a890 |
1071 | return ($x,$rem) if wantarray; # including remainder? |
1072 | return $x; |
990fb837 |
1073 | } |
aef458a0 |
1074 | # $x >= $y, so proceed normally |
b68b7ab1 |
1075 | |
990fb837 |
1076 | } |
990fb837 |
1077 | } |
1078 | |
1079 | # all other cases: |
1080 | |
1081 | my $y = [ @$yorg ]; # always make copy to preserve |
61f5c3f5 |
1082 | |
1083 | my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0); |
1084 | |
1085 | $car = $bar = $prd = 0; |
2ebb273f |
1086 | if (($dd = int($BASE/($y->[-1]+1))) != 1) |
ee15d750 |
1087 | { |
1088 | for $xi (@$x) |
1089 | { |
1090 | $xi = $xi * $dd + $car; |
2ebb273f |
1091 | $xi -= ($car = int($xi / $BASE)) * $BASE; |
ee15d750 |
1092 | } |
1093 | push(@$x, $car); $car = 0; |
1094 | for $yi (@$y) |
1095 | { |
1096 | $yi = $yi * $dd + $car; |
2ebb273f |
1097 | $yi -= ($car = int($yi / $BASE)) * $BASE; |
ee15d750 |
1098 | } |
1099 | } |
1100 | else |
1101 | { |
1102 | push(@$x, 0); |
1103 | } |
aef458a0 |
1104 | |
1105 | # @q will accumulate the final result, $q contains the current computed |
1106 | # part of the final result |
1107 | |
ee15d750 |
1108 | @q = (); ($v2,$v1) = @$y[-2,-1]; |
1109 | $v2 = 0 unless $v2; |
1110 | while ($#$x > $#$y) |
1111 | { |
1112 | ($u2,$u1,$u0) = @$x[-3..-1]; |
1113 | $u2 = 0 unless $u2; |
1114 | #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" |
1115 | # if $v1 == 0; |
2ebb273f |
1116 | $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$BASE+$u1)/$v1)); |
1117 | --$q while ($v2*$q > ($u0*$BASE+$u1-$q*$v1)*$BASE+$u2); |
ee15d750 |
1118 | if ($q) |
1119 | { |
1120 | ($car, $bar) = (0,0); |
1121 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
1122 | { |
1123 | $prd = $q * $y->[$yi] + $car; |
2ebb273f |
1124 | $prd -= ($car = int($prd / $BASE)) * $BASE; |
1125 | $x->[$xi] += $BASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0)); |
ee15d750 |
1126 | } |
1127 | if ($x->[-1] < $car + $bar) |
1128 | { |
1129 | $car = 0; --$q; |
1130 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
1131 | { |
2ebb273f |
1132 | $x->[$xi] -= $BASE |
1133 | if ($car = (($x->[$xi] += $y->[$yi] + $car) >= $BASE)); |
ee15d750 |
1134 | } |
1135 | } |
1136 | } |
61f5c3f5 |
1137 | pop(@$x); unshift(@q, $q); |
ee15d750 |
1138 | } |
1139 | if (wantarray) |
1140 | { |
1141 | @d = (); |
1142 | if ($dd != 1) |
1143 | { |
1144 | $car = 0; |
1145 | for $xi (reverse @$x) |
1146 | { |
2ebb273f |
1147 | $prd = $car * $BASE + $xi; |
ee15d750 |
1148 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; |
1149 | unshift(@d, $tmp); |
1150 | } |
1151 | } |
1152 | else |
1153 | { |
1154 | @d = @$x; |
1155 | } |
1156 | @$x = @q; |
61f5c3f5 |
1157 | my $d = \@d; |
990fb837 |
1158 | __strip_zeros($x); |
1159 | __strip_zeros($d); |
61f5c3f5 |
1160 | return ($x,$d); |
ee15d750 |
1161 | } |
1162 | @$x = @q; |
990fb837 |
1163 | __strip_zeros($x); |
61f5c3f5 |
1164 | $x; |
ee15d750 |
1165 | } |
1166 | |
394e6ffb |
1167 | ############################################################################## |
1168 | # testing |
1169 | |
1170 | sub _acmp |
1171 | { |
1172 | # internal absolute post-normalized compare (ignore signs) |
1173 | # ref to array, ref to array, return <0, 0, >0 |
1174 | # arrays must have at least one entry; this is not checked for |
394e6ffb |
1175 | my ($c,$cx,$cy) = @_; |
091c87b1 |
1176 | |
1177 | # shortcut for short numbers |
1178 | return (($cx->[0] <=> $cy->[0]) <=> 0) |
1179 | if scalar @$cx == scalar @$cy && scalar @$cx == 1; |
394e6ffb |
1180 | |
f9a08e12 |
1181 | # fast comp based on number of array elements (aka pseudo-length) |
091c87b1 |
1182 | my $lxy = (scalar @$cx - scalar @$cy) |
1183 | # or length of first element if same number of elements (aka difference 0) |
1184 | || |
1185 | # need int() here because sometimes the last element is '00018' vs '18' |
1186 | (length(int($cx->[-1])) - length(int($cy->[-1]))); |
394e6ffb |
1187 | return -1 if $lxy < 0; # already differs, ret |
1188 | return 1 if $lxy > 0; # ditto |
56d9de68 |
1189 | |
394e6ffb |
1190 | # manual way (abort if unequal, good for early ne) |
091c87b1 |
1191 | my $a; my $j = scalar @$cx; |
1192 | while (--$j >= 0) |
9393ace2 |
1193 | { |
091c87b1 |
1194 | last if ($a = $cx->[$j] - $cy->[$j]); |
9393ace2 |
1195 | } |
091c87b1 |
1196 | $a <=> 0; |
394e6ffb |
1197 | } |
1198 | |
1199 | sub _len |
1200 | { |
50109ad0 |
1201 | # compute number of digits in base 10 |
394e6ffb |
1202 | |
1203 | # int() because add/sub sometimes leaves strings (like '00005') instead of |
1204 | # '5' in this place, thus causing length() to report wrong length |
1205 | my $cx = $_[1]; |
1206 | |
56d9de68 |
1207 | (@$cx-1)*$BASE_LEN+length(int($cx->[-1])); |
394e6ffb |
1208 | } |
1209 | |
1210 | sub _digit |
1211 | { |
1212 | # return the nth digit, negative values count backward |
1213 | # zero is rightmost, so _digit(123,0) will give 3 |
1214 | my ($c,$x,$n) = @_; |
1215 | |
1216 | my $len = _len('',$x); |
1217 | |
1218 | $n = $len+$n if $n < 0; # -1 last, -2 second-to-last |
1219 | $n = abs($n); # if negative was too big |
1220 | $len--; $n = $len if $n > $len; # n to big? |
1221 | |
1222 | my $elem = int($n / $BASE_LEN); # which array element |
1223 | my $digit = $n % $BASE_LEN; # which digit in this element |
52edfb59 |
1224 | $elem = '0' x $BASE_LEN . @$x[$elem]; # get element padded with 0's |
56d9de68 |
1225 | substr($elem,-$digit-1,1); |
394e6ffb |
1226 | } |
1227 | |
1228 | sub _zeros |
1229 | { |
1230 | # return amount of trailing zeros in decimal |
1231 | # check each array elem in _m for having 0 at end as long as elem == 0 |
1232 | # Upon finding a elem != 0, stop |
1233 | my $x = $_[1]; |
9b924220 |
1234 | |
1235 | return 0 if scalar @$x == 1 && $x->[0] == 0; |
1236 | |
394e6ffb |
1237 | my $zeros = 0; my $elem; |
1238 | foreach my $e (@$x) |
1239 | { |
1240 | if ($e != 0) |
1241 | { |
1242 | $elem = "$e"; # preserve x |
1243 | $elem =~ s/.*?(0*$)/$1/; # strip anything not zero |
1244 | $zeros *= $BASE_LEN; # elems * 5 |
61f5c3f5 |
1245 | $zeros += length($elem); # count trailing zeros |
394e6ffb |
1246 | last; # early out |
1247 | } |
1248 | $zeros ++; # real else branch: 50% slower! |
1249 | } |
61f5c3f5 |
1250 | $zeros; |
394e6ffb |
1251 | } |
1252 | |
1253 | ############################################################################## |
1254 | # _is_* routines |
1255 | |
1256 | sub _is_zero |
1257 | { |
9b924220 |
1258 | # return true if arg is zero |
1259 | (((scalar @{$_[1]} == 1) && ($_[1]->[0] == 0))) <=> 0; |
394e6ffb |
1260 | } |
1261 | |
1262 | sub _is_even |
1263 | { |
9b924220 |
1264 | # return true if arg is even |
1265 | (!($_[1]->[0] & 1)) <=> 0; |
394e6ffb |
1266 | } |
1267 | |
1268 | sub _is_odd |
1269 | { |
9b924220 |
1270 | # return true if arg is even |
1271 | (($_[1]->[0] & 1)) <=> 0; |
394e6ffb |
1272 | } |
1273 | |
1274 | sub _is_one |
1275 | { |
9b924220 |
1276 | # return true if arg is one |
1277 | (scalar @{$_[1]} == 1) && ($_[1]->[0] == 1) <=> 0; |
1278 | } |
1279 | |
1280 | sub _is_two |
1281 | { |
1282 | # return true if arg is two |
1283 | (scalar @{$_[1]} == 1) && ($_[1]->[0] == 2) <=> 0; |
1284 | } |
61f5c3f5 |
1285 | |
9b924220 |
1286 | sub _is_ten |
1287 | { |
1288 | # return true if arg is ten |
1289 | (scalar @{$_[1]} == 1) && ($_[1]->[0] == 10) <=> 0; |
394e6ffb |
1290 | } |
1291 | |
1292 | sub __strip_zeros |
1293 | { |
1294 | # internal normalization function that strips leading zeros from the array |
1295 | # args: ref to array |
1296 | my $s = shift; |
1297 | |
1298 | my $cnt = scalar @$s; # get count of parts |
1299 | my $i = $cnt-1; |
1300 | push @$s,0 if $i < 0; # div might return empty results, so fix it |
1301 | |
61f5c3f5 |
1302 | return $s if @$s == 1; # early out |
1303 | |
394e6ffb |
1304 | #print "strip: cnt $cnt i $i\n"; |
1305 | # '0', '3', '4', '0', '0', |
1306 | # 0 1 2 3 4 |
1307 | # cnt = 5, i = 4 |
1308 | # i = 4 |
1309 | # i = 3 |
1310 | # => fcnt = cnt - i (5-2 => 3, cnt => 5-1 = 4, throw away from 4th pos) |
1311 | # >= 1: skip first part (this can be zero) |
1312 | while ($i > 0) { last if $s->[$i] != 0; $i--; } |
1313 | $i++; splice @$s,$i if ($i < $cnt); # $i cant be 0 |
1314 | $s; |
1315 | } |
1316 | |
1317 | ############################################################################### |
3a427a11 |
1318 | # check routine to test internal state for corruptions |
394e6ffb |
1319 | |
1320 | sub _check |
1321 | { |
1322 | # used by the test suite |
1323 | my $x = $_[1]; |
1324 | |
1325 | return "$x is not a reference" if !ref($x); |
1326 | |
1327 | # are all parts are valid? |
1328 | my $i = 0; my $j = scalar @$x; my ($e,$try); |
1329 | while ($i < $j) |
1330 | { |
1331 | $e = $x->[$i]; $e = 'undef' unless defined $e; |
1332 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e)"; |
1333 | last if $e !~ /^[+]?[0-9]+$/; |
1334 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (stringify)"; |
1335 | last if "$e" !~ /^[+]?[0-9]+$/; |
1336 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (cat-stringify)"; |
1337 | last if '' . "$e" !~ /^[+]?[0-9]+$/; |
1338 | $try = ' < 0 || >= $BASE; '."($x, $e)"; |
1339 | last if $e <0 || $e >= $BASE; |
1340 | # this test is disabled, since new/bnorm and certain ops (like early out |
1341 | # in add/sub) are allowed/expected to leave '00000' in some elements |
1342 | #$try = '=~ /^00+/; '."($x, $e)"; |
1343 | #last if $e =~ /^00+/; |
1344 | $i++; |
1345 | } |
1346 | return "Illegal part '$e' at pos $i (tested: $try)" if $i < $j; |
3a427a11 |
1347 | 0; |
394e6ffb |
1348 | } |
1349 | |
1350 | |
1351 | ############################################################################### |
394e6ffb |
1352 | |
dccbb853 |
1353 | sub _mod |
1354 | { |
1355 | # if possible, use mod shortcut |
1356 | my ($c,$x,$yo) = @_; |
1357 | |
1358 | # slow way since $y to big |
1359 | if (scalar @$yo > 1) |
1360 | { |
1361 | my ($xo,$rem) = _div($c,$x,$yo); |
1362 | return $rem; |
1363 | } |
aef458a0 |
1364 | |
dccbb853 |
1365 | my $y = $yo->[0]; |
027dc388 |
1366 | # both are single element arrays |
dccbb853 |
1367 | if (scalar @$x == 1) |
1368 | { |
1369 | $x->[0] %= $y; |
1370 | return $x; |
1371 | } |
1372 | |
aef458a0 |
1373 | # @y is a single element, but @x has more than one element |
dccbb853 |
1374 | my $b = $BASE % $y; |
1375 | if ($b == 0) |
1376 | { |
1377 | # when BASE % Y == 0 then (B * BASE) % Y == 0 |
1378 | # (B * BASE) % $y + A % Y => A % Y |
1379 | # so need to consider only last element: O(1) |
1380 | $x->[0] %= $y; |
1381 | } |
027dc388 |
1382 | elsif ($b == 1) |
1383 | { |
3a427a11 |
1384 | # else need to go through all elements: O(N), but loop is a bit simplified |
027dc388 |
1385 | my $r = 0; |
1386 | foreach (@$x) |
1387 | { |
28df3e88 |
1388 | $r = ($r + $_) % $y; # not much faster, but heh... |
1389 | #$r += $_ % $y; $r %= $y; |
027dc388 |
1390 | } |
1391 | $r = 0 if $r == $y; |
1392 | $x->[0] = $r; |
1393 | } |
dccbb853 |
1394 | else |
1395 | { |
3a427a11 |
1396 | # else need to go through all elements: O(N) |
027dc388 |
1397 | my $r = 0; my $bm = 1; |
1398 | foreach (@$x) |
1399 | { |
28df3e88 |
1400 | $r = ($_ * $bm + $r) % $y; |
1401 | $bm = ($bm * $b) % $y; |
1402 | |
1403 | #$r += ($_ % $y) * $bm; |
1404 | #$bm *= $b; |
1405 | #$bm %= $y; |
1406 | #$r %= $y; |
027dc388 |
1407 | } |
1408 | $r = 0 if $r == $y; |
1409 | $x->[0] = $r; |
dccbb853 |
1410 | } |
091c87b1 |
1411 | splice (@$x,1); # keep one element of $x |
61f5c3f5 |
1412 | $x; |
dccbb853 |
1413 | } |
1414 | |
0716bf9b |
1415 | ############################################################################## |
574bacfe |
1416 | # shifts |
1417 | |
1418 | sub _rsft |
1419 | { |
1420 | my ($c,$x,$y,$n) = @_; |
1421 | |
1422 | if ($n != 10) |
1423 | { |
9b924220 |
1424 | $n = _new($c,$n); return _div($c,$x, _pow($c,$n,$y)); |
61f5c3f5 |
1425 | } |
1426 | |
1427 | # shortcut (faster) for shifting by 10) |
1428 | # multiples of $BASE_LEN |
1429 | my $dst = 0; # destination |
1430 | my $src = _num($c,$y); # as normal int |
56d9de68 |
1431 | my $xlen = (@$x-1)*$BASE_LEN+length(int($x->[-1])); # len of x in digits |
90d1b129 |
1432 | if ($src >= $xlen or ($src == $xlen and ! defined $x->[1])) |
56d9de68 |
1433 | { |
1434 | # 12345 67890 shifted right by more than 10 digits => 0 |
1435 | splice (@$x,1); # leave only one element |
1436 | $x->[0] = 0; # set to zero |
1437 | return $x; |
1438 | } |
61f5c3f5 |
1439 | my $rem = $src % $BASE_LEN; # remainder to shift |
1440 | $src = int($src / $BASE_LEN); # source |
1441 | if ($rem == 0) |
1442 | { |
1443 | splice (@$x,0,$src); # even faster, 38.4 => 39.3 |
574bacfe |
1444 | } |
1445 | else |
1446 | { |
61f5c3f5 |
1447 | my $len = scalar @$x - $src; # elems to go |
1448 | my $vd; my $z = '0'x $BASE_LEN; |
1449 | $x->[scalar @$x] = 0; # avoid || 0 test inside loop |
1450 | while ($dst < $len) |
574bacfe |
1451 | { |
61f5c3f5 |
1452 | $vd = $z.$x->[$src]; |
1453 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN-$rem); |
1454 | $src++; |
1455 | $vd = substr($z.$x->[$src],-$rem,$rem) . $vd; |
1456 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN; |
1457 | $x->[$dst] = int($vd); |
1458 | $dst++; |
574bacfe |
1459 | } |
61f5c3f5 |
1460 | splice (@$x,$dst) if $dst > 0; # kill left-over array elems |
56b9c951 |
1461 | pop @$x if $x->[-1] == 0 && @$x > 1; # kill last element if 0 |
61f5c3f5 |
1462 | } # else rem == 0 |
574bacfe |
1463 | $x; |
1464 | } |
1465 | |
1466 | sub _lsft |
1467 | { |
1468 | my ($c,$x,$y,$n) = @_; |
1469 | |
1470 | if ($n != 10) |
1471 | { |
9b924220 |
1472 | $n = _new($c,$n); return _mul($c,$x, _pow($c,$n,$y)); |
574bacfe |
1473 | } |
61f5c3f5 |
1474 | |
1475 | # shortcut (faster) for shifting by 10) since we are in base 10eX |
1476 | # multiples of $BASE_LEN: |
1477 | my $src = scalar @$x; # source |
1478 | my $len = _num($c,$y); # shift-len as normal int |
1479 | my $rem = $len % $BASE_LEN; # remainder to shift |
1480 | my $dst = $src + int($len/$BASE_LEN); # destination |
1481 | my $vd; # further speedup |
1482 | $x->[$src] = 0; # avoid first ||0 for speed |
1483 | my $z = '0' x $BASE_LEN; |
1484 | while ($src >= 0) |
574bacfe |
1485 | { |
61f5c3f5 |
1486 | $vd = $x->[$src]; $vd = $z.$vd; |
1487 | $vd = substr($vd,-$BASE_LEN+$rem,$BASE_LEN-$rem); |
1488 | $vd .= $src > 0 ? substr($z.$x->[$src-1],-$BASE_LEN,$rem) : '0' x $rem; |
1489 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN; |
1490 | $x->[$dst] = int($vd); |
1491 | $dst--; $src--; |
574bacfe |
1492 | } |
61f5c3f5 |
1493 | # set lowest parts to 0 |
1494 | while ($dst >= 0) { $x->[$dst--] = 0; } |
1495 | # fix spurios last zero element |
1496 | splice @$x,-1 if $x->[-1] == 0; |
574bacfe |
1497 | $x; |
1498 | } |
1499 | |
027dc388 |
1500 | sub _pow |
1501 | { |
1502 | # power of $x to $y |
1503 | # ref to array, ref to array, return ref to array |
1504 | my ($c,$cx,$cy) = @_; |
1505 | |
b282a552 |
1506 | if (scalar @$cy == 1 && $cy->[0] == 0) |
1507 | { |
1508 | splice (@$cx,1); $cx->[0] = 1; # y == 0 => x => 1 |
1509 | return $cx; |
1510 | } |
1511 | if ((scalar @$cx == 1 && $cx->[0] == 1) || # x == 1 |
1512 | (scalar @$cy == 1 && $cy->[0] == 1)) # or y == 1 |
1513 | { |
1514 | return $cx; |
1515 | } |
1516 | if (scalar @$cx == 1 && $cx->[0] == 0) |
1517 | { |
1518 | splice (@$cx,1); $cx->[0] = 0; # 0 ** y => 0 (if not y <= 0) |
1519 | return $cx; |
1520 | } |
1521 | |
027dc388 |
1522 | my $pow2 = _one(); |
1ddff52a |
1523 | |
9b924220 |
1524 | my $y_bin = _as_bin($c,$cy); $y_bin =~ s/^0b//; |
1ddff52a |
1525 | my $len = length($y_bin); |
1526 | while (--$len > 0) |
027dc388 |
1527 | { |
1ddff52a |
1528 | _mul($c,$pow2,$cx) if substr($y_bin,$len,1) eq '1'; # is odd? |
027dc388 |
1529 | _mul($c,$cx,$cx); |
1530 | } |
1ddff52a |
1531 | |
1532 | _mul($c,$cx,$pow2); |
61f5c3f5 |
1533 | $cx; |
027dc388 |
1534 | } |
1535 | |
50109ad0 |
1536 | sub _nok |
1537 | { |
1538 | # n over k |
1539 | # ref to array, return ref to array |
1540 | my ($c,$n,$k) = @_; |
1541 | |
1542 | # ( 7 ) 7! 7*6*5 * 4*3*2*1 7 * 6 * 5 |
1543 | # ( - ) = --------- = --------------- = --------- |
1544 | # ( 3 ) 3! (7-3)! 3*2*1 * 4*3*2*1 3 * 2 * 1 |
1545 | |
1546 | # compute n - k + 2 (so we start with 5 in the example above) |
1547 | my $x = _copy($c,$n); |
1548 | |
1549 | _sub($c,$n,$k); |
1550 | if (!_is_one($c,$n)) |
1551 | { |
1552 | _inc($c,$n); |
1553 | my $f = _copy($c,$n); _inc($c,$f); # n = 5, f = 6, d = 2 |
1554 | my $d = _two($c); |
1555 | while (_acmp($c,$f,$x) <= 0) # f < n ? |
1556 | { |
1557 | # n = (n * f / d) == 5 * 6 / 2 => n == 3 |
1558 | $n = _mul($c,$n,$f); $n = _div($c,$n,$d); |
1559 | # f = 7, d = 3 |
1560 | _inc($c,$f); _inc($c,$d); |
1561 | } |
1562 | } |
1563 | else |
1564 | { |
1565 | # keep ref to $n and set it to 1 |
1566 | splice (@$n,1); $n->[0] = 1; |
1567 | } |
1568 | $n; |
1569 | } |
1570 | |
1571 | my @factorials = ( |
1572 | 1, |
1573 | 1, |
1574 | 2, |
1575 | 2*3, |
1576 | 2*3*4, |
1577 | 2*3*4*5, |
1578 | 2*3*4*5*6, |
1579 | 2*3*4*5*6*7, |
1580 | ); |
1581 | |
b3abae2a |
1582 | sub _fac |
1583 | { |
1584 | # factorial of $x |
1585 | # ref to array, return ref to array |
1586 | my ($c,$cx) = @_; |
1587 | |
50109ad0 |
1588 | if ((@$cx == 1) && ($cx->[0] <= 7)) |
b3abae2a |
1589 | { |
50109ad0 |
1590 | $cx->[0] = $factorials[$cx->[0]]; # 0 => 1, 1 => 1, 2 => 2 etc. |
b3abae2a |
1591 | return $cx; |
1592 | } |
1593 | |
50109ad0 |
1594 | if ((@$cx == 1) && # we do this only if $x >= 12 and $x <= 7000 |
1595 | ($cx->[0] >= 12 && $cx->[0] < 7000)) |
1596 | { |
1597 | |
1598 | # Calculate (k-j) * (k-j+1) ... k .. (k+j-1) * (k + j) |
1599 | # See http://blogten.blogspot.com/2007/01/calculating-n.html |
1600 | # The above series can be expressed as factors: |
1601 | # k * k - (j - i) * 2 |
1602 | # We cache k*k, and calculate (j * j) as the sum of the first j odd integers |
1603 | |
1604 | # This will not work when N exceeds the storage of a Perl scalar, however, |
1605 | # in this case the algorithm would be way to slow to terminate, anyway. |
1606 | |
1607 | # As soon as the last element of $cx is 0, we split it up and remember |
1608 | # how many zeors we got so far. The reason is that n! will accumulate |
1609 | # zeros at the end rather fast. |
1610 | my $zero_elements = 0; |
1611 | |
1612 | # If n is even, set n = n -1 |
1613 | my $k = _num($c,$cx); my $even = 1; |
1614 | if (($k & 1) == 0) |
1615 | { |
1616 | $even = $k; $k --; |
1617 | } |
1618 | # set k to the center point |
1619 | $k = ($k + 1) / 2; |
1620 | # print "k $k even: $even\n"; |
1621 | # now calculate k * k |
1622 | my $k2 = $k * $k; |
1623 | my $odd = 1; my $sum = 1; |
1624 | my $i = $k - 1; |
1625 | # keep reference to x |
1626 | my $new_x = _new($c, $k * $even); |
1627 | @$cx = @$new_x; |
1628 | if ($cx->[0] == 0) |
1629 | { |
1630 | $zero_elements ++; shift @$cx; |
1631 | } |
1632 | # print STDERR "x = ", _str($c,$cx),"\n"; |
1633 | my $BASE2 = int(sqrt($BASE))-1; |
1634 | my $j = 1; |
1635 | while ($j <= $i) |
1636 | { |
1637 | my $m = ($k2 - $sum); $odd += 2; $sum += $odd; $j++; |
1638 | while ($j <= $i && ($m < $BASE2) && (($k2 - $sum) < $BASE2)) |
1639 | { |
1640 | $m *= ($k2 - $sum); |
1641 | $odd += 2; $sum += $odd; $j++; |
1642 | # print STDERR "\n k2 $k2 m $m sum $sum odd $odd\n"; sleep(1); |
1643 | } |
1644 | if ($m < $BASE) |
1645 | { |
1646 | _mul($c,$cx,[$m]); |
1647 | } |
1648 | else |
1649 | { |
1650 | _mul($c,$cx,$c->_new($m)); |
1651 | } |
1652 | if ($cx->[0] == 0) |
1653 | { |
1654 | $zero_elements ++; shift @$cx; |
1655 | } |
1656 | # print STDERR "Calculate $k2 - $sum = $m (x = ", _str($c,$cx),")\n"; |
1657 | } |
1658 | # multiply in the zeros again |
1659 | unshift @$cx, (0) x $zero_elements; |
1660 | return $cx; |
1661 | } |
1662 | |
b3abae2a |
1663 | # go forward until $base is exceeded |
091c87b1 |
1664 | # limit is either $x steps (steps == 100 means a result always too high) or |
1665 | # $base. |
b3abae2a |
1666 | my $steps = 100; $steps = $cx->[0] if @$cx == 1; |
091c87b1 |
1667 | my $r = 2; my $cf = 3; my $step = 2; my $last = $r; |
1668 | while ($r*$cf < $BASE && $step < $steps) |
b3abae2a |
1669 | { |
1670 | $last = $r; $r *= $cf++; $step++; |
1671 | } |
091c87b1 |
1672 | if ((@$cx == 1) && $step == $cx->[0]) |
b3abae2a |
1673 | { |
091c87b1 |
1674 | # completely done, so keep reference to $x and return |
1675 | $cx->[0] = $r; |
b3abae2a |
1676 | return $cx; |
1677 | } |
091c87b1 |
1678 | |
990fb837 |
1679 | # now we must do the left over steps |
091c87b1 |
1680 | my $n; # steps still to do |
1681 | if (scalar @$cx == 1) |
1682 | { |
1683 | $n = $cx->[0]; |
1684 | } |
1685 | else |
1686 | { |
1687 | $n = _copy($c,$cx); |
1688 | } |
b3abae2a |
1689 | |
50109ad0 |
1690 | # Set $cx to the last result below $BASE (but keep ref to $x) |
1691 | $cx->[0] = $last; splice (@$cx,1); |
1692 | # As soon as the last element of $cx is 0, we split it up and remember |
1693 | # how many zeors we got so far. The reason is that n! will accumulate |
1694 | # zeros at the end rather fast. |
990fb837 |
1695 | my $zero_elements = 0; |
091c87b1 |
1696 | |
1697 | # do left-over steps fit into a scalar? |
1698 | if (ref $n eq 'ARRAY') |
b3abae2a |
1699 | { |
091c87b1 |
1700 | # No, so use slower inc() & cmp() |
50109ad0 |
1701 | # ($n is at least $BASE here) |
1702 | my $base_2 = int(sqrt($BASE)) - 1; |
1703 | #print STDERR "base_2: $base_2\n"; |
1704 | while ($step < $base_2) |
1705 | { |
1706 | if ($cx->[0] == 0) |
1707 | { |
1708 | $zero_elements ++; shift @$cx; |
1709 | } |
1710 | my $b = $step * ($step + 1); $step += 2; |
1711 | _mul($c,$cx,[$b]); |
1712 | } |
091c87b1 |
1713 | $step = [$step]; |
50109ad0 |
1714 | while (_acmp($c,$step,$n) <= 0) |
990fb837 |
1715 | { |
1716 | if ($cx->[0] == 0) |
1717 | { |
1718 | $zero_elements ++; shift @$cx; |
1719 | } |
091c87b1 |
1720 | _mul($c,$cx,$step); _inc($c,$step); |
990fb837 |
1721 | } |
990fb837 |
1722 | } |
091c87b1 |
1723 | else |
990fb837 |
1724 | { |
091c87b1 |
1725 | # Yes, so we can speed it up slightly |
50109ad0 |
1726 | |
1727 | # print "# left over steps $n\n"; |
1728 | |
1729 | my $base_4 = int(sqrt(sqrt($BASE))) - 2; |
1730 | #print STDERR "base_4: $base_4\n"; |
1731 | my $n4 = $n - 4; |
1732 | while ($step < $n4 && $step < $base_4) |
990fb837 |
1733 | { |
091c87b1 |
1734 | if ($cx->[0] == 0) |
1735 | { |
1736 | $zero_elements ++; shift @$cx; |
1737 | } |
50109ad0 |
1738 | my $b = $step * ($step + 1); $step += 2; $b *= $step * ($step + 1); $step += 2; |
1739 | _mul($c,$cx,[$b]); |
1740 | } |
1741 | my $base_2 = int(sqrt($BASE)) - 1; |
1742 | my $n2 = $n - 2; |
1743 | #print STDERR "base_2: $base_2\n"; |
1744 | while ($step < $n2 && $step < $base_2) |
1745 | { |
1746 | if ($cx->[0] == 0) |
1747 | { |
1748 | $zero_elements ++; shift @$cx; |
1749 | } |
1750 | my $b = $step * ($step + 1); $step += 2; |
1751 | _mul($c,$cx,[$b]); |
1752 | } |
1753 | # do what's left over |
1754 | while ($step <= $n) |
1755 | { |
091c87b1 |
1756 | _mul($c,$cx,[$step]); $step++; |
50109ad0 |
1757 | if ($cx->[0] == 0) |
1758 | { |
1759 | $zero_elements ++; shift @$cx; |
1760 | } |
990fb837 |
1761 | } |
990fb837 |
1762 | } |
1763 | # multiply in the zeros again |
50109ad0 |
1764 | unshift @$cx, (0) x $zero_elements; |
091c87b1 |
1765 | $cx; # return result |
1766 | } |
1767 | |
9b924220 |
1768 | ############################################################################# |
1769 | |
091c87b1 |
1770 | sub _log_int |
1771 | { |
1772 | # calculate integer log of $x to base $base |
1773 | # ref to array, ref to array - return ref to array |
1774 | my ($c,$x,$base) = @_; |
1775 | |
1776 | # X == 0 => NaN |
1777 | return if (scalar @$x == 1 && $x->[0] == 0); |
1778 | # BASE 0 or 1 => NaN |
1779 | return if (scalar @$base == 1 && $base->[0] < 2); |
b282a552 |
1780 | my $cmp = _acmp($c,$x,$base); # X == BASE => 1 |
091c87b1 |
1781 | if ($cmp == 0) |
1782 | { |
1783 | splice (@$x,1); $x->[0] = 1; |
8df1e0a2 |
1784 | return ($x,1) |
091c87b1 |
1785 | } |
1786 | # X < BASE |
1787 | if ($cmp < 0) |
1788 | { |
1789 | splice (@$x,1); $x->[0] = 0; |
8df1e0a2 |
1790 | return ($x,undef); |
091c87b1 |
1791 | } |
1792 | |
091c87b1 |
1793 | my $x_org = _copy($c,$x); # preserve x |
8df1e0a2 |
1794 | splice(@$x,1); $x->[0] = 1; # keep ref to $x |
1795 | |
86b76201 |
1796 | # Compute a guess for the result based on: |
1797 | # $guess = int ( length_in_base_10(X) / ( log(base) / log(10) ) ) |
2ebb273f |
1798 | my $len = _len($c,$x_org); |
1799 | my $log = log($base->[-1]) / log(10); |
b282a552 |
1800 | |
2ebb273f |
1801 | # for each additional element in $base, we add $BASE_LEN to the result, |
1802 | # based on the observation that log($BASE,10) is BASE_LEN and |
1803 | # log(x*y) == log(x) + log(y): |
1804 | $log += ((scalar @$base)-1) * $BASE_LEN; |
b282a552 |
1805 | |
2ebb273f |
1806 | # calculate now a guess based on the values obtained above: |
1807 | my $res = int($len / $log); |
1808 | |
1809 | $x->[0] = $res; |
1810 | my $trial = _pow ($c, _copy($c, $base), $x); |
1811 | my $a = _acmp($c,$trial,$x_org); |
1812 | |
1813 | # print STDERR "# trial ", _str($c,$x)," was: $a (0 = exact, -1 too small, +1 too big)\n"; |
1814 | |
1815 | # found an exact result? |
1816 | return ($x,1) if $a == 0; |
1817 | |
1818 | if ($a > 0) |
1819 | { |
1820 | # or too big |
1821 | _div($c,$trial,$base); _dec($c, $x); |
1822 | while (($a = _acmp($c,$trial,$x_org)) > 0) |
b282a552 |
1823 | { |
2ebb273f |
1824 | # print STDERR "# big _log_int at ", _str($c,$x), "\n"; |
1825 | _div($c,$trial,$base); _dec($c, $x); |
b282a552 |
1826 | } |
2ebb273f |
1827 | # result is now exact (a == 0), or too small (a < 0) |
1828 | return ($x, $a == 0 ? 1 : 0); |
1829 | } |
1830 | |
1831 | # else: result was to small |
1832 | _mul($c,$trial,$base); |
1833 | |
1834 | # did we now get the right result? |
1835 | $a = _acmp($c,$trial,$x_org); |
1836 | |
1837 | if ($a == 0) # yes, exactly |
1838 | { |
1839 | _inc($c, $x); |
1840 | return ($x,1); |
1841 | } |
1842 | return ($x,0) if $a > 0; |
1843 | |
1844 | # Result still too small (we should come here only if the estimate above |
1845 | # was very off base): |
1846 | |
1847 | # Now let the normal trial run obtain the real result |
1848 | # Simple loop that increments $x by 2 in each step, possible overstepping |
1849 | # the real result |
091c87b1 |
1850 | |
2ebb273f |
1851 | my $base_mul = _mul($c, _copy($c,$base), $base); # $base * $base |
8df1e0a2 |
1852 | |
9b924220 |
1853 | while (($a = _acmp($c,$trial,$x_org)) < 0) |
091c87b1 |
1854 | { |
2ebb273f |
1855 | # print STDERR "# small _log_int at ", _str($c,$x), "\n"; |
8df1e0a2 |
1856 | _mul($c,$trial,$base_mul); _add($c, $x, [2]); |
091c87b1 |
1857 | } |
8df1e0a2 |
1858 | |
1859 | my $exact = 1; |
1860 | if ($a > 0) |
091c87b1 |
1861 | { |
8df1e0a2 |
1862 | # overstepped the result |
1863 | _dec($c, $x); |
1864 | _div($c,$trial,$base); |
9b924220 |
1865 | $a = _acmp($c,$trial,$x_org); |
8df1e0a2 |
1866 | if ($a > 0) |
091c87b1 |
1867 | { |
8df1e0a2 |
1868 | _dec($c, $x); |
091c87b1 |
1869 | } |
2ebb273f |
1870 | $exact = 0 if $a != 0; # a = -1 => not exact result, a = 0 => exact |
091c87b1 |
1871 | } |
1872 | |
8df1e0a2 |
1873 | ($x,$exact); # return result |
b3abae2a |
1874 | } |
1875 | |
56d9de68 |
1876 | # for debugging: |
1877 | use constant DEBUG => 0; |
1878 | my $steps = 0; |
1879 | sub steps { $steps }; |
b3abae2a |
1880 | |
1881 | sub _sqrt |
0716bf9b |
1882 | { |
56d9de68 |
1883 | # square-root of $x in place |
990fb837 |
1884 | # Compute a guess of the result (by rule of thumb), then improve it via |
56d9de68 |
1885 | # Newton's method. |
394e6ffb |
1886 | my ($c,$x) = @_; |
0716bf9b |
1887 | |
394e6ffb |
1888 | if (scalar @$x == 1) |
1889 | { |
50109ad0 |
1890 | # fits into one Perl scalar, so result can be computed directly |
394e6ffb |
1891 | $x->[0] = int(sqrt($x->[0])); |
1892 | return $x; |
1893 | } |
1894 | my $y = _copy($c,$x); |
b3abae2a |
1895 | # hopefully _len/2 is < $BASE, the -1 is to always undershot the guess |
1896 | # since our guess will "grow" |
1897 | my $l = int((_len($c,$x)-1) / 2); |
1898 | |
56d9de68 |
1899 | my $lastelem = $x->[-1]; # for guess |
b3abae2a |
1900 | my $elems = scalar @$x - 1; |
1901 | # not enough digits, but could have more? |
56d9de68 |
1902 | if ((length($lastelem) <= 3) && ($elems > 1)) |
b3abae2a |
1903 | { |
1904 | # right-align with zero pad |
1905 | my $len = length($lastelem) & 1; |
1906 | print "$lastelem => " if DEBUG; |
1907 | $lastelem .= substr($x->[-2] . '0' x $BASE_LEN,0,$BASE_LEN); |
1908 | # former odd => make odd again, or former even to even again |
56d9de68 |
1909 | $lastelem = $lastelem / 10 if (length($lastelem) & 1) != $len; |
b3abae2a |
1910 | print "$lastelem\n" if DEBUG; |
1911 | } |
0716bf9b |
1912 | |
61f5c3f5 |
1913 | # construct $x (instead of _lsft($c,$x,$l,10) |
1914 | my $r = $l % $BASE_LEN; # 10000 00000 00000 00000 ($BASE_LEN=5) |
1915 | $l = int($l / $BASE_LEN); |
b3abae2a |
1916 | print "l = $l " if DEBUG; |
56d9de68 |
1917 | |
1918 | splice @$x,$l; # keep ref($x), but modify it |
1919 | |
b3abae2a |
1920 | # we make the first part of the guess not '1000...0' but int(sqrt($lastelem)) |
1921 | # that gives us: |
56d9de68 |
1922 | # 14400 00000 => sqrt(14400) => guess first digits to be 120 |
1923 | # 144000 000000 => sqrt(144000) => guess 379 |
b3abae2a |
1924 | |
b3abae2a |
1925 | print "$lastelem (elems $elems) => " if DEBUG; |
1926 | $lastelem = $lastelem / 10 if ($elems & 1 == 1); # odd or even? |
1927 | my $g = sqrt($lastelem); $g =~ s/\.//; # 2.345 => 2345 |
1928 | $r -= 1 if $elems & 1 == 0; # 70 => 7 |
1929 | |
1930 | # padd with zeros if result is too short |
1931 | $x->[$l--] = int(substr($g . '0' x $r,0,$r+1)); |
1932 | print "now ",$x->[-1] if DEBUG; |
1933 | print " would have been ", int('1' . '0' x $r),"\n" if DEBUG; |
56d9de68 |
1934 | |
b3abae2a |
1935 | # If @$x > 1, we could compute the second elem of the guess, too, to create |
56d9de68 |
1936 | # an even better guess. Not implemented yet. Does it improve performance? |
b3abae2a |
1937 | $x->[$l--] = 0 while ($l >= 0); # all other digits of guess are zero |
56d9de68 |
1938 | |
9b924220 |
1939 | print "start x= ",_str($c,$x),"\n" if DEBUG; |
394e6ffb |
1940 | my $two = _two(); |
1941 | my $last = _zero(); |
1942 | my $lastlast = _zero(); |
b3abae2a |
1943 | $steps = 0 if DEBUG; |
394e6ffb |
1944 | while (_acmp($c,$last,$x) != 0 && _acmp($c,$lastlast,$x) != 0) |
1945 | { |
b3abae2a |
1946 | $steps++ if DEBUG; |
394e6ffb |
1947 | $lastlast = _copy($c,$last); |
1948 | $last = _copy($c,$x); |
1949 | _add($c,$x, _div($c,_copy($c,$y),$x)); |
1950 | _div($c,$x, $two ); |
9b924220 |
1951 | print " x= ",_str($c,$x),"\n" if DEBUG; |
394e6ffb |
1952 | } |
b3abae2a |
1953 | print "\nsteps in sqrt: $steps, " if DEBUG; |
394e6ffb |
1954 | _dec($c,$x) if _acmp($c,$y,_mul($c,_copy($c,$x),$x)) < 0; # overshot? |
b3abae2a |
1955 | print " final ",$x->[-1],"\n" if DEBUG; |
394e6ffb |
1956 | $x; |
0716bf9b |
1957 | } |
1958 | |
990fb837 |
1959 | sub _root |
1960 | { |
1961 | # take n'th root of $x in place (n >= 3) |
990fb837 |
1962 | my ($c,$x,$n) = @_; |
1963 | |
1964 | if (scalar @$x == 1) |
1965 | { |
1966 | if (scalar @$n > 1) |
1967 | { |
1968 | # result will always be smaller than 2 so trunc to 1 at once |
1969 | $x->[0] = 1; |
1970 | } |
1971 | else |
1972 | { |
50109ad0 |
1973 | # fits into one Perl scalar, so result can be computed directly |
091c87b1 |
1974 | # cannot use int() here, because it rounds wrongly (try |
1975 | # (81 ** 3) ** (1/3) to see what I mean) |
1976 | #$x->[0] = int( $x->[0] ** (1 / $n->[0]) ); |
1977 | # round to 8 digits, then truncate result to integer |
1978 | $x->[0] = int ( sprintf ("%.8f", $x->[0] ** (1 / $n->[0]) ) ); |
990fb837 |
1979 | } |
1980 | return $x; |
1981 | } |
1982 | |
3a427a11 |
1983 | # we know now that X is more than one element long |
1984 | |
c38b2de2 |
1985 | # if $n is a power of two, we can repeatedly take sqrt($X) and find the |
1986 | # proper result, because sqrt(sqrt($x)) == root($x,4) |
1987 | my $b = _as_bin($c,$n); |
9b924220 |
1988 | if ($b =~ /0b1(0+)$/) |
c38b2de2 |
1989 | { |
1990 | my $count = CORE::length($1); # 0b100 => len('00') => 2 |
1991 | my $cnt = $count; # counter for loop |
1992 | unshift (@$x, 0); # add one element, together with one |
1993 | # more below in the loop this makes 2 |
1994 | while ($cnt-- > 0) |
1995 | { |
1996 | # 'inflate' $X by adding one element, basically computing |
1997 | # $x * $BASE * $BASE. This gives us more $BASE_LEN digits for result |
1998 | # since len(sqrt($X)) approx == len($x) / 2. |
1999 | unshift (@$x, 0); |
2000 | # calculate sqrt($x), $x is now one element to big, again. In the next |
2001 | # round we make that two, again. |
2002 | _sqrt($c,$x); |
2003 | } |
2004 | # $x is now one element to big, so truncate result by removing it |
2005 | splice (@$x,0,1); |
2006 | } |
2007 | else |
2008 | { |
091c87b1 |
2009 | # trial computation by starting with 2,4,8,16 etc until we overstep |
3a427a11 |
2010 | my $step; |
091c87b1 |
2011 | my $trial = _two(); |
2012 | |
3a427a11 |
2013 | # while still to do more than X steps |
2014 | do |
091c87b1 |
2015 | { |
3a427a11 |
2016 | $step = _two(); |
2017 | while (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) < 0) |
2018 | { |
2019 | _mul ($c, $step, [2]); |
2020 | _add ($c, $trial, $step); |
2021 | } |
2022 | |
2023 | # hit exactly? |
2024 | if (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) == 0) |
2025 | { |
2026 | @$x = @$trial; # make copy while preserving ref to $x |
2027 | return $x; |
2028 | } |
2029 | # overstepped, so go back on step |
2030 | _sub($c, $trial, $step); |
2031 | } while (scalar @$step > 1 || $step->[0] > 128); |
091c87b1 |
2032 | |
3a427a11 |
2033 | # reset step to 2 |
2034 | $step = _two(); |
091c87b1 |
2035 | # add two, because $trial cannot be exactly the result (otherwise we would |
2036 | # alrady have found it) |
2037 | _add($c, $trial, $step); |
2038 | |
3a427a11 |
2039 | # and now add more and more (2,4,6,8,10 etc) |
2040 | while (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) < 0) |
2041 | { |
2042 | _add ($c, $trial, $step); |
2043 | } |
091c87b1 |
2044 | |
2045 | # hit not exactly? (overstepped) |
091c87b1 |
2046 | if (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) > 0) |
2047 | { |
2048 | _dec($c,$trial); |
2049 | } |
3a427a11 |
2050 | |
2051 | # hit not exactly? (overstepped) |
2052 | # 80 too small, 81 slightly too big, 82 too big |
091c87b1 |
2053 | if (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) > 0) |
2054 | { |
3a427a11 |
2055 | _dec ($c, $trial); |
091c87b1 |
2056 | } |
3a427a11 |
2057 | |
091c87b1 |
2058 | @$x = @$trial; # make copy while preserving ref to $x |
2059 | return $x; |
c38b2de2 |
2060 | } |
990fb837 |
2061 | $x; |
2062 | } |
2063 | |
394e6ffb |
2064 | ############################################################################## |
2065 | # binary stuff |
0716bf9b |
2066 | |
394e6ffb |
2067 | sub _and |
2068 | { |
2069 | my ($c,$x,$y) = @_; |
0716bf9b |
2070 | |
394e6ffb |
2071 | # the shortcut makes equal, large numbers _really_ fast, and makes only a |
2072 | # very small performance drop for small numbers (e.g. something with less |
2073 | # than 32 bit) Since we optimize for large numbers, this is enabled. |
2074 | return $x if _acmp($c,$x,$y) == 0; # shortcut |
0716bf9b |
2075 | |
394e6ffb |
2076 | my $m = _one(); my ($xr,$yr); |
2077 | my $mask = $AND_MASK; |
2078 | |
2079 | my $x1 = $x; |
2080 | my $y1 = _copy($c,$y); # make copy |
2081 | $x = _zero(); |
2082 | my ($b,$xrr,$yrr); |
2083 | use integer; |
2084 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
2085 | { |
2086 | ($x1, $xr) = _div($c,$x1,$mask); |
2087 | ($y1, $yr) = _div($c,$y1,$mask); |
2088 | |
2089 | # make ints() from $xr, $yr |
9b924220 |
2090 | # this is when the AND_BITS are greater than $BASE and is slower for |
394e6ffb |
2091 | # small (<256 bits) numbers, but faster for large numbers. Disabled |
2092 | # due to KISS principle |
2093 | |
2094 | # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
2095 | # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
9b924220 |
2096 | # _add($c,$x, _mul($c, _new( $c, ($xrr & $yrr) ), $m) ); |
394e6ffb |
2097 | |
61f5c3f5 |
2098 | # 0+ due to '&' doesn't work in strings |
2099 | _add($c,$x, _mul($c, [ 0+$xr->[0] & 0+$yr->[0] ], $m) ); |
394e6ffb |
2100 | _mul($c,$m,$mask); |
2101 | } |
2102 | $x; |
0716bf9b |
2103 | } |
2104 | |
394e6ffb |
2105 | sub _xor |
0716bf9b |
2106 | { |
394e6ffb |
2107 | my ($c,$x,$y) = @_; |
2108 | |
2109 | return _zero() if _acmp($c,$x,$y) == 0; # shortcut (see -and) |
2110 | |
2111 | my $m = _one(); my ($xr,$yr); |
2112 | my $mask = $XOR_MASK; |
2113 | |
2114 | my $x1 = $x; |
2115 | my $y1 = _copy($c,$y); # make copy |
2116 | $x = _zero(); |
2117 | my ($b,$xrr,$yrr); |
2118 | use integer; |
2119 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
0716bf9b |
2120 | { |
394e6ffb |
2121 | ($x1, $xr) = _div($c,$x1,$mask); |
2122 | ($y1, $yr) = _div($c,$y1,$mask); |
2123 | # make ints() from $xr, $yr (see _and()) |
2124 | #$b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
2125 | #$b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
9b924220 |
2126 | #_add($c,$x, _mul($c, _new( $c, ($xrr ^ $yrr) ), $m) ); |
61f5c3f5 |
2127 | |
2128 | # 0+ due to '^' doesn't work in strings |
2129 | _add($c,$x, _mul($c, [ 0+$xr->[0] ^ 0+$yr->[0] ], $m) ); |
394e6ffb |
2130 | _mul($c,$m,$mask); |
0716bf9b |
2131 | } |
394e6ffb |
2132 | # the loop stops when the shorter of the two numbers is exhausted |
2133 | # the remainder of the longer one will survive bit-by-bit, so we simple |
2134 | # multiply-add it in |
2135 | _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1); |
2136 | _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1); |
2137 | |
2138 | $x; |
0716bf9b |
2139 | } |
2140 | |
394e6ffb |
2141 | sub _or |
0716bf9b |
2142 | { |
394e6ffb |
2143 | my ($c,$x,$y) = @_; |
0716bf9b |
2144 | |
394e6ffb |
2145 | return $x if _acmp($c,$x,$y) == 0; # shortcut (see _and) |
0716bf9b |
2146 | |
394e6ffb |
2147 | my $m = _one(); my ($xr,$yr); |
2148 | my $mask = $OR_MASK; |
0716bf9b |
2149 | |
394e6ffb |
2150 | my $x1 = $x; |
2151 | my $y1 = _copy($c,$y); # make copy |
2152 | $x = _zero(); |
2153 | my ($b,$xrr,$yrr); |
2154 | use integer; |
2155 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
2156 | { |
2157 | ($x1, $xr) = _div($c,$x1,$mask); |
2158 | ($y1, $yr) = _div($c,$y1,$mask); |
2159 | # make ints() from $xr, $yr (see _and()) |
2160 | # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
2161 | # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
9b924220 |
2162 | # _add($c,$x, _mul($c, _new( $c, ($xrr | $yrr) ), $m) ); |
394e6ffb |
2163 | |
61f5c3f5 |
2164 | # 0+ due to '|' doesn't work in strings |
2165 | _add($c,$x, _mul($c, [ 0+$xr->[0] | 0+$yr->[0] ], $m) ); |
394e6ffb |
2166 | _mul($c,$m,$mask); |
2167 | } |
2168 | # the loop stops when the shorter of the two numbers is exhausted |
2169 | # the remainder of the longer one will survive bit-by-bit, so we simple |
2170 | # multiply-add it in |
2171 | _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1); |
2172 | _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1); |
2173 | |
2174 | $x; |
0716bf9b |
2175 | } |
2176 | |
61f5c3f5 |
2177 | sub _as_hex |
2178 | { |
2179 | # convert a decimal number to hex (ref to array, return ref to string) |
2180 | my ($c,$x) = @_; |
2181 | |
50109ad0 |
2182 | # fits into one element (handle also 0x0 case) |
03874afe |
2183 | return sprintf("0x%x",$x->[0]) if @$x == 1; |
990fb837 |
2184 | |
61f5c3f5 |
2185 | my $x1 = _copy($c,$x); |
2186 | |
2187 | my $es = ''; |
1ddff52a |
2188 | my ($xr, $h, $x10000); |
2189 | if ($] >= 5.006) |
2190 | { |
2191 | $x10000 = [ 0x10000 ]; $h = 'h4'; |
2192 | } |
2193 | else |
2194 | { |
2195 | $x10000 = [ 0x1000 ]; $h = 'h3'; |
2196 | } |
091c87b1 |
2197 | while (@$x1 != 1 || $x1->[0] != 0) # _is_zero() |
61f5c3f5 |
2198 | { |
2199 | ($x1, $xr) = _div($c,$x1,$x10000); |
7b29e1e6 |
2200 | $es .= unpack($h,pack('V',$xr->[0])); |
61f5c3f5 |
2201 | } |
2202 | $es = reverse $es; |
2203 | $es =~ s/^[0]+//; # strip leading zeros |
03874afe |
2204 | '0x' . $es; # return result prepended with 0x |
61f5c3f5 |
2205 | } |
2206 | |
2207 | sub _as_bin |
2208 | { |
2209 | # convert a decimal number to bin (ref to array, return ref to string) |
2210 | my ($c,$x) = @_; |
2211 | |
50109ad0 |
2212 | # fits into one element (and Perl recent enough), handle also 0b0 case |
091c87b1 |
2213 | # handle zero case for older Perls |
2214 | if ($] <= 5.005 && @$x == 1 && $x->[0] == 0) |
2215 | { |
9b924220 |
2216 | my $t = '0b0'; return $t; |
091c87b1 |
2217 | } |
2218 | if (@$x == 1 && $] >= 5.006) |
990fb837 |
2219 | { |
091c87b1 |
2220 | my $t = sprintf("0b%b",$x->[0]); |
9b924220 |
2221 | return $t; |
990fb837 |
2222 | } |
61f5c3f5 |
2223 | my $x1 = _copy($c,$x); |
2224 | |
2225 | my $es = ''; |
1ddff52a |
2226 | my ($xr, $b, $x10000); |
2227 | if ($] >= 5.006) |
2228 | { |
2229 | $x10000 = [ 0x10000 ]; $b = 'b16'; |
2230 | } |
2231 | else |
2232 | { |
2233 | $x10000 = [ 0x1000 ]; $b = 'b12'; |
2234 | } |
091c87b1 |
2235 | while (!(@$x1 == 1 && $x1->[0] == 0)) # _is_zero() |
61f5c3f5 |
2236 | { |
2237 | ($x1, $xr) = _div($c,$x1,$x10000); |
7b29e1e6 |
2238 | $es .= unpack($b,pack('v',$xr->[0])); |
61f5c3f5 |
2239 | } |
2240 | $es = reverse $es; |
2241 | $es =~ s/^[0]+//; # strip leading zeros |
03874afe |
2242 | '0b' . $es; # return result prepended with 0b |
61f5c3f5 |
2243 | } |
2244 | |
7b29e1e6 |
2245 | sub _as_oct |
2246 | { |
2247 | # convert a decimal number to octal (ref to array, return ref to string) |
2248 | my ($c,$x) = @_; |
2249 | |
50109ad0 |
2250 | # fits into one element (handle also 0 case) |
7b29e1e6 |
2251 | return sprintf("0%o",$x->[0]) if @$x == 1; |
2252 | |
2253 | my $x1 = _copy($c,$x); |
2254 | |
2255 | my $es = ''; |
2256 | my $xr; |
2257 | my $x1000 = [ 0100000 ]; |
2258 | while (@$x1 != 1 || $x1->[0] != 0) # _is_zero() |
2259 | { |
2260 | ($x1, $xr) = _div($c,$x1,$x1000); |
2261 | $es .= reverse sprintf("%05o", $xr->[0]); |
2262 | } |
2263 | $es = reverse $es; |
2264 | $es =~ s/^[0]+//; # strip leading zeros |
2265 | '0' . $es; # return result prepended with 0 |
2266 | } |
2267 | |
2268 | sub _from_oct |
2269 | { |
50109ad0 |
2270 | # convert a octal number to decimal (string, return ref to array) |
7b29e1e6 |
2271 | my ($c,$os) = @_; |
2272 | |
2273 | # for older Perls, play safe |
2274 | my $m = [ 0100000 ]; |
2275 | my $d = 5; # 5 digits at a time |
2276 | |
2277 | my $mul = _one(); |
2278 | my $x = _zero(); |
2279 | |
2280 | my $len = int( (length($os)-1)/$d ); # $d digit parts, w/o the '0' |
2281 | my $val; my $i = -$d; |
2282 | while ($len >= 0) |
2283 | { |
2284 | $val = substr($os,$i,$d); # get oct digits |
2285 | $val = oct($val); |
2286 | $i -= $d; $len --; |
2287 | my $adder = [ $val ]; |
2288 | _add ($c, $x, _mul ($c, $adder, $mul ) ) if $val != 0; |
2289 | _mul ($c, $mul, $m ) if $len >= 0; # skip last mul |
2290 | } |
2291 | $x; |
2292 | } |
2293 | |
394e6ffb |
2294 | sub _from_hex |
0716bf9b |
2295 | { |
50109ad0 |
2296 | # convert a hex number to decimal (string, return ref to array) |
394e6ffb |
2297 | my ($c,$hs) = @_; |
0716bf9b |
2298 | |
2d2b2744 |
2299 | my $m = _new($c, 0x10000000); # 28 bit at a time (<32 bit!) |
03874afe |
2300 | my $d = 7; # 7 digits at a time |
2301 | if ($] <= 5.006) |
2302 | { |
2303 | # for older Perls, play safe |
2304 | $m = [ 0x10000 ]; # 16 bit at a time (<32 bit!) |
2305 | $d = 4; # 4 digits at a time |
2306 | } |
2307 | |
394e6ffb |
2308 | my $mul = _one(); |
394e6ffb |
2309 | my $x = _zero(); |
0716bf9b |
2310 | |
03874afe |
2311 | my $len = int( (length($hs)-2)/$d ); # $d digit parts, w/o the '0x' |
2312 | my $val; my $i = -$d; |
394e6ffb |
2313 | while ($len >= 0) |
2314 | { |
03874afe |
2315 | $val = substr($hs,$i,$d); # get hex digits |
7b29e1e6 |
2316 | $val =~ s/^0x// if $len == 0; # for last part only because |
394e6ffb |
2317 | $val = hex($val); # hex does not like wrong chars |
03874afe |
2318 | $i -= $d; $len --; |
2d2b2744 |
2319 | my $adder = [ $val ]; |
2320 | # if the resulting number was to big to fit into one element, create a |
2321 | # two-element version (bug found by Mark Lakata - Thanx!) |
2322 | if (CORE::length($val) > $BASE_LEN) |
2323 | { |
2324 | $adder = _new($c,$val); |
2325 | } |
2326 | _add ($c, $x, _mul ($c, $adder, $mul ) ) if $val != 0; |
394e6ffb |
2327 | _mul ($c, $mul, $m ) if $len >= 0; # skip last mul |
2328 | } |
2329 | $x; |
2330 | } |
2331 | |
2332 | sub _from_bin |
0716bf9b |
2333 | { |
50109ad0 |
2334 | # convert a hex number to decimal (string, return ref to array) |
394e6ffb |
2335 | my ($c,$bs) = @_; |
0716bf9b |
2336 | |
091c87b1 |
2337 | # instead of converting X (8) bit at a time, it is faster to "convert" the |
13a12e00 |
2338 | # number to hex, and then call _from_hex. |
2339 | |
9b924220 |
2340 | my $hs = $bs; |
13a12e00 |
2341 | $hs =~ s/^[+-]?0b//; # remove sign and 0b |
2342 | my $l = length($hs); # bits |
2343 | $hs = '0' x (8-($l % 8)) . $hs if ($l % 8) != 0; # padd left side w/ 0 |
03874afe |
2344 | my $h = '0x' . unpack('H*', pack ('B*', $hs)); # repack as hex |
091c87b1 |
2345 | |
03874afe |
2346 | $c->_from_hex($h); |
0716bf9b |
2347 | } |
2348 | |
07d34614 |
2349 | ############################################################################## |
2350 | # special modulus functions |
2351 | |
56d9de68 |
2352 | sub _modinv |
d614cd8b |
2353 | { |
56d9de68 |
2354 | # modular inverse |
2355 | my ($c,$x,$y) = @_; |
1ddff52a |
2356 | |
56d9de68 |
2357 | my $u = _zero($c); my $u1 = _one($c); |
2358 | my $a = _copy($c,$y); my $b = _copy($c,$x); |
1ddff52a |
2359 | |
2360 | # Euclid's Algorithm for bgcd(), only that we calc bgcd() ($a) and the |
56d9de68 |
2361 | # result ($u) at the same time. See comments in BigInt for why this works. |
2362 | my $q; |
2363 | ($a, $q, $b) = ($b, _div($c,$a,$b)); # step 1 |
2364 | my $sign = 1; |
1ddff52a |
2365 | while (!_is_zero($c,$b)) |
2366 | { |
56d9de68 |
2367 | my $t = _add($c, # step 2: |
2368 | _mul($c,_copy($c,$u1), $q) , # t = u1 * q |
2369 | $u ); # + u |
2370 | $u = $u1; # u = u1, u1 = t |
2371 | $u1 = $t; |
2372 | $sign = -$sign; |
2373 | ($a, $q, $b) = ($b, _div($c,$a,$b)); # step 1 |
1ddff52a |
2374 | } |
2375 | |
2376 | # if the gcd is not 1, then return NaN |
56d9de68 |
2377 | return (undef,undef) unless _is_one($c,$a); |
2378 | |
03874afe |
2379 | ($u1, $sign == 1 ? '+' : '-'); |
d614cd8b |
2380 | } |
2381 | |
2382 | sub _modpow |
2383 | { |
2384 | # modulus of power ($x ** $y) % $z |
07d34614 |
2385 | my ($c,$num,$exp,$mod) = @_; |
2386 | |
2387 | # in the trivial case, |
2388 | if (_is_one($c,$mod)) |
2389 | { |
2390 | splice @$num,0,1; $num->[0] = 0; |
2391 | return $num; |
2392 | } |
2393 | if ((scalar @$num == 1) && (($num->[0] == 0) || ($num->[0] == 1))) |
2394 | { |
2395 | $num->[0] = 1; |
2396 | return $num; |
2397 | } |
1ddff52a |
2398 | |
2399 | # $num = _mod($c,$num,$mod); # this does not make it faster |
07d34614 |
2400 | |
2401 | my $acc = _copy($c,$num); my $t = _one(); |
2402 | |
9b924220 |
2403 | my $expbin = _as_bin($c,$exp); $expbin =~ s/^0b//; |
1ddff52a |
2404 | my $len = length($expbin); |
2405 | while (--$len >= 0) |
07d34614 |
2406 | { |
1ddff52a |
2407 | if ( substr($expbin,$len,1) eq '1') # is_odd |
07d34614 |
2408 | { |
2409 | _mul($c,$t,$acc); |
2410 | $t = _mod($c,$t,$mod); |
2411 | } |
2412 | _mul($c,$acc,$acc); |
2413 | $acc = _mod($c,$acc,$mod); |
07d34614 |
2414 | } |
2415 | @$num = @$t; |
2416 | $num; |
d614cd8b |
2417 | } |
2418 | |
9b924220 |
2419 | sub _gcd |
2420 | { |
2421 | # greatest common divisor |
2422 | my ($c,$x,$y) = @_; |
2423 | |
b68b7ab1 |
2424 | while ( (scalar @$y != 1) || ($y->[0] != 0) ) # while ($y != 0) |
9b924220 |
2425 | { |
2426 | my $t = _copy($c,$y); |
2427 | $y = _mod($c, $x, $y); |
2428 | $x = $t; |
2429 | } |
2430 | $x; |
2431 | } |
2432 | |
394e6ffb |
2433 | ############################################################################## |
2434 | ############################################################################## |
2435 | |
0716bf9b |
2436 | 1; |
2437 | __END__ |
2438 | |
2439 | =head1 NAME |
2440 | |
2441 | Math::BigInt::Calc - Pure Perl module to support Math::BigInt |
2442 | |
2443 | =head1 SYNOPSIS |
2444 | |
ee15d750 |
2445 | Provides support for big integer calculations. Not intended to be used by other |
091c87b1 |
2446 | modules. Other modules which sport the same functions can also be used to support |
2447 | Math::BigInt, like Math::BigInt::GMP or Math::BigInt::Pari. |
0716bf9b |
2448 | |
2449 | =head1 DESCRIPTION |
2450 | |
027dc388 |
2451 | In order to allow for multiple big integer libraries, Math::BigInt was |
2452 | rewritten to use library modules for core math routines. Any module which |
2453 | follows the same API as this can be used instead by using the following: |
0716bf9b |
2454 | |
ee15d750 |
2455 | use Math::BigInt lib => 'libname'; |
0716bf9b |
2456 | |
027dc388 |
2457 | 'libname' is either the long name ('Math::BigInt::Pari'), or only the short |
2458 | version like 'Pari'. |
2459 | |
990fb837 |
2460 | =head1 STORAGE |
2461 | |
2462 | =head1 METHODS |
0716bf9b |
2463 | |
027dc388 |
2464 | The following functions MUST be defined in order to support the use by |
9b924220 |
2465 | Math::BigInt v1.70 or later: |
0716bf9b |
2466 | |
50109ad0 |
2467 | api_version() return API version, 1 for v1.70, 2 for v1.83 |
0716bf9b |
2468 | _new(string) return ref to new object from ref to decimal string |
2469 | _zero() return a new object with value 0 |
2470 | _one() return a new object with value 1 |
9b924220 |
2471 | _two() return a new object with value 2 |
2472 | _ten() return a new object with value 10 |
0716bf9b |
2473 | |
2474 | _str(obj) return ref to a string representing the object |
2475 | _num(obj) returns a Perl integer/floating point number |
2476 | NOTE: because of Perl numeric notation defaults, |
2477 | the _num'ified obj may lose accuracy due to |
3c4b39be |
2478 | machine-dependent floating point size limitations |
0716bf9b |
2479 | |
2480 | _add(obj,obj) Simple addition of two objects |
2481 | _mul(obj,obj) Multiplication of two objects |
2482 | _div(obj,obj) Division of the 1st object by the 2nd |
b22b3e31 |
2483 | In list context, returns (result,remainder). |
2484 | NOTE: this is integer math, so no |
2485 | fractional part will be returned. |
990fb837 |
2486 | The second operand will be not be 0, so no need to |
2487 | check for that. |
b22b3e31 |
2488 | _sub(obj,obj) Simple subtraction of 1 object from another |
0716bf9b |
2489 | a third, optional parameter indicates that the params |
2490 | are swapped. In this case, the first param needs to |
2491 | be preserved, while you can destroy the second. |
2492 | sub (x,y,1) => return x - y and keep x intact! |
3c4b39be |
2493 | _dec(obj) decrement object by one (input is guaranteed to be > 0) |
e745a66c |
2494 | _inc(obj) increment object by one |
2495 | |
0716bf9b |
2496 | |
2497 | _acmp(obj,obj) <=> operator for objects (return -1, 0 or 1) |
2498 | |
2499 | _len(obj) returns count of the decimal digits of the object |
2500 | _digit(obj,n) returns the n'th decimal digit of object |
2501 | |
9b924220 |
2502 | _is_one(obj) return true if argument is 1 |
2503 | _is_two(obj) return true if argument is 2 |
2504 | _is_ten(obj) return true if argument is 10 |
0716bf9b |
2505 | _is_zero(obj) return true if argument is 0 |
2506 | _is_even(obj) return true if argument is even (0,2,4,6..) |
2507 | _is_odd(obj) return true if argument is odd (1,3,5,7..) |
2508 | |
2509 | _copy return a ref to a true copy of the object |
2510 | |
2511 | _check(obj) check whether internal representation is still intact |
2512 | return 0 for ok, otherwise error message as string |
2513 | |
50109ad0 |
2514 | _from_hex(str) return new object from a hexadecimal string |
2515 | _from_bin(str) return new object from a binary string |
2516 | _from_oct(str) return new object from an octal string |
0716bf9b |
2517 | |
9b924220 |
2518 | _as_hex(str) return string containing the value as |
ee15d750 |
2519 | unsigned hex string, with the '0x' prepended. |
2520 | Leading zeros must be stripped. |
2521 | _as_bin(str) Like as_hex, only as binary string containing only |
2522 | zeros and ones. Leading zeros must be stripped and a |
2523 | '0b' must be prepended. |
2524 | |
0716bf9b |
2525 | _rsft(obj,N,B) shift object in base B by N 'digits' right |
2526 | _lsft(obj,N,B) shift object in base B by N 'digits' left |
2527 | |
2528 | _xor(obj1,obj2) XOR (bit-wise) object 1 with object 2 |
dccbb853 |
2529 | Note: XOR, AND and OR pad with zeros if size mismatches |
0716bf9b |
2530 | _and(obj1,obj2) AND (bit-wise) object 1 with object 2 |
2531 | _or(obj1,obj2) OR (bit-wise) object 1 with object 2 |
2532 | |
50109ad0 |
2533 | _mod(obj1,obj2) Return remainder of div of the 1st by the 2nd object |
990fb837 |
2534 | _sqrt(obj) return the square root of object (truncated to int) |
2535 | _root(obj) return the n'th (n >= 3) root of obj (truncated to int) |
b3abae2a |
2536 | _fac(obj) return factorial of object 1 (1*2*3*4..) |
50109ad0 |
2537 | _pow(obj1,obj2) return object 1 to the power of object 2 |
b282a552 |
2538 | return undef for NaN |
b22b3e31 |
2539 | _zeros(obj) return number of trailing decimal zeros |
d614cd8b |
2540 | _modinv return inverse modulus |
2541 | _modpow return modulus of power ($x ** $y) % $z |
091c87b1 |
2542 | _log_int(X,N) calculate integer log() of X in base N |
2543 | X >= 0, N >= 0 (return undef for NaN) |
8df1e0a2 |
2544 | returns (RESULT, EXACT) where EXACT is: |
2545 | 1 : result is exactly RESULT |
2546 | 0 : result was truncated to RESULT |
2547 | undef : unknown whether result is exactly RESULT |
9b924220 |
2548 | _gcd(obj,obj) return Greatest Common Divisor of two objects |
2549 | |
50109ad0 |
2550 | The following functions are REQUIRED for an api_version of 2 or greater: |
2551 | |
2552 | _1ex($x) create the number 1Ex where x >= 0 |
2553 | _alen(obj) returns approximate count of the decimal digits of the |
2554 | object. This estimate MUST always be greater or equal |
2555 | to what _len() returns. |
2556 | _nok(n,k) calculate n over k (binomial coefficient) |
2557 | |
9b924220 |
2558 | The following functions are optional, and can be defined if the underlying lib |
2559 | has a fast way to do them. If undefined, Math::BigInt will use pure Perl (hence |
2560 | slow) fallback routines to emulate these: |
2561 | |
2562 | _signed_or |
2563 | _signed_and |
2564 | _signed_xor |
2565 | |
b22b3e31 |
2566 | Input strings come in as unsigned but with prefix (i.e. as '123', '0xabc' |
0716bf9b |
2567 | or '0b1101'). |
2568 | |
990fb837 |
2569 | So the library needs only to deal with unsigned big integers. Testing of input |
2570 | parameter validity is done by the caller, so you need not worry about |
2571 | underflow (f.i. in C<_sub()>, C<_dec()>) nor about division by zero or similar |
2572 | cases. |
574bacfe |
2573 | |
2574 | The first parameter can be modified, that includes the possibility that you |
2575 | return a reference to a completely different object instead. Although keeping |
50109ad0 |
2576 | the reference and just changing its contents is preferred over creating and |
dccbb853 |
2577 | returning a different reference. |
574bacfe |
2578 | |
990fb837 |
2579 | Return values are always references to objects, strings, or true/false for |
3c4b39be |
2580 | comparison routines. |
990fb837 |
2581 | |
574bacfe |
2582 | =head1 WRAP YOUR OWN |
2583 | |
2584 | If you want to port your own favourite c-lib for big numbers to the |
2585 | Math::BigInt interface, you can take any of the already existing modules as |
2586 | a rough guideline. You should really wrap up the latest BigInt and BigFloat |
bd05a461 |
2587 | testsuites with your module, and replace in them any of the following: |
574bacfe |
2588 | |
2589 | use Math::BigInt; |
2590 | |
bd05a461 |
2591 | by this: |
574bacfe |
2592 | |
2593 | use Math::BigInt lib => 'yourlib'; |
2594 | |
2595 | This way you ensure that your library really works 100% within Math::BigInt. |
0716bf9b |
2596 | |
2597 | =head1 LICENSE |
2598 | |
2599 | This program is free software; you may redistribute it and/or modify it under |
2600 | the same terms as Perl itself. |
2601 | |
2602 | =head1 AUTHORS |
2603 | |
2604 | Original math code by Mark Biggar, rewritten by Tels L<http://bloodgate.com/> |
990fb837 |
2605 | in late 2000. |
0716bf9b |
2606 | Seperated from BigInt and shaped API with the help of John Peacock. |
b68b7ab1 |
2607 | |
7b29e1e6 |
2608 | Fixed, speed-up, streamlined and enhanced by Tels 2001 - 2007. |
0716bf9b |
2609 | |
2610 | =head1 SEE ALSO |
2611 | |
50109ad0 |
2612 | L<Math::BigInt>, L<Math::BigFloat>, |
990fb837 |
2613 | L<Math::BigInt::GMP>, L<Math::BigInt::FastCalc> and L<Math::BigInt::Pari>. |
0716bf9b |
2614 | |
2615 | =cut |