Commit | Line | Data |
0716bf9b |
1 | package Math::BigInt::Calc; |
2 | |
3 | use 5.005; |
4 | use strict; |
574bacfe |
5 | # use warnings; # dont use warnings for older Perls |
0716bf9b |
6 | |
7 | require Exporter; |
bd05a461 |
8 | use vars qw/@ISA $VERSION/; |
0716bf9b |
9 | @ISA = qw(Exporter); |
10 | |
091c87b1 |
11 | $VERSION = '0.37'; |
0716bf9b |
12 | |
13 | # Package to store unsigned big integers in decimal and do math with them |
14 | |
15 | # Internally the numbers are stored in an array with at least 1 element, no |
027dc388 |
16 | # leading zero parts (except the first) and in base 1eX where X is determined |
17 | # automatically at loading time to be the maximum possible value |
0716bf9b |
18 | |
19 | # todo: |
20 | # - fully remove funky $# stuff (maybe) |
0716bf9b |
21 | |
22 | # USE_MUL: due to problems on certain os (os390, posix-bc) "* 1e-5" is used |
ee15d750 |
23 | # instead of "/ 1e5" at some places, (marked with USE_MUL). Other platforms |
24 | # BS2000, some Crays need USE_DIV instead. |
bd05a461 |
25 | # The BEGIN block is used to determine which of the two variants gives the |
26 | # correct result. |
0716bf9b |
27 | |
990fb837 |
28 | # Beware of things like: |
29 | # $i = $i * $y + $car; $car = int($i / $MBASE); $i = $i % $MBASE; |
93c87d9d |
30 | # This works on x86, but fails on ARM (SA1100, iPAQ) due to whoknows what |
990fb837 |
31 | # reasons. So, use this instead (slower, but correct): |
32 | # $i = $i * $y + $car; $car = int($i / $MBASE); $i -= $MBASE * $car; |
33 | |
0716bf9b |
34 | ############################################################################## |
35 | # global constants, flags and accessory |
36 | |
37 | # constants for easier life |
38 | my $nan = 'NaN'; |
61f5c3f5 |
39 | my ($MBASE,$BASE,$RBASE,$BASE_LEN,$MAX_VAL,$BASE_LEN2,$BASE_LEN_SMALL); |
394e6ffb |
40 | my ($AND_BITS,$XOR_BITS,$OR_BITS); |
41 | my ($AND_MASK,$XOR_MASK,$OR_MASK); |
ee15d750 |
42 | |
43 | sub _base_len |
44 | { |
dccbb853 |
45 | # set/get the BASE_LEN and assorted other, connected values |
46 | # used only be the testsuite, set is used only by the BEGIN block below |
394e6ffb |
47 | shift; |
48 | |
ee15d750 |
49 | my $b = shift; |
50 | if (defined $b) |
51 | { |
61f5c3f5 |
52 | # find whether we can use mul or div or none in mul()/div() |
53 | # (in last case reduce BASE_LEN_SMALL) |
54 | $BASE_LEN_SMALL = $b+1; |
55 | my $caught = 0; |
56 | while (--$BASE_LEN_SMALL > 5) |
394e6ffb |
57 | { |
61f5c3f5 |
58 | $MBASE = int("1e".$BASE_LEN_SMALL); |
59 | $RBASE = abs('1e-'.$BASE_LEN_SMALL); # see USE_MUL |
394e6ffb |
60 | $caught = 0; |
61f5c3f5 |
61 | $caught += 1 if (int($MBASE * $RBASE) != 1); # should be 1 |
62 | $caught += 2 if (int($MBASE / $MBASE) != 1); # should be 1 |
394e6ffb |
63 | last if $caught != 3; |
64 | } |
61f5c3f5 |
65 | # BASE_LEN is used for anything else than mul()/div() |
66 | $BASE_LEN = $BASE_LEN_SMALL; |
67 | $BASE_LEN = shift if (defined $_[0]); # one more arg? |
ee15d750 |
68 | $BASE = int("1e".$BASE_LEN); |
61f5c3f5 |
69 | |
70 | $BASE_LEN2 = int($BASE_LEN_SMALL / 2); # for mul shortcut |
71 | $MBASE = int("1e".$BASE_LEN_SMALL); |
72 | $RBASE = abs('1e-'.$BASE_LEN_SMALL); # see USE_MUL |
73 | $MAX_VAL = $MBASE-1; |
990fb837 |
74 | |
61f5c3f5 |
75 | #print "BASE_LEN: $BASE_LEN MAX_VAL: $MAX_VAL BASE: $BASE RBASE: $RBASE "; |
76 | #print "BASE_LEN_SMALL: $BASE_LEN_SMALL MBASE: $MBASE\n"; |
77 | |
b05afeb3 |
78 | undef &_mul; |
79 | undef &_div; |
1ddff52a |
80 | |
990fb837 |
81 | # $caught & 1 != 0 => cannot use MUL |
82 | # $caught & 2 != 0 => cannot use DIV |
83 | # The parens around ($caught & 1) were important, indeed, if we would use |
84 | # & here. |
85 | if ($caught == 2) # 2 |
ee15d750 |
86 | { |
990fb837 |
87 | # print "# use mul\n"; |
88 | # must USE_MUL since we cannot use DIV |
ee15d750 |
89 | *{_mul} = \&_mul_use_mul; |
90 | *{_div} = \&_div_use_mul; |
91 | } |
990fb837 |
92 | else # 0 or 1 |
ee15d750 |
93 | { |
990fb837 |
94 | # print "# use div\n"; |
ee15d750 |
95 | # can USE_DIV instead |
96 | *{_mul} = \&_mul_use_div; |
97 | *{_div} = \&_div_use_div; |
98 | } |
99 | } |
61f5c3f5 |
100 | return $BASE_LEN unless wantarray; |
101 | return ($BASE_LEN, $AND_BITS, $XOR_BITS, $OR_BITS, $BASE_LEN_SMALL, $MAX_VAL); |
ee15d750 |
102 | } |
574bacfe |
103 | |
104 | BEGIN |
105 | { |
bd05a461 |
106 | # from Daniel Pfeiffer: determine largest group of digits that is precisely |
574bacfe |
107 | # multipliable with itself plus carry |
dccbb853 |
108 | # Test now changed to expect the proper pattern, not a result off by 1 or 2 |
109 | my ($e, $num) = 3; # lowest value we will use is 3+1-1 = 3 |
bd05a461 |
110 | do |
111 | { |
112 | $num = ('9' x ++$e) + 0; |
394e6ffb |
113 | $num *= $num + 1.0; |
394e6ffb |
114 | } while ("$num" =~ /9{$e}0{$e}/); # must be a certain pattern |
115 | $e--; # last test failed, so retract one step |
116 | # the limits below brush the problems with the test above under the rug: |
117 | # the test should be able to find the proper $e automatically |
118 | $e = 5 if $^O =~ /^uts/; # UTS get's some special treatment |
119 | $e = 5 if $^O =~ /^unicos/; # unicos is also problematic (6 seems to work |
120 | # there, but we play safe) |
07d34614 |
121 | $e = 5 if $] < 5.006; # cap, for older Perls |
2e507a43 |
122 | $e = 7 if $e > 7; # cap, for VMS, OS/390 and other 64 bit systems |
123 | # 8 fails inside random testsuite, so take 7 |
394e6ffb |
124 | |
61f5c3f5 |
125 | # determine how many digits fit into an integer and can be safely added |
126 | # together plus carry w/o causing an overflow |
127 | |
61f5c3f5 |
128 | use integer; |
c38b2de2 |
129 | |
130 | ############################################################################ |
131 | # the next block is no longer important |
132 | |
133 | ## this below detects 15 on a 64 bit system, because after that it becomes |
134 | ## 1e16 and not 1000000 :/ I can make it detect 18, but then I get a lot of |
135 | ## test failures. Ugh! (Tomake detect 18: uncomment lines marked with *) |
136 | |
137 | #my $bi = 5; # approx. 16 bit |
138 | #$num = int('9' x $bi); |
139 | ## $num = 99999; # * |
140 | ## while ( ($num+$num+1) eq '1' . '9' x $bi) # * |
141 | #while ( int($num+$num+1) eq '1' . '9' x $bi) |
142 | # { |
143 | # $bi++; $num = int('9' x $bi); |
144 | # # $bi++; $num *= 10; $num += 9; # * |
145 | # } |
146 | #$bi--; # back off one step |
61f5c3f5 |
147 | # by setting them equal, we ignore the findings and use the default |
148 | # one-size-fits-all approach from former versions |
c38b2de2 |
149 | my $bi = $e; # XXX, this should work always |
61f5c3f5 |
150 | |
151 | __PACKAGE__->_base_len($e,$bi); # set and store |
394e6ffb |
152 | |
153 | # find out how many bits _and, _or and _xor can take (old default = 16) |
154 | # I don't think anybody has yet 128 bit scalars, so let's play safe. |
394e6ffb |
155 | local $^W = 0; # don't warn about 'nonportable number' |
c38b2de2 |
156 | $AND_BITS = 15; $XOR_BITS = 15; $OR_BITS = 15; |
394e6ffb |
157 | |
158 | # find max bits, we will not go higher than numberofbits that fit into $BASE |
159 | # to make _and etc simpler (and faster for smaller, slower for large numbers) |
160 | my $max = 16; |
161 | while (2 ** $max < $BASE) { $max++; } |
1ddff52a |
162 | { |
163 | no integer; |
164 | $max = 16 if $] < 5.006; # older Perls might not take >16 too well |
165 | } |
394e6ffb |
166 | my ($x,$y,$z); |
167 | do { |
168 | $AND_BITS++; |
169 | $x = oct('0b' . '1' x $AND_BITS); $y = $x & $x; |
170 | $z = (2 ** $AND_BITS) - 1; |
171 | } while ($AND_BITS < $max && $x == $z && $y == $x); |
172 | $AND_BITS --; # retreat one step |
173 | do { |
174 | $XOR_BITS++; |
175 | $x = oct('0b' . '1' x $XOR_BITS); $y = $x ^ 0; |
176 | $z = (2 ** $XOR_BITS) - 1; |
177 | } while ($XOR_BITS < $max && $x == $z && $y == $x); |
178 | $XOR_BITS --; # retreat one step |
179 | do { |
180 | $OR_BITS++; |
181 | $x = oct('0b' . '1' x $OR_BITS); $y = $x | $x; |
182 | $z = (2 ** $OR_BITS) - 1; |
183 | } while ($OR_BITS < $max && $x == $z && $y == $x); |
184 | $OR_BITS --; # retreat one step |
185 | |
574bacfe |
186 | } |
187 | |
61f5c3f5 |
188 | ############################################################################### |
0716bf9b |
189 | |
190 | sub _new |
191 | { |
394e6ffb |
192 | # (ref to string) return ref to num_array |
9393ace2 |
193 | # Convert a number from string format (without sign) to internal base |
194 | # 1ex format. Assumes normalized value as input. |
574bacfe |
195 | my $d = $_[1]; |
61f5c3f5 |
196 | my $il = length($$d)-1; |
197 | # this leaves '00000' instead of int 0 and will be corrected after any op |
198 | [ reverse(unpack("a" . ($il % $BASE_LEN+1) |
574bacfe |
199 | . ("a$BASE_LEN" x ($il / $BASE_LEN)), $$d)) ]; |
0716bf9b |
200 | } |
394e6ffb |
201 | |
202 | BEGIN |
203 | { |
204 | $AND_MASK = __PACKAGE__->_new( \( 2 ** $AND_BITS )); |
205 | $XOR_MASK = __PACKAGE__->_new( \( 2 ** $XOR_BITS )); |
206 | $OR_MASK = __PACKAGE__->_new( \( 2 ** $OR_BITS )); |
207 | } |
0716bf9b |
208 | |
209 | sub _zero |
210 | { |
211 | # create a zero |
61f5c3f5 |
212 | [ 0 ]; |
0716bf9b |
213 | } |
214 | |
215 | sub _one |
216 | { |
217 | # create a one |
61f5c3f5 |
218 | [ 1 ]; |
0716bf9b |
219 | } |
220 | |
027dc388 |
221 | sub _two |
222 | { |
1ddff52a |
223 | # create a two (used internally for shifting) |
61f5c3f5 |
224 | [ 2 ]; |
027dc388 |
225 | } |
226 | |
0716bf9b |
227 | sub _copy |
228 | { |
091c87b1 |
229 | # make a true copy |
61f5c3f5 |
230 | [ @{$_[1]} ]; |
0716bf9b |
231 | } |
232 | |
bd05a461 |
233 | # catch and throw away |
234 | sub import { } |
235 | |
0716bf9b |
236 | ############################################################################## |
237 | # convert back to string and number |
238 | |
239 | sub _str |
240 | { |
241 | # (ref to BINT) return num_str |
242 | # Convert number from internal base 100000 format to string format. |
243 | # internal format is always normalized (no leading zeros, "-0" => "+0") |
574bacfe |
244 | my $ar = $_[1]; |
0716bf9b |
245 | my $ret = ""; |
61f5c3f5 |
246 | |
247 | my $l = scalar @$ar; # number of parts |
248 | return $nan if $l < 1; # should not happen |
249 | |
0716bf9b |
250 | # handle first one different to strip leading zeros from it (there are no |
251 | # leading zero parts in internal representation) |
61f5c3f5 |
252 | $l --; $ret .= int($ar->[$l]); $l--; |
0716bf9b |
253 | # Interestingly, the pre-padd method uses more time |
091c87b1 |
254 | # the old grep variant takes longer (14 vs. 10 sec) |
574bacfe |
255 | my $z = '0' x ($BASE_LEN-1); |
0716bf9b |
256 | while ($l >= 0) |
257 | { |
574bacfe |
258 | $ret .= substr($z.$ar->[$l],-$BASE_LEN); # fastest way I could think of |
0716bf9b |
259 | $l--; |
260 | } |
61f5c3f5 |
261 | \$ret; |
0716bf9b |
262 | } |
263 | |
264 | sub _num |
265 | { |
266 | # Make a number (scalar int/float) from a BigInt object |
574bacfe |
267 | my $x = $_[1]; |
0716bf9b |
268 | return $x->[0] if scalar @$x == 1; # below $BASE |
269 | my $fac = 1; |
270 | my $num = 0; |
271 | foreach (@$x) |
272 | { |
273 | $num += $fac*$_; $fac *= $BASE; |
274 | } |
61f5c3f5 |
275 | $num; |
0716bf9b |
276 | } |
277 | |
278 | ############################################################################## |
279 | # actual math code |
280 | |
281 | sub _add |
282 | { |
283 | # (ref to int_num_array, ref to int_num_array) |
574bacfe |
284 | # routine to add two base 1eX numbers |
0716bf9b |
285 | # stolen from Knuth Vol 2 Algorithm A pg 231 |
b22b3e31 |
286 | # there are separate routines to add and sub as per Knuth pg 233 |
0716bf9b |
287 | # This routine clobbers up array x, but not y. |
288 | |
574bacfe |
289 | my ($c,$x,$y) = @_; |
b3abae2a |
290 | |
291 | return $x if (@$y == 1) && $y->[0] == 0; # $x + 0 => $x |
292 | if ((@$x == 1) && $x->[0] == 0) # 0 + $y => $y->copy |
293 | { |
294 | # twice as slow as $x = [ @$y ], but necc. to retain $x as ref :( |
295 | @$x = @$y; return $x; |
296 | } |
0716bf9b |
297 | |
298 | # for each in Y, add Y to X and carry. If after that, something is left in |
299 | # X, foreach in X add carry to X and then return X, carry |
091c87b1 |
300 | # Trades one "$j++" for having to shift arrays |
0716bf9b |
301 | my $i; my $car = 0; my $j = 0; |
302 | for $i (@$y) |
303 | { |
e745a66c |
304 | $x->[$j] -= $BASE if $car = (($x->[$j] += $i + $car) >= $BASE) ? 1 : 0; |
0716bf9b |
305 | $j++; |
306 | } |
307 | while ($car != 0) |
308 | { |
309 | $x->[$j] -= $BASE if $car = (($x->[$j] += $car) >= $BASE) ? 1 : 0; $j++; |
310 | } |
61f5c3f5 |
311 | $x; |
e745a66c |
312 | } |
313 | |
314 | sub _inc |
315 | { |
316 | # (ref to int_num_array, ref to int_num_array) |
091c87b1 |
317 | # Add 1 to $x, modify $x in place |
e745a66c |
318 | my ($c,$x) = @_; |
319 | |
320 | for my $i (@$x) |
321 | { |
322 | return $x if (($i += 1) < $BASE); # early out |
61f5c3f5 |
323 | $i = 0; # overflow, next |
e745a66c |
324 | } |
61f5c3f5 |
325 | push @$x,1 if ($x->[-1] == 0); # last overflowed, so extend |
326 | $x; |
e745a66c |
327 | } |
328 | |
329 | sub _dec |
330 | { |
331 | # (ref to int_num_array, ref to int_num_array) |
091c87b1 |
332 | # Sub 1 from $x, modify $x in place |
e745a66c |
333 | my ($c,$x) = @_; |
334 | |
61f5c3f5 |
335 | my $MAX = $BASE-1; # since MAX_VAL based on MBASE |
e745a66c |
336 | for my $i (@$x) |
337 | { |
338 | last if (($i -= 1) >= 0); # early out |
091c87b1 |
339 | $i = $MAX; # underflow, next |
e745a66c |
340 | } |
091c87b1 |
341 | pop @$x if $x->[-1] == 0 && @$x > 1; # last underflowed (but leave 0) |
61f5c3f5 |
342 | $x; |
0716bf9b |
343 | } |
344 | |
345 | sub _sub |
346 | { |
9393ace2 |
347 | # (ref to int_num_array, ref to int_num_array, swap) |
574bacfe |
348 | # subtract base 1eX numbers -- stolen from Knuth Vol 2 pg 232, $x > $y |
56b9c951 |
349 | # subtract Y from X by modifying x in place |
574bacfe |
350 | my ($c,$sx,$sy,$s) = @_; |
0716bf9b |
351 | |
352 | my $car = 0; my $i; my $j = 0; |
353 | if (!$s) |
354 | { |
355 | #print "case 2\n"; |
356 | for $i (@$sx) |
357 | { |
358 | last unless defined $sy->[$j] || $car; |
0716bf9b |
359 | $i += $BASE if $car = (($i -= ($sy->[$j] || 0) + $car) < 0); $j++; |
0716bf9b |
360 | } |
361 | # might leave leading zeros, so fix that |
394e6ffb |
362 | return __strip_zeros($sx); |
0716bf9b |
363 | } |
394e6ffb |
364 | #print "case 1 (swap)\n"; |
365 | for $i (@$sx) |
0716bf9b |
366 | { |
07d34614 |
367 | # we can't do an early out if $x is < than $y, since we |
56b9c951 |
368 | # need to copy the high chunks from $y. Found by Bob Mathews. |
369 | #last unless defined $sy->[$j] || $car; |
394e6ffb |
370 | $sy->[$j] += $BASE |
371 | if $car = (($sy->[$j] = $i-($sy->[$j]||0) - $car) < 0); |
372 | $j++; |
0716bf9b |
373 | } |
394e6ffb |
374 | # might leave leading zeros, so fix that |
375 | __strip_zeros($sy); |
0716bf9b |
376 | } |
377 | |
ee15d750 |
378 | sub _mul_use_mul |
0716bf9b |
379 | { |
9393ace2 |
380 | # (ref to int_num_array, ref to int_num_array) |
0716bf9b |
381 | # multiply two numbers in internal representation |
b22b3e31 |
382 | # modifies first arg, second need not be different from first |
574bacfe |
383 | my ($c,$xv,$yv) = @_; |
dccbb853 |
384 | |
990fb837 |
385 | if (@$yv == 1) |
b3abae2a |
386 | { |
990fb837 |
387 | # shortcut for two very short numbers (improved by Nathan Zook) |
388 | # works also if xv and yv are the same reference, and handles also $x == 0 |
389 | if (@$xv == 1) |
390 | { |
391 | if (($xv->[0] *= $yv->[0]) >= $MBASE) |
392 | { |
393 | $xv->[0] = $xv->[0] - ($xv->[1] = int($xv->[0] * $RBASE)) * $MBASE; |
394 | }; |
395 | return $xv; |
396 | } |
397 | # $x * 0 => 0 |
398 | if ($yv->[0] == 0) |
399 | { |
400 | @$xv = (0); |
401 | return $xv; |
402 | } |
403 | # multiply a large number a by a single element one, so speed up |
404 | my $y = $yv->[0]; my $car = 0; |
405 | foreach my $i (@$xv) |
406 | { |
407 | $i = $i * $y + $car; $car = int($i * $RBASE); $i -= $car * $MBASE; |
408 | } |
409 | push @$xv, $car if $car != 0; |
b3abae2a |
410 | return $xv; |
411 | } |
990fb837 |
412 | # shortcut for result $x == 0 => result = 0 |
413 | return $xv if ( ((@$xv == 1) && ($xv->[0] == 0)) ); |
b3abae2a |
414 | |
0716bf9b |
415 | # since multiplying $x with $x fails, make copy in this case |
d614cd8b |
416 | $yv = [@$xv] if $xv == $yv; # same references? |
9393ace2 |
417 | |
61f5c3f5 |
418 | my @prod = (); my ($prod,$car,$cty,$xi,$yi); |
419 | |
0716bf9b |
420 | for $xi (@$xv) |
421 | { |
422 | $car = 0; $cty = 0; |
574bacfe |
423 | |
424 | # slow variant |
425 | # for $yi (@$yv) |
426 | # { |
427 | # $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
428 | # $prod[$cty++] = |
61f5c3f5 |
429 | # $prod - ($car = int($prod * RBASE)) * $MBASE; # see USE_MUL |
574bacfe |
430 | # } |
431 | # $prod[$cty] += $car if $car; # need really to check for 0? |
432 | # $xi = shift @prod; |
433 | |
434 | # faster variant |
435 | # looping through this if $xi == 0 is silly - so optimize it away! |
436 | $xi = (shift @prod || 0), next if $xi == 0; |
0716bf9b |
437 | for $yi (@$yv) |
438 | { |
439 | $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
574bacfe |
440 | ## this is actually a tad slower |
441 | ## $prod = $prod[$cty]; $prod += ($car + $xi * $yi); # no ||0 here |
0716bf9b |
442 | $prod[$cty++] = |
61f5c3f5 |
443 | $prod - ($car = int($prod * $RBASE)) * $MBASE; # see USE_MUL |
0716bf9b |
444 | } |
445 | $prod[$cty] += $car if $car; # need really to check for 0? |
027dc388 |
446 | $xi = shift @prod || 0; # || 0 makes v5.005_3 happy |
0716bf9b |
447 | } |
0716bf9b |
448 | push @$xv, @prod; |
990fb837 |
449 | __strip_zeros($xv); |
61f5c3f5 |
450 | $xv; |
0716bf9b |
451 | } |
452 | |
ee15d750 |
453 | sub _mul_use_div |
454 | { |
9393ace2 |
455 | # (ref to int_num_array, ref to int_num_array) |
ee15d750 |
456 | # multiply two numbers in internal representation |
457 | # modifies first arg, second need not be different from first |
458 | my ($c,$xv,$yv) = @_; |
459 | |
990fb837 |
460 | if (@$yv == 1) |
b3abae2a |
461 | { |
990fb837 |
462 | # shortcut for two small numbers, also handles $x == 0 |
463 | if (@$xv == 1) |
464 | { |
465 | # shortcut for two very short numbers (improved by Nathan Zook) |
466 | # works also if xv and yv are the same reference, and handles also $x == 0 |
467 | if (($xv->[0] *= $yv->[0]) >= $MBASE) |
468 | { |
469 | $xv->[0] = |
470 | $xv->[0] - ($xv->[1] = int($xv->[0] / $MBASE)) * $MBASE; |
471 | }; |
472 | return $xv; |
473 | } |
474 | # $x * 0 => 0 |
475 | if ($yv->[0] == 0) |
476 | { |
477 | @$xv = (0); |
478 | return $xv; |
479 | } |
480 | # multiply a large number a by a single element one, so speed up |
481 | my $y = $yv->[0]; my $car = 0; |
482 | foreach my $i (@$xv) |
483 | { |
484 | $i = $i * $y + $car; $car = int($i / $MBASE); $i -= $car * $MBASE; |
485 | } |
486 | push @$xv, $car if $car != 0; |
b3abae2a |
487 | return $xv; |
488 | } |
990fb837 |
489 | # shortcut for result $x == 0 => result = 0 |
490 | return $xv if ( ((@$xv == 1) && ($xv->[0] == 0)) ); |
b3abae2a |
491 | |
ee15d750 |
492 | # since multiplying $x with $x fails, make copy in this case |
d614cd8b |
493 | $yv = [@$xv] if $xv == $yv; # same references? |
9393ace2 |
494 | |
61f5c3f5 |
495 | my @prod = (); my ($prod,$car,$cty,$xi,$yi); |
ee15d750 |
496 | for $xi (@$xv) |
497 | { |
498 | $car = 0; $cty = 0; |
499 | # looping through this if $xi == 0 is silly - so optimize it away! |
500 | $xi = (shift @prod || 0), next if $xi == 0; |
501 | for $yi (@$yv) |
502 | { |
503 | $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
504 | $prod[$cty++] = |
61f5c3f5 |
505 | $prod - ($car = int($prod / $MBASE)) * $MBASE; |
ee15d750 |
506 | } |
507 | $prod[$cty] += $car if $car; # need really to check for 0? |
027dc388 |
508 | $xi = shift @prod || 0; # || 0 makes v5.005_3 happy |
ee15d750 |
509 | } |
510 | push @$xv, @prod; |
990fb837 |
511 | __strip_zeros($xv); |
61f5c3f5 |
512 | $xv; |
ee15d750 |
513 | } |
514 | |
515 | sub _div_use_mul |
0716bf9b |
516 | { |
b22b3e31 |
517 | # ref to array, ref to array, modify first array and return remainder if |
0716bf9b |
518 | # in list context |
aef458a0 |
519 | |
520 | # see comments in _div_use_div() for more explanations |
521 | |
574bacfe |
522 | my ($c,$x,$yorg) = @_; |
aef458a0 |
523 | |
524 | # the general div algorithmn here is about O(N*N) and thus quite slow, so |
525 | # we first check for some special cases and use shortcuts to handle them. |
0716bf9b |
526 | |
aef458a0 |
527 | # This works, because we store the numbers in a chunked format where each |
528 | # element contains 5..7 digits (depending on system). |
529 | |
530 | # if both numbers have only one element: |
61f5c3f5 |
531 | if (@$x == 1 && @$yorg == 1) |
532 | { |
13a12e00 |
533 | # shortcut, $yorg and $x are two small numbers |
61f5c3f5 |
534 | if (wantarray) |
535 | { |
536 | my $r = [ $x->[0] % $yorg->[0] ]; |
537 | $x->[0] = int($x->[0] / $yorg->[0]); |
538 | return ($x,$r); |
539 | } |
540 | else |
541 | { |
542 | $x->[0] = int($x->[0] / $yorg->[0]); |
543 | return $x; |
544 | } |
545 | } |
aef458a0 |
546 | |
547 | # if x has more than one, but y has only one element: |
28df3e88 |
548 | if (@$yorg == 1) |
549 | { |
550 | my $rem; |
551 | $rem = _mod($c,[ @$x ],$yorg) if wantarray; |
13a12e00 |
552 | |
28df3e88 |
553 | # shortcut, $y is < $BASE |
554 | my $j = scalar @$x; my $r = 0; |
555 | my $y = $yorg->[0]; my $b; |
556 | while ($j-- > 0) |
557 | { |
558 | $b = $r * $MBASE + $x->[$j]; |
559 | $x->[$j] = int($b/$y); |
560 | $r = $b % $y; |
561 | } |
562 | pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero |
563 | return ($x,$rem) if wantarray; |
564 | return $x; |
565 | } |
0716bf9b |
566 | |
aef458a0 |
567 | # now x and y have more than one element |
568 | |
569 | # check whether y has more elements than x, if yet, the result will be 0 |
570 | if (@$yorg > @$x) |
571 | { |
572 | my $rem; |
573 | $rem = [@$x] if wantarray; # make copy |
574 | splice (@$x,1); # keep ref to original array |
575 | $x->[0] = 0; # set to 0 |
576 | return ($x,$rem) if wantarray; # including remainder? |
577 | return $x; # only x, which is [0] now |
578 | } |
579 | # check whether the numbers have the same number of elements, in that case |
580 | # the result will fit into one element and can be computed efficiently |
581 | if (@$yorg == @$x) |
582 | { |
583 | my $rem; |
584 | # if $yorg has more digits than $x (it's leading element is longer than |
585 | # the one from $x), the result will also be 0: |
586 | if (length(int($yorg->[-1])) > length(int($x->[-1]))) |
587 | { |
588 | $rem = [@$x] if wantarray; # make copy |
589 | splice (@$x,1); # keep ref to org array |
590 | $x->[0] = 0; # set to 0 |
591 | return ($x,$rem) if wantarray; # including remainder? |
592 | return $x; |
593 | } |
594 | # now calculate $x / $yorg |
595 | if (length(int($yorg->[-1])) == length(int($x->[-1]))) |
596 | { |
597 | # same length, so make full compare, and if equal, return 1 |
598 | # hm, same lengths, but same contents? So we need to check all parts: |
599 | my $a = 0; my $j = scalar @$x - 1; |
600 | # manual way (abort if unequal, good for early ne) |
601 | while ($j >= 0) |
602 | { |
603 | last if ($a = $x->[$j] - $yorg->[$j]); $j--; |
604 | } |
605 | # $a contains the result of the compare between X and Y |
606 | # a < 0: x < y, a == 0 => x == y, a > 0: x > y |
607 | if ($a <= 0) |
608 | { |
609 | if (wantarray) |
610 | { |
611 | $rem = [ 0 ]; # a = 0 => x == y => rem 1 |
612 | $rem = [@$x] if $a != 0; # a < 0 => x < y => rem = x |
613 | } |
614 | splice(@$x,1); # keep single element |
615 | $x->[0] = 0; # if $a < 0 |
616 | if ($a == 0) |
617 | { |
618 | # $x == $y |
619 | $x->[0] = 1; |
620 | } |
621 | return ($x,$rem) if wantarray; |
622 | return $x; |
623 | } |
624 | # $x >= $y, proceed normally |
625 | } |
626 | } |
627 | |
628 | # all other cases: |
629 | |
d614cd8b |
630 | my $y = [ @$yorg ]; # always make copy to preserve |
61f5c3f5 |
631 | |
632 | my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0); |
633 | |
634 | $car = $bar = $prd = 0; |
635 | if (($dd = int($MBASE/($y->[-1]+1))) != 1) |
0716bf9b |
636 | { |
637 | for $xi (@$x) |
638 | { |
639 | $xi = $xi * $dd + $car; |
61f5c3f5 |
640 | $xi -= ($car = int($xi * $RBASE)) * $MBASE; # see USE_MUL |
0716bf9b |
641 | } |
642 | push(@$x, $car); $car = 0; |
643 | for $yi (@$y) |
644 | { |
645 | $yi = $yi * $dd + $car; |
61f5c3f5 |
646 | $yi -= ($car = int($yi * $RBASE)) * $MBASE; # see USE_MUL |
0716bf9b |
647 | } |
648 | } |
649 | else |
650 | { |
651 | push(@$x, 0); |
652 | } |
653 | @q = (); ($v2,$v1) = @$y[-2,-1]; |
654 | $v2 = 0 unless $v2; |
655 | while ($#$x > $#$y) |
656 | { |
657 | ($u2,$u1,$u0) = @$x[-3..-1]; |
658 | $u2 = 0 unless $u2; |
659 | #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" |
660 | # if $v1 == 0; |
aef458a0 |
661 | $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$MBASE+$u1)/$v1)); |
61f5c3f5 |
662 | --$q while ($v2*$q > ($u0*$MBASE+$u1-$q*$v1)*$MBASE+$u2); |
0716bf9b |
663 | if ($q) |
664 | { |
665 | ($car, $bar) = (0,0); |
666 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
667 | { |
668 | $prd = $q * $y->[$yi] + $car; |
61f5c3f5 |
669 | $prd -= ($car = int($prd * $RBASE)) * $MBASE; # see USE_MUL |
670 | $x->[$xi] += $MBASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0)); |
0716bf9b |
671 | } |
672 | if ($x->[-1] < $car + $bar) |
673 | { |
674 | $car = 0; --$q; |
675 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
676 | { |
61f5c3f5 |
677 | $x->[$xi] -= $MBASE |
aef458a0 |
678 | if ($car = (($x->[$xi] += $y->[$yi] + $car) >= $MBASE)); |
0716bf9b |
679 | } |
680 | } |
681 | } |
aef458a0 |
682 | pop(@$x); |
683 | unshift(@q, $q); |
0716bf9b |
684 | } |
685 | if (wantarray) |
686 | { |
687 | @d = (); |
688 | if ($dd != 1) |
689 | { |
690 | $car = 0; |
691 | for $xi (reverse @$x) |
692 | { |
61f5c3f5 |
693 | $prd = $car * $MBASE + $xi; |
0716bf9b |
694 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; # see USE_MUL |
695 | unshift(@d, $tmp); |
696 | } |
697 | } |
698 | else |
699 | { |
700 | @d = @$x; |
701 | } |
702 | @$x = @q; |
61f5c3f5 |
703 | my $d = \@d; |
990fb837 |
704 | __strip_zeros($x); |
705 | __strip_zeros($d); |
61f5c3f5 |
706 | return ($x,$d); |
0716bf9b |
707 | } |
708 | @$x = @q; |
990fb837 |
709 | __strip_zeros($x); |
61f5c3f5 |
710 | $x; |
0716bf9b |
711 | } |
712 | |
ee15d750 |
713 | sub _div_use_div |
714 | { |
715 | # ref to array, ref to array, modify first array and return remainder if |
716 | # in list context |
ee15d750 |
717 | my ($c,$x,$yorg) = @_; |
ee15d750 |
718 | |
990fb837 |
719 | # the general div algorithmn here is about O(N*N) and thus quite slow, so |
720 | # we first check for some special cases and use shortcuts to handle them. |
721 | |
722 | # This works, because we store the numbers in a chunked format where each |
723 | # element contains 5..7 digits (depending on system). |
724 | |
725 | # if both numbers have only one element: |
61f5c3f5 |
726 | if (@$x == 1 && @$yorg == 1) |
727 | { |
13a12e00 |
728 | # shortcut, $yorg and $x are two small numbers |
61f5c3f5 |
729 | if (wantarray) |
730 | { |
731 | my $r = [ $x->[0] % $yorg->[0] ]; |
732 | $x->[0] = int($x->[0] / $yorg->[0]); |
733 | return ($x,$r); |
734 | } |
735 | else |
736 | { |
737 | $x->[0] = int($x->[0] / $yorg->[0]); |
738 | return $x; |
739 | } |
740 | } |
990fb837 |
741 | # if x has more than one, but y has only one element: |
28df3e88 |
742 | if (@$yorg == 1) |
743 | { |
744 | my $rem; |
745 | $rem = _mod($c,[ @$x ],$yorg) if wantarray; |
746 | |
747 | # shortcut, $y is < $BASE |
748 | my $j = scalar @$x; my $r = 0; |
749 | my $y = $yorg->[0]; my $b; |
750 | while ($j-- > 0) |
751 | { |
752 | $b = $r * $MBASE + $x->[$j]; |
753 | $x->[$j] = int($b/$y); |
754 | $r = $b % $y; |
755 | } |
756 | pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero |
757 | return ($x,$rem) if wantarray; |
758 | return $x; |
759 | } |
990fb837 |
760 | # now x and y have more than one element |
ee15d750 |
761 | |
990fb837 |
762 | # check whether y has more elements than x, if yet, the result will be 0 |
763 | if (@$yorg > @$x) |
61f5c3f5 |
764 | { |
990fb837 |
765 | my $rem; |
766 | $rem = [@$x] if wantarray; # make copy |
767 | splice (@$x,1); # keep ref to original array |
768 | $x->[0] = 0; # set to 0 |
769 | return ($x,$rem) if wantarray; # including remainder? |
aef458a0 |
770 | return $x; # only x, which is [0] now |
61f5c3f5 |
771 | } |
990fb837 |
772 | # check whether the numbers have the same number of elements, in that case |
773 | # the result will fit into one element and can be computed efficiently |
774 | if (@$yorg == @$x) |
775 | { |
776 | my $rem; |
777 | # if $yorg has more digits than $x (it's leading element is longer than |
778 | # the one from $x), the result will also be 0: |
779 | if (length(int($yorg->[-1])) > length(int($x->[-1]))) |
780 | { |
781 | $rem = [@$x] if wantarray; # make copy |
782 | splice (@$x,1); # keep ref to org array |
783 | $x->[0] = 0; # set to 0 |
784 | return ($x,$rem) if wantarray; # including remainder? |
785 | return $x; |
786 | } |
787 | # now calculate $x / $yorg |
091c87b1 |
788 | |
990fb837 |
789 | if (length(int($yorg->[-1])) == length(int($x->[-1]))) |
790 | { |
791 | # same length, so make full compare, and if equal, return 1 |
aef458a0 |
792 | # hm, same lengths, but same contents? So we need to check all parts: |
990fb837 |
793 | my $a = 0; my $j = scalar @$x - 1; |
794 | # manual way (abort if unequal, good for early ne) |
795 | while ($j >= 0) |
796 | { |
797 | last if ($a = $x->[$j] - $yorg->[$j]); $j--; |
798 | } |
aef458a0 |
799 | # $a contains the result of the compare between X and Y |
990fb837 |
800 | # a < 0: x < y, a == 0 => x == y, a > 0: x > y |
801 | if ($a <= 0) |
802 | { |
aef458a0 |
803 | if (wantarray) |
804 | { |
805 | $rem = [ 0 ]; # a = 0 => x == y => rem 1 |
806 | $rem = [@$x] if $a != 0; # a < 0 => x < y => rem = x |
807 | } |
808 | splice(@$x,1); # keep single element |
990fb837 |
809 | $x->[0] = 0; # if $a < 0 |
810 | if ($a == 0) |
811 | { |
812 | # $x == $y |
813 | $x->[0] = 1; |
814 | } |
815 | return ($x,$rem) if wantarray; |
816 | return $x; |
817 | } |
aef458a0 |
818 | # $x >= $y, so proceed normally |
990fb837 |
819 | } |
990fb837 |
820 | } |
821 | |
822 | # all other cases: |
823 | |
824 | my $y = [ @$yorg ]; # always make copy to preserve |
61f5c3f5 |
825 | |
826 | my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0); |
827 | |
828 | $car = $bar = $prd = 0; |
829 | if (($dd = int($MBASE/($y->[-1]+1))) != 1) |
ee15d750 |
830 | { |
831 | for $xi (@$x) |
832 | { |
833 | $xi = $xi * $dd + $car; |
61f5c3f5 |
834 | $xi -= ($car = int($xi / $MBASE)) * $MBASE; |
ee15d750 |
835 | } |
836 | push(@$x, $car); $car = 0; |
837 | for $yi (@$y) |
838 | { |
839 | $yi = $yi * $dd + $car; |
61f5c3f5 |
840 | $yi -= ($car = int($yi / $MBASE)) * $MBASE; |
ee15d750 |
841 | } |
842 | } |
843 | else |
844 | { |
845 | push(@$x, 0); |
846 | } |
aef458a0 |
847 | |
848 | # @q will accumulate the final result, $q contains the current computed |
849 | # part of the final result |
850 | |
ee15d750 |
851 | @q = (); ($v2,$v1) = @$y[-2,-1]; |
852 | $v2 = 0 unless $v2; |
853 | while ($#$x > $#$y) |
854 | { |
855 | ($u2,$u1,$u0) = @$x[-3..-1]; |
856 | $u2 = 0 unless $u2; |
857 | #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" |
858 | # if $v1 == 0; |
aef458a0 |
859 | $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$MBASE+$u1)/$v1)); |
61f5c3f5 |
860 | --$q while ($v2*$q > ($u0*$MBASE+$u1-$q*$v1)*$MBASE+$u2); |
ee15d750 |
861 | if ($q) |
862 | { |
863 | ($car, $bar) = (0,0); |
864 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
865 | { |
866 | $prd = $q * $y->[$yi] + $car; |
61f5c3f5 |
867 | $prd -= ($car = int($prd / $MBASE)) * $MBASE; |
868 | $x->[$xi] += $MBASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0)); |
ee15d750 |
869 | } |
870 | if ($x->[-1] < $car + $bar) |
871 | { |
872 | $car = 0; --$q; |
873 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
874 | { |
61f5c3f5 |
875 | $x->[$xi] -= $MBASE |
aef458a0 |
876 | if ($car = (($x->[$xi] += $y->[$yi] + $car) >= $MBASE)); |
ee15d750 |
877 | } |
878 | } |
879 | } |
61f5c3f5 |
880 | pop(@$x); unshift(@q, $q); |
ee15d750 |
881 | } |
882 | if (wantarray) |
883 | { |
884 | @d = (); |
885 | if ($dd != 1) |
886 | { |
887 | $car = 0; |
888 | for $xi (reverse @$x) |
889 | { |
61f5c3f5 |
890 | $prd = $car * $MBASE + $xi; |
ee15d750 |
891 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; |
892 | unshift(@d, $tmp); |
893 | } |
894 | } |
895 | else |
896 | { |
897 | @d = @$x; |
898 | } |
899 | @$x = @q; |
61f5c3f5 |
900 | my $d = \@d; |
990fb837 |
901 | __strip_zeros($x); |
902 | __strip_zeros($d); |
61f5c3f5 |
903 | return ($x,$d); |
ee15d750 |
904 | } |
905 | @$x = @q; |
990fb837 |
906 | __strip_zeros($x); |
61f5c3f5 |
907 | $x; |
ee15d750 |
908 | } |
909 | |
394e6ffb |
910 | ############################################################################## |
911 | # testing |
912 | |
913 | sub _acmp |
914 | { |
915 | # internal absolute post-normalized compare (ignore signs) |
916 | # ref to array, ref to array, return <0, 0, >0 |
917 | # arrays must have at least one entry; this is not checked for |
394e6ffb |
918 | my ($c,$cx,$cy) = @_; |
091c87b1 |
919 | |
920 | # shortcut for short numbers |
921 | return (($cx->[0] <=> $cy->[0]) <=> 0) |
922 | if scalar @$cx == scalar @$cy && scalar @$cx == 1; |
394e6ffb |
923 | |
f9a08e12 |
924 | # fast comp based on number of array elements (aka pseudo-length) |
091c87b1 |
925 | my $lxy = (scalar @$cx - scalar @$cy) |
926 | # or length of first element if same number of elements (aka difference 0) |
927 | || |
928 | # need int() here because sometimes the last element is '00018' vs '18' |
929 | (length(int($cx->[-1])) - length(int($cy->[-1]))); |
394e6ffb |
930 | return -1 if $lxy < 0; # already differs, ret |
931 | return 1 if $lxy > 0; # ditto |
56d9de68 |
932 | |
394e6ffb |
933 | # manual way (abort if unequal, good for early ne) |
091c87b1 |
934 | my $a; my $j = scalar @$cx; |
935 | while (--$j >= 0) |
9393ace2 |
936 | { |
091c87b1 |
937 | last if ($a = $cx->[$j] - $cy->[$j]); |
9393ace2 |
938 | } |
091c87b1 |
939 | $a <=> 0; |
394e6ffb |
940 | } |
941 | |
942 | sub _len |
943 | { |
56d9de68 |
944 | # compute number of digits |
394e6ffb |
945 | |
946 | # int() because add/sub sometimes leaves strings (like '00005') instead of |
947 | # '5' in this place, thus causing length() to report wrong length |
948 | my $cx = $_[1]; |
949 | |
56d9de68 |
950 | (@$cx-1)*$BASE_LEN+length(int($cx->[-1])); |
394e6ffb |
951 | } |
952 | |
953 | sub _digit |
954 | { |
955 | # return the nth digit, negative values count backward |
956 | # zero is rightmost, so _digit(123,0) will give 3 |
957 | my ($c,$x,$n) = @_; |
958 | |
959 | my $len = _len('',$x); |
960 | |
961 | $n = $len+$n if $n < 0; # -1 last, -2 second-to-last |
962 | $n = abs($n); # if negative was too big |
963 | $len--; $n = $len if $n > $len; # n to big? |
964 | |
965 | my $elem = int($n / $BASE_LEN); # which array element |
966 | my $digit = $n % $BASE_LEN; # which digit in this element |
967 | $elem = '0000'.@$x[$elem]; # get element padded with 0's |
56d9de68 |
968 | substr($elem,-$digit-1,1); |
394e6ffb |
969 | } |
970 | |
971 | sub _zeros |
972 | { |
973 | # return amount of trailing zeros in decimal |
974 | # check each array elem in _m for having 0 at end as long as elem == 0 |
975 | # Upon finding a elem != 0, stop |
976 | my $x = $_[1]; |
977 | my $zeros = 0; my $elem; |
978 | foreach my $e (@$x) |
979 | { |
980 | if ($e != 0) |
981 | { |
982 | $elem = "$e"; # preserve x |
983 | $elem =~ s/.*?(0*$)/$1/; # strip anything not zero |
984 | $zeros *= $BASE_LEN; # elems * 5 |
61f5c3f5 |
985 | $zeros += length($elem); # count trailing zeros |
394e6ffb |
986 | last; # early out |
987 | } |
988 | $zeros ++; # real else branch: 50% slower! |
989 | } |
61f5c3f5 |
990 | $zeros; |
394e6ffb |
991 | } |
992 | |
993 | ############################################################################## |
994 | # _is_* routines |
995 | |
996 | sub _is_zero |
997 | { |
998 | # return true if arg (BINT or num_str) is zero (array '+', '0') |
999 | my $x = $_[1]; |
61f5c3f5 |
1000 | |
1001 | (((scalar @$x == 1) && ($x->[0] == 0))) <=> 0; |
394e6ffb |
1002 | } |
1003 | |
1004 | sub _is_even |
1005 | { |
1006 | # return true if arg (BINT or num_str) is even |
1007 | my $x = $_[1]; |
61f5c3f5 |
1008 | (!($x->[0] & 1)) <=> 0; |
394e6ffb |
1009 | } |
1010 | |
1011 | sub _is_odd |
1012 | { |
1013 | # return true if arg (BINT or num_str) is even |
1014 | my $x = $_[1]; |
61f5c3f5 |
1015 | |
1016 | (($x->[0] & 1)) <=> 0; |
394e6ffb |
1017 | } |
1018 | |
1019 | sub _is_one |
1020 | { |
1021 | # return true if arg (BINT or num_str) is one (array '+', '1') |
1022 | my $x = $_[1]; |
61f5c3f5 |
1023 | |
1024 | (scalar @$x == 1) && ($x->[0] == 1) <=> 0; |
394e6ffb |
1025 | } |
1026 | |
1027 | sub __strip_zeros |
1028 | { |
1029 | # internal normalization function that strips leading zeros from the array |
1030 | # args: ref to array |
1031 | my $s = shift; |
1032 | |
1033 | my $cnt = scalar @$s; # get count of parts |
1034 | my $i = $cnt-1; |
1035 | push @$s,0 if $i < 0; # div might return empty results, so fix it |
1036 | |
61f5c3f5 |
1037 | return $s if @$s == 1; # early out |
1038 | |
394e6ffb |
1039 | #print "strip: cnt $cnt i $i\n"; |
1040 | # '0', '3', '4', '0', '0', |
1041 | # 0 1 2 3 4 |
1042 | # cnt = 5, i = 4 |
1043 | # i = 4 |
1044 | # i = 3 |
1045 | # => fcnt = cnt - i (5-2 => 3, cnt => 5-1 = 4, throw away from 4th pos) |
1046 | # >= 1: skip first part (this can be zero) |
1047 | while ($i > 0) { last if $s->[$i] != 0; $i--; } |
1048 | $i++; splice @$s,$i if ($i < $cnt); # $i cant be 0 |
1049 | $s; |
1050 | } |
1051 | |
1052 | ############################################################################### |
1053 | # check routine to test internal state of corruptions |
1054 | |
1055 | sub _check |
1056 | { |
1057 | # used by the test suite |
1058 | my $x = $_[1]; |
1059 | |
1060 | return "$x is not a reference" if !ref($x); |
1061 | |
1062 | # are all parts are valid? |
1063 | my $i = 0; my $j = scalar @$x; my ($e,$try); |
1064 | while ($i < $j) |
1065 | { |
1066 | $e = $x->[$i]; $e = 'undef' unless defined $e; |
1067 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e)"; |
1068 | last if $e !~ /^[+]?[0-9]+$/; |
1069 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (stringify)"; |
1070 | last if "$e" !~ /^[+]?[0-9]+$/; |
1071 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (cat-stringify)"; |
1072 | last if '' . "$e" !~ /^[+]?[0-9]+$/; |
1073 | $try = ' < 0 || >= $BASE; '."($x, $e)"; |
1074 | last if $e <0 || $e >= $BASE; |
1075 | # this test is disabled, since new/bnorm and certain ops (like early out |
1076 | # in add/sub) are allowed/expected to leave '00000' in some elements |
1077 | #$try = '=~ /^00+/; '."($x, $e)"; |
1078 | #last if $e =~ /^00+/; |
1079 | $i++; |
1080 | } |
1081 | return "Illegal part '$e' at pos $i (tested: $try)" if $i < $j; |
1082 | return 0; |
1083 | } |
1084 | |
1085 | |
1086 | ############################################################################### |
1087 | ############################################################################### |
1088 | # some optional routines to make BigInt faster |
1089 | |
dccbb853 |
1090 | sub _mod |
1091 | { |
1092 | # if possible, use mod shortcut |
1093 | my ($c,$x,$yo) = @_; |
1094 | |
1095 | # slow way since $y to big |
1096 | if (scalar @$yo > 1) |
1097 | { |
1098 | my ($xo,$rem) = _div($c,$x,$yo); |
1099 | return $rem; |
1100 | } |
aef458a0 |
1101 | |
dccbb853 |
1102 | my $y = $yo->[0]; |
027dc388 |
1103 | # both are single element arrays |
dccbb853 |
1104 | if (scalar @$x == 1) |
1105 | { |
1106 | $x->[0] %= $y; |
1107 | return $x; |
1108 | } |
1109 | |
aef458a0 |
1110 | # @y is a single element, but @x has more than one element |
dccbb853 |
1111 | my $b = $BASE % $y; |
1112 | if ($b == 0) |
1113 | { |
1114 | # when BASE % Y == 0 then (B * BASE) % Y == 0 |
1115 | # (B * BASE) % $y + A % Y => A % Y |
1116 | # so need to consider only last element: O(1) |
1117 | $x->[0] %= $y; |
1118 | } |
027dc388 |
1119 | elsif ($b == 1) |
1120 | { |
28df3e88 |
1121 | # else need to go trough all elements: O(N), but loop is a bit simplified |
027dc388 |
1122 | my $r = 0; |
1123 | foreach (@$x) |
1124 | { |
28df3e88 |
1125 | $r = ($r + $_) % $y; # not much faster, but heh... |
1126 | #$r += $_ % $y; $r %= $y; |
027dc388 |
1127 | } |
1128 | $r = 0 if $r == $y; |
1129 | $x->[0] = $r; |
1130 | } |
dccbb853 |
1131 | else |
1132 | { |
027dc388 |
1133 | # else need to go trough all elements: O(N) |
1134 | my $r = 0; my $bm = 1; |
1135 | foreach (@$x) |
1136 | { |
28df3e88 |
1137 | $r = ($_ * $bm + $r) % $y; |
1138 | $bm = ($bm * $b) % $y; |
1139 | |
1140 | #$r += ($_ % $y) * $bm; |
1141 | #$bm *= $b; |
1142 | #$bm %= $y; |
1143 | #$r %= $y; |
027dc388 |
1144 | } |
1145 | $r = 0 if $r == $y; |
1146 | $x->[0] = $r; |
dccbb853 |
1147 | } |
091c87b1 |
1148 | splice (@$x,1); # keep one element of $x |
61f5c3f5 |
1149 | $x; |
dccbb853 |
1150 | } |
1151 | |
0716bf9b |
1152 | ############################################################################## |
574bacfe |
1153 | # shifts |
1154 | |
1155 | sub _rsft |
1156 | { |
1157 | my ($c,$x,$y,$n) = @_; |
1158 | |
1159 | if ($n != 10) |
1160 | { |
61f5c3f5 |
1161 | $n = _new($c,\$n); return _div($c,$x, _pow($c,$n,$y)); |
1162 | } |
1163 | |
1164 | # shortcut (faster) for shifting by 10) |
1165 | # multiples of $BASE_LEN |
1166 | my $dst = 0; # destination |
1167 | my $src = _num($c,$y); # as normal int |
56d9de68 |
1168 | my $xlen = (@$x-1)*$BASE_LEN+length(int($x->[-1])); # len of x in digits |
990fb837 |
1169 | if ($src > $xlen or ($src == $xlen and ! defined $x->[1])) |
56d9de68 |
1170 | { |
1171 | # 12345 67890 shifted right by more than 10 digits => 0 |
1172 | splice (@$x,1); # leave only one element |
1173 | $x->[0] = 0; # set to zero |
1174 | return $x; |
1175 | } |
61f5c3f5 |
1176 | my $rem = $src % $BASE_LEN; # remainder to shift |
1177 | $src = int($src / $BASE_LEN); # source |
1178 | if ($rem == 0) |
1179 | { |
1180 | splice (@$x,0,$src); # even faster, 38.4 => 39.3 |
574bacfe |
1181 | } |
1182 | else |
1183 | { |
61f5c3f5 |
1184 | my $len = scalar @$x - $src; # elems to go |
1185 | my $vd; my $z = '0'x $BASE_LEN; |
1186 | $x->[scalar @$x] = 0; # avoid || 0 test inside loop |
1187 | while ($dst < $len) |
574bacfe |
1188 | { |
61f5c3f5 |
1189 | $vd = $z.$x->[$src]; |
1190 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN-$rem); |
1191 | $src++; |
1192 | $vd = substr($z.$x->[$src],-$rem,$rem) . $vd; |
1193 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN; |
1194 | $x->[$dst] = int($vd); |
1195 | $dst++; |
574bacfe |
1196 | } |
61f5c3f5 |
1197 | splice (@$x,$dst) if $dst > 0; # kill left-over array elems |
56b9c951 |
1198 | pop @$x if $x->[-1] == 0 && @$x > 1; # kill last element if 0 |
61f5c3f5 |
1199 | } # else rem == 0 |
574bacfe |
1200 | $x; |
1201 | } |
1202 | |
1203 | sub _lsft |
1204 | { |
1205 | my ($c,$x,$y,$n) = @_; |
1206 | |
1207 | if ($n != 10) |
1208 | { |
61f5c3f5 |
1209 | $n = _new($c,\$n); return _mul($c,$x, _pow($c,$n,$y)); |
574bacfe |
1210 | } |
61f5c3f5 |
1211 | |
1212 | # shortcut (faster) for shifting by 10) since we are in base 10eX |
1213 | # multiples of $BASE_LEN: |
1214 | my $src = scalar @$x; # source |
1215 | my $len = _num($c,$y); # shift-len as normal int |
1216 | my $rem = $len % $BASE_LEN; # remainder to shift |
1217 | my $dst = $src + int($len/$BASE_LEN); # destination |
1218 | my $vd; # further speedup |
1219 | $x->[$src] = 0; # avoid first ||0 for speed |
1220 | my $z = '0' x $BASE_LEN; |
1221 | while ($src >= 0) |
574bacfe |
1222 | { |
61f5c3f5 |
1223 | $vd = $x->[$src]; $vd = $z.$vd; |
1224 | $vd = substr($vd,-$BASE_LEN+$rem,$BASE_LEN-$rem); |
1225 | $vd .= $src > 0 ? substr($z.$x->[$src-1],-$BASE_LEN,$rem) : '0' x $rem; |
1226 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN; |
1227 | $x->[$dst] = int($vd); |
1228 | $dst--; $src--; |
574bacfe |
1229 | } |
61f5c3f5 |
1230 | # set lowest parts to 0 |
1231 | while ($dst >= 0) { $x->[$dst--] = 0; } |
1232 | # fix spurios last zero element |
1233 | splice @$x,-1 if $x->[-1] == 0; |
574bacfe |
1234 | $x; |
1235 | } |
1236 | |
027dc388 |
1237 | sub _pow |
1238 | { |
1239 | # power of $x to $y |
1240 | # ref to array, ref to array, return ref to array |
1241 | my ($c,$cx,$cy) = @_; |
1242 | |
1243 | my $pow2 = _one(); |
1ddff52a |
1244 | |
1245 | my $y_bin = ${_as_bin($c,$cy)}; $y_bin =~ s/^0b//; |
1246 | my $len = length($y_bin); |
1247 | while (--$len > 0) |
027dc388 |
1248 | { |
1ddff52a |
1249 | _mul($c,$pow2,$cx) if substr($y_bin,$len,1) eq '1'; # is odd? |
027dc388 |
1250 | _mul($c,$cx,$cx); |
1251 | } |
1ddff52a |
1252 | |
1253 | _mul($c,$cx,$pow2); |
61f5c3f5 |
1254 | $cx; |
027dc388 |
1255 | } |
1256 | |
b3abae2a |
1257 | sub _fac |
1258 | { |
1259 | # factorial of $x |
1260 | # ref to array, return ref to array |
1261 | my ($c,$cx) = @_; |
1262 | |
1263 | if ((@$cx == 1) && ($cx->[0] <= 2)) |
1264 | { |
091c87b1 |
1265 | $cx->[0] ||= 1; # 0 => 1, 1 => 1, 2 => 2 |
b3abae2a |
1266 | return $cx; |
1267 | } |
1268 | |
1269 | # go forward until $base is exceeded |
091c87b1 |
1270 | # limit is either $x steps (steps == 100 means a result always too high) or |
1271 | # $base. |
b3abae2a |
1272 | my $steps = 100; $steps = $cx->[0] if @$cx == 1; |
091c87b1 |
1273 | my $r = 2; my $cf = 3; my $step = 2; my $last = $r; |
1274 | while ($r*$cf < $BASE && $step < $steps) |
b3abae2a |
1275 | { |
1276 | $last = $r; $r *= $cf++; $step++; |
1277 | } |
091c87b1 |
1278 | if ((@$cx == 1) && $step == $cx->[0]) |
b3abae2a |
1279 | { |
091c87b1 |
1280 | # completely done, so keep reference to $x and return |
1281 | $cx->[0] = $r; |
b3abae2a |
1282 | return $cx; |
1283 | } |
091c87b1 |
1284 | |
990fb837 |
1285 | # now we must do the left over steps |
091c87b1 |
1286 | my $n; # steps still to do |
1287 | if (scalar @$cx == 1) |
1288 | { |
1289 | $n = $cx->[0]; |
1290 | } |
1291 | else |
1292 | { |
1293 | $n = _copy($c,$cx); |
1294 | } |
b3abae2a |
1295 | |
091c87b1 |
1296 | $cx->[0] = $last; splice (@$cx,1); # keep ref to $x |
990fb837 |
1297 | my $zero_elements = 0; |
091c87b1 |
1298 | |
1299 | # do left-over steps fit into a scalar? |
1300 | if (ref $n eq 'ARRAY') |
b3abae2a |
1301 | { |
091c87b1 |
1302 | # No, so use slower inc() & cmp() |
1303 | $step = [$step]; |
1304 | while (_acmp($step,$n) <= 0) |
990fb837 |
1305 | { |
091c87b1 |
1306 | # as soon as the last element of $cx is 0, we split it up and remember |
1307 | # how many zeors we got so far. The reason is that n! will accumulate |
1308 | # zeros at the end rather fast. |
990fb837 |
1309 | if ($cx->[0] == 0) |
1310 | { |
1311 | $zero_elements ++; shift @$cx; |
1312 | } |
091c87b1 |
1313 | _mul($c,$cx,$step); _inc($c,$step); |
990fb837 |
1314 | } |
990fb837 |
1315 | } |
091c87b1 |
1316 | else |
990fb837 |
1317 | { |
091c87b1 |
1318 | # Yes, so we can speed it up slightly |
1319 | while ($step <= $n) |
990fb837 |
1320 | { |
091c87b1 |
1321 | # When the last element of $cx is 0, we split it up and remember |
1322 | # how many we got so far. The reason is that n! will accumulate |
1323 | # zeros at the end rather fast. |
1324 | if ($cx->[0] == 0) |
1325 | { |
1326 | $zero_elements ++; shift @$cx; |
1327 | } |
1328 | _mul($c,$cx,[$step]); $step++; |
990fb837 |
1329 | } |
990fb837 |
1330 | } |
1331 | # multiply in the zeros again |
1332 | while ($zero_elements-- > 0) |
1333 | { |
1334 | unshift @$cx, 0; |
b3abae2a |
1335 | } |
091c87b1 |
1336 | $cx; # return result |
1337 | } |
1338 | |
1339 | sub _log_int |
1340 | { |
1341 | # calculate integer log of $x to base $base |
1342 | # ref to array, ref to array - return ref to array |
1343 | my ($c,$x,$base) = @_; |
1344 | |
1345 | # X == 0 => NaN |
1346 | return if (scalar @$x == 1 && $x->[0] == 0); |
1347 | # BASE 0 or 1 => NaN |
1348 | return if (scalar @$base == 1 && $base->[0] < 2); |
1349 | my $cmp = _acmp($c,$x,$base); |
1350 | # X == BASE => 1 |
1351 | if ($cmp == 0) |
1352 | { |
1353 | splice (@$x,1); $x->[0] = 1; |
8df1e0a2 |
1354 | return ($x,1) |
091c87b1 |
1355 | } |
1356 | # X < BASE |
1357 | if ($cmp < 0) |
1358 | { |
1359 | splice (@$x,1); $x->[0] = 0; |
8df1e0a2 |
1360 | return ($x,undef); |
091c87b1 |
1361 | } |
1362 | |
1363 | # this trial multiplication is very fast, even for large counts (like for |
1364 | # 2 ** 1024, since this still requires only 1024 very fast steps |
1365 | # (multiplication of a large number by a very small number is very fast)) |
1366 | my $x_org = _copy($c,$x); # preserve x |
8df1e0a2 |
1367 | splice(@$x,1); $x->[0] = 1; # keep ref to $x |
1368 | |
1369 | # simple loop that increments $x by two in each step, possible overstepping |
1370 | # the real result by one |
091c87b1 |
1371 | |
1372 | # use a loop that keeps $x as scalar as long as possible (this is faster) |
8df1e0a2 |
1373 | my $trial = _copy($c,$base); my $a; |
1374 | my $base_mul = _mul($c, _copy($c,$base), $base); |
1375 | |
1376 | while (($a = _acmp($x,$trial,$x_org)) < 0) |
091c87b1 |
1377 | { |
8df1e0a2 |
1378 | _mul($c,$trial,$base_mul); _add($c, $x, [2]); |
091c87b1 |
1379 | } |
8df1e0a2 |
1380 | |
1381 | my $exact = 1; |
1382 | if ($a > 0) |
091c87b1 |
1383 | { |
8df1e0a2 |
1384 | # overstepped the result |
1385 | _dec($c, $x); |
1386 | _div($c,$trial,$base); |
1387 | $a = _acmp($x,$trial,$x_org); |
1388 | if ($a > 0) |
091c87b1 |
1389 | { |
8df1e0a2 |
1390 | _dec($c, $x); |
091c87b1 |
1391 | } |
8df1e0a2 |
1392 | $exact = 0 if $a != 0; |
091c87b1 |
1393 | } |
1394 | |
8df1e0a2 |
1395 | ($x,$exact); # return result |
b3abae2a |
1396 | } |
1397 | |
56d9de68 |
1398 | # for debugging: |
1399 | use constant DEBUG => 0; |
1400 | my $steps = 0; |
1401 | sub steps { $steps }; |
b3abae2a |
1402 | |
1403 | sub _sqrt |
0716bf9b |
1404 | { |
56d9de68 |
1405 | # square-root of $x in place |
990fb837 |
1406 | # Compute a guess of the result (by rule of thumb), then improve it via |
56d9de68 |
1407 | # Newton's method. |
394e6ffb |
1408 | my ($c,$x) = @_; |
0716bf9b |
1409 | |
394e6ffb |
1410 | if (scalar @$x == 1) |
1411 | { |
56d9de68 |
1412 | # fit's into one Perl scalar, so result can be computed directly |
394e6ffb |
1413 | $x->[0] = int(sqrt($x->[0])); |
1414 | return $x; |
1415 | } |
1416 | my $y = _copy($c,$x); |
b3abae2a |
1417 | # hopefully _len/2 is < $BASE, the -1 is to always undershot the guess |
1418 | # since our guess will "grow" |
1419 | my $l = int((_len($c,$x)-1) / 2); |
1420 | |
56d9de68 |
1421 | my $lastelem = $x->[-1]; # for guess |
b3abae2a |
1422 | my $elems = scalar @$x - 1; |
1423 | # not enough digits, but could have more? |
56d9de68 |
1424 | if ((length($lastelem) <= 3) && ($elems > 1)) |
b3abae2a |
1425 | { |
1426 | # right-align with zero pad |
1427 | my $len = length($lastelem) & 1; |
1428 | print "$lastelem => " if DEBUG; |
1429 | $lastelem .= substr($x->[-2] . '0' x $BASE_LEN,0,$BASE_LEN); |
1430 | # former odd => make odd again, or former even to even again |
56d9de68 |
1431 | $lastelem = $lastelem / 10 if (length($lastelem) & 1) != $len; |
b3abae2a |
1432 | print "$lastelem\n" if DEBUG; |
1433 | } |
0716bf9b |
1434 | |
61f5c3f5 |
1435 | # construct $x (instead of _lsft($c,$x,$l,10) |
1436 | my $r = $l % $BASE_LEN; # 10000 00000 00000 00000 ($BASE_LEN=5) |
1437 | $l = int($l / $BASE_LEN); |
b3abae2a |
1438 | print "l = $l " if DEBUG; |
56d9de68 |
1439 | |
1440 | splice @$x,$l; # keep ref($x), but modify it |
1441 | |
b3abae2a |
1442 | # we make the first part of the guess not '1000...0' but int(sqrt($lastelem)) |
1443 | # that gives us: |
56d9de68 |
1444 | # 14400 00000 => sqrt(14400) => guess first digits to be 120 |
1445 | # 144000 000000 => sqrt(144000) => guess 379 |
b3abae2a |
1446 | |
b3abae2a |
1447 | print "$lastelem (elems $elems) => " if DEBUG; |
1448 | $lastelem = $lastelem / 10 if ($elems & 1 == 1); # odd or even? |
1449 | my $g = sqrt($lastelem); $g =~ s/\.//; # 2.345 => 2345 |
1450 | $r -= 1 if $elems & 1 == 0; # 70 => 7 |
1451 | |
1452 | # padd with zeros if result is too short |
1453 | $x->[$l--] = int(substr($g . '0' x $r,0,$r+1)); |
1454 | print "now ",$x->[-1] if DEBUG; |
1455 | print " would have been ", int('1' . '0' x $r),"\n" if DEBUG; |
56d9de68 |
1456 | |
b3abae2a |
1457 | # If @$x > 1, we could compute the second elem of the guess, too, to create |
56d9de68 |
1458 | # an even better guess. Not implemented yet. Does it improve performance? |
b3abae2a |
1459 | $x->[$l--] = 0 while ($l >= 0); # all other digits of guess are zero |
56d9de68 |
1460 | |
b3abae2a |
1461 | print "start x= ",${_str($c,$x)},"\n" if DEBUG; |
394e6ffb |
1462 | my $two = _two(); |
1463 | my $last = _zero(); |
1464 | my $lastlast = _zero(); |
b3abae2a |
1465 | $steps = 0 if DEBUG; |
394e6ffb |
1466 | while (_acmp($c,$last,$x) != 0 && _acmp($c,$lastlast,$x) != 0) |
1467 | { |
b3abae2a |
1468 | $steps++ if DEBUG; |
394e6ffb |
1469 | $lastlast = _copy($c,$last); |
1470 | $last = _copy($c,$x); |
1471 | _add($c,$x, _div($c,_copy($c,$y),$x)); |
1472 | _div($c,$x, $two ); |
56d9de68 |
1473 | print " x= ",${_str($c,$x)},"\n" if DEBUG; |
394e6ffb |
1474 | } |
b3abae2a |
1475 | print "\nsteps in sqrt: $steps, " if DEBUG; |
394e6ffb |
1476 | _dec($c,$x) if _acmp($c,$y,_mul($c,_copy($c,$x),$x)) < 0; # overshot? |
b3abae2a |
1477 | print " final ",$x->[-1],"\n" if DEBUG; |
394e6ffb |
1478 | $x; |
0716bf9b |
1479 | } |
1480 | |
990fb837 |
1481 | sub _root |
1482 | { |
1483 | # take n'th root of $x in place (n >= 3) |
990fb837 |
1484 | my ($c,$x,$n) = @_; |
1485 | |
1486 | if (scalar @$x == 1) |
1487 | { |
1488 | if (scalar @$n > 1) |
1489 | { |
1490 | # result will always be smaller than 2 so trunc to 1 at once |
1491 | $x->[0] = 1; |
1492 | } |
1493 | else |
1494 | { |
1495 | # fit's into one Perl scalar, so result can be computed directly |
091c87b1 |
1496 | # cannot use int() here, because it rounds wrongly (try |
1497 | # (81 ** 3) ** (1/3) to see what I mean) |
1498 | #$x->[0] = int( $x->[0] ** (1 / $n->[0]) ); |
1499 | # round to 8 digits, then truncate result to integer |
1500 | $x->[0] = int ( sprintf ("%.8f", $x->[0] ** (1 / $n->[0]) ) ); |
990fb837 |
1501 | } |
1502 | return $x; |
1503 | } |
1504 | |
c38b2de2 |
1505 | # X is more than one element |
1506 | # if $n is a power of two, we can repeatedly take sqrt($X) and find the |
1507 | # proper result, because sqrt(sqrt($x)) == root($x,4) |
1508 | my $b = _as_bin($c,$n); |
1509 | if ($$b =~ /0b1(0+)/) |
1510 | { |
1511 | my $count = CORE::length($1); # 0b100 => len('00') => 2 |
1512 | my $cnt = $count; # counter for loop |
1513 | unshift (@$x, 0); # add one element, together with one |
1514 | # more below in the loop this makes 2 |
1515 | while ($cnt-- > 0) |
1516 | { |
1517 | # 'inflate' $X by adding one element, basically computing |
1518 | # $x * $BASE * $BASE. This gives us more $BASE_LEN digits for result |
1519 | # since len(sqrt($X)) approx == len($x) / 2. |
1520 | unshift (@$x, 0); |
1521 | # calculate sqrt($x), $x is now one element to big, again. In the next |
1522 | # round we make that two, again. |
1523 | _sqrt($c,$x); |
1524 | } |
1525 | # $x is now one element to big, so truncate result by removing it |
1526 | splice (@$x,0,1); |
1527 | } |
1528 | else |
1529 | { |
091c87b1 |
1530 | # trial computation by starting with 2,4,8,16 etc until we overstep |
1531 | |
1532 | my $step = _two(); |
1533 | my $trial = _two(); |
1534 | |
1535 | _mul($c, $trial, $step) |
1536 | while (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) < 0); |
1537 | |
1538 | # hit exactly? |
1539 | if (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) == 0) |
1540 | { |
1541 | @$x = @$trial; # make copy while preserving ref to $x |
1542 | return $x; |
1543 | } |
1544 | # overstepped, so go back on step |
1545 | _div($c, $trial, $step); |
1546 | |
1547 | # add two, because $trial cannot be exactly the result (otherwise we would |
1548 | # alrady have found it) |
1549 | _add($c, $trial, $step); |
1550 | |
1551 | # and now add more and more (2,4,6,8, etc) |
1552 | _add($c, $trial, $step) |
1553 | while (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) < 0); |
1554 | |
1555 | # hit not exactly? (overstepped) |
1556 | # 80 too small, 81 slightly too big, 82 too big |
1557 | if (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) > 0) |
1558 | { |
1559 | _dec($c,$trial); |
1560 | } |
1561 | # 80 too small, 81 slightly too big |
1562 | if (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) > 0) |
1563 | { |
1564 | _dec($c,$trial); |
1565 | } |
1566 | |
1567 | @$x = @$trial; # make copy while preserving ref to $x |
1568 | return $x; |
c38b2de2 |
1569 | } |
990fb837 |
1570 | $x; |
1571 | } |
1572 | |
394e6ffb |
1573 | ############################################################################## |
1574 | # binary stuff |
0716bf9b |
1575 | |
394e6ffb |
1576 | sub _and |
1577 | { |
1578 | my ($c,$x,$y) = @_; |
0716bf9b |
1579 | |
394e6ffb |
1580 | # the shortcut makes equal, large numbers _really_ fast, and makes only a |
1581 | # very small performance drop for small numbers (e.g. something with less |
1582 | # than 32 bit) Since we optimize for large numbers, this is enabled. |
1583 | return $x if _acmp($c,$x,$y) == 0; # shortcut |
0716bf9b |
1584 | |
394e6ffb |
1585 | my $m = _one(); my ($xr,$yr); |
1586 | my $mask = $AND_MASK; |
1587 | |
1588 | my $x1 = $x; |
1589 | my $y1 = _copy($c,$y); # make copy |
1590 | $x = _zero(); |
1591 | my ($b,$xrr,$yrr); |
1592 | use integer; |
1593 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
1594 | { |
1595 | ($x1, $xr) = _div($c,$x1,$mask); |
1596 | ($y1, $yr) = _div($c,$y1,$mask); |
1597 | |
1598 | # make ints() from $xr, $yr |
1599 | # this is when the AND_BITS are greater tahn $BASE and is slower for |
1600 | # small (<256 bits) numbers, but faster for large numbers. Disabled |
1601 | # due to KISS principle |
1602 | |
1603 | # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
1604 | # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
1605 | # _add($c,$x, _mul($c, _new( $c, \($xrr & $yrr) ), $m) ); |
1606 | |
61f5c3f5 |
1607 | # 0+ due to '&' doesn't work in strings |
1608 | _add($c,$x, _mul($c, [ 0+$xr->[0] & 0+$yr->[0] ], $m) ); |
394e6ffb |
1609 | _mul($c,$m,$mask); |
1610 | } |
1611 | $x; |
0716bf9b |
1612 | } |
1613 | |
394e6ffb |
1614 | sub _xor |
0716bf9b |
1615 | { |
394e6ffb |
1616 | my ($c,$x,$y) = @_; |
1617 | |
1618 | return _zero() if _acmp($c,$x,$y) == 0; # shortcut (see -and) |
1619 | |
1620 | my $m = _one(); my ($xr,$yr); |
1621 | my $mask = $XOR_MASK; |
1622 | |
1623 | my $x1 = $x; |
1624 | my $y1 = _copy($c,$y); # make copy |
1625 | $x = _zero(); |
1626 | my ($b,$xrr,$yrr); |
1627 | use integer; |
1628 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
0716bf9b |
1629 | { |
394e6ffb |
1630 | ($x1, $xr) = _div($c,$x1,$mask); |
1631 | ($y1, $yr) = _div($c,$y1,$mask); |
1632 | # make ints() from $xr, $yr (see _and()) |
1633 | #$b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
1634 | #$b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
1635 | #_add($c,$x, _mul($c, _new( $c, \($xrr ^ $yrr) ), $m) ); |
61f5c3f5 |
1636 | |
1637 | # 0+ due to '^' doesn't work in strings |
1638 | _add($c,$x, _mul($c, [ 0+$xr->[0] ^ 0+$yr->[0] ], $m) ); |
394e6ffb |
1639 | _mul($c,$m,$mask); |
0716bf9b |
1640 | } |
394e6ffb |
1641 | # the loop stops when the shorter of the two numbers is exhausted |
1642 | # the remainder of the longer one will survive bit-by-bit, so we simple |
1643 | # multiply-add it in |
1644 | _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1); |
1645 | _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1); |
1646 | |
1647 | $x; |
0716bf9b |
1648 | } |
1649 | |
394e6ffb |
1650 | sub _or |
0716bf9b |
1651 | { |
394e6ffb |
1652 | my ($c,$x,$y) = @_; |
0716bf9b |
1653 | |
394e6ffb |
1654 | return $x if _acmp($c,$x,$y) == 0; # shortcut (see _and) |
0716bf9b |
1655 | |
394e6ffb |
1656 | my $m = _one(); my ($xr,$yr); |
1657 | my $mask = $OR_MASK; |
0716bf9b |
1658 | |
394e6ffb |
1659 | my $x1 = $x; |
1660 | my $y1 = _copy($c,$y); # make copy |
1661 | $x = _zero(); |
1662 | my ($b,$xrr,$yrr); |
1663 | use integer; |
1664 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
1665 | { |
1666 | ($x1, $xr) = _div($c,$x1,$mask); |
1667 | ($y1, $yr) = _div($c,$y1,$mask); |
1668 | # make ints() from $xr, $yr (see _and()) |
1669 | # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
1670 | # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
1671 | # _add($c,$x, _mul($c, _new( $c, \($xrr | $yrr) ), $m) ); |
1672 | |
61f5c3f5 |
1673 | # 0+ due to '|' doesn't work in strings |
1674 | _add($c,$x, _mul($c, [ 0+$xr->[0] | 0+$yr->[0] ], $m) ); |
394e6ffb |
1675 | _mul($c,$m,$mask); |
1676 | } |
1677 | # the loop stops when the shorter of the two numbers is exhausted |
1678 | # the remainder of the longer one will survive bit-by-bit, so we simple |
1679 | # multiply-add it in |
1680 | _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1); |
1681 | _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1); |
1682 | |
1683 | $x; |
0716bf9b |
1684 | } |
1685 | |
61f5c3f5 |
1686 | sub _as_hex |
1687 | { |
1688 | # convert a decimal number to hex (ref to array, return ref to string) |
1689 | my ($c,$x) = @_; |
1690 | |
091c87b1 |
1691 | # fit's into one element (handle also 0x0 case) |
990fb837 |
1692 | if (@$x == 1) |
1693 | { |
091c87b1 |
1694 | my $t = sprintf("0x%x",$x->[0]); |
990fb837 |
1695 | return \$t; |
1696 | } |
1697 | |
61f5c3f5 |
1698 | my $x1 = _copy($c,$x); |
1699 | |
1700 | my $es = ''; |
1ddff52a |
1701 | my ($xr, $h, $x10000); |
1702 | if ($] >= 5.006) |
1703 | { |
1704 | $x10000 = [ 0x10000 ]; $h = 'h4'; |
1705 | } |
1706 | else |
1707 | { |
1708 | $x10000 = [ 0x1000 ]; $h = 'h3'; |
1709 | } |
091c87b1 |
1710 | # while (! _is_zero($c,$x1)) |
1711 | while (@$x1 != 1 || $x1->[0] != 0) # _is_zero() |
61f5c3f5 |
1712 | { |
1713 | ($x1, $xr) = _div($c,$x1,$x10000); |
091c87b1 |
1714 | $es .= unpack($h,pack('v',$xr->[0])); # XXX TODO: why pack('v',...)? |
61f5c3f5 |
1715 | } |
1716 | $es = reverse $es; |
1717 | $es =~ s/^[0]+//; # strip leading zeros |
1718 | $es = '0x' . $es; |
1719 | \$es; |
1720 | } |
1721 | |
1722 | sub _as_bin |
1723 | { |
1724 | # convert a decimal number to bin (ref to array, return ref to string) |
1725 | my ($c,$x) = @_; |
1726 | |
091c87b1 |
1727 | # fit's into one element (and Perl recent enough), handle also 0b0 case |
1728 | # handle zero case for older Perls |
1729 | if ($] <= 5.005 && @$x == 1 && $x->[0] == 0) |
1730 | { |
1731 | my $t = '0b0'; return \$t; |
1732 | } |
1733 | if (@$x == 1 && $] >= 5.006) |
990fb837 |
1734 | { |
091c87b1 |
1735 | my $t = sprintf("0b%b",$x->[0]); |
990fb837 |
1736 | return \$t; |
1737 | } |
61f5c3f5 |
1738 | my $x1 = _copy($c,$x); |
1739 | |
1740 | my $es = ''; |
1ddff52a |
1741 | my ($xr, $b, $x10000); |
1742 | if ($] >= 5.006) |
1743 | { |
1744 | $x10000 = [ 0x10000 ]; $b = 'b16'; |
1745 | } |
1746 | else |
1747 | { |
1748 | $x10000 = [ 0x1000 ]; $b = 'b12'; |
1749 | } |
091c87b1 |
1750 | # while (! _is_zero($c,$x1)) |
1751 | while (!(@$x1 == 1 && $x1->[0] == 0)) # _is_zero() |
61f5c3f5 |
1752 | { |
1753 | ($x1, $xr) = _div($c,$x1,$x10000); |
091c87b1 |
1754 | $es .= unpack($b,pack('v',$xr->[0])); # XXX TODO: why pack('v',...)? |
1755 | # $es .= unpack($b,$xr->[0]); |
61f5c3f5 |
1756 | } |
1757 | $es = reverse $es; |
1758 | $es =~ s/^[0]+//; # strip leading zeros |
1759 | $es = '0b' . $es; |
1760 | \$es; |
1761 | } |
1762 | |
394e6ffb |
1763 | sub _from_hex |
0716bf9b |
1764 | { |
394e6ffb |
1765 | # convert a hex number to decimal (ref to string, return ref to array) |
1766 | my ($c,$hs) = @_; |
0716bf9b |
1767 | |
394e6ffb |
1768 | my $mul = _one(); |
1769 | my $m = [ 0x10000 ]; # 16 bit at a time |
1770 | my $x = _zero(); |
0716bf9b |
1771 | |
61f5c3f5 |
1772 | my $len = length($$hs)-2; |
394e6ffb |
1773 | $len = int($len/4); # 4-digit parts, w/o '0x' |
1774 | my $val; my $i = -4; |
1775 | while ($len >= 0) |
1776 | { |
1777 | $val = substr($$hs,$i,4); |
1778 | $val =~ s/^[+-]?0x// if $len == 0; # for last part only because |
1779 | $val = hex($val); # hex does not like wrong chars |
1780 | $i -= 4; $len --; |
1781 | _add ($c, $x, _mul ($c, [ $val ], $mul ) ) if $val != 0; |
1782 | _mul ($c, $mul, $m ) if $len >= 0; # skip last mul |
1783 | } |
1784 | $x; |
1785 | } |
1786 | |
1787 | sub _from_bin |
0716bf9b |
1788 | { |
394e6ffb |
1789 | # convert a hex number to decimal (ref to string, return ref to array) |
1790 | my ($c,$bs) = @_; |
0716bf9b |
1791 | |
091c87b1 |
1792 | # instead of converting X (8) bit at a time, it is faster to "convert" the |
13a12e00 |
1793 | # number to hex, and then call _from_hex. |
1794 | |
1795 | my $hs = $$bs; |
1796 | $hs =~ s/^[+-]?0b//; # remove sign and 0b |
1797 | my $l = length($hs); # bits |
1798 | $hs = '0' x (8-($l % 8)) . $hs if ($l % 8) != 0; # padd left side w/ 0 |
1799 | my $h = unpack('H*', pack ('B*', $hs)); # repack as hex |
091c87b1 |
1800 | |
1801 | $c->_from_hex(\('0x'.$h)); |
0716bf9b |
1802 | } |
1803 | |
07d34614 |
1804 | ############################################################################## |
1805 | # special modulus functions |
1806 | |
56d9de68 |
1807 | sub _modinv |
d614cd8b |
1808 | { |
56d9de68 |
1809 | # modular inverse |
1810 | my ($c,$x,$y) = @_; |
1ddff52a |
1811 | |
56d9de68 |
1812 | my $u = _zero($c); my $u1 = _one($c); |
1813 | my $a = _copy($c,$y); my $b = _copy($c,$x); |
1ddff52a |
1814 | |
1815 | # Euclid's Algorithm for bgcd(), only that we calc bgcd() ($a) and the |
56d9de68 |
1816 | # result ($u) at the same time. See comments in BigInt for why this works. |
1817 | my $q; |
1818 | ($a, $q, $b) = ($b, _div($c,$a,$b)); # step 1 |
1819 | my $sign = 1; |
1ddff52a |
1820 | while (!_is_zero($c,$b)) |
1821 | { |
56d9de68 |
1822 | my $t = _add($c, # step 2: |
1823 | _mul($c,_copy($c,$u1), $q) , # t = u1 * q |
1824 | $u ); # + u |
1825 | $u = $u1; # u = u1, u1 = t |
1826 | $u1 = $t; |
1827 | $sign = -$sign; |
1828 | ($a, $q, $b) = ($b, _div($c,$a,$b)); # step 1 |
1ddff52a |
1829 | } |
1830 | |
1831 | # if the gcd is not 1, then return NaN |
56d9de68 |
1832 | return (undef,undef) unless _is_one($c,$a); |
1833 | |
1834 | $sign = $sign == 1 ? '+' : '-'; |
1835 | ($u1,$sign); |
d614cd8b |
1836 | } |
1837 | |
1838 | sub _modpow |
1839 | { |
1840 | # modulus of power ($x ** $y) % $z |
07d34614 |
1841 | my ($c,$num,$exp,$mod) = @_; |
1842 | |
1843 | # in the trivial case, |
1844 | if (_is_one($c,$mod)) |
1845 | { |
1846 | splice @$num,0,1; $num->[0] = 0; |
1847 | return $num; |
1848 | } |
1849 | if ((scalar @$num == 1) && (($num->[0] == 0) || ($num->[0] == 1))) |
1850 | { |
1851 | $num->[0] = 1; |
1852 | return $num; |
1853 | } |
1ddff52a |
1854 | |
1855 | # $num = _mod($c,$num,$mod); # this does not make it faster |
07d34614 |
1856 | |
1857 | my $acc = _copy($c,$num); my $t = _one(); |
1858 | |
1ddff52a |
1859 | my $expbin = ${_as_bin($c,$exp)}; $expbin =~ s/^0b//; |
1860 | my $len = length($expbin); |
1861 | while (--$len >= 0) |
07d34614 |
1862 | { |
1ddff52a |
1863 | if ( substr($expbin,$len,1) eq '1') # is_odd |
07d34614 |
1864 | { |
1865 | _mul($c,$t,$acc); |
1866 | $t = _mod($c,$t,$mod); |
1867 | } |
1868 | _mul($c,$acc,$acc); |
1869 | $acc = _mod($c,$acc,$mod); |
07d34614 |
1870 | } |
1871 | @$num = @$t; |
1872 | $num; |
d614cd8b |
1873 | } |
1874 | |
394e6ffb |
1875 | ############################################################################## |
1876 | ############################################################################## |
1877 | |
0716bf9b |
1878 | 1; |
1879 | __END__ |
1880 | |
1881 | =head1 NAME |
1882 | |
1883 | Math::BigInt::Calc - Pure Perl module to support Math::BigInt |
1884 | |
1885 | =head1 SYNOPSIS |
1886 | |
ee15d750 |
1887 | Provides support for big integer calculations. Not intended to be used by other |
091c87b1 |
1888 | modules. Other modules which sport the same functions can also be used to support |
1889 | Math::BigInt, like Math::BigInt::GMP or Math::BigInt::Pari. |
0716bf9b |
1890 | |
1891 | =head1 DESCRIPTION |
1892 | |
027dc388 |
1893 | In order to allow for multiple big integer libraries, Math::BigInt was |
1894 | rewritten to use library modules for core math routines. Any module which |
1895 | follows the same API as this can be used instead by using the following: |
0716bf9b |
1896 | |
ee15d750 |
1897 | use Math::BigInt lib => 'libname'; |
0716bf9b |
1898 | |
027dc388 |
1899 | 'libname' is either the long name ('Math::BigInt::Pari'), or only the short |
1900 | version like 'Pari'. |
1901 | |
990fb837 |
1902 | =head1 STORAGE |
1903 | |
1904 | =head1 METHODS |
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1905 | |
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1906 | The following functions MUST be defined in order to support the use by |
1907 | Math::BigInt: |
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1908 | |
1909 | _new(string) return ref to new object from ref to decimal string |
1910 | _zero() return a new object with value 0 |
1911 | _one() return a new object with value 1 |
1912 | |
1913 | _str(obj) return ref to a string representing the object |
1914 | _num(obj) returns a Perl integer/floating point number |
1915 | NOTE: because of Perl numeric notation defaults, |
1916 | the _num'ified obj may lose accuracy due to |
1917 | machine-dependend floating point size limitations |
1918 | |
1919 | _add(obj,obj) Simple addition of two objects |
1920 | _mul(obj,obj) Multiplication of two objects |
1921 | _div(obj,obj) Division of the 1st object by the 2nd |
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1922 | In list context, returns (result,remainder). |
1923 | NOTE: this is integer math, so no |
1924 | fractional part will be returned. |
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1925 | The second operand will be not be 0, so no need to |
1926 | check for that. |
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1927 | _sub(obj,obj) Simple subtraction of 1 object from another |
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1928 | a third, optional parameter indicates that the params |
1929 | are swapped. In this case, the first param needs to |
1930 | be preserved, while you can destroy the second. |
1931 | sub (x,y,1) => return x - y and keep x intact! |
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1932 | _dec(obj) decrement object by one (input is garant. to be > 0) |
1933 | _inc(obj) increment object by one |
1934 | |
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1935 | |
1936 | _acmp(obj,obj) <=> operator for objects (return -1, 0 or 1) |
1937 | |
1938 | _len(obj) returns count of the decimal digits of the object |
1939 | _digit(obj,n) returns the n'th decimal digit of object |
1940 | |
1941 | _is_one(obj) return true if argument is +1 |
1942 | _is_zero(obj) return true if argument is 0 |
1943 | _is_even(obj) return true if argument is even (0,2,4,6..) |
1944 | _is_odd(obj) return true if argument is odd (1,3,5,7..) |
1945 | |
1946 | _copy return a ref to a true copy of the object |
1947 | |
1948 | _check(obj) check whether internal representation is still intact |
1949 | return 0 for ok, otherwise error message as string |
1950 | |
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1951 | The following functions are optional, and can be defined if the underlying lib |
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1952 | has a fast way to do them. If undefined, Math::BigInt will use pure Perl (hence |
1953 | slow) fallback routines to emulate these: |
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1954 | |
1955 | _from_hex(str) return ref to new object from ref to hexadecimal string |
1956 | _from_bin(str) return ref to new object from ref to binary string |
1957 | |
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1958 | _as_hex(str) return ref to scalar string containing the value as |
1959 | unsigned hex string, with the '0x' prepended. |
1960 | Leading zeros must be stripped. |
1961 | _as_bin(str) Like as_hex, only as binary string containing only |
1962 | zeros and ones. Leading zeros must be stripped and a |
1963 | '0b' must be prepended. |
1964 | |
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1965 | _rsft(obj,N,B) shift object in base B by N 'digits' right |
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1966 | For unsupported bases B, return undef to signal failure |
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1967 | _lsft(obj,N,B) shift object in base B by N 'digits' left |
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1968 | For unsupported bases B, return undef to signal failure |
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1969 | |
1970 | _xor(obj1,obj2) XOR (bit-wise) object 1 with object 2 |
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1971 | Note: XOR, AND and OR pad with zeros if size mismatches |
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1972 | _and(obj1,obj2) AND (bit-wise) object 1 with object 2 |
1973 | _or(obj1,obj2) OR (bit-wise) object 1 with object 2 |
1974 | |
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1975 | _signed_or |
1976 | _signed_and |
1977 | _signed_xor |
1978 | |
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1979 | _mod(obj,obj) Return remainder of div of the 1st by the 2nd object |
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1980 | _sqrt(obj) return the square root of object (truncated to int) |
1981 | _root(obj) return the n'th (n >= 3) root of obj (truncated to int) |
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1982 | _fac(obj) return factorial of object 1 (1*2*3*4..) |
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1983 | _pow(obj,obj) return object 1 to the power of object 2 |
1984 | _gcd(obj,obj) return Greatest Common Divisor of two objects |
1985 | |
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1986 | _zeros(obj) return number of trailing decimal zeros |
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1987 | _modinv return inverse modulus |
1988 | _modpow return modulus of power ($x ** $y) % $z |
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1989 | _log_int(X,N) calculate integer log() of X in base N |
1990 | X >= 0, N >= 0 (return undef for NaN) |
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1991 | returns (RESULT, EXACT) where EXACT is: |
1992 | 1 : result is exactly RESULT |
1993 | 0 : result was truncated to RESULT |
1994 | undef : unknown whether result is exactly RESULT |
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1995 | |
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1996 | Input strings come in as unsigned but with prefix (i.e. as '123', '0xabc' |
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1997 | or '0b1101'). |
1998 | |
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1999 | So the library needs only to deal with unsigned big integers. Testing of input |
2000 | parameter validity is done by the caller, so you need not worry about |
2001 | underflow (f.i. in C<_sub()>, C<_dec()>) nor about division by zero or similar |
2002 | cases. |
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2003 | |
2004 | The first parameter can be modified, that includes the possibility that you |
2005 | return a reference to a completely different object instead. Although keeping |
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2006 | the reference and just changing it's contents is prefered over creating and |
2007 | returning a different reference. |
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2008 | |
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2009 | Return values are always references to objects, strings, or true/false for |
2010 | comparisation routines. |
2011 | |
2012 | Exceptions are C<_lsft()> and C<_rsft()>, which return undef if they can not |
2013 | shift the argument. This is used to delegate shifting of bases different than |
2014 | the one you can support back to Math::BigInt, which will use some generic code |
2015 | to calculate the result. |
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2016 | |
2017 | =head1 WRAP YOUR OWN |
2018 | |
2019 | If you want to port your own favourite c-lib for big numbers to the |
2020 | Math::BigInt interface, you can take any of the already existing modules as |
2021 | a rough guideline. You should really wrap up the latest BigInt and BigFloat |
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2022 | testsuites with your module, and replace in them any of the following: |
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2023 | |
2024 | use Math::BigInt; |
2025 | |
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2026 | by this: |
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2027 | |
2028 | use Math::BigInt lib => 'yourlib'; |
2029 | |
2030 | This way you ensure that your library really works 100% within Math::BigInt. |
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2031 | |
2032 | =head1 LICENSE |
2033 | |
2034 | This program is free software; you may redistribute it and/or modify it under |
2035 | the same terms as Perl itself. |
2036 | |
2037 | =head1 AUTHORS |
2038 | |
2039 | Original math code by Mark Biggar, rewritten by Tels L<http://bloodgate.com/> |
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2040 | in late 2000. |
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2041 | Seperated from BigInt and shaped API with the help of John Peacock. |
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2042 | Fixed, sped-up and enhanced by Tels http://bloodgate.com 2001-2003. |
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2043 | |
2044 | =head1 SEE ALSO |
2045 | |
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2046 | L<Math::BigInt>, L<Math::BigFloat>, L<Math::BigInt::BitVect>, |
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2047 | L<Math::BigInt::GMP>, L<Math::BigInt::FastCalc> and L<Math::BigInt::Pari>. |
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2048 | |
2049 | =cut |