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 | |
b282a552 |
7 | use vars qw/$VERSION/; |
0716bf9b |
8 | |
9b924220 |
9 | $VERSION = '0.40'; |
0716bf9b |
10 | |
11 | # Package to store unsigned big integers in decimal and do math with them |
12 | |
13 | # Internally the numbers are stored in an array with at least 1 element, no |
027dc388 |
14 | # leading zero parts (except the first) and in base 1eX where X is determined |
15 | # automatically at loading time to be the maximum possible value |
0716bf9b |
16 | |
17 | # todo: |
18 | # - fully remove funky $# stuff (maybe) |
0716bf9b |
19 | |
20 | # USE_MUL: due to problems on certain os (os390, posix-bc) "* 1e-5" is used |
ee15d750 |
21 | # instead of "/ 1e5" at some places, (marked with USE_MUL). Other platforms |
22 | # BS2000, some Crays need USE_DIV instead. |
bd05a461 |
23 | # The BEGIN block is used to determine which of the two variants gives the |
24 | # correct result. |
0716bf9b |
25 | |
990fb837 |
26 | # Beware of things like: |
27 | # $i = $i * $y + $car; $car = int($i / $MBASE); $i = $i % $MBASE; |
93c87d9d |
28 | # This works on x86, but fails on ARM (SA1100, iPAQ) due to whoknows what |
990fb837 |
29 | # reasons. So, use this instead (slower, but correct): |
30 | # $i = $i * $y + $car; $car = int($i / $MBASE); $i -= $MBASE * $car; |
31 | |
0716bf9b |
32 | ############################################################################## |
33 | # global constants, flags and accessory |
9b924220 |
34 | |
35 | # announce that we are compatible with MBI v1.70 and up |
36 | sub api_version () { 1; } |
0716bf9b |
37 | |
38 | # constants for easier life |
39 | my $nan = 'NaN'; |
61f5c3f5 |
40 | my ($MBASE,$BASE,$RBASE,$BASE_LEN,$MAX_VAL,$BASE_LEN2,$BASE_LEN_SMALL); |
394e6ffb |
41 | my ($AND_BITS,$XOR_BITS,$OR_BITS); |
42 | my ($AND_MASK,$XOR_MASK,$OR_MASK); |
ee15d750 |
43 | |
44 | sub _base_len |
45 | { |
dccbb853 |
46 | # set/get the BASE_LEN and assorted other, connected values |
47 | # used only be the testsuite, set is used only by the BEGIN block below |
394e6ffb |
48 | shift; |
49 | |
ee15d750 |
50 | my $b = shift; |
51 | if (defined $b) |
52 | { |
61f5c3f5 |
53 | # find whether we can use mul or div or none in mul()/div() |
54 | # (in last case reduce BASE_LEN_SMALL) |
55 | $BASE_LEN_SMALL = $b+1; |
56 | my $caught = 0; |
57 | while (--$BASE_LEN_SMALL > 5) |
394e6ffb |
58 | { |
61f5c3f5 |
59 | $MBASE = int("1e".$BASE_LEN_SMALL); |
60 | $RBASE = abs('1e-'.$BASE_LEN_SMALL); # see USE_MUL |
394e6ffb |
61 | $caught = 0; |
61f5c3f5 |
62 | $caught += 1 if (int($MBASE * $RBASE) != 1); # should be 1 |
63 | $caught += 2 if (int($MBASE / $MBASE) != 1); # should be 1 |
394e6ffb |
64 | last if $caught != 3; |
65 | } |
61f5c3f5 |
66 | # BASE_LEN is used for anything else than mul()/div() |
67 | $BASE_LEN = $BASE_LEN_SMALL; |
68 | $BASE_LEN = shift if (defined $_[0]); # one more arg? |
ee15d750 |
69 | $BASE = int("1e".$BASE_LEN); |
61f5c3f5 |
70 | |
71 | $BASE_LEN2 = int($BASE_LEN_SMALL / 2); # for mul shortcut |
72 | $MBASE = int("1e".$BASE_LEN_SMALL); |
73 | $RBASE = abs('1e-'.$BASE_LEN_SMALL); # see USE_MUL |
74 | $MAX_VAL = $MBASE-1; |
990fb837 |
75 | |
b05afeb3 |
76 | undef &_mul; |
77 | undef &_div; |
1ddff52a |
78 | |
990fb837 |
79 | # $caught & 1 != 0 => cannot use MUL |
80 | # $caught & 2 != 0 => cannot use DIV |
81 | # The parens around ($caught & 1) were important, indeed, if we would use |
82 | # & here. |
83 | if ($caught == 2) # 2 |
ee15d750 |
84 | { |
990fb837 |
85 | # must USE_MUL since we cannot use DIV |
ee15d750 |
86 | *{_mul} = \&_mul_use_mul; |
87 | *{_div} = \&_div_use_mul; |
88 | } |
990fb837 |
89 | else # 0 or 1 |
ee15d750 |
90 | { |
ee15d750 |
91 | # can USE_DIV instead |
92 | *{_mul} = \&_mul_use_div; |
93 | *{_div} = \&_div_use_div; |
94 | } |
95 | } |
61f5c3f5 |
96 | return $BASE_LEN unless wantarray; |
97 | return ($BASE_LEN, $AND_BITS, $XOR_BITS, $OR_BITS, $BASE_LEN_SMALL, $MAX_VAL); |
ee15d750 |
98 | } |
574bacfe |
99 | |
100 | BEGIN |
101 | { |
bd05a461 |
102 | # from Daniel Pfeiffer: determine largest group of digits that is precisely |
574bacfe |
103 | # multipliable with itself plus carry |
dccbb853 |
104 | # Test now changed to expect the proper pattern, not a result off by 1 or 2 |
105 | my ($e, $num) = 3; # lowest value we will use is 3+1-1 = 3 |
bd05a461 |
106 | do |
107 | { |
108 | $num = ('9' x ++$e) + 0; |
394e6ffb |
109 | $num *= $num + 1.0; |
394e6ffb |
110 | } while ("$num" =~ /9{$e}0{$e}/); # must be a certain pattern |
111 | $e--; # last test failed, so retract one step |
112 | # the limits below brush the problems with the test above under the rug: |
113 | # the test should be able to find the proper $e automatically |
114 | $e = 5 if $^O =~ /^uts/; # UTS get's some special treatment |
115 | $e = 5 if $^O =~ /^unicos/; # unicos is also problematic (6 seems to work |
116 | # there, but we play safe) |
07d34614 |
117 | $e = 5 if $] < 5.006; # cap, for older Perls |
2e507a43 |
118 | $e = 7 if $e > 7; # cap, for VMS, OS/390 and other 64 bit systems |
119 | # 8 fails inside random testsuite, so take 7 |
394e6ffb |
120 | |
61f5c3f5 |
121 | # determine how many digits fit into an integer and can be safely added |
122 | # together plus carry w/o causing an overflow |
123 | |
61f5c3f5 |
124 | use integer; |
c38b2de2 |
125 | |
126 | ############################################################################ |
127 | # the next block is no longer important |
128 | |
129 | ## this below detects 15 on a 64 bit system, because after that it becomes |
130 | ## 1e16 and not 1000000 :/ I can make it detect 18, but then I get a lot of |
131 | ## test failures. Ugh! (Tomake detect 18: uncomment lines marked with *) |
132 | |
133 | #my $bi = 5; # approx. 16 bit |
134 | #$num = int('9' x $bi); |
135 | ## $num = 99999; # * |
136 | ## while ( ($num+$num+1) eq '1' . '9' x $bi) # * |
137 | #while ( int($num+$num+1) eq '1' . '9' x $bi) |
138 | # { |
139 | # $bi++; $num = int('9' x $bi); |
140 | # # $bi++; $num *= 10; $num += 9; # * |
141 | # } |
142 | #$bi--; # back off one step |
61f5c3f5 |
143 | # by setting them equal, we ignore the findings and use the default |
144 | # one-size-fits-all approach from former versions |
c38b2de2 |
145 | my $bi = $e; # XXX, this should work always |
61f5c3f5 |
146 | |
147 | __PACKAGE__->_base_len($e,$bi); # set and store |
394e6ffb |
148 | |
149 | # find out how many bits _and, _or and _xor can take (old default = 16) |
150 | # I don't think anybody has yet 128 bit scalars, so let's play safe. |
394e6ffb |
151 | local $^W = 0; # don't warn about 'nonportable number' |
c38b2de2 |
152 | $AND_BITS = 15; $XOR_BITS = 15; $OR_BITS = 15; |
394e6ffb |
153 | |
154 | # find max bits, we will not go higher than numberofbits that fit into $BASE |
155 | # to make _and etc simpler (and faster for smaller, slower for large numbers) |
156 | my $max = 16; |
157 | while (2 ** $max < $BASE) { $max++; } |
1ddff52a |
158 | { |
159 | no integer; |
160 | $max = 16 if $] < 5.006; # older Perls might not take >16 too well |
161 | } |
394e6ffb |
162 | my ($x,$y,$z); |
163 | do { |
164 | $AND_BITS++; |
165 | $x = oct('0b' . '1' x $AND_BITS); $y = $x & $x; |
166 | $z = (2 ** $AND_BITS) - 1; |
167 | } while ($AND_BITS < $max && $x == $z && $y == $x); |
168 | $AND_BITS --; # retreat one step |
169 | do { |
170 | $XOR_BITS++; |
171 | $x = oct('0b' . '1' x $XOR_BITS); $y = $x ^ 0; |
172 | $z = (2 ** $XOR_BITS) - 1; |
173 | } while ($XOR_BITS < $max && $x == $z && $y == $x); |
174 | $XOR_BITS --; # retreat one step |
175 | do { |
176 | $OR_BITS++; |
177 | $x = oct('0b' . '1' x $OR_BITS); $y = $x | $x; |
178 | $z = (2 ** $OR_BITS) - 1; |
179 | } while ($OR_BITS < $max && $x == $z && $y == $x); |
180 | $OR_BITS --; # retreat one step |
181 | |
574bacfe |
182 | } |
183 | |
61f5c3f5 |
184 | ############################################################################### |
0716bf9b |
185 | |
186 | sub _new |
187 | { |
394e6ffb |
188 | # (ref to string) return ref to num_array |
9393ace2 |
189 | # Convert a number from string format (without sign) to internal base |
190 | # 1ex format. Assumes normalized value as input. |
9b924220 |
191 | my $il = length($_[1])-1; |
b282a552 |
192 | |
193 | # < BASE_LEN due len-1 above |
9b924220 |
194 | return [ int($_[1]) ] if $il < $BASE_LEN; # shortcut for short numbers |
b282a552 |
195 | |
61f5c3f5 |
196 | # this leaves '00000' instead of int 0 and will be corrected after any op |
197 | [ reverse(unpack("a" . ($il % $BASE_LEN+1) |
9b924220 |
198 | . ("a$BASE_LEN" x ($il / $BASE_LEN)), $_[1])) ]; |
0716bf9b |
199 | } |
394e6ffb |
200 | |
201 | BEGIN |
202 | { |
9b924220 |
203 | $AND_MASK = __PACKAGE__->_new( ( 2 ** $AND_BITS )); |
204 | $XOR_MASK = __PACKAGE__->_new( ( 2 ** $XOR_BITS )); |
205 | $OR_MASK = __PACKAGE__->_new( ( 2 ** $OR_BITS )); |
394e6ffb |
206 | } |
0716bf9b |
207 | |
208 | sub _zero |
209 | { |
210 | # create a zero |
61f5c3f5 |
211 | [ 0 ]; |
0716bf9b |
212 | } |
213 | |
214 | sub _one |
215 | { |
216 | # create a one |
61f5c3f5 |
217 | [ 1 ]; |
0716bf9b |
218 | } |
219 | |
027dc388 |
220 | sub _two |
221 | { |
1ddff52a |
222 | # create a two (used internally for shifting) |
61f5c3f5 |
223 | [ 2 ]; |
027dc388 |
224 | } |
225 | |
9b924220 |
226 | sub _ten |
227 | { |
228 | # create a 10 (used internally for shifting) |
229 | [ 10 ]; |
230 | } |
231 | |
0716bf9b |
232 | sub _copy |
233 | { |
091c87b1 |
234 | # make a true copy |
61f5c3f5 |
235 | [ @{$_[1]} ]; |
0716bf9b |
236 | } |
237 | |
bd05a461 |
238 | # catch and throw away |
239 | sub import { } |
240 | |
0716bf9b |
241 | ############################################################################## |
242 | # convert back to string and number |
243 | |
244 | sub _str |
245 | { |
246 | # (ref to BINT) return num_str |
247 | # Convert number from internal base 100000 format to string format. |
248 | # internal format is always normalized (no leading zeros, "-0" => "+0") |
574bacfe |
249 | my $ar = $_[1]; |
0716bf9b |
250 | my $ret = ""; |
61f5c3f5 |
251 | |
252 | my $l = scalar @$ar; # number of parts |
253 | return $nan if $l < 1; # should not happen |
254 | |
0716bf9b |
255 | # handle first one different to strip leading zeros from it (there are no |
256 | # leading zero parts in internal representation) |
61f5c3f5 |
257 | $l --; $ret .= int($ar->[$l]); $l--; |
0716bf9b |
258 | # Interestingly, the pre-padd method uses more time |
091c87b1 |
259 | # the old grep variant takes longer (14 vs. 10 sec) |
574bacfe |
260 | my $z = '0' x ($BASE_LEN-1); |
0716bf9b |
261 | while ($l >= 0) |
262 | { |
574bacfe |
263 | $ret .= substr($z.$ar->[$l],-$BASE_LEN); # fastest way I could think of |
0716bf9b |
264 | $l--; |
265 | } |
9b924220 |
266 | $ret; |
0716bf9b |
267 | } |
268 | |
269 | sub _num |
270 | { |
9b924220 |
271 | # Make a number (scalar int/float) from a BigInt object |
574bacfe |
272 | my $x = $_[1]; |
9b924220 |
273 | |
274 | return 0+$x->[0] if scalar @$x == 1; # below $BASE |
0716bf9b |
275 | my $fac = 1; |
276 | my $num = 0; |
277 | foreach (@$x) |
278 | { |
279 | $num += $fac*$_; $fac *= $BASE; |
280 | } |
61f5c3f5 |
281 | $num; |
0716bf9b |
282 | } |
283 | |
284 | ############################################################################## |
285 | # actual math code |
286 | |
287 | sub _add |
288 | { |
289 | # (ref to int_num_array, ref to int_num_array) |
574bacfe |
290 | # routine to add two base 1eX numbers |
0716bf9b |
291 | # stolen from Knuth Vol 2 Algorithm A pg 231 |
b22b3e31 |
292 | # there are separate routines to add and sub as per Knuth pg 233 |
0716bf9b |
293 | # This routine clobbers up array x, but not y. |
294 | |
574bacfe |
295 | my ($c,$x,$y) = @_; |
b3abae2a |
296 | |
297 | return $x if (@$y == 1) && $y->[0] == 0; # $x + 0 => $x |
298 | if ((@$x == 1) && $x->[0] == 0) # 0 + $y => $y->copy |
299 | { |
300 | # twice as slow as $x = [ @$y ], but necc. to retain $x as ref :( |
301 | @$x = @$y; return $x; |
302 | } |
0716bf9b |
303 | |
304 | # for each in Y, add Y to X and carry. If after that, something is left in |
305 | # X, foreach in X add carry to X and then return X, carry |
091c87b1 |
306 | # Trades one "$j++" for having to shift arrays |
0716bf9b |
307 | my $i; my $car = 0; my $j = 0; |
308 | for $i (@$y) |
309 | { |
e745a66c |
310 | $x->[$j] -= $BASE if $car = (($x->[$j] += $i + $car) >= $BASE) ? 1 : 0; |
0716bf9b |
311 | $j++; |
312 | } |
313 | while ($car != 0) |
314 | { |
315 | $x->[$j] -= $BASE if $car = (($x->[$j] += $car) >= $BASE) ? 1 : 0; $j++; |
316 | } |
61f5c3f5 |
317 | $x; |
e745a66c |
318 | } |
319 | |
320 | sub _inc |
321 | { |
322 | # (ref to int_num_array, ref to int_num_array) |
091c87b1 |
323 | # Add 1 to $x, modify $x in place |
e745a66c |
324 | my ($c,$x) = @_; |
325 | |
326 | for my $i (@$x) |
327 | { |
328 | return $x if (($i += 1) < $BASE); # early out |
61f5c3f5 |
329 | $i = 0; # overflow, next |
e745a66c |
330 | } |
61f5c3f5 |
331 | push @$x,1 if ($x->[-1] == 0); # last overflowed, so extend |
332 | $x; |
e745a66c |
333 | } |
334 | |
335 | sub _dec |
336 | { |
337 | # (ref to int_num_array, ref to int_num_array) |
091c87b1 |
338 | # Sub 1 from $x, modify $x in place |
e745a66c |
339 | my ($c,$x) = @_; |
340 | |
61f5c3f5 |
341 | my $MAX = $BASE-1; # since MAX_VAL based on MBASE |
e745a66c |
342 | for my $i (@$x) |
343 | { |
344 | last if (($i -= 1) >= 0); # early out |
091c87b1 |
345 | $i = $MAX; # underflow, next |
e745a66c |
346 | } |
091c87b1 |
347 | pop @$x if $x->[-1] == 0 && @$x > 1; # last underflowed (but leave 0) |
61f5c3f5 |
348 | $x; |
0716bf9b |
349 | } |
350 | |
351 | sub _sub |
352 | { |
9393ace2 |
353 | # (ref to int_num_array, ref to int_num_array, swap) |
574bacfe |
354 | # subtract base 1eX numbers -- stolen from Knuth Vol 2 pg 232, $x > $y |
56b9c951 |
355 | # subtract Y from X by modifying x in place |
574bacfe |
356 | my ($c,$sx,$sy,$s) = @_; |
0716bf9b |
357 | |
358 | my $car = 0; my $i; my $j = 0; |
359 | if (!$s) |
360 | { |
0716bf9b |
361 | for $i (@$sx) |
362 | { |
363 | last unless defined $sy->[$j] || $car; |
0716bf9b |
364 | $i += $BASE if $car = (($i -= ($sy->[$j] || 0) + $car) < 0); $j++; |
0716bf9b |
365 | } |
366 | # might leave leading zeros, so fix that |
394e6ffb |
367 | return __strip_zeros($sx); |
0716bf9b |
368 | } |
394e6ffb |
369 | for $i (@$sx) |
0716bf9b |
370 | { |
07d34614 |
371 | # we can't do an early out if $x is < than $y, since we |
56b9c951 |
372 | # need to copy the high chunks from $y. Found by Bob Mathews. |
373 | #last unless defined $sy->[$j] || $car; |
394e6ffb |
374 | $sy->[$j] += $BASE |
375 | if $car = (($sy->[$j] = $i-($sy->[$j]||0) - $car) < 0); |
376 | $j++; |
0716bf9b |
377 | } |
394e6ffb |
378 | # might leave leading zeros, so fix that |
379 | __strip_zeros($sy); |
0716bf9b |
380 | } |
381 | |
ee15d750 |
382 | sub _mul_use_mul |
0716bf9b |
383 | { |
9393ace2 |
384 | # (ref to int_num_array, ref to int_num_array) |
0716bf9b |
385 | # multiply two numbers in internal representation |
b22b3e31 |
386 | # modifies first arg, second need not be different from first |
574bacfe |
387 | my ($c,$xv,$yv) = @_; |
dccbb853 |
388 | |
990fb837 |
389 | if (@$yv == 1) |
b3abae2a |
390 | { |
990fb837 |
391 | # shortcut for two very short numbers (improved by Nathan Zook) |
392 | # works also if xv and yv are the same reference, and handles also $x == 0 |
393 | if (@$xv == 1) |
394 | { |
395 | if (($xv->[0] *= $yv->[0]) >= $MBASE) |
396 | { |
397 | $xv->[0] = $xv->[0] - ($xv->[1] = int($xv->[0] * $RBASE)) * $MBASE; |
398 | }; |
399 | return $xv; |
400 | } |
401 | # $x * 0 => 0 |
402 | if ($yv->[0] == 0) |
403 | { |
404 | @$xv = (0); |
405 | return $xv; |
406 | } |
407 | # multiply a large number a by a single element one, so speed up |
408 | my $y = $yv->[0]; my $car = 0; |
409 | foreach my $i (@$xv) |
410 | { |
411 | $i = $i * $y + $car; $car = int($i * $RBASE); $i -= $car * $MBASE; |
412 | } |
413 | push @$xv, $car if $car != 0; |
b3abae2a |
414 | return $xv; |
415 | } |
990fb837 |
416 | # shortcut for result $x == 0 => result = 0 |
417 | return $xv if ( ((@$xv == 1) && ($xv->[0] == 0)) ); |
b3abae2a |
418 | |
0716bf9b |
419 | # since multiplying $x with $x fails, make copy in this case |
d614cd8b |
420 | $yv = [@$xv] if $xv == $yv; # same references? |
9393ace2 |
421 | |
61f5c3f5 |
422 | my @prod = (); my ($prod,$car,$cty,$xi,$yi); |
423 | |
0716bf9b |
424 | for $xi (@$xv) |
425 | { |
426 | $car = 0; $cty = 0; |
574bacfe |
427 | |
428 | # slow variant |
429 | # for $yi (@$yv) |
430 | # { |
431 | # $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
432 | # $prod[$cty++] = |
61f5c3f5 |
433 | # $prod - ($car = int($prod * RBASE)) * $MBASE; # see USE_MUL |
574bacfe |
434 | # } |
435 | # $prod[$cty] += $car if $car; # need really to check for 0? |
436 | # $xi = shift @prod; |
437 | |
438 | # faster variant |
439 | # looping through this if $xi == 0 is silly - so optimize it away! |
440 | $xi = (shift @prod || 0), next if $xi == 0; |
0716bf9b |
441 | for $yi (@$yv) |
442 | { |
443 | $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
574bacfe |
444 | ## this is actually a tad slower |
445 | ## $prod = $prod[$cty]; $prod += ($car + $xi * $yi); # no ||0 here |
0716bf9b |
446 | $prod[$cty++] = |
61f5c3f5 |
447 | $prod - ($car = int($prod * $RBASE)) * $MBASE; # see USE_MUL |
0716bf9b |
448 | } |
449 | $prod[$cty] += $car if $car; # need really to check for 0? |
027dc388 |
450 | $xi = shift @prod || 0; # || 0 makes v5.005_3 happy |
0716bf9b |
451 | } |
0716bf9b |
452 | push @$xv, @prod; |
990fb837 |
453 | __strip_zeros($xv); |
61f5c3f5 |
454 | $xv; |
0716bf9b |
455 | } |
456 | |
ee15d750 |
457 | sub _mul_use_div |
458 | { |
9393ace2 |
459 | # (ref to int_num_array, ref to int_num_array) |
ee15d750 |
460 | # multiply two numbers in internal representation |
461 | # modifies first arg, second need not be different from first |
462 | my ($c,$xv,$yv) = @_; |
463 | |
990fb837 |
464 | if (@$yv == 1) |
b3abae2a |
465 | { |
990fb837 |
466 | # shortcut for two small numbers, also handles $x == 0 |
467 | if (@$xv == 1) |
468 | { |
469 | # shortcut for two very short numbers (improved by Nathan Zook) |
470 | # works also if xv and yv are the same reference, and handles also $x == 0 |
471 | if (($xv->[0] *= $yv->[0]) >= $MBASE) |
472 | { |
473 | $xv->[0] = |
474 | $xv->[0] - ($xv->[1] = int($xv->[0] / $MBASE)) * $MBASE; |
475 | }; |
476 | return $xv; |
477 | } |
478 | # $x * 0 => 0 |
479 | if ($yv->[0] == 0) |
480 | { |
481 | @$xv = (0); |
482 | return $xv; |
483 | } |
484 | # multiply a large number a by a single element one, so speed up |
485 | my $y = $yv->[0]; my $car = 0; |
486 | foreach my $i (@$xv) |
487 | { |
488 | $i = $i * $y + $car; $car = int($i / $MBASE); $i -= $car * $MBASE; |
489 | } |
490 | push @$xv, $car if $car != 0; |
b3abae2a |
491 | return $xv; |
492 | } |
990fb837 |
493 | # shortcut for result $x == 0 => result = 0 |
494 | return $xv if ( ((@$xv == 1) && ($xv->[0] == 0)) ); |
b3abae2a |
495 | |
ee15d750 |
496 | # since multiplying $x with $x fails, make copy in this case |
d614cd8b |
497 | $yv = [@$xv] if $xv == $yv; # same references? |
9393ace2 |
498 | |
61f5c3f5 |
499 | my @prod = (); my ($prod,$car,$cty,$xi,$yi); |
ee15d750 |
500 | for $xi (@$xv) |
501 | { |
502 | $car = 0; $cty = 0; |
503 | # looping through this if $xi == 0 is silly - so optimize it away! |
504 | $xi = (shift @prod || 0), next if $xi == 0; |
505 | for $yi (@$yv) |
506 | { |
507 | $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
508 | $prod[$cty++] = |
61f5c3f5 |
509 | $prod - ($car = int($prod / $MBASE)) * $MBASE; |
ee15d750 |
510 | } |
511 | $prod[$cty] += $car if $car; # need really to check for 0? |
027dc388 |
512 | $xi = shift @prod || 0; # || 0 makes v5.005_3 happy |
ee15d750 |
513 | } |
514 | push @$xv, @prod; |
990fb837 |
515 | __strip_zeros($xv); |
61f5c3f5 |
516 | $xv; |
ee15d750 |
517 | } |
518 | |
519 | sub _div_use_mul |
0716bf9b |
520 | { |
b22b3e31 |
521 | # ref to array, ref to array, modify first array and return remainder if |
0716bf9b |
522 | # in list context |
aef458a0 |
523 | |
524 | # see comments in _div_use_div() for more explanations |
525 | |
574bacfe |
526 | my ($c,$x,$yorg) = @_; |
aef458a0 |
527 | |
528 | # the general div algorithmn here is about O(N*N) and thus quite slow, so |
529 | # we first check for some special cases and use shortcuts to handle them. |
0716bf9b |
530 | |
aef458a0 |
531 | # This works, because we store the numbers in a chunked format where each |
532 | # element contains 5..7 digits (depending on system). |
533 | |
534 | # if both numbers have only one element: |
61f5c3f5 |
535 | if (@$x == 1 && @$yorg == 1) |
536 | { |
13a12e00 |
537 | # shortcut, $yorg and $x are two small numbers |
61f5c3f5 |
538 | if (wantarray) |
539 | { |
540 | my $r = [ $x->[0] % $yorg->[0] ]; |
541 | $x->[0] = int($x->[0] / $yorg->[0]); |
542 | return ($x,$r); |
543 | } |
544 | else |
545 | { |
546 | $x->[0] = int($x->[0] / $yorg->[0]); |
547 | return $x; |
548 | } |
549 | } |
aef458a0 |
550 | |
551 | # if x has more than one, but y has only one element: |
28df3e88 |
552 | if (@$yorg == 1) |
553 | { |
554 | my $rem; |
555 | $rem = _mod($c,[ @$x ],$yorg) if wantarray; |
13a12e00 |
556 | |
28df3e88 |
557 | # shortcut, $y is < $BASE |
558 | my $j = scalar @$x; my $r = 0; |
559 | my $y = $yorg->[0]; my $b; |
560 | while ($j-- > 0) |
561 | { |
562 | $b = $r * $MBASE + $x->[$j]; |
563 | $x->[$j] = int($b/$y); |
564 | $r = $b % $y; |
565 | } |
566 | pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero |
567 | return ($x,$rem) if wantarray; |
568 | return $x; |
569 | } |
0716bf9b |
570 | |
aef458a0 |
571 | # now x and y have more than one element |
572 | |
573 | # check whether y has more elements than x, if yet, the result will be 0 |
574 | if (@$yorg > @$x) |
575 | { |
576 | my $rem; |
577 | $rem = [@$x] if wantarray; # make copy |
578 | splice (@$x,1); # keep ref to original array |
579 | $x->[0] = 0; # set to 0 |
580 | return ($x,$rem) if wantarray; # including remainder? |
581 | return $x; # only x, which is [0] now |
582 | } |
583 | # check whether the numbers have the same number of elements, in that case |
584 | # the result will fit into one element and can be computed efficiently |
585 | if (@$yorg == @$x) |
586 | { |
587 | my $rem; |
588 | # if $yorg has more digits than $x (it's leading element is longer than |
589 | # the one from $x), the result will also be 0: |
590 | if (length(int($yorg->[-1])) > length(int($x->[-1]))) |
591 | { |
592 | $rem = [@$x] if wantarray; # make copy |
593 | splice (@$x,1); # keep ref to org array |
594 | $x->[0] = 0; # set to 0 |
595 | return ($x,$rem) if wantarray; # including remainder? |
596 | return $x; |
597 | } |
598 | # now calculate $x / $yorg |
599 | if (length(int($yorg->[-1])) == length(int($x->[-1]))) |
600 | { |
601 | # same length, so make full compare, and if equal, return 1 |
602 | # hm, same lengths, but same contents? So we need to check all parts: |
603 | my $a = 0; my $j = scalar @$x - 1; |
604 | # manual way (abort if unequal, good for early ne) |
605 | while ($j >= 0) |
606 | { |
607 | last if ($a = $x->[$j] - $yorg->[$j]); $j--; |
608 | } |
609 | # $a contains the result of the compare between X and Y |
610 | # a < 0: x < y, a == 0 => x == y, a > 0: x > y |
611 | if ($a <= 0) |
612 | { |
613 | if (wantarray) |
614 | { |
615 | $rem = [ 0 ]; # a = 0 => x == y => rem 1 |
616 | $rem = [@$x] if $a != 0; # a < 0 => x < y => rem = x |
617 | } |
618 | splice(@$x,1); # keep single element |
619 | $x->[0] = 0; # if $a < 0 |
620 | if ($a == 0) |
621 | { |
622 | # $x == $y |
623 | $x->[0] = 1; |
624 | } |
625 | return ($x,$rem) if wantarray; |
626 | return $x; |
627 | } |
628 | # $x >= $y, proceed normally |
629 | } |
630 | } |
631 | |
632 | # all other cases: |
633 | |
d614cd8b |
634 | my $y = [ @$yorg ]; # always make copy to preserve |
61f5c3f5 |
635 | |
636 | my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0); |
637 | |
638 | $car = $bar = $prd = 0; |
639 | if (($dd = int($MBASE/($y->[-1]+1))) != 1) |
0716bf9b |
640 | { |
641 | for $xi (@$x) |
642 | { |
643 | $xi = $xi * $dd + $car; |
61f5c3f5 |
644 | $xi -= ($car = int($xi * $RBASE)) * $MBASE; # see USE_MUL |
0716bf9b |
645 | } |
646 | push(@$x, $car); $car = 0; |
647 | for $yi (@$y) |
648 | { |
649 | $yi = $yi * $dd + $car; |
61f5c3f5 |
650 | $yi -= ($car = int($yi * $RBASE)) * $MBASE; # see USE_MUL |
0716bf9b |
651 | } |
652 | } |
653 | else |
654 | { |
655 | push(@$x, 0); |
656 | } |
657 | @q = (); ($v2,$v1) = @$y[-2,-1]; |
658 | $v2 = 0 unless $v2; |
659 | while ($#$x > $#$y) |
660 | { |
661 | ($u2,$u1,$u0) = @$x[-3..-1]; |
662 | $u2 = 0 unless $u2; |
663 | #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" |
664 | # if $v1 == 0; |
aef458a0 |
665 | $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$MBASE+$u1)/$v1)); |
61f5c3f5 |
666 | --$q while ($v2*$q > ($u0*$MBASE+$u1-$q*$v1)*$MBASE+$u2); |
0716bf9b |
667 | if ($q) |
668 | { |
669 | ($car, $bar) = (0,0); |
670 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
671 | { |
672 | $prd = $q * $y->[$yi] + $car; |
61f5c3f5 |
673 | $prd -= ($car = int($prd * $RBASE)) * $MBASE; # see USE_MUL |
674 | $x->[$xi] += $MBASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0)); |
0716bf9b |
675 | } |
676 | if ($x->[-1] < $car + $bar) |
677 | { |
678 | $car = 0; --$q; |
679 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
680 | { |
61f5c3f5 |
681 | $x->[$xi] -= $MBASE |
aef458a0 |
682 | if ($car = (($x->[$xi] += $y->[$yi] + $car) >= $MBASE)); |
0716bf9b |
683 | } |
684 | } |
685 | } |
aef458a0 |
686 | pop(@$x); |
687 | unshift(@q, $q); |
0716bf9b |
688 | } |
689 | if (wantarray) |
690 | { |
691 | @d = (); |
692 | if ($dd != 1) |
693 | { |
694 | $car = 0; |
695 | for $xi (reverse @$x) |
696 | { |
61f5c3f5 |
697 | $prd = $car * $MBASE + $xi; |
0716bf9b |
698 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; # see USE_MUL |
699 | unshift(@d, $tmp); |
700 | } |
701 | } |
702 | else |
703 | { |
704 | @d = @$x; |
705 | } |
706 | @$x = @q; |
61f5c3f5 |
707 | my $d = \@d; |
990fb837 |
708 | __strip_zeros($x); |
709 | __strip_zeros($d); |
61f5c3f5 |
710 | return ($x,$d); |
0716bf9b |
711 | } |
712 | @$x = @q; |
990fb837 |
713 | __strip_zeros($x); |
61f5c3f5 |
714 | $x; |
0716bf9b |
715 | } |
716 | |
ee15d750 |
717 | sub _div_use_div |
718 | { |
719 | # ref to array, ref to array, modify first array and return remainder if |
720 | # in list context |
ee15d750 |
721 | my ($c,$x,$yorg) = @_; |
ee15d750 |
722 | |
990fb837 |
723 | # the general div algorithmn here is about O(N*N) and thus quite slow, so |
724 | # we first check for some special cases and use shortcuts to handle them. |
725 | |
726 | # This works, because we store the numbers in a chunked format where each |
727 | # element contains 5..7 digits (depending on system). |
728 | |
729 | # if both numbers have only one element: |
61f5c3f5 |
730 | if (@$x == 1 && @$yorg == 1) |
731 | { |
13a12e00 |
732 | # shortcut, $yorg and $x are two small numbers |
61f5c3f5 |
733 | if (wantarray) |
734 | { |
735 | my $r = [ $x->[0] % $yorg->[0] ]; |
736 | $x->[0] = int($x->[0] / $yorg->[0]); |
737 | return ($x,$r); |
738 | } |
739 | else |
740 | { |
741 | $x->[0] = int($x->[0] / $yorg->[0]); |
742 | return $x; |
743 | } |
744 | } |
990fb837 |
745 | # if x has more than one, but y has only one element: |
28df3e88 |
746 | if (@$yorg == 1) |
747 | { |
748 | my $rem; |
749 | $rem = _mod($c,[ @$x ],$yorg) if wantarray; |
750 | |
751 | # shortcut, $y is < $BASE |
752 | my $j = scalar @$x; my $r = 0; |
753 | my $y = $yorg->[0]; my $b; |
754 | while ($j-- > 0) |
755 | { |
756 | $b = $r * $MBASE + $x->[$j]; |
757 | $x->[$j] = int($b/$y); |
758 | $r = $b % $y; |
759 | } |
760 | pop @$x if @$x > 1 && $x->[-1] == 0; # splice up a leading zero |
761 | return ($x,$rem) if wantarray; |
762 | return $x; |
763 | } |
990fb837 |
764 | # now x and y have more than one element |
ee15d750 |
765 | |
990fb837 |
766 | # check whether y has more elements than x, if yet, the result will be 0 |
767 | if (@$yorg > @$x) |
61f5c3f5 |
768 | { |
990fb837 |
769 | my $rem; |
770 | $rem = [@$x] if wantarray; # make copy |
771 | splice (@$x,1); # keep ref to original array |
772 | $x->[0] = 0; # set to 0 |
773 | return ($x,$rem) if wantarray; # including remainder? |
aef458a0 |
774 | return $x; # only x, which is [0] now |
61f5c3f5 |
775 | } |
990fb837 |
776 | # check whether the numbers have the same number of elements, in that case |
777 | # the result will fit into one element and can be computed efficiently |
778 | if (@$yorg == @$x) |
779 | { |
780 | my $rem; |
781 | # if $yorg has more digits than $x (it's leading element is longer than |
782 | # the one from $x), the result will also be 0: |
783 | if (length(int($yorg->[-1])) > length(int($x->[-1]))) |
784 | { |
785 | $rem = [@$x] if wantarray; # make copy |
786 | splice (@$x,1); # keep ref to org array |
787 | $x->[0] = 0; # set to 0 |
788 | return ($x,$rem) if wantarray; # including remainder? |
789 | return $x; |
790 | } |
791 | # now calculate $x / $yorg |
091c87b1 |
792 | |
990fb837 |
793 | if (length(int($yorg->[-1])) == length(int($x->[-1]))) |
794 | { |
795 | # same length, so make full compare, and if equal, return 1 |
aef458a0 |
796 | # hm, same lengths, but same contents? So we need to check all parts: |
990fb837 |
797 | my $a = 0; my $j = scalar @$x - 1; |
798 | # manual way (abort if unequal, good for early ne) |
799 | while ($j >= 0) |
800 | { |
801 | last if ($a = $x->[$j] - $yorg->[$j]); $j--; |
802 | } |
aef458a0 |
803 | # $a contains the result of the compare between X and Y |
990fb837 |
804 | # a < 0: x < y, a == 0 => x == y, a > 0: x > y |
805 | if ($a <= 0) |
806 | { |
aef458a0 |
807 | if (wantarray) |
808 | { |
809 | $rem = [ 0 ]; # a = 0 => x == y => rem 1 |
810 | $rem = [@$x] if $a != 0; # a < 0 => x < y => rem = x |
811 | } |
812 | splice(@$x,1); # keep single element |
990fb837 |
813 | $x->[0] = 0; # if $a < 0 |
814 | if ($a == 0) |
815 | { |
816 | # $x == $y |
817 | $x->[0] = 1; |
818 | } |
819 | return ($x,$rem) if wantarray; |
820 | return $x; |
821 | } |
aef458a0 |
822 | # $x >= $y, so proceed normally |
990fb837 |
823 | } |
990fb837 |
824 | } |
825 | |
826 | # all other cases: |
827 | |
828 | my $y = [ @$yorg ]; # always make copy to preserve |
61f5c3f5 |
829 | |
830 | my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1,@d,$tmp,$q,$u2,$u1,$u0); |
831 | |
832 | $car = $bar = $prd = 0; |
833 | if (($dd = int($MBASE/($y->[-1]+1))) != 1) |
ee15d750 |
834 | { |
835 | for $xi (@$x) |
836 | { |
837 | $xi = $xi * $dd + $car; |
61f5c3f5 |
838 | $xi -= ($car = int($xi / $MBASE)) * $MBASE; |
ee15d750 |
839 | } |
840 | push(@$x, $car); $car = 0; |
841 | for $yi (@$y) |
842 | { |
843 | $yi = $yi * $dd + $car; |
61f5c3f5 |
844 | $yi -= ($car = int($yi / $MBASE)) * $MBASE; |
ee15d750 |
845 | } |
846 | } |
847 | else |
848 | { |
849 | push(@$x, 0); |
850 | } |
aef458a0 |
851 | |
852 | # @q will accumulate the final result, $q contains the current computed |
853 | # part of the final result |
854 | |
ee15d750 |
855 | @q = (); ($v2,$v1) = @$y[-2,-1]; |
856 | $v2 = 0 unless $v2; |
857 | while ($#$x > $#$y) |
858 | { |
859 | ($u2,$u1,$u0) = @$x[-3..-1]; |
860 | $u2 = 0 unless $u2; |
861 | #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" |
862 | # if $v1 == 0; |
aef458a0 |
863 | $q = (($u0 == $v1) ? $MAX_VAL : int(($u0*$MBASE+$u1)/$v1)); |
61f5c3f5 |
864 | --$q while ($v2*$q > ($u0*$MBASE+$u1-$q*$v1)*$MBASE+$u2); |
ee15d750 |
865 | if ($q) |
866 | { |
867 | ($car, $bar) = (0,0); |
868 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
869 | { |
870 | $prd = $q * $y->[$yi] + $car; |
61f5c3f5 |
871 | $prd -= ($car = int($prd / $MBASE)) * $MBASE; |
872 | $x->[$xi] += $MBASE if ($bar = (($x->[$xi] -= $prd + $bar) < 0)); |
ee15d750 |
873 | } |
874 | if ($x->[-1] < $car + $bar) |
875 | { |
876 | $car = 0; --$q; |
877 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
878 | { |
61f5c3f5 |
879 | $x->[$xi] -= $MBASE |
aef458a0 |
880 | if ($car = (($x->[$xi] += $y->[$yi] + $car) >= $MBASE)); |
ee15d750 |
881 | } |
882 | } |
883 | } |
61f5c3f5 |
884 | pop(@$x); unshift(@q, $q); |
ee15d750 |
885 | } |
886 | if (wantarray) |
887 | { |
888 | @d = (); |
889 | if ($dd != 1) |
890 | { |
891 | $car = 0; |
892 | for $xi (reverse @$x) |
893 | { |
61f5c3f5 |
894 | $prd = $car * $MBASE + $xi; |
ee15d750 |
895 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; |
896 | unshift(@d, $tmp); |
897 | } |
898 | } |
899 | else |
900 | { |
901 | @d = @$x; |
902 | } |
903 | @$x = @q; |
61f5c3f5 |
904 | my $d = \@d; |
990fb837 |
905 | __strip_zeros($x); |
906 | __strip_zeros($d); |
61f5c3f5 |
907 | return ($x,$d); |
ee15d750 |
908 | } |
909 | @$x = @q; |
990fb837 |
910 | __strip_zeros($x); |
61f5c3f5 |
911 | $x; |
ee15d750 |
912 | } |
913 | |
394e6ffb |
914 | ############################################################################## |
915 | # testing |
916 | |
917 | sub _acmp |
918 | { |
919 | # internal absolute post-normalized compare (ignore signs) |
920 | # ref to array, ref to array, return <0, 0, >0 |
921 | # arrays must have at least one entry; this is not checked for |
394e6ffb |
922 | my ($c,$cx,$cy) = @_; |
091c87b1 |
923 | |
924 | # shortcut for short numbers |
925 | return (($cx->[0] <=> $cy->[0]) <=> 0) |
926 | if scalar @$cx == scalar @$cy && scalar @$cx == 1; |
394e6ffb |
927 | |
f9a08e12 |
928 | # fast comp based on number of array elements (aka pseudo-length) |
091c87b1 |
929 | my $lxy = (scalar @$cx - scalar @$cy) |
930 | # or length of first element if same number of elements (aka difference 0) |
931 | || |
932 | # need int() here because sometimes the last element is '00018' vs '18' |
933 | (length(int($cx->[-1])) - length(int($cy->[-1]))); |
394e6ffb |
934 | return -1 if $lxy < 0; # already differs, ret |
935 | return 1 if $lxy > 0; # ditto |
56d9de68 |
936 | |
394e6ffb |
937 | # manual way (abort if unequal, good for early ne) |
091c87b1 |
938 | my $a; my $j = scalar @$cx; |
939 | while (--$j >= 0) |
9393ace2 |
940 | { |
091c87b1 |
941 | last if ($a = $cx->[$j] - $cy->[$j]); |
9393ace2 |
942 | } |
091c87b1 |
943 | $a <=> 0; |
394e6ffb |
944 | } |
945 | |
946 | sub _len |
947 | { |
56d9de68 |
948 | # compute number of digits |
394e6ffb |
949 | |
950 | # int() because add/sub sometimes leaves strings (like '00005') instead of |
951 | # '5' in this place, thus causing length() to report wrong length |
952 | my $cx = $_[1]; |
953 | |
56d9de68 |
954 | (@$cx-1)*$BASE_LEN+length(int($cx->[-1])); |
394e6ffb |
955 | } |
956 | |
957 | sub _digit |
958 | { |
959 | # return the nth digit, negative values count backward |
960 | # zero is rightmost, so _digit(123,0) will give 3 |
961 | my ($c,$x,$n) = @_; |
962 | |
963 | my $len = _len('',$x); |
964 | |
965 | $n = $len+$n if $n < 0; # -1 last, -2 second-to-last |
966 | $n = abs($n); # if negative was too big |
967 | $len--; $n = $len if $n > $len; # n to big? |
968 | |
969 | my $elem = int($n / $BASE_LEN); # which array element |
970 | my $digit = $n % $BASE_LEN; # which digit in this element |
971 | $elem = '0000'.@$x[$elem]; # get element padded with 0's |
56d9de68 |
972 | substr($elem,-$digit-1,1); |
394e6ffb |
973 | } |
974 | |
975 | sub _zeros |
976 | { |
977 | # return amount of trailing zeros in decimal |
978 | # check each array elem in _m for having 0 at end as long as elem == 0 |
979 | # Upon finding a elem != 0, stop |
980 | my $x = $_[1]; |
9b924220 |
981 | |
982 | return 0 if scalar @$x == 1 && $x->[0] == 0; |
983 | |
394e6ffb |
984 | my $zeros = 0; my $elem; |
985 | foreach my $e (@$x) |
986 | { |
987 | if ($e != 0) |
988 | { |
989 | $elem = "$e"; # preserve x |
990 | $elem =~ s/.*?(0*$)/$1/; # strip anything not zero |
991 | $zeros *= $BASE_LEN; # elems * 5 |
61f5c3f5 |
992 | $zeros += length($elem); # count trailing zeros |
394e6ffb |
993 | last; # early out |
994 | } |
995 | $zeros ++; # real else branch: 50% slower! |
996 | } |
61f5c3f5 |
997 | $zeros; |
394e6ffb |
998 | } |
999 | |
1000 | ############################################################################## |
1001 | # _is_* routines |
1002 | |
1003 | sub _is_zero |
1004 | { |
9b924220 |
1005 | # return true if arg is zero |
1006 | (((scalar @{$_[1]} == 1) && ($_[1]->[0] == 0))) <=> 0; |
394e6ffb |
1007 | } |
1008 | |
1009 | sub _is_even |
1010 | { |
9b924220 |
1011 | # return true if arg is even |
1012 | (!($_[1]->[0] & 1)) <=> 0; |
394e6ffb |
1013 | } |
1014 | |
1015 | sub _is_odd |
1016 | { |
9b924220 |
1017 | # return true if arg is even |
1018 | (($_[1]->[0] & 1)) <=> 0; |
394e6ffb |
1019 | } |
1020 | |
1021 | sub _is_one |
1022 | { |
9b924220 |
1023 | # return true if arg is one |
1024 | (scalar @{$_[1]} == 1) && ($_[1]->[0] == 1) <=> 0; |
1025 | } |
1026 | |
1027 | sub _is_two |
1028 | { |
1029 | # return true if arg is two |
1030 | (scalar @{$_[1]} == 1) && ($_[1]->[0] == 2) <=> 0; |
1031 | } |
61f5c3f5 |
1032 | |
9b924220 |
1033 | sub _is_ten |
1034 | { |
1035 | # return true if arg is ten |
1036 | (scalar @{$_[1]} == 1) && ($_[1]->[0] == 10) <=> 0; |
394e6ffb |
1037 | } |
1038 | |
1039 | sub __strip_zeros |
1040 | { |
1041 | # internal normalization function that strips leading zeros from the array |
1042 | # args: ref to array |
1043 | my $s = shift; |
1044 | |
1045 | my $cnt = scalar @$s; # get count of parts |
1046 | my $i = $cnt-1; |
1047 | push @$s,0 if $i < 0; # div might return empty results, so fix it |
1048 | |
61f5c3f5 |
1049 | return $s if @$s == 1; # early out |
1050 | |
394e6ffb |
1051 | #print "strip: cnt $cnt i $i\n"; |
1052 | # '0', '3', '4', '0', '0', |
1053 | # 0 1 2 3 4 |
1054 | # cnt = 5, i = 4 |
1055 | # i = 4 |
1056 | # i = 3 |
1057 | # => fcnt = cnt - i (5-2 => 3, cnt => 5-1 = 4, throw away from 4th pos) |
1058 | # >= 1: skip first part (this can be zero) |
1059 | while ($i > 0) { last if $s->[$i] != 0; $i--; } |
1060 | $i++; splice @$s,$i if ($i < $cnt); # $i cant be 0 |
1061 | $s; |
1062 | } |
1063 | |
1064 | ############################################################################### |
3a427a11 |
1065 | # check routine to test internal state for corruptions |
394e6ffb |
1066 | |
1067 | sub _check |
1068 | { |
1069 | # used by the test suite |
1070 | my $x = $_[1]; |
1071 | |
1072 | return "$x is not a reference" if !ref($x); |
1073 | |
1074 | # are all parts are valid? |
1075 | my $i = 0; my $j = scalar @$x; my ($e,$try); |
1076 | while ($i < $j) |
1077 | { |
1078 | $e = $x->[$i]; $e = 'undef' unless defined $e; |
1079 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e)"; |
1080 | last if $e !~ /^[+]?[0-9]+$/; |
1081 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (stringify)"; |
1082 | last if "$e" !~ /^[+]?[0-9]+$/; |
1083 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e) (cat-stringify)"; |
1084 | last if '' . "$e" !~ /^[+]?[0-9]+$/; |
1085 | $try = ' < 0 || >= $BASE; '."($x, $e)"; |
1086 | last if $e <0 || $e >= $BASE; |
1087 | # this test is disabled, since new/bnorm and certain ops (like early out |
1088 | # in add/sub) are allowed/expected to leave '00000' in some elements |
1089 | #$try = '=~ /^00+/; '."($x, $e)"; |
1090 | #last if $e =~ /^00+/; |
1091 | $i++; |
1092 | } |
1093 | return "Illegal part '$e' at pos $i (tested: $try)" if $i < $j; |
3a427a11 |
1094 | 0; |
394e6ffb |
1095 | } |
1096 | |
1097 | |
1098 | ############################################################################### |
394e6ffb |
1099 | |
dccbb853 |
1100 | sub _mod |
1101 | { |
1102 | # if possible, use mod shortcut |
1103 | my ($c,$x,$yo) = @_; |
1104 | |
1105 | # slow way since $y to big |
1106 | if (scalar @$yo > 1) |
1107 | { |
1108 | my ($xo,$rem) = _div($c,$x,$yo); |
1109 | return $rem; |
1110 | } |
aef458a0 |
1111 | |
dccbb853 |
1112 | my $y = $yo->[0]; |
027dc388 |
1113 | # both are single element arrays |
dccbb853 |
1114 | if (scalar @$x == 1) |
1115 | { |
1116 | $x->[0] %= $y; |
1117 | return $x; |
1118 | } |
1119 | |
aef458a0 |
1120 | # @y is a single element, but @x has more than one element |
dccbb853 |
1121 | my $b = $BASE % $y; |
1122 | if ($b == 0) |
1123 | { |
1124 | # when BASE % Y == 0 then (B * BASE) % Y == 0 |
1125 | # (B * BASE) % $y + A % Y => A % Y |
1126 | # so need to consider only last element: O(1) |
1127 | $x->[0] %= $y; |
1128 | } |
027dc388 |
1129 | elsif ($b == 1) |
1130 | { |
3a427a11 |
1131 | # else need to go through all elements: O(N), but loop is a bit simplified |
027dc388 |
1132 | my $r = 0; |
1133 | foreach (@$x) |
1134 | { |
28df3e88 |
1135 | $r = ($r + $_) % $y; # not much faster, but heh... |
1136 | #$r += $_ % $y; $r %= $y; |
027dc388 |
1137 | } |
1138 | $r = 0 if $r == $y; |
1139 | $x->[0] = $r; |
1140 | } |
dccbb853 |
1141 | else |
1142 | { |
3a427a11 |
1143 | # else need to go through all elements: O(N) |
027dc388 |
1144 | my $r = 0; my $bm = 1; |
1145 | foreach (@$x) |
1146 | { |
28df3e88 |
1147 | $r = ($_ * $bm + $r) % $y; |
1148 | $bm = ($bm * $b) % $y; |
1149 | |
1150 | #$r += ($_ % $y) * $bm; |
1151 | #$bm *= $b; |
1152 | #$bm %= $y; |
1153 | #$r %= $y; |
027dc388 |
1154 | } |
1155 | $r = 0 if $r == $y; |
1156 | $x->[0] = $r; |
dccbb853 |
1157 | } |
091c87b1 |
1158 | splice (@$x,1); # keep one element of $x |
61f5c3f5 |
1159 | $x; |
dccbb853 |
1160 | } |
1161 | |
0716bf9b |
1162 | ############################################################################## |
574bacfe |
1163 | # shifts |
1164 | |
1165 | sub _rsft |
1166 | { |
1167 | my ($c,$x,$y,$n) = @_; |
1168 | |
1169 | if ($n != 10) |
1170 | { |
9b924220 |
1171 | $n = _new($c,$n); return _div($c,$x, _pow($c,$n,$y)); |
61f5c3f5 |
1172 | } |
1173 | |
1174 | # shortcut (faster) for shifting by 10) |
1175 | # multiples of $BASE_LEN |
1176 | my $dst = 0; # destination |
1177 | my $src = _num($c,$y); # as normal int |
56d9de68 |
1178 | my $xlen = (@$x-1)*$BASE_LEN+length(int($x->[-1])); # len of x in digits |
990fb837 |
1179 | if ($src > $xlen or ($src == $xlen and ! defined $x->[1])) |
56d9de68 |
1180 | { |
1181 | # 12345 67890 shifted right by more than 10 digits => 0 |
1182 | splice (@$x,1); # leave only one element |
1183 | $x->[0] = 0; # set to zero |
1184 | return $x; |
1185 | } |
61f5c3f5 |
1186 | my $rem = $src % $BASE_LEN; # remainder to shift |
1187 | $src = int($src / $BASE_LEN); # source |
1188 | if ($rem == 0) |
1189 | { |
1190 | splice (@$x,0,$src); # even faster, 38.4 => 39.3 |
574bacfe |
1191 | } |
1192 | else |
1193 | { |
61f5c3f5 |
1194 | my $len = scalar @$x - $src; # elems to go |
1195 | my $vd; my $z = '0'x $BASE_LEN; |
1196 | $x->[scalar @$x] = 0; # avoid || 0 test inside loop |
1197 | while ($dst < $len) |
574bacfe |
1198 | { |
61f5c3f5 |
1199 | $vd = $z.$x->[$src]; |
1200 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN-$rem); |
1201 | $src++; |
1202 | $vd = substr($z.$x->[$src],-$rem,$rem) . $vd; |
1203 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN; |
1204 | $x->[$dst] = int($vd); |
1205 | $dst++; |
574bacfe |
1206 | } |
61f5c3f5 |
1207 | splice (@$x,$dst) if $dst > 0; # kill left-over array elems |
56b9c951 |
1208 | pop @$x if $x->[-1] == 0 && @$x > 1; # kill last element if 0 |
61f5c3f5 |
1209 | } # else rem == 0 |
574bacfe |
1210 | $x; |
1211 | } |
1212 | |
1213 | sub _lsft |
1214 | { |
1215 | my ($c,$x,$y,$n) = @_; |
1216 | |
1217 | if ($n != 10) |
1218 | { |
9b924220 |
1219 | $n = _new($c,$n); return _mul($c,$x, _pow($c,$n,$y)); |
574bacfe |
1220 | } |
61f5c3f5 |
1221 | |
1222 | # shortcut (faster) for shifting by 10) since we are in base 10eX |
1223 | # multiples of $BASE_LEN: |
1224 | my $src = scalar @$x; # source |
1225 | my $len = _num($c,$y); # shift-len as normal int |
1226 | my $rem = $len % $BASE_LEN; # remainder to shift |
1227 | my $dst = $src + int($len/$BASE_LEN); # destination |
1228 | my $vd; # further speedup |
1229 | $x->[$src] = 0; # avoid first ||0 for speed |
1230 | my $z = '0' x $BASE_LEN; |
1231 | while ($src >= 0) |
574bacfe |
1232 | { |
61f5c3f5 |
1233 | $vd = $x->[$src]; $vd = $z.$vd; |
1234 | $vd = substr($vd,-$BASE_LEN+$rem,$BASE_LEN-$rem); |
1235 | $vd .= $src > 0 ? substr($z.$x->[$src-1],-$BASE_LEN,$rem) : '0' x $rem; |
1236 | $vd = substr($vd,-$BASE_LEN,$BASE_LEN) if length($vd) > $BASE_LEN; |
1237 | $x->[$dst] = int($vd); |
1238 | $dst--; $src--; |
574bacfe |
1239 | } |
61f5c3f5 |
1240 | # set lowest parts to 0 |
1241 | while ($dst >= 0) { $x->[$dst--] = 0; } |
1242 | # fix spurios last zero element |
1243 | splice @$x,-1 if $x->[-1] == 0; |
574bacfe |
1244 | $x; |
1245 | } |
1246 | |
027dc388 |
1247 | sub _pow |
1248 | { |
1249 | # power of $x to $y |
1250 | # ref to array, ref to array, return ref to array |
1251 | my ($c,$cx,$cy) = @_; |
1252 | |
b282a552 |
1253 | if (scalar @$cy == 1 && $cy->[0] == 0) |
1254 | { |
1255 | splice (@$cx,1); $cx->[0] = 1; # y == 0 => x => 1 |
1256 | return $cx; |
1257 | } |
1258 | if ((scalar @$cx == 1 && $cx->[0] == 1) || # x == 1 |
1259 | (scalar @$cy == 1 && $cy->[0] == 1)) # or y == 1 |
1260 | { |
1261 | return $cx; |
1262 | } |
1263 | if (scalar @$cx == 1 && $cx->[0] == 0) |
1264 | { |
1265 | splice (@$cx,1); $cx->[0] = 0; # 0 ** y => 0 (if not y <= 0) |
1266 | return $cx; |
1267 | } |
1268 | |
027dc388 |
1269 | my $pow2 = _one(); |
1ddff52a |
1270 | |
9b924220 |
1271 | my $y_bin = _as_bin($c,$cy); $y_bin =~ s/^0b//; |
1ddff52a |
1272 | my $len = length($y_bin); |
1273 | while (--$len > 0) |
027dc388 |
1274 | { |
1ddff52a |
1275 | _mul($c,$pow2,$cx) if substr($y_bin,$len,1) eq '1'; # is odd? |
027dc388 |
1276 | _mul($c,$cx,$cx); |
1277 | } |
1ddff52a |
1278 | |
1279 | _mul($c,$cx,$pow2); |
61f5c3f5 |
1280 | $cx; |
027dc388 |
1281 | } |
1282 | |
b3abae2a |
1283 | sub _fac |
1284 | { |
1285 | # factorial of $x |
1286 | # ref to array, return ref to array |
1287 | my ($c,$cx) = @_; |
1288 | |
1289 | if ((@$cx == 1) && ($cx->[0] <= 2)) |
1290 | { |
091c87b1 |
1291 | $cx->[0] ||= 1; # 0 => 1, 1 => 1, 2 => 2 |
b3abae2a |
1292 | return $cx; |
1293 | } |
1294 | |
1295 | # go forward until $base is exceeded |
091c87b1 |
1296 | # limit is either $x steps (steps == 100 means a result always too high) or |
1297 | # $base. |
b3abae2a |
1298 | my $steps = 100; $steps = $cx->[0] if @$cx == 1; |
091c87b1 |
1299 | my $r = 2; my $cf = 3; my $step = 2; my $last = $r; |
1300 | while ($r*$cf < $BASE && $step < $steps) |
b3abae2a |
1301 | { |
1302 | $last = $r; $r *= $cf++; $step++; |
1303 | } |
091c87b1 |
1304 | if ((@$cx == 1) && $step == $cx->[0]) |
b3abae2a |
1305 | { |
091c87b1 |
1306 | # completely done, so keep reference to $x and return |
1307 | $cx->[0] = $r; |
b3abae2a |
1308 | return $cx; |
1309 | } |
091c87b1 |
1310 | |
990fb837 |
1311 | # now we must do the left over steps |
091c87b1 |
1312 | my $n; # steps still to do |
1313 | if (scalar @$cx == 1) |
1314 | { |
1315 | $n = $cx->[0]; |
1316 | } |
1317 | else |
1318 | { |
1319 | $n = _copy($c,$cx); |
1320 | } |
b3abae2a |
1321 | |
091c87b1 |
1322 | $cx->[0] = $last; splice (@$cx,1); # keep ref to $x |
990fb837 |
1323 | my $zero_elements = 0; |
091c87b1 |
1324 | |
1325 | # do left-over steps fit into a scalar? |
1326 | if (ref $n eq 'ARRAY') |
b3abae2a |
1327 | { |
091c87b1 |
1328 | # No, so use slower inc() & cmp() |
1329 | $step = [$step]; |
1330 | while (_acmp($step,$n) <= 0) |
990fb837 |
1331 | { |
091c87b1 |
1332 | # as soon as the last element of $cx is 0, we split it up and remember |
1333 | # how many zeors we got so far. The reason is that n! will accumulate |
1334 | # zeros at the end rather fast. |
990fb837 |
1335 | if ($cx->[0] == 0) |
1336 | { |
1337 | $zero_elements ++; shift @$cx; |
1338 | } |
091c87b1 |
1339 | _mul($c,$cx,$step); _inc($c,$step); |
990fb837 |
1340 | } |
990fb837 |
1341 | } |
091c87b1 |
1342 | else |
990fb837 |
1343 | { |
091c87b1 |
1344 | # Yes, so we can speed it up slightly |
1345 | while ($step <= $n) |
990fb837 |
1346 | { |
091c87b1 |
1347 | # When the last element of $cx is 0, we split it up and remember |
1348 | # how many we got so far. The reason is that n! will accumulate |
1349 | # zeros at the end rather fast. |
1350 | if ($cx->[0] == 0) |
1351 | { |
1352 | $zero_elements ++; shift @$cx; |
1353 | } |
1354 | _mul($c,$cx,[$step]); $step++; |
990fb837 |
1355 | } |
990fb837 |
1356 | } |
1357 | # multiply in the zeros again |
1358 | while ($zero_elements-- > 0) |
1359 | { |
1360 | unshift @$cx, 0; |
b3abae2a |
1361 | } |
091c87b1 |
1362 | $cx; # return result |
1363 | } |
1364 | |
9b924220 |
1365 | ############################################################################# |
1366 | |
091c87b1 |
1367 | sub _log_int |
1368 | { |
1369 | # calculate integer log of $x to base $base |
1370 | # ref to array, ref to array - return ref to array |
1371 | my ($c,$x,$base) = @_; |
1372 | |
1373 | # X == 0 => NaN |
1374 | return if (scalar @$x == 1 && $x->[0] == 0); |
1375 | # BASE 0 or 1 => NaN |
1376 | return if (scalar @$base == 1 && $base->[0] < 2); |
b282a552 |
1377 | my $cmp = _acmp($c,$x,$base); # X == BASE => 1 |
091c87b1 |
1378 | if ($cmp == 0) |
1379 | { |
1380 | splice (@$x,1); $x->[0] = 1; |
8df1e0a2 |
1381 | return ($x,1) |
091c87b1 |
1382 | } |
1383 | # X < BASE |
1384 | if ($cmp < 0) |
1385 | { |
1386 | splice (@$x,1); $x->[0] = 0; |
8df1e0a2 |
1387 | return ($x,undef); |
091c87b1 |
1388 | } |
1389 | |
1390 | # this trial multiplication is very fast, even for large counts (like for |
1391 | # 2 ** 1024, since this still requires only 1024 very fast steps |
1392 | # (multiplication of a large number by a very small number is very fast)) |
1393 | my $x_org = _copy($c,$x); # preserve x |
8df1e0a2 |
1394 | splice(@$x,1); $x->[0] = 1; # keep ref to $x |
1395 | |
b282a552 |
1396 | my $trial = _copy($c,$base); |
1397 | |
1398 | # XXX TODO this only works if $base has only one element |
1399 | if (scalar @$base == 1) |
1400 | { |
1401 | # compute int ( length_in_base_10(X) / ( log(base) / log(10) ) ) |
1402 | my $len = _len($c,$x_org); |
1403 | my $res = int($len / (log($base->[0]) / log(10))) || 1; # avoid $res == 0 |
1404 | |
1405 | $x->[0] = $res; |
1406 | $trial = _pow ($c, _copy($c, $base), $x); |
1407 | my $a = _acmp($x,$trial,$x_org); |
1408 | return ($x,1) if $a == 0; |
3a427a11 |
1409 | # we now know that $res is too small |
b282a552 |
1410 | if ($res < 0) |
1411 | { |
1412 | _mul($c,$trial,$base); _add($c, $x, [1]); |
1413 | } |
1414 | else |
1415 | { |
1416 | # or too big |
1417 | _div($c,$trial,$base); _sub($c, $x, [1]); |
1418 | } |
1419 | # did we now get the right result? |
1420 | $a = _acmp($x,$trial,$x_org); |
1421 | return ($x,1) if $a == 0; # yes, exactly |
1422 | # still too big |
1423 | if ($a > 0) |
1424 | { |
1425 | _div($c,$trial,$base); _sub($c, $x, [1]); |
1426 | } |
1427 | } |
1428 | |
8df1e0a2 |
1429 | # simple loop that increments $x by two in each step, possible overstepping |
1430 | # the real result by one |
091c87b1 |
1431 | |
b282a552 |
1432 | my $a; |
8df1e0a2 |
1433 | my $base_mul = _mul($c, _copy($c,$base), $base); |
1434 | |
9b924220 |
1435 | while (($a = _acmp($c,$trial,$x_org)) < 0) |
091c87b1 |
1436 | { |
8df1e0a2 |
1437 | _mul($c,$trial,$base_mul); _add($c, $x, [2]); |
091c87b1 |
1438 | } |
8df1e0a2 |
1439 | |
1440 | my $exact = 1; |
1441 | if ($a > 0) |
091c87b1 |
1442 | { |
8df1e0a2 |
1443 | # overstepped the result |
1444 | _dec($c, $x); |
1445 | _div($c,$trial,$base); |
9b924220 |
1446 | $a = _acmp($c,$trial,$x_org); |
8df1e0a2 |
1447 | if ($a > 0) |
091c87b1 |
1448 | { |
8df1e0a2 |
1449 | _dec($c, $x); |
091c87b1 |
1450 | } |
8df1e0a2 |
1451 | $exact = 0 if $a != 0; |
091c87b1 |
1452 | } |
1453 | |
8df1e0a2 |
1454 | ($x,$exact); # return result |
b3abae2a |
1455 | } |
1456 | |
56d9de68 |
1457 | # for debugging: |
1458 | use constant DEBUG => 0; |
1459 | my $steps = 0; |
1460 | sub steps { $steps }; |
b3abae2a |
1461 | |
1462 | sub _sqrt |
0716bf9b |
1463 | { |
56d9de68 |
1464 | # square-root of $x in place |
990fb837 |
1465 | # Compute a guess of the result (by rule of thumb), then improve it via |
56d9de68 |
1466 | # Newton's method. |
394e6ffb |
1467 | my ($c,$x) = @_; |
0716bf9b |
1468 | |
394e6ffb |
1469 | if (scalar @$x == 1) |
1470 | { |
56d9de68 |
1471 | # fit's into one Perl scalar, so result can be computed directly |
394e6ffb |
1472 | $x->[0] = int(sqrt($x->[0])); |
1473 | return $x; |
1474 | } |
1475 | my $y = _copy($c,$x); |
b3abae2a |
1476 | # hopefully _len/2 is < $BASE, the -1 is to always undershot the guess |
1477 | # since our guess will "grow" |
1478 | my $l = int((_len($c,$x)-1) / 2); |
1479 | |
56d9de68 |
1480 | my $lastelem = $x->[-1]; # for guess |
b3abae2a |
1481 | my $elems = scalar @$x - 1; |
1482 | # not enough digits, but could have more? |
56d9de68 |
1483 | if ((length($lastelem) <= 3) && ($elems > 1)) |
b3abae2a |
1484 | { |
1485 | # right-align with zero pad |
1486 | my $len = length($lastelem) & 1; |
1487 | print "$lastelem => " if DEBUG; |
1488 | $lastelem .= substr($x->[-2] . '0' x $BASE_LEN,0,$BASE_LEN); |
1489 | # former odd => make odd again, or former even to even again |
56d9de68 |
1490 | $lastelem = $lastelem / 10 if (length($lastelem) & 1) != $len; |
b3abae2a |
1491 | print "$lastelem\n" if DEBUG; |
1492 | } |
0716bf9b |
1493 | |
61f5c3f5 |
1494 | # construct $x (instead of _lsft($c,$x,$l,10) |
1495 | my $r = $l % $BASE_LEN; # 10000 00000 00000 00000 ($BASE_LEN=5) |
1496 | $l = int($l / $BASE_LEN); |
b3abae2a |
1497 | print "l = $l " if DEBUG; |
56d9de68 |
1498 | |
1499 | splice @$x,$l; # keep ref($x), but modify it |
1500 | |
b3abae2a |
1501 | # we make the first part of the guess not '1000...0' but int(sqrt($lastelem)) |
1502 | # that gives us: |
56d9de68 |
1503 | # 14400 00000 => sqrt(14400) => guess first digits to be 120 |
1504 | # 144000 000000 => sqrt(144000) => guess 379 |
b3abae2a |
1505 | |
b3abae2a |
1506 | print "$lastelem (elems $elems) => " if DEBUG; |
1507 | $lastelem = $lastelem / 10 if ($elems & 1 == 1); # odd or even? |
1508 | my $g = sqrt($lastelem); $g =~ s/\.//; # 2.345 => 2345 |
1509 | $r -= 1 if $elems & 1 == 0; # 70 => 7 |
1510 | |
1511 | # padd with zeros if result is too short |
1512 | $x->[$l--] = int(substr($g . '0' x $r,0,$r+1)); |
1513 | print "now ",$x->[-1] if DEBUG; |
1514 | print " would have been ", int('1' . '0' x $r),"\n" if DEBUG; |
56d9de68 |
1515 | |
b3abae2a |
1516 | # If @$x > 1, we could compute the second elem of the guess, too, to create |
56d9de68 |
1517 | # an even better guess. Not implemented yet. Does it improve performance? |
b3abae2a |
1518 | $x->[$l--] = 0 while ($l >= 0); # all other digits of guess are zero |
56d9de68 |
1519 | |
9b924220 |
1520 | print "start x= ",_str($c,$x),"\n" if DEBUG; |
394e6ffb |
1521 | my $two = _two(); |
1522 | my $last = _zero(); |
1523 | my $lastlast = _zero(); |
b3abae2a |
1524 | $steps = 0 if DEBUG; |
394e6ffb |
1525 | while (_acmp($c,$last,$x) != 0 && _acmp($c,$lastlast,$x) != 0) |
1526 | { |
b3abae2a |
1527 | $steps++ if DEBUG; |
394e6ffb |
1528 | $lastlast = _copy($c,$last); |
1529 | $last = _copy($c,$x); |
1530 | _add($c,$x, _div($c,_copy($c,$y),$x)); |
1531 | _div($c,$x, $two ); |
9b924220 |
1532 | print " x= ",_str($c,$x),"\n" if DEBUG; |
394e6ffb |
1533 | } |
b3abae2a |
1534 | print "\nsteps in sqrt: $steps, " if DEBUG; |
394e6ffb |
1535 | _dec($c,$x) if _acmp($c,$y,_mul($c,_copy($c,$x),$x)) < 0; # overshot? |
b3abae2a |
1536 | print " final ",$x->[-1],"\n" if DEBUG; |
394e6ffb |
1537 | $x; |
0716bf9b |
1538 | } |
1539 | |
990fb837 |
1540 | sub _root |
1541 | { |
1542 | # take n'th root of $x in place (n >= 3) |
990fb837 |
1543 | my ($c,$x,$n) = @_; |
1544 | |
1545 | if (scalar @$x == 1) |
1546 | { |
1547 | if (scalar @$n > 1) |
1548 | { |
1549 | # result will always be smaller than 2 so trunc to 1 at once |
1550 | $x->[0] = 1; |
1551 | } |
1552 | else |
1553 | { |
1554 | # fit's into one Perl scalar, so result can be computed directly |
091c87b1 |
1555 | # cannot use int() here, because it rounds wrongly (try |
1556 | # (81 ** 3) ** (1/3) to see what I mean) |
1557 | #$x->[0] = int( $x->[0] ** (1 / $n->[0]) ); |
1558 | # round to 8 digits, then truncate result to integer |
1559 | $x->[0] = int ( sprintf ("%.8f", $x->[0] ** (1 / $n->[0]) ) ); |
990fb837 |
1560 | } |
1561 | return $x; |
1562 | } |
1563 | |
3a427a11 |
1564 | # we know now that X is more than one element long |
1565 | |
c38b2de2 |
1566 | # if $n is a power of two, we can repeatedly take sqrt($X) and find the |
1567 | # proper result, because sqrt(sqrt($x)) == root($x,4) |
1568 | my $b = _as_bin($c,$n); |
9b924220 |
1569 | if ($b =~ /0b1(0+)$/) |
c38b2de2 |
1570 | { |
1571 | my $count = CORE::length($1); # 0b100 => len('00') => 2 |
1572 | my $cnt = $count; # counter for loop |
1573 | unshift (@$x, 0); # add one element, together with one |
1574 | # more below in the loop this makes 2 |
1575 | while ($cnt-- > 0) |
1576 | { |
1577 | # 'inflate' $X by adding one element, basically computing |
1578 | # $x * $BASE * $BASE. This gives us more $BASE_LEN digits for result |
1579 | # since len(sqrt($X)) approx == len($x) / 2. |
1580 | unshift (@$x, 0); |
1581 | # calculate sqrt($x), $x is now one element to big, again. In the next |
1582 | # round we make that two, again. |
1583 | _sqrt($c,$x); |
1584 | } |
1585 | # $x is now one element to big, so truncate result by removing it |
1586 | splice (@$x,0,1); |
1587 | } |
1588 | else |
1589 | { |
091c87b1 |
1590 | # trial computation by starting with 2,4,8,16 etc until we overstep |
3a427a11 |
1591 | my $step; |
091c87b1 |
1592 | my $trial = _two(); |
1593 | |
3a427a11 |
1594 | # while still to do more than X steps |
1595 | do |
091c87b1 |
1596 | { |
3a427a11 |
1597 | $step = _two(); |
1598 | while (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) < 0) |
1599 | { |
1600 | _mul ($c, $step, [2]); |
1601 | _add ($c, $trial, $step); |
1602 | } |
1603 | |
1604 | # hit exactly? |
1605 | if (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) == 0) |
1606 | { |
1607 | @$x = @$trial; # make copy while preserving ref to $x |
1608 | return $x; |
1609 | } |
1610 | # overstepped, so go back on step |
1611 | _sub($c, $trial, $step); |
1612 | } while (scalar @$step > 1 || $step->[0] > 128); |
091c87b1 |
1613 | |
3a427a11 |
1614 | # reset step to 2 |
1615 | $step = _two(); |
091c87b1 |
1616 | # add two, because $trial cannot be exactly the result (otherwise we would |
1617 | # alrady have found it) |
1618 | _add($c, $trial, $step); |
1619 | |
3a427a11 |
1620 | # and now add more and more (2,4,6,8,10 etc) |
1621 | while (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) < 0) |
1622 | { |
1623 | _add ($c, $trial, $step); |
1624 | } |
091c87b1 |
1625 | |
1626 | # hit not exactly? (overstepped) |
091c87b1 |
1627 | if (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) > 0) |
1628 | { |
1629 | _dec($c,$trial); |
1630 | } |
3a427a11 |
1631 | |
1632 | # hit not exactly? (overstepped) |
1633 | # 80 too small, 81 slightly too big, 82 too big |
091c87b1 |
1634 | if (_acmp($c, _pow($c, _copy($c, $trial), $n), $x) > 0) |
1635 | { |
3a427a11 |
1636 | _dec ($c, $trial); |
091c87b1 |
1637 | } |
3a427a11 |
1638 | |
091c87b1 |
1639 | @$x = @$trial; # make copy while preserving ref to $x |
1640 | return $x; |
c38b2de2 |
1641 | } |
990fb837 |
1642 | $x; |
1643 | } |
1644 | |
394e6ffb |
1645 | ############################################################################## |
1646 | # binary stuff |
0716bf9b |
1647 | |
394e6ffb |
1648 | sub _and |
1649 | { |
1650 | my ($c,$x,$y) = @_; |
0716bf9b |
1651 | |
394e6ffb |
1652 | # the shortcut makes equal, large numbers _really_ fast, and makes only a |
1653 | # very small performance drop for small numbers (e.g. something with less |
1654 | # than 32 bit) Since we optimize for large numbers, this is enabled. |
1655 | return $x if _acmp($c,$x,$y) == 0; # shortcut |
0716bf9b |
1656 | |
394e6ffb |
1657 | my $m = _one(); my ($xr,$yr); |
1658 | my $mask = $AND_MASK; |
1659 | |
1660 | my $x1 = $x; |
1661 | my $y1 = _copy($c,$y); # make copy |
1662 | $x = _zero(); |
1663 | my ($b,$xrr,$yrr); |
1664 | use integer; |
1665 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
1666 | { |
1667 | ($x1, $xr) = _div($c,$x1,$mask); |
1668 | ($y1, $yr) = _div($c,$y1,$mask); |
1669 | |
1670 | # make ints() from $xr, $yr |
9b924220 |
1671 | # this is when the AND_BITS are greater than $BASE and is slower for |
394e6ffb |
1672 | # small (<256 bits) numbers, but faster for large numbers. Disabled |
1673 | # due to KISS principle |
1674 | |
1675 | # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
1676 | # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
9b924220 |
1677 | # _add($c,$x, _mul($c, _new( $c, ($xrr & $yrr) ), $m) ); |
394e6ffb |
1678 | |
61f5c3f5 |
1679 | # 0+ due to '&' doesn't work in strings |
1680 | _add($c,$x, _mul($c, [ 0+$xr->[0] & 0+$yr->[0] ], $m) ); |
394e6ffb |
1681 | _mul($c,$m,$mask); |
1682 | } |
1683 | $x; |
0716bf9b |
1684 | } |
1685 | |
394e6ffb |
1686 | sub _xor |
0716bf9b |
1687 | { |
394e6ffb |
1688 | my ($c,$x,$y) = @_; |
1689 | |
1690 | return _zero() if _acmp($c,$x,$y) == 0; # shortcut (see -and) |
1691 | |
1692 | my $m = _one(); my ($xr,$yr); |
1693 | my $mask = $XOR_MASK; |
1694 | |
1695 | my $x1 = $x; |
1696 | my $y1 = _copy($c,$y); # make copy |
1697 | $x = _zero(); |
1698 | my ($b,$xrr,$yrr); |
1699 | use integer; |
1700 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
0716bf9b |
1701 | { |
394e6ffb |
1702 | ($x1, $xr) = _div($c,$x1,$mask); |
1703 | ($y1, $yr) = _div($c,$y1,$mask); |
1704 | # make ints() from $xr, $yr (see _and()) |
1705 | #$b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
1706 | #$b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
9b924220 |
1707 | #_add($c,$x, _mul($c, _new( $c, ($xrr ^ $yrr) ), $m) ); |
61f5c3f5 |
1708 | |
1709 | # 0+ due to '^' doesn't work in strings |
1710 | _add($c,$x, _mul($c, [ 0+$xr->[0] ^ 0+$yr->[0] ], $m) ); |
394e6ffb |
1711 | _mul($c,$m,$mask); |
0716bf9b |
1712 | } |
394e6ffb |
1713 | # the loop stops when the shorter of the two numbers is exhausted |
1714 | # the remainder of the longer one will survive bit-by-bit, so we simple |
1715 | # multiply-add it in |
1716 | _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1); |
1717 | _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1); |
1718 | |
1719 | $x; |
0716bf9b |
1720 | } |
1721 | |
394e6ffb |
1722 | sub _or |
0716bf9b |
1723 | { |
394e6ffb |
1724 | my ($c,$x,$y) = @_; |
0716bf9b |
1725 | |
394e6ffb |
1726 | return $x if _acmp($c,$x,$y) == 0; # shortcut (see _and) |
0716bf9b |
1727 | |
394e6ffb |
1728 | my $m = _one(); my ($xr,$yr); |
1729 | my $mask = $OR_MASK; |
0716bf9b |
1730 | |
394e6ffb |
1731 | my $x1 = $x; |
1732 | my $y1 = _copy($c,$y); # make copy |
1733 | $x = _zero(); |
1734 | my ($b,$xrr,$yrr); |
1735 | use integer; |
1736 | while (!_is_zero($c,$x1) && !_is_zero($c,$y1)) |
1737 | { |
1738 | ($x1, $xr) = _div($c,$x1,$mask); |
1739 | ($y1, $yr) = _div($c,$y1,$mask); |
1740 | # make ints() from $xr, $yr (see _and()) |
1741 | # $b = 1; $xrr = 0; foreach (@$xr) { $xrr += $_ * $b; $b *= $BASE; } |
1742 | # $b = 1; $yrr = 0; foreach (@$yr) { $yrr += $_ * $b; $b *= $BASE; } |
9b924220 |
1743 | # _add($c,$x, _mul($c, _new( $c, ($xrr | $yrr) ), $m) ); |
394e6ffb |
1744 | |
61f5c3f5 |
1745 | # 0+ due to '|' doesn't work in strings |
1746 | _add($c,$x, _mul($c, [ 0+$xr->[0] | 0+$yr->[0] ], $m) ); |
394e6ffb |
1747 | _mul($c,$m,$mask); |
1748 | } |
1749 | # the loop stops when the shorter of the two numbers is exhausted |
1750 | # the remainder of the longer one will survive bit-by-bit, so we simple |
1751 | # multiply-add it in |
1752 | _add($c,$x, _mul($c, $x1, $m) ) if !_is_zero($c,$x1); |
1753 | _add($c,$x, _mul($c, $y1, $m) ) if !_is_zero($c,$y1); |
1754 | |
1755 | $x; |
0716bf9b |
1756 | } |
1757 | |
61f5c3f5 |
1758 | sub _as_hex |
1759 | { |
1760 | # convert a decimal number to hex (ref to array, return ref to string) |
1761 | my ($c,$x) = @_; |
1762 | |
091c87b1 |
1763 | # fit's into one element (handle also 0x0 case) |
990fb837 |
1764 | if (@$x == 1) |
1765 | { |
091c87b1 |
1766 | my $t = sprintf("0x%x",$x->[0]); |
9b924220 |
1767 | return $t; |
990fb837 |
1768 | } |
1769 | |
61f5c3f5 |
1770 | my $x1 = _copy($c,$x); |
1771 | |
1772 | my $es = ''; |
1ddff52a |
1773 | my ($xr, $h, $x10000); |
1774 | if ($] >= 5.006) |
1775 | { |
1776 | $x10000 = [ 0x10000 ]; $h = 'h4'; |
1777 | } |
1778 | else |
1779 | { |
1780 | $x10000 = [ 0x1000 ]; $h = 'h3'; |
1781 | } |
091c87b1 |
1782 | # while (! _is_zero($c,$x1)) |
1783 | while (@$x1 != 1 || $x1->[0] != 0) # _is_zero() |
61f5c3f5 |
1784 | { |
1785 | ($x1, $xr) = _div($c,$x1,$x10000); |
091c87b1 |
1786 | $es .= unpack($h,pack('v',$xr->[0])); # XXX TODO: why pack('v',...)? |
61f5c3f5 |
1787 | } |
1788 | $es = reverse $es; |
1789 | $es =~ s/^[0]+//; # strip leading zeros |
1790 | $es = '0x' . $es; |
9b924220 |
1791 | $es; |
61f5c3f5 |
1792 | } |
1793 | |
1794 | sub _as_bin |
1795 | { |
1796 | # convert a decimal number to bin (ref to array, return ref to string) |
1797 | my ($c,$x) = @_; |
1798 | |
091c87b1 |
1799 | # fit's into one element (and Perl recent enough), handle also 0b0 case |
1800 | # handle zero case for older Perls |
1801 | if ($] <= 5.005 && @$x == 1 && $x->[0] == 0) |
1802 | { |
9b924220 |
1803 | my $t = '0b0'; return $t; |
091c87b1 |
1804 | } |
1805 | if (@$x == 1 && $] >= 5.006) |
990fb837 |
1806 | { |
091c87b1 |
1807 | my $t = sprintf("0b%b",$x->[0]); |
9b924220 |
1808 | return $t; |
990fb837 |
1809 | } |
61f5c3f5 |
1810 | my $x1 = _copy($c,$x); |
1811 | |
1812 | my $es = ''; |
1ddff52a |
1813 | my ($xr, $b, $x10000); |
1814 | if ($] >= 5.006) |
1815 | { |
1816 | $x10000 = [ 0x10000 ]; $b = 'b16'; |
1817 | } |
1818 | else |
1819 | { |
1820 | $x10000 = [ 0x1000 ]; $b = 'b12'; |
1821 | } |
091c87b1 |
1822 | # while (! _is_zero($c,$x1)) |
1823 | while (!(@$x1 == 1 && $x1->[0] == 0)) # _is_zero() |
61f5c3f5 |
1824 | { |
1825 | ($x1, $xr) = _div($c,$x1,$x10000); |
091c87b1 |
1826 | $es .= unpack($b,pack('v',$xr->[0])); # XXX TODO: why pack('v',...)? |
1827 | # $es .= unpack($b,$xr->[0]); |
61f5c3f5 |
1828 | } |
1829 | $es = reverse $es; |
1830 | $es =~ s/^[0]+//; # strip leading zeros |
1831 | $es = '0b' . $es; |
9b924220 |
1832 | $es; |
61f5c3f5 |
1833 | } |
1834 | |
394e6ffb |
1835 | sub _from_hex |
0716bf9b |
1836 | { |
394e6ffb |
1837 | # convert a hex number to decimal (ref to string, return ref to array) |
1838 | my ($c,$hs) = @_; |
0716bf9b |
1839 | |
394e6ffb |
1840 | my $mul = _one(); |
1841 | my $m = [ 0x10000 ]; # 16 bit at a time |
1842 | my $x = _zero(); |
0716bf9b |
1843 | |
9b924220 |
1844 | my $len = length($hs)-2; |
394e6ffb |
1845 | $len = int($len/4); # 4-digit parts, w/o '0x' |
1846 | my $val; my $i = -4; |
1847 | while ($len >= 0) |
1848 | { |
9b924220 |
1849 | $val = substr($hs,$i,4); |
394e6ffb |
1850 | $val =~ s/^[+-]?0x// if $len == 0; # for last part only because |
1851 | $val = hex($val); # hex does not like wrong chars |
1852 | $i -= 4; $len --; |
1853 | _add ($c, $x, _mul ($c, [ $val ], $mul ) ) if $val != 0; |
1854 | _mul ($c, $mul, $m ) if $len >= 0; # skip last mul |
1855 | } |
1856 | $x; |
1857 | } |
1858 | |
1859 | sub _from_bin |
0716bf9b |
1860 | { |
394e6ffb |
1861 | # convert a hex number to decimal (ref to string, return ref to array) |
1862 | my ($c,$bs) = @_; |
0716bf9b |
1863 | |
091c87b1 |
1864 | # instead of converting X (8) bit at a time, it is faster to "convert" the |
13a12e00 |
1865 | # number to hex, and then call _from_hex. |
1866 | |
9b924220 |
1867 | my $hs = $bs; |
13a12e00 |
1868 | $hs =~ s/^[+-]?0b//; # remove sign and 0b |
1869 | my $l = length($hs); # bits |
1870 | $hs = '0' x (8-($l % 8)) . $hs if ($l % 8) != 0; # padd left side w/ 0 |
1871 | my $h = unpack('H*', pack ('B*', $hs)); # repack as hex |
091c87b1 |
1872 | |
9b924220 |
1873 | $c->_from_hex('0x'.$h); |
0716bf9b |
1874 | } |
1875 | |
07d34614 |
1876 | ############################################################################## |
1877 | # special modulus functions |
1878 | |
56d9de68 |
1879 | sub _modinv |
d614cd8b |
1880 | { |
56d9de68 |
1881 | # modular inverse |
1882 | my ($c,$x,$y) = @_; |
1ddff52a |
1883 | |
56d9de68 |
1884 | my $u = _zero($c); my $u1 = _one($c); |
1885 | my $a = _copy($c,$y); my $b = _copy($c,$x); |
1ddff52a |
1886 | |
1887 | # Euclid's Algorithm for bgcd(), only that we calc bgcd() ($a) and the |
56d9de68 |
1888 | # result ($u) at the same time. See comments in BigInt for why this works. |
1889 | my $q; |
1890 | ($a, $q, $b) = ($b, _div($c,$a,$b)); # step 1 |
1891 | my $sign = 1; |
1ddff52a |
1892 | while (!_is_zero($c,$b)) |
1893 | { |
56d9de68 |
1894 | my $t = _add($c, # step 2: |
1895 | _mul($c,_copy($c,$u1), $q) , # t = u1 * q |
1896 | $u ); # + u |
1897 | $u = $u1; # u = u1, u1 = t |
1898 | $u1 = $t; |
1899 | $sign = -$sign; |
1900 | ($a, $q, $b) = ($b, _div($c,$a,$b)); # step 1 |
1ddff52a |
1901 | } |
1902 | |
1903 | # if the gcd is not 1, then return NaN |
56d9de68 |
1904 | return (undef,undef) unless _is_one($c,$a); |
1905 | |
1906 | $sign = $sign == 1 ? '+' : '-'; |
1907 | ($u1,$sign); |
d614cd8b |
1908 | } |
1909 | |
1910 | sub _modpow |
1911 | { |
1912 | # modulus of power ($x ** $y) % $z |
07d34614 |
1913 | my ($c,$num,$exp,$mod) = @_; |
1914 | |
1915 | # in the trivial case, |
1916 | if (_is_one($c,$mod)) |
1917 | { |
1918 | splice @$num,0,1; $num->[0] = 0; |
1919 | return $num; |
1920 | } |
1921 | if ((scalar @$num == 1) && (($num->[0] == 0) || ($num->[0] == 1))) |
1922 | { |
1923 | $num->[0] = 1; |
1924 | return $num; |
1925 | } |
1ddff52a |
1926 | |
1927 | # $num = _mod($c,$num,$mod); # this does not make it faster |
07d34614 |
1928 | |
1929 | my $acc = _copy($c,$num); my $t = _one(); |
1930 | |
9b924220 |
1931 | my $expbin = _as_bin($c,$exp); $expbin =~ s/^0b//; |
1ddff52a |
1932 | my $len = length($expbin); |
1933 | while (--$len >= 0) |
07d34614 |
1934 | { |
1ddff52a |
1935 | if ( substr($expbin,$len,1) eq '1') # is_odd |
07d34614 |
1936 | { |
1937 | _mul($c,$t,$acc); |
1938 | $t = _mod($c,$t,$mod); |
1939 | } |
1940 | _mul($c,$acc,$acc); |
1941 | $acc = _mod($c,$acc,$mod); |
07d34614 |
1942 | } |
1943 | @$num = @$t; |
1944 | $num; |
d614cd8b |
1945 | } |
1946 | |
9b924220 |
1947 | sub _gcd |
1948 | { |
1949 | # greatest common divisor |
1950 | my ($c,$x,$y) = @_; |
1951 | |
1952 | while (! _is_zero($c,$y)) |
1953 | { |
1954 | my $t = _copy($c,$y); |
1955 | $y = _mod($c, $x, $y); |
1956 | $x = $t; |
1957 | } |
1958 | $x; |
1959 | } |
1960 | |
394e6ffb |
1961 | ############################################################################## |
1962 | ############################################################################## |
1963 | |
0716bf9b |
1964 | 1; |
1965 | __END__ |
1966 | |
1967 | =head1 NAME |
1968 | |
1969 | Math::BigInt::Calc - Pure Perl module to support Math::BigInt |
1970 | |
1971 | =head1 SYNOPSIS |
1972 | |
ee15d750 |
1973 | Provides support for big integer calculations. Not intended to be used by other |
091c87b1 |
1974 | modules. Other modules which sport the same functions can also be used to support |
1975 | Math::BigInt, like Math::BigInt::GMP or Math::BigInt::Pari. |
0716bf9b |
1976 | |
1977 | =head1 DESCRIPTION |
1978 | |
027dc388 |
1979 | In order to allow for multiple big integer libraries, Math::BigInt was |
1980 | rewritten to use library modules for core math routines. Any module which |
1981 | follows the same API as this can be used instead by using the following: |
0716bf9b |
1982 | |
ee15d750 |
1983 | use Math::BigInt lib => 'libname'; |
0716bf9b |
1984 | |
027dc388 |
1985 | 'libname' is either the long name ('Math::BigInt::Pari'), or only the short |
1986 | version like 'Pari'. |
1987 | |
990fb837 |
1988 | =head1 STORAGE |
1989 | |
1990 | =head1 METHODS |
0716bf9b |
1991 | |
027dc388 |
1992 | The following functions MUST be defined in order to support the use by |
9b924220 |
1993 | Math::BigInt v1.70 or later: |
0716bf9b |
1994 | |
9b924220 |
1995 | api_version() return API version, minimum 1 for v1.70 |
0716bf9b |
1996 | _new(string) return ref to new object from ref to decimal string |
1997 | _zero() return a new object with value 0 |
1998 | _one() return a new object with value 1 |
9b924220 |
1999 | _two() return a new object with value 2 |
2000 | _ten() return a new object with value 10 |
0716bf9b |
2001 | |
2002 | _str(obj) return ref to a string representing the object |
2003 | _num(obj) returns a Perl integer/floating point number |
2004 | NOTE: because of Perl numeric notation defaults, |
2005 | the _num'ified obj may lose accuracy due to |
2006 | machine-dependend floating point size limitations |
2007 | |
2008 | _add(obj,obj) Simple addition of two objects |
2009 | _mul(obj,obj) Multiplication of two objects |
2010 | _div(obj,obj) Division of the 1st object by the 2nd |
b22b3e31 |
2011 | In list context, returns (result,remainder). |
2012 | NOTE: this is integer math, so no |
2013 | fractional part will be returned. |
990fb837 |
2014 | The second operand will be not be 0, so no need to |
2015 | check for that. |
b22b3e31 |
2016 | _sub(obj,obj) Simple subtraction of 1 object from another |
0716bf9b |
2017 | a third, optional parameter indicates that the params |
2018 | are swapped. In this case, the first param needs to |
2019 | be preserved, while you can destroy the second. |
2020 | sub (x,y,1) => return x - y and keep x intact! |
e745a66c |
2021 | _dec(obj) decrement object by one (input is garant. to be > 0) |
2022 | _inc(obj) increment object by one |
2023 | |
0716bf9b |
2024 | |
2025 | _acmp(obj,obj) <=> operator for objects (return -1, 0 or 1) |
2026 | |
2027 | _len(obj) returns count of the decimal digits of the object |
2028 | _digit(obj,n) returns the n'th decimal digit of object |
2029 | |
9b924220 |
2030 | _is_one(obj) return true if argument is 1 |
2031 | _is_two(obj) return true if argument is 2 |
2032 | _is_ten(obj) return true if argument is 10 |
0716bf9b |
2033 | _is_zero(obj) return true if argument is 0 |
2034 | _is_even(obj) return true if argument is even (0,2,4,6..) |
2035 | _is_odd(obj) return true if argument is odd (1,3,5,7..) |
2036 | |
2037 | _copy return a ref to a true copy of the object |
2038 | |
2039 | _check(obj) check whether internal representation is still intact |
2040 | return 0 for ok, otherwise error message as string |
2041 | |
0716bf9b |
2042 | _from_hex(str) return ref to new object from ref to hexadecimal string |
2043 | _from_bin(str) return ref to new object from ref to binary string |
2044 | |
9b924220 |
2045 | _as_hex(str) return string containing the value as |
ee15d750 |
2046 | unsigned hex string, with the '0x' prepended. |
2047 | Leading zeros must be stripped. |
2048 | _as_bin(str) Like as_hex, only as binary string containing only |
2049 | zeros and ones. Leading zeros must be stripped and a |
2050 | '0b' must be prepended. |
2051 | |
0716bf9b |
2052 | _rsft(obj,N,B) shift object in base B by N 'digits' right |
2053 | _lsft(obj,N,B) shift object in base B by N 'digits' left |
2054 | |
2055 | _xor(obj1,obj2) XOR (bit-wise) object 1 with object 2 |
dccbb853 |
2056 | Note: XOR, AND and OR pad with zeros if size mismatches |
0716bf9b |
2057 | _and(obj1,obj2) AND (bit-wise) object 1 with object 2 |
2058 | _or(obj1,obj2) OR (bit-wise) object 1 with object 2 |
2059 | |
dccbb853 |
2060 | _mod(obj,obj) Return remainder of div of the 1st by the 2nd object |
990fb837 |
2061 | _sqrt(obj) return the square root of object (truncated to int) |
2062 | _root(obj) return the n'th (n >= 3) root of obj (truncated to int) |
b3abae2a |
2063 | _fac(obj) return factorial of object 1 (1*2*3*4..) |
0716bf9b |
2064 | _pow(obj,obj) return object 1 to the power of object 2 |
b282a552 |
2065 | return undef for NaN |
b22b3e31 |
2066 | _zeros(obj) return number of trailing decimal zeros |
d614cd8b |
2067 | _modinv return inverse modulus |
2068 | _modpow return modulus of power ($x ** $y) % $z |
091c87b1 |
2069 | _log_int(X,N) calculate integer log() of X in base N |
2070 | X >= 0, N >= 0 (return undef for NaN) |
8df1e0a2 |
2071 | returns (RESULT, EXACT) where EXACT is: |
2072 | 1 : result is exactly RESULT |
2073 | 0 : result was truncated to RESULT |
2074 | undef : unknown whether result is exactly RESULT |
9b924220 |
2075 | _gcd(obj,obj) return Greatest Common Divisor of two objects |
2076 | |
2077 | The following functions are optional, and can be defined if the underlying lib |
2078 | has a fast way to do them. If undefined, Math::BigInt will use pure Perl (hence |
2079 | slow) fallback routines to emulate these: |
2080 | |
2081 | _signed_or |
2082 | _signed_and |
2083 | _signed_xor |
2084 | |
0716bf9b |
2085 | |
b22b3e31 |
2086 | Input strings come in as unsigned but with prefix (i.e. as '123', '0xabc' |
0716bf9b |
2087 | or '0b1101'). |
2088 | |
990fb837 |
2089 | So the library needs only to deal with unsigned big integers. Testing of input |
2090 | parameter validity is done by the caller, so you need not worry about |
2091 | underflow (f.i. in C<_sub()>, C<_dec()>) nor about division by zero or similar |
2092 | cases. |
574bacfe |
2093 | |
2094 | The first parameter can be modified, that includes the possibility that you |
2095 | return a reference to a completely different object instead. Although keeping |
dccbb853 |
2096 | the reference and just changing it's contents is prefered over creating and |
2097 | returning a different reference. |
574bacfe |
2098 | |
990fb837 |
2099 | Return values are always references to objects, strings, or true/false for |
2100 | comparisation routines. |
2101 | |
574bacfe |
2102 | =head1 WRAP YOUR OWN |
2103 | |
2104 | If you want to port your own favourite c-lib for big numbers to the |
2105 | Math::BigInt interface, you can take any of the already existing modules as |
2106 | a rough guideline. You should really wrap up the latest BigInt and BigFloat |
bd05a461 |
2107 | testsuites with your module, and replace in them any of the following: |
574bacfe |
2108 | |
2109 | use Math::BigInt; |
2110 | |
bd05a461 |
2111 | by this: |
574bacfe |
2112 | |
2113 | use Math::BigInt lib => 'yourlib'; |
2114 | |
2115 | This way you ensure that your library really works 100% within Math::BigInt. |
0716bf9b |
2116 | |
2117 | =head1 LICENSE |
2118 | |
2119 | This program is free software; you may redistribute it and/or modify it under |
2120 | the same terms as Perl itself. |
2121 | |
2122 | =head1 AUTHORS |
2123 | |
2124 | Original math code by Mark Biggar, rewritten by Tels L<http://bloodgate.com/> |
990fb837 |
2125 | in late 2000. |
0716bf9b |
2126 | Seperated from BigInt and shaped API with the help of John Peacock. |
091c87b1 |
2127 | Fixed, sped-up and enhanced by Tels http://bloodgate.com 2001-2003. |
9b924220 |
2128 | Further streamlining (api_version 1) by Tels 2004. |
0716bf9b |
2129 | |
2130 | =head1 SEE ALSO |
2131 | |
ee15d750 |
2132 | L<Math::BigInt>, L<Math::BigFloat>, L<Math::BigInt::BitVect>, |
990fb837 |
2133 | L<Math::BigInt::GMP>, L<Math::BigInt::FastCalc> and L<Math::BigInt::Pari>. |
0716bf9b |
2134 | |
2135 | =cut |