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0716bf9b |
1 | package Math::BigInt::Calc; |
2 | |
3 | use 5.005; |
4 | use strict; |
5 | use warnings; |
6 | |
7 | require Exporter; |
8 | |
9 | use vars qw/ @ISA @EXPORT $VERSION/; |
10 | @ISA = qw(Exporter); |
11 | |
12 | @EXPORT = qw( |
13 | _add _mul _div _mod _sub |
14 | _new |
15 | _str _num _acmp _len |
16 | _digit |
17 | _is_zero _is_one |
18 | _is_even _is_odd |
19 | _check _zero _one _copy _zeros |
20 | ); |
21 | $VERSION = '0.06'; |
22 | |
23 | # Package to store unsigned big integers in decimal and do math with them |
24 | |
25 | # Internally the numbers are stored in an array with at least 1 element, no |
26 | # leading zero parts (except the first) and in base 100000 |
27 | |
28 | # todo: |
29 | # - fully remove funky $# stuff (maybe) |
30 | # - use integer; vs 1e7 as base |
31 | |
32 | # USE_MUL: due to problems on certain os (os390, posix-bc) "* 1e-5" is used |
33 | # instead of "/ 1e5" at some places, (marked with USE_MUL). But instead of |
34 | # using the reverse only on problematic machines, I used it everytime to avoid |
b22b3e31 |
35 | # the costly comparisons. This _should_ work everywhere. Thanx Peter Prymmer |
0716bf9b |
36 | |
37 | ############################################################################## |
38 | # global constants, flags and accessory |
39 | |
40 | # constants for easier life |
41 | my $nan = 'NaN'; |
42 | my $BASE_LEN = 5; |
43 | my $BASE = int("1e".$BASE_LEN); # var for trying to change it to 1e7 |
44 | my $RBASE = 1e-5; # see USE_MUL |
45 | my $class = 'Math::BigInt::Calc'; |
46 | |
47 | ############################################################################## |
48 | # create objects from various representations |
49 | |
50 | sub _new |
51 | { |
52 | # (string) return ref to num_array |
53 | # Convert a number from string format to internal base 100000 format. |
54 | # Assumes normalized value as input. |
55 | shift @_ if $_[0] eq $class; |
56 | my $d = shift; |
57 | # print "_new $d $$d\n"; |
58 | my $il = CORE::length($$d)-1; |
59 | # these leaves '00000' instead of int 0 and will be corrected after any op |
60 | return [ reverse(unpack("a" . ($il%5+1) . ("a5" x ($il/5)), $$d)) ]; |
61 | } |
62 | |
63 | sub _zero |
64 | { |
65 | # create a zero |
66 | return [ 0 ]; |
67 | } |
68 | |
69 | sub _one |
70 | { |
71 | # create a one |
72 | return [ 1 ]; |
73 | } |
74 | |
75 | sub _copy |
76 | { |
77 | shift @_ if $_[0] eq $class; |
78 | my $x = shift; |
79 | return [ @$x ]; |
80 | } |
81 | |
82 | ############################################################################## |
83 | # convert back to string and number |
84 | |
85 | sub _str |
86 | { |
87 | # (ref to BINT) return num_str |
88 | # Convert number from internal base 100000 format to string format. |
89 | # internal format is always normalized (no leading zeros, "-0" => "+0") |
90 | shift @_ if $_[0] eq $class; |
91 | my $ar = shift; |
92 | my $ret = ""; |
93 | my $l = scalar @$ar; # number of parts |
94 | return $nan if $l < 1; # should not happen |
95 | # handle first one different to strip leading zeros from it (there are no |
96 | # leading zero parts in internal representation) |
97 | $l --; $ret .= $ar->[$l]; $l--; |
98 | # Interestingly, the pre-padd method uses more time |
99 | # the old grep variant takes longer (14 to 10 sec) |
100 | while ($l >= 0) |
101 | { |
102 | $ret .= substr('0000'.$ar->[$l],-5); # fastest way I could think of |
103 | $l--; |
104 | } |
105 | return \$ret; |
106 | } |
107 | |
108 | sub _num |
109 | { |
110 | # Make a number (scalar int/float) from a BigInt object |
111 | shift @_ if $_[0] eq $class; |
112 | my $x = shift; |
113 | return $x->[0] if scalar @$x == 1; # below $BASE |
114 | my $fac = 1; |
115 | my $num = 0; |
116 | foreach (@$x) |
117 | { |
118 | $num += $fac*$_; $fac *= $BASE; |
119 | } |
120 | return $num; |
121 | } |
122 | |
123 | ############################################################################## |
124 | # actual math code |
125 | |
126 | sub _add |
127 | { |
128 | # (ref to int_num_array, ref to int_num_array) |
129 | # routine to add two base 1e5 numbers |
130 | # stolen from Knuth Vol 2 Algorithm A pg 231 |
b22b3e31 |
131 | # there are separate routines to add and sub as per Knuth pg 233 |
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132 | # This routine clobbers up array x, but not y. |
133 | |
134 | shift @_ if $_[0] eq $class; |
135 | my ($x,$y) = @_; |
136 | |
137 | # for each in Y, add Y to X and carry. If after that, something is left in |
138 | # X, foreach in X add carry to X and then return X, carry |
139 | # Trades one "$j++" for having to shift arrays, $j could be made integer |
b22b3e31 |
140 | # but this would impose a limit to number-length of 2**32. |
0716bf9b |
141 | my $i; my $car = 0; my $j = 0; |
142 | for $i (@$y) |
143 | { |
144 | $x->[$j] -= $BASE |
145 | if $car = (($x->[$j] += $i + $car) >= $BASE) ? 1 : 0; |
146 | $j++; |
147 | } |
148 | while ($car != 0) |
149 | { |
150 | $x->[$j] -= $BASE if $car = (($x->[$j] += $car) >= $BASE) ? 1 : 0; $j++; |
151 | } |
152 | return $x; |
153 | } |
154 | |
155 | sub _sub |
156 | { |
157 | # (ref to int_num_array, ref to int_num_array) |
158 | # subtract base 1e5 numbers -- stolen from Knuth Vol 2 pg 232, $x > $y |
b22b3e31 |
159 | # subtract Y from X (X is always greater/equal!) by modifying x in place |
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160 | shift @_ if $_[0] eq $class; |
161 | my ($sx,$sy,$s) = @_; |
162 | |
163 | my $car = 0; my $i; my $j = 0; |
164 | if (!$s) |
165 | { |
166 | #print "case 2\n"; |
167 | for $i (@$sx) |
168 | { |
169 | last unless defined $sy->[$j] || $car; |
170 | #print "x: $i y: $sy->[$j] c: $car\n"; |
171 | $i += $BASE if $car = (($i -= ($sy->[$j] || 0) + $car) < 0); $j++; |
172 | #print "x: $i y: $sy->[$j-1] c: $car\n"; |
173 | } |
174 | # might leave leading zeros, so fix that |
175 | __strip_zeros($sx); |
176 | return $sx; |
177 | } |
178 | else |
179 | { |
180 | #print "case 1 (swap)\n"; |
181 | for $i (@$sx) |
182 | { |
183 | last unless defined $sy->[$j] || $car; |
184 | #print "$sy->[$j] $i $car => $sx->[$j]\n"; |
185 | $sy->[$j] += $BASE |
186 | if $car = (($sy->[$j] = $i-($sy->[$j]||0) - $car) < 0); |
187 | #print "$sy->[$j] $i $car => $sy->[$j]\n"; |
188 | $j++; |
189 | } |
190 | # might leave leading zeros, so fix that |
191 | __strip_zeros($sy); |
192 | return $sy; |
193 | } |
194 | } |
195 | |
196 | sub _mul |
197 | { |
198 | # (BINT, BINT) return nothing |
199 | # multiply two numbers in internal representation |
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200 | # modifies first arg, second need not be different from first |
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201 | shift @_ if $_[0] eq $class; |
202 | my ($xv,$yv) = @_; |
203 | |
204 | my @prod = (); my ($prod,$car,$cty,$xi,$yi); |
205 | # since multiplying $x with $x fails, make copy in this case |
206 | $yv = [@$xv] if "$xv" eq "$yv"; |
207 | # looping through @$y if $xi == 0 is silly! optimize it! |
208 | for $xi (@$xv) |
209 | { |
210 | $car = 0; $cty = 0; |
211 | for $yi (@$yv) |
212 | { |
213 | $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
214 | $prod[$cty++] = |
215 | $prod - ($car = int($prod * 1e-5)) * $BASE; # see USE_MUL |
216 | } |
217 | $prod[$cty] += $car if $car; # need really to check for 0? |
218 | $xi = shift @prod; |
219 | } |
220 | # for $xi (@$xv) |
221 | # { |
222 | # $car = 0; $cty = 0; |
223 | # # looping through this if $xi == 0 is silly! optimize it! |
224 | # if (($xi||0) != 0) |
225 | # { |
226 | # for $yi (@$yv) |
227 | # { |
228 | # $prod = $prod[$cty]; $prod += ($car + $xi * $yi); # no ||0 here |
229 | # $prod[$cty++] = |
230 | # $prod - ($car = int($prod * 1e-5)) * $BASE; # see USE_MUL |
231 | # } |
232 | # } |
233 | # $prod[$cty] += $car if $car; # need really to check for 0? |
234 | # $xi = shift @prod; |
235 | # } |
236 | push @$xv, @prod; |
237 | __strip_zeros($xv); |
238 | # normalize (handled last to save check for $y->is_zero() |
239 | return $xv; |
240 | } |
241 | |
242 | sub _div |
243 | { |
b22b3e31 |
244 | # ref to array, ref to array, modify first array and return remainder if |
0716bf9b |
245 | # in list context |
b22b3e31 |
246 | # no longer handles sign |
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247 | shift @_ if $_[0] eq $class; |
248 | my ($x,$yorg) = @_; |
249 | my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1); |
250 | |
251 | my (@d,$tmp,$q,$u2,$u1,$u0); |
252 | |
253 | $car = $bar = $prd = 0; |
254 | |
255 | my $y = [ @$yorg ]; |
256 | if (($dd = int($BASE/($y->[-1]+1))) != 1) |
257 | { |
258 | for $xi (@$x) |
259 | { |
260 | $xi = $xi * $dd + $car; |
261 | $xi -= ($car = int($xi * $RBASE)) * $BASE; # see USE_MUL |
262 | } |
263 | push(@$x, $car); $car = 0; |
264 | for $yi (@$y) |
265 | { |
266 | $yi = $yi * $dd + $car; |
267 | $yi -= ($car = int($yi * $RBASE)) * $BASE; # see USE_MUL |
268 | } |
269 | } |
270 | else |
271 | { |
272 | push(@$x, 0); |
273 | } |
274 | @q = (); ($v2,$v1) = @$y[-2,-1]; |
275 | $v2 = 0 unless $v2; |
276 | while ($#$x > $#$y) |
277 | { |
278 | ($u2,$u1,$u0) = @$x[-3..-1]; |
279 | $u2 = 0 unless $u2; |
280 | #warn "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" |
281 | # if $v1 == 0; |
282 | $q = (($u0 == $v1) ? 99999 : int(($u0*$BASE+$u1)/$v1)); |
283 | --$q while ($v2*$q > ($u0*1e5+$u1-$q*$v1)*$BASE+$u2); |
284 | if ($q) |
285 | { |
286 | ($car, $bar) = (0,0); |
287 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
288 | { |
289 | $prd = $q * $y->[$yi] + $car; |
290 | $prd -= ($car = int($prd * $RBASE)) * $BASE; # see USE_MUL |
291 | $x->[$xi] += 1e5 if ($bar = (($x->[$xi] -= $prd + $bar) < 0)); |
292 | } |
293 | if ($x->[-1] < $car + $bar) |
294 | { |
295 | $car = 0; --$q; |
296 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
297 | { |
298 | $x->[$xi] -= 1e5 |
299 | if ($car = (($x->[$xi] += $y->[$yi] + $car) > $BASE)); |
300 | } |
301 | } |
302 | } |
303 | pop(@$x); unshift(@q, $q); |
304 | } |
305 | if (wantarray) |
306 | { |
307 | @d = (); |
308 | if ($dd != 1) |
309 | { |
310 | $car = 0; |
311 | for $xi (reverse @$x) |
312 | { |
313 | $prd = $car * $BASE + $xi; |
314 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; # see USE_MUL |
315 | unshift(@d, $tmp); |
316 | } |
317 | } |
318 | else |
319 | { |
320 | @d = @$x; |
321 | } |
322 | @$x = @q; |
323 | __strip_zeros($x); |
324 | __strip_zeros(\@d); |
325 | return ($x,\@d); |
326 | } |
327 | @$x = @q; |
328 | __strip_zeros($x); |
329 | return $x; |
330 | } |
331 | |
332 | ############################################################################## |
333 | # testing |
334 | |
335 | sub _acmp |
336 | { |
337 | # internal absolute post-normalized compare (ignore signs) |
338 | # ref to array, ref to array, return <0, 0, >0 |
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339 | # arrays must have at least one entry; this is not checked for |
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340 | |
341 | shift @_ if $_[0] eq $class; |
342 | my ($cx, $cy) = @_; |
343 | |
344 | #print "$cx $cy\n"; |
345 | my ($i,$a,$x,$y,$k); |
346 | # calculate length based on digits, not parts |
347 | $x = _len($cx); $y = _len($cy); |
348 | # print "length: ",($x-$y),"\n"; |
349 | return $x-$y if ($x - $y); # if different in length |
350 | #print "full compare\n"; |
351 | $i = 0; $a = 0; |
352 | # first way takes 5.49 sec instead of 4.87, but has the early out advantage |
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353 | # so grep is slightly faster, but more inflexible. hm. $_ instead of $k |
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354 | # yields 5.6 instead of 5.5 sec huh? |
355 | # manual way (abort if unequal, good for early ne) |
356 | my $j = scalar @$cx - 1; |
357 | while ($j >= 0) |
358 | { |
359 | # print "$cx->[$j] $cy->[$j] $a",$cx->[$j]-$cy->[$j],"\n"; |
360 | last if ($a = $cx->[$j] - $cy->[$j]); $j--; |
361 | } |
362 | return $a; |
363 | # while it early aborts, it is even slower than the manual variant |
364 | #grep { return $a if ($a = $_ - $cy->[$i++]); } @$cx; |
365 | # grep way, go trough all (bad for early ne) |
366 | #grep { $a = $_ - $cy->[$i++]; } @$cx; |
367 | #return $a; |
368 | } |
369 | |
370 | sub _len |
371 | { |
372 | # computer number of digits in bigint, minus the sign |
b22b3e31 |
373 | # int() because add/sub sometimes leaves strings (like '00005') instead of |
0716bf9b |
374 | # int ('5') in this place, causing length to fail |
375 | shift @_ if $_[0] eq $class; |
376 | my $cx = shift; |
377 | |
378 | return (@$cx-1)*5+length(int($cx->[-1])); |
379 | } |
380 | |
381 | sub _digit |
382 | { |
383 | # return the nth digit, negative values count backward |
384 | # zero is rightmost, so _digit(123,0) will give 3 |
385 | shift @_ if $_[0] eq $class; |
386 | my $x = shift; |
387 | my $n = shift || 0; |
388 | |
389 | my $len = _len($x); |
390 | |
391 | $n = $len+$n if $n < 0; # -1 last, -2 second-to-last |
392 | $n = abs($n); # if negative was too big |
393 | $len--; $n = $len if $n > $len; # n to big? |
394 | |
395 | my $elem = int($n / 5); # which array element |
396 | my $digit = $n % 5; # which digit in this element |
397 | $elem = '0000'.@$x[$elem]; # get element padded with 0's |
398 | return substr($elem,-$digit-1,1); |
399 | } |
400 | |
401 | sub _zeros |
402 | { |
403 | # return amount of trailing zeros in decimal |
404 | # check each array elem in _m for having 0 at end as long as elem == 0 |
405 | # Upon finding a elem != 0, stop |
406 | shift @_ if $_[0] eq $class; |
407 | my $x = shift; |
408 | my $zeros = 0; my $elem; |
409 | foreach my $e (@$x) |
410 | { |
411 | if ($e != 0) |
412 | { |
413 | $elem = "$e"; # preserve x |
414 | $elem =~ s/.*?(0*$)/$1/; # strip anything not zero |
415 | $zeros *= 5; # elems * 5 |
416 | $zeros += CORE::length($elem); # count trailing zeros |
417 | last; # early out |
418 | } |
419 | $zeros ++; # real else branch: 50% slower! |
420 | } |
421 | return $zeros; |
422 | } |
423 | |
424 | ############################################################################## |
425 | # _is_* routines |
426 | |
427 | sub _is_zero |
428 | { |
429 | # return true if arg (BINT or num_str) is zero (array '+', '0') |
430 | shift @_ if $_[0] eq $class; |
431 | my ($x) = shift; |
432 | return (((scalar @$x == 1) && ($x->[0] == 0))) <=> 0; |
433 | } |
434 | |
435 | sub _is_even |
436 | { |
437 | # return true if arg (BINT or num_str) is even |
438 | shift @_ if $_[0] eq $class; |
439 | my ($x) = shift; |
440 | return (!($x->[0] & 1)) <=> 0; |
441 | } |
442 | |
443 | sub _is_odd |
444 | { |
445 | # return true if arg (BINT or num_str) is even |
446 | shift @_ if $_[0] eq $class; |
447 | my ($x) = shift; |
448 | return (($x->[0] & 1)) <=> 0; |
449 | } |
450 | |
451 | sub _is_one |
452 | { |
453 | # return true if arg (BINT or num_str) is one (array '+', '1') |
454 | shift @_ if $_[0] eq $class; |
455 | my ($x) = shift; |
456 | return (scalar @$x == 1) && ($x->[0] == 1) <=> 0; |
457 | } |
458 | |
459 | sub __strip_zeros |
460 | { |
461 | # internal normalization function that strips leading zeros from the array |
462 | # args: ref to array |
463 | #trace(@_); |
464 | shift @_ if $_[0] eq $class; |
465 | my $s = shift; |
466 | |
467 | my $cnt = scalar @$s; # get count of parts |
468 | my $i = $cnt-1; |
469 | #print "strip: cnt $cnt i $i\n"; |
470 | # '0', '3', '4', '0', '0', |
471 | # 0 1 2 3 4 |
472 | # cnt = 5, i = 4 |
473 | # i = 4 |
474 | # i = 3 |
475 | # => fcnt = cnt - i (5-2 => 3, cnt => 5-1 = 4, throw away from 4th pos) |
476 | # >= 1: skip first part (this can be zero) |
477 | while ($i > 0) { last if $s->[$i] != 0; $i--; } |
478 | $i++; splice @$s,$i if ($i < $cnt); # $i cant be 0 |
479 | return $s; |
480 | } |
481 | |
482 | ############################################################################### |
483 | # check routine to test internal state of corruptions |
484 | |
485 | sub _check |
486 | { |
487 | # no checks yet, pull it out from the test suite |
488 | shift @_ if $_[0] eq $class; |
489 | |
490 | my ($x) = shift; |
491 | return "$x is not a reference" if !ref($x); |
492 | |
493 | # are all parts are valid? |
494 | my $i = 0; my $j = scalar @$x; my ($e,$try); |
495 | while ($i < $j) |
496 | { |
497 | $e = $x->[$i]; $e = 'undef' unless defined $e; |
498 | $try = '=~ /^[\+]?[0-9]+\$/; '."($x, $e)"; |
499 | last if $e !~ /^[+]?[0-9]+$/; |
500 | $try = ' < 0 || >= $BASE; '."($x, $e)"; |
501 | last if $e <0 || $e >= $BASE; |
502 | # this test is disabled, since new/bnorm and certain ops (like early out |
503 | # in add/sub) are allowed/expected to leave '00000' in some elements |
504 | #$try = '=~ /^00+/; '."($x, $e)"; |
505 | #last if $e =~ /^00+/; |
506 | $i++; |
507 | } |
508 | return "Illegal part '$e' at pos $i (tested: $try)" if $i < $j; |
509 | return 0; |
510 | } |
511 | |
512 | 1; |
513 | __END__ |
514 | |
515 | =head1 NAME |
516 | |
517 | Math::BigInt::Calc - Pure Perl module to support Math::BigInt |
518 | |
519 | =head1 SYNOPSIS |
520 | |
521 | Provides support for big integer calculations. Not intended |
522 | to be used by other modules. Other modules which export the |
523 | same functions can also be used to support Math::Bigint |
524 | |
525 | =head1 DESCRIPTION |
526 | |
527 | In order to allow for multiple big integer libraries, Math::BigInt |
528 | was rewritten to use library modules for core math routines. Any |
529 | module which follows the same API as this can be used instead by |
530 | using the following call: |
531 | |
532 | use Math::BigInt Calc => BigNum; |
533 | |
534 | =head1 EXPORT |
535 | |
536 | The following functions MUST be exported in order to support |
537 | the use by Math::BigInt: |
538 | |
539 | _new(string) return ref to new object from ref to decimal string |
540 | _zero() return a new object with value 0 |
541 | _one() return a new object with value 1 |
542 | |
543 | _str(obj) return ref to a string representing the object |
544 | _num(obj) returns a Perl integer/floating point number |
545 | NOTE: because of Perl numeric notation defaults, |
546 | the _num'ified obj may lose accuracy due to |
547 | machine-dependend floating point size limitations |
548 | |
549 | _add(obj,obj) Simple addition of two objects |
550 | _mul(obj,obj) Multiplication of two objects |
551 | _div(obj,obj) Division of the 1st object by the 2nd |
b22b3e31 |
552 | In list context, returns (result,remainder). |
553 | NOTE: this is integer math, so no |
554 | fractional part will be returned. |
555 | _sub(obj,obj) Simple subtraction of 1 object from another |
0716bf9b |
556 | a third, optional parameter indicates that the params |
557 | are swapped. In this case, the first param needs to |
558 | be preserved, while you can destroy the second. |
559 | sub (x,y,1) => return x - y and keep x intact! |
560 | |
561 | _acmp(obj,obj) <=> operator for objects (return -1, 0 or 1) |
562 | |
563 | _len(obj) returns count of the decimal digits of the object |
564 | _digit(obj,n) returns the n'th decimal digit of object |
565 | |
566 | _is_one(obj) return true if argument is +1 |
567 | _is_zero(obj) return true if argument is 0 |
568 | _is_even(obj) return true if argument is even (0,2,4,6..) |
569 | _is_odd(obj) return true if argument is odd (1,3,5,7..) |
570 | |
571 | _copy return a ref to a true copy of the object |
572 | |
573 | _check(obj) check whether internal representation is still intact |
574 | return 0 for ok, otherwise error message as string |
575 | |
576 | The following functions are optional, and can be exported if the underlying lib |
577 | has a fast way to do them. If not defined, Math::BigInt will use a pure, but |
b22b3e31 |
578 | slow, Perl function as fallback to emulate these: |
0716bf9b |
579 | |
580 | _from_hex(str) return ref to new object from ref to hexadecimal string |
581 | _from_bin(str) return ref to new object from ref to binary string |
582 | |
583 | _rsft(obj,N,B) shift object in base B by N 'digits' right |
584 | _lsft(obj,N,B) shift object in base B by N 'digits' left |
585 | |
586 | _xor(obj1,obj2) XOR (bit-wise) object 1 with object 2 |
587 | Mote: XOR, AND and OR pad with zeros if size mismatches |
588 | _and(obj1,obj2) AND (bit-wise) object 1 with object 2 |
589 | _or(obj1,obj2) OR (bit-wise) object 1 with object 2 |
590 | |
591 | _sqrt(obj) return the square root of object |
592 | _pow(obj,obj) return object 1 to the power of object 2 |
593 | _gcd(obj,obj) return Greatest Common Divisor of two objects |
594 | |
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595 | _zeros(obj) return number of trailing decimal zeros |
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596 | |
597 | _dec(obj) decrement object by one (input is >= 1) |
598 | _inc(obj) increment object by one |
599 | |
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600 | Input strings come in as unsigned but with prefix (i.e. as '123', '0xabc' |
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601 | or '0b1101'). |
602 | |
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603 | Testing of input parameter validity is done by the caller, so you need not |
604 | worry about underflow (C<_sub()>, C<_dec()>) nor about division by zero or |
605 | similar cases. |
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606 | |
607 | =head1 LICENSE |
608 | |
609 | This program is free software; you may redistribute it and/or modify it under |
610 | the same terms as Perl itself. |
611 | |
612 | =head1 AUTHORS |
613 | |
614 | Original math code by Mark Biggar, rewritten by Tels L<http://bloodgate.com/> |
615 | in late 2000, 2001. |
616 | Seperated from BigInt and shaped API with the help of John Peacock. |
617 | |
618 | =head1 SEE ALSO |
619 | |
620 | L<Math::BigInt>, L<Math::BigFloat>. |
621 | |
622 | =cut |