Commit | Line | Data |
58cde26e |
1 | #!/usr/bin/perl -w |
2 | |
3 | # mark.biggar@TrustedSysLabs.com |
4 | # eay@mincom.com is dead (math::BigInteger) |
5 | # see: http://www.cypherspace.org/~adam/rsa/pureperl.html (contacted c. adam |
6 | # on 2000/11/13 - but email is dead |
7 | |
8 | # todo: |
9 | # - fully remove funky $# stuff (maybe) |
10 | # - use integer; vs 1e7 as base |
11 | # - speed issues (XS? Bit::Vector?) |
12 | # - split out actual math code to Math::BigNumber |
13 | |
14 | # Qs: what exactly happens on numify of HUGE numbers? overflow? |
15 | # $a = -$a is much slower (making copy of $a) than $a->bneg(), hm!? |
16 | # (copy_on_write will help there, but that is not yet implemented) |
17 | |
18 | # The following hash values are used: |
19 | # value: the internal array, base 100000 |
20 | # sign : +,-,NaN,+inf,-inf |
21 | # _a : accuracy |
22 | # _p : precision |
23 | # _cow : copy on write: number of objects that share the data (NRY) |
24 | # Internally the numbers are stored in an array with at least 1 element, no |
25 | # leading zero parts (except the first) and in base 100000 |
26 | |
27 | # USE_MUL: due to problems on certain os (os390, posix-bc) "* 1e-5" is used |
28 | # instead of "/ 1e5" at some places, (marked with USE_MUL). But instead of |
29 | # using the reverse only on problematic machines, I used it everytime to avoid |
30 | # the costly comparisations. This _should_ work everywhere. Thanx Peter Prymmer |
b4f14daa |
31 | |
58cde26e |
32 | package Math::BigInt; |
33 | my $class = "Math::BigInt"; |
34 | |
35 | $VERSION = 1.35; |
36 | use Exporter; |
37 | @ISA = qw( Exporter ); |
38 | @EXPORT_OK = qw( bneg babs bcmp badd bmul bdiv bmod bnorm bsub |
39 | bgcd blcm |
40 | bround |
41 | blsft brsft band bior bxor bnot bpow bnan bzero |
42 | bacmp bstr bsstr binc bdec bint binf bfloor bceil |
43 | is_odd is_even is_zero is_one is_nan is_inf sign |
44 | length as_number |
45 | trace objectify _swap |
46 | ); |
47 | |
48 | #@EXPORT = qw( ); |
49 | use vars qw/$rnd_mode $accuracy $precision $div_scale/; |
50 | use strict; |
51 | |
52 | # Inside overload, the first arg is always an object. If the original code had |
53 | # it reversed (like $x = 2 * $y), then the third paramater indicates this |
54 | # swapping. To make it work, we use a helper routine which not only reswaps the |
55 | # params, but also makes a new object in this case. See _swap() for details, |
56 | # especially the cases of operators with different classes. |
57 | |
58 | # For overloaded ops with only one argument we simple use $_[0]->copy() to |
59 | # preserve the argument. |
60 | |
61 | # Thus inheritance of overload operators becomes possible and transparent for |
62 | # our subclasses without the need to repeat the entire overload section there. |
a0d0e21e |
63 | |
a5f75d66 |
64 | use overload |
58cde26e |
65 | '=' => sub { $_[0]->copy(); }, |
66 | |
67 | # '+' and '-' do not use _swap, since it is a triffle slower. If you want to |
68 | # override _swap (if ever), then override overload of '+' and '-', too! |
69 | # for sub it is a bit tricky to keep b: b-a => -a+b |
70 | '-' => sub { my $c = $_[0]->copy; $_[2] ? |
71 | $c->bneg()->badd($_[1]) : |
72 | $c->bsub( $_[1]) }, |
73 | '+' => sub { $_[0]->copy()->badd($_[1]); }, |
74 | |
75 | # some shortcuts for speed (assumes that reversed order of arguments is routed |
76 | # to normal '+' and we thus can always modify first arg. If this is changed, |
77 | # this breaks and must be adjusted.) |
78 | '+=' => sub { $_[0]->badd($_[1]); }, |
79 | '-=' => sub { $_[0]->bsub($_[1]); }, |
80 | '*=' => sub { $_[0]->bmul($_[1]); }, |
81 | '/=' => sub { scalar $_[0]->bdiv($_[1]); }, |
82 | '**=' => sub { $_[0]->bpow($_[1]); }, |
83 | |
84 | '<=>' => sub { $_[2] ? |
85 | $class->bcmp($_[1],$_[0]) : |
86 | $class->bcmp($_[0],$_[1])}, |
87 | 'cmp' => sub { |
88 | $_[2] ? |
89 | $_[1] cmp $_[0]->bstr() : |
90 | $_[0]->bstr() cmp $_[1] }, |
91 | |
92 | 'int' => sub { $_[0]->copy(); }, |
93 | 'neg' => sub { $_[0]->copy()->bneg(); }, |
94 | 'abs' => sub { $_[0]->copy()->babs(); }, |
95 | '~' => sub { $_[0]->copy()->bnot(); }, |
96 | |
97 | '*' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bmul($a[1]); }, |
98 | '/' => sub { my @a = ref($_[0])->_swap(@_);scalar $a[0]->bdiv($a[1]);}, |
99 | '%' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bmod($a[1]); }, |
100 | '**' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bpow($a[1]); }, |
101 | '<<' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->blsft($a[1]); }, |
102 | '>>' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->brsft($a[1]); }, |
103 | |
104 | '&' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->band($a[1]); }, |
105 | '|' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bior($a[1]); }, |
106 | '^' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bxor($a[1]); }, |
107 | |
108 | # can modify arg of ++ and --, so avoid a new-copy for speed, but don't |
109 | # use $_[0]->_one(), it modifies $_[0] to be 1! |
110 | '++' => sub { $_[0]->binc() }, |
111 | '--' => sub { $_[0]->bdec() }, |
112 | |
113 | # if overloaded, O(1) instead of O(N) and twice as fast for small numbers |
114 | 'bool' => sub { |
115 | # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/ |
116 | # v5.6.1 dumps on that: return !$_[0]->is_zero() || undef; :-( |
117 | my $t = !$_[0]->is_zero(); |
118 | undef $t if $t == 0; |
119 | return $t; |
120 | }, |
a0d0e21e |
121 | |
122 | qw( |
58cde26e |
123 | "" bstr |
124 | 0+ numify), # Order of arguments unsignificant |
a5f75d66 |
125 | ; |
a0d0e21e |
126 | |
58cde26e |
127 | ############################################################################## |
128 | # global constants, flags and accessory |
129 | |
130 | # are NaNs ok? |
131 | my $NaNOK=1; |
132 | # set to 1 for tracing |
133 | my $trace = 0; |
134 | # constants for easier life |
135 | my $nan = 'NaN'; |
136 | my $BASE_LEN = 5; |
137 | my $BASE = int("1e".$BASE_LEN); # var for trying to change it to 1e7 |
138 | my $RBASE = 1e-5; # see USE_MUL |
139 | |
140 | # Rounding modes one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc' |
141 | $rnd_mode = 'even'; |
142 | $accuracy = undef; |
143 | $precision = undef; |
144 | $div_scale = 40; |
145 | |
146 | sub round_mode |
147 | { |
148 | # make Class->round_mode() work |
149 | my $self = shift || $class; |
150 | # shift @_ if defined $_[0] && $_[0] eq $class; |
151 | if (defined $_[0]) |
152 | { |
153 | my $m = shift; |
154 | die "Unknown round mode $m" |
155 | if $m !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/; |
156 | $rnd_mode = $m; return; |
157 | } |
158 | return $rnd_mode; |
159 | } |
160 | |
161 | sub accuracy |
162 | { |
163 | # $x->accuracy($a); ref($x) a |
164 | # $x->accuracy(); ref($x); |
165 | # Class::accuracy(); # not supported |
166 | #print "MBI @_ ($class)\n"; |
167 | my $x = shift; |
168 | |
169 | die ("accuracy() needs reference to object as first parameter.") |
170 | if !ref $x; |
171 | |
172 | if (@_ > 0) |
173 | { |
174 | $x->{_a} = shift; |
175 | $x->round() if defined $x->{_a}; |
176 | } |
177 | return $x->{_a}; |
178 | } |
179 | |
180 | sub precision |
181 | { |
182 | my $x = shift; |
183 | |
184 | die ("precision() needs reference to object as first parameter.") |
185 | unless ref $x; |
186 | |
187 | if (@_ > 0) |
188 | { |
189 | $x->{_p} = shift; |
190 | $x->round() if defined $x->{_p}; |
191 | } |
192 | return $x->{_p}; |
193 | } |
194 | |
195 | sub _scale_a |
196 | { |
197 | # select accuracy parameter based on precedence, |
198 | # used by bround() and bfround(), may return undef for scale (means no op) |
199 | my ($x,$s,$m,$scale,$mode) = @_; |
200 | $scale = $x->{_a} if !defined $scale; |
201 | $scale = $s if (!defined $scale); |
202 | $mode = $m if !defined $mode; |
203 | return ($scale,$mode); |
204 | } |
205 | |
206 | sub _scale_p |
207 | { |
208 | # select precision parameter based on precedence, |
209 | # used by bround() and bfround(), may return undef for scale (means no op) |
210 | my ($x,$s,$m,$scale,$mode) = @_; |
211 | $scale = $x->{_p} if !defined $scale; |
212 | $scale = $s if (!defined $scale); |
213 | $mode = $m if !defined $mode; |
214 | return ($scale,$mode); |
215 | } |
216 | |
217 | ############################################################################## |
218 | # constructors |
219 | |
220 | sub copy |
221 | { |
222 | my ($c,$x); |
223 | if (@_ > 1) |
224 | { |
225 | # if two arguments, the first one is the class to "swallow" subclasses |
226 | ($c,$x) = @_; |
227 | } |
228 | else |
229 | { |
230 | $x = shift; |
231 | $c = ref($x); |
232 | } |
233 | return unless ref($x); # only for objects |
234 | |
235 | my $self = {}; bless $self,$c; |
236 | foreach my $k (keys %$x) |
237 | { |
238 | if (ref($x->{$k}) eq 'ARRAY') |
239 | { |
240 | $self->{$k} = [ @{$x->{$k}} ]; |
241 | } |
242 | elsif (ref($x->{$k}) eq 'HASH') |
243 | { |
244 | # only one level deep! |
245 | foreach my $h (keys %{$x->{$k}}) |
246 | { |
247 | $self->{$k}->{$h} = $x->{$k}->{$h}; |
248 | } |
249 | } |
250 | elsif (ref($x->{$k})) |
251 | { |
252 | my $c = ref($x->{$k}); |
253 | $self->{$k} = $c->new($x->{$k}); # no copy() due to deep rec |
254 | } |
255 | else |
256 | { |
257 | $self->{$k} = $x->{$k}; |
258 | } |
259 | } |
260 | $self; |
261 | } |
262 | |
263 | sub new |
264 | { |
265 | # create a new BigInts object from a string or another bigint object. |
266 | # value => internal array representation |
267 | # sign => sign (+/-), or "NaN" |
268 | |
269 | # the argument could be an object, so avoid ||, && etc on it, this would |
270 | # cause costly overloaded code to be called. The only allowed op are ref() |
271 | # and definend. |
272 | |
273 | trace (@_); |
274 | my $class = shift; |
275 | |
276 | my $wanted = shift; # avoid numify call by not using || here |
277 | return $class->bzero() if !defined $wanted; # default to 0 |
278 | return $class->copy($wanted) if ref($wanted); |
279 | |
280 | my $self = {}; bless $self, $class; |
281 | # handle '+inf', '-inf' first |
282 | if ($wanted =~ /^[+-]inf$/) |
283 | { |
284 | $self->{value} = [ 0 ]; |
285 | $self->{sign} = $wanted; |
286 | return $self; |
287 | } |
288 | # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign |
289 | my ($mis,$miv,$mfv,$es,$ev) = _split(\$wanted); |
290 | if (ref $mis && !ref $miv) |
291 | { |
292 | # _from_hex |
293 | $self->{value} = $mis->{value}; |
294 | $self->{sign} = $mis->{sign}; |
295 | return $self; |
296 | } |
297 | if (!ref $mis) |
298 | { |
299 | die "$wanted is not a number initialized to $class" if !$NaNOK; |
300 | #print "NaN 1\n"; |
301 | $self->{value} = [ 0 ]; |
302 | $self->{sign} = $nan; |
303 | return $self; |
304 | } |
305 | # make integer from mantissa by adjusting exp, then convert to bigint |
306 | $self->{sign} = $$mis; # store sign |
307 | $self->{value} = [ 0 ]; # for all the NaN cases |
308 | my $e = int("$$es$$ev"); # exponent (avoid recursion) |
309 | if ($e > 0) |
310 | { |
311 | my $diff = $e - CORE::length($$mfv); |
312 | if ($diff < 0) # Not integer |
313 | { |
314 | #print "NOI 1\n"; |
315 | $self->{sign} = $nan; |
316 | } |
317 | else # diff >= 0 |
318 | { |
319 | # adjust fraction and add it to value |
320 | # print "diff > 0 $$miv\n"; |
321 | $$miv = $$miv . ($$mfv . '0' x $diff); |
322 | } |
323 | } |
324 | else |
325 | { |
326 | if ($$mfv ne '') # e <= 0 |
327 | { |
328 | # fraction and negative/zero E => NOI |
329 | #print "NOI 2 \$\$mfv '$$mfv'\n"; |
330 | $self->{sign} = $nan; |
331 | } |
332 | elsif ($e < 0) |
333 | { |
334 | # xE-y, and empty mfv |
335 | #print "xE-y\n"; |
336 | $e = abs($e); |
337 | if ($$miv !~ s/0{$e}$//) # can strip so many zero's? |
338 | { |
339 | #print "NOI 3\n"; |
340 | $self->{sign} = $nan; |
341 | } |
342 | } |
343 | } |
344 | $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0 |
345 | $self->_internal($miv) if $self->{sign} ne $nan; # as internal array |
346 | #print "$wanted => $self->{sign} $self->{value}->[0]\n"; |
347 | # if any of the globals is set, round to them and thus store them insid $self |
348 | $self->round($accuracy,$precision,$rnd_mode) |
349 | if defined $accuracy || defined $precision; |
350 | return $self; |
351 | } |
352 | |
353 | # some shortcuts for easier life |
354 | sub bint |
355 | { |
356 | # exportable version of new |
357 | trace(@_); |
358 | return $class->new(@_); |
359 | } |
360 | |
361 | sub bnan |
362 | { |
363 | # create a bigint 'NaN', if given a BigInt, set it to 'NaN' |
b4f14daa |
364 | my $self = shift; |
58cde26e |
365 | $self = $class if !defined $self; |
366 | if (!ref($self)) |
367 | { |
368 | my $c = $self; $self = {}; bless $self, $c; |
369 | } |
370 | return if $self->modify('bnan'); |
371 | $self->{value} = [ 0 ]; |
372 | $self->{sign} = $nan; |
373 | trace('NaN'); |
374 | return $self; |
b4f14daa |
375 | } |
58cde26e |
376 | |
377 | sub binf |
378 | { |
379 | # create a bigint '+-inf', if given a BigInt, set it to '+-inf' |
380 | # the sign is either '+', or if given, used from there |
381 | my $self = shift; |
382 | my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-'; |
383 | $self = $class if !defined $self; |
384 | if (!ref($self)) |
385 | { |
386 | my $c = $self; $self = {}; bless $self, $c; |
387 | } |
388 | return if $self->modify('binf'); |
389 | $self->{value} = [ 0 ]; |
390 | $self->{sign} = $sign.'inf'; |
391 | trace('inf'); |
392 | return $self; |
393 | } |
394 | |
395 | sub bzero |
396 | { |
397 | # create a bigint '+0', if given a BigInt, set it to 0 |
398 | my $self = shift; |
399 | $self = $class if !defined $self; |
400 | if (!ref($self)) |
401 | { |
402 | my $c = $self; $self = {}; bless $self, $c; |
403 | } |
404 | return if $self->modify('bzero'); |
405 | $self->{value} = [ 0 ]; |
406 | $self->{sign} = '+'; |
407 | trace('0'); |
408 | return $self; |
409 | } |
410 | |
411 | ############################################################################## |
412 | # string conversation |
413 | |
414 | sub bsstr |
415 | { |
416 | # (ref to BFLOAT or num_str ) return num_str |
417 | # Convert number from internal format to scientific string format. |
418 | # internal format is always normalized (no leading zeros, "-0E0" => "+0E0") |
419 | trace(@_); |
420 | my ($self,$x) = objectify(1,@_); |
421 | |
422 | return $x->{sign} if $x->{sign} !~ /^[+-]$/; |
423 | my ($m,$e) = $x->parts(); |
424 | # can be only '+', so |
425 | my $sign = 'e+'; |
426 | # MBF: my $s = $e->{sign}; $s = '' if $s eq '-'; my $sep = 'e'.$s; |
427 | return $m->bstr().$sign.$e->bstr(); |
428 | } |
429 | |
430 | sub bstr |
431 | { |
432 | # (ref to BINT or num_str ) return num_str |
433 | # Convert number from internal base 100000 format to string format. |
434 | # internal format is always normalized (no leading zeros, "-0" => "+0") |
435 | trace(@_); |
436 | my $x = shift; $x = $class->new($x) unless ref $x; |
437 | # my ($self,$x) = objectify(1,@_); |
438 | |
439 | return $x->{sign} if $x->{sign} !~ /^[+-]$/; |
440 | my $ar = $x->{value} || return $nan; # should not happen |
441 | my $es = ""; |
442 | $es = $x->{sign} if $x->{sign} eq '-'; # get sign, but not '+' |
443 | my $l = scalar @$ar; # number of parts |
444 | return $nan if $l < 1; # should not happen |
445 | # handle first one different to strip leading zeros from it (there are no |
446 | # leading zero parts in internal representation) |
447 | $l --; $es .= $ar->[$l]; $l--; |
448 | # Interestingly, the pre-padd method uses more time |
449 | # the old grep variant takes longer (14 to 10 sec) |
450 | while ($l >= 0) |
451 | { |
452 | $es .= substr('0000'.$ar->[$l],-5); # fastest way I could think of |
453 | $l--; |
a0d0e21e |
454 | } |
58cde26e |
455 | return $es; |
456 | } |
457 | |
458 | sub numify |
459 | { |
460 | # Make a number from a BigInt object |
461 | # old: simple return string and let Perl's atoi() handle the rest |
462 | # new: calc because it is faster than bstr()+atoi() |
463 | #trace (@_); |
464 | #my ($self,$x) = objectify(1,@_); |
465 | #return $x->bstr(); # ref($x); |
466 | my $x = shift; $x = $class->new($x) unless ref $x; |
467 | |
468 | return $nan if $x->{sign} eq $nan; |
469 | my $fac = 1; $fac = -1 if $x->{sign} eq '-'; |
470 | return $fac*$x->{value}->[0] if @{$x->{value}} == 1; # below $BASE |
471 | my $num = 0; |
472 | foreach (@{$x->{value}}) |
473 | { |
474 | $num += $fac*$_; $fac *= $BASE; |
475 | } |
476 | return $num; |
477 | } |
478 | |
479 | ############################################################################## |
480 | # public stuff (usually prefixed with "b") |
481 | |
482 | sub sign |
483 | { |
484 | # return the sign of the number: +/-/NaN |
485 | my ($self,$x) = objectify(1,@_); |
486 | return $x->{sign}; |
487 | } |
488 | |
489 | sub round |
490 | { |
491 | # After any operation or when calling round(), the result is rounded by |
492 | # regarding the A & P from arguments, local parameters, or globals. |
493 | # The result's A or P are set by the rounding, but not inspected beforehand |
494 | # (aka only the arguments enter into it). This works because the given |
495 | # 'first' argument is both the result and true first argument with unchanged |
496 | # A and P settings. |
497 | # This does not yet handle $x with A, and $y with P (which should be an |
498 | # error). |
499 | my $self = shift; |
500 | my $a = shift; # accuracy, if given by caller |
501 | my $p = shift; # precision, if given by caller |
502 | my $r = shift; # round_mode, if given by caller |
503 | my @args = @_; # all 'other' arguments (0 for unary, 1 for binary ops) |
504 | |
505 | unshift @args,$self; # add 'first' argument |
506 | |
507 | $self = new($self) unless ref($self); # if not object, make one |
508 | |
509 | # find out class of argument to round |
510 | my $c = ref($args[0]); |
511 | |
512 | # now pick $a or $p, but only if we have got "arguments" |
513 | if ((!defined $a) && (!defined $p) && (@args > 0)) |
514 | { |
515 | foreach (@args) |
516 | { |
517 | # take the defined one, or if both defined, the one that is smaller |
518 | $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a); |
519 | } |
520 | if (!defined $a) # if it still is not defined, take p |
521 | { |
522 | foreach (@args) |
523 | { |
524 | # take the defined one, or if both defined, the one that is smaller |
525 | $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} < $p); |
1f45ae4a |
526 | } |
58cde26e |
527 | # if none defined, use globals (#2) |
528 | if (!defined $p) |
529 | { |
530 | no strict 'refs'; |
531 | my $z = "$c\::accuracy"; $a = $$z; |
532 | if (!defined $a) |
533 | { |
534 | $z = "$c\::precision"; $p = $$z; |
535 | } |
1f45ae4a |
536 | } |
58cde26e |
537 | } # endif !$a |
538 | } # endif !$a || !$P && args > 0 |
539 | # for clearity, this is not merged at place (#2) |
540 | # now round, by calling fround or ffround: |
541 | if (defined $a) |
542 | { |
543 | $self->{_a} = $a; $self->bround($a,$r); |
544 | } |
545 | elsif (defined $p) |
546 | { |
547 | $self->{_p} = $p; $self->bfround($p,$r); |
548 | } |
549 | return $self->bnorm(); |
550 | } |
551 | |
552 | sub bnorm |
553 | { |
554 | # (num_str or BINT) return BINT |
555 | # Normalize number -- no-op here |
556 | my $self = shift; |
557 | |
558 | return $self; |
559 | } |
560 | |
561 | sub babs |
562 | { |
563 | # (BINT or num_str) return BINT |
564 | # make number absolute, or return absolute BINT from string |
565 | #my ($self,$x) = objectify(1,@_); |
566 | my $x = shift; $x = $class->new($x) unless ref $x; |
567 | return $x if $x->modify('babs'); |
568 | # post-normalized abs for internal use (does nothing for NaN) |
569 | $x->{sign} =~ s/^-/+/; |
570 | $x; |
571 | } |
572 | |
573 | sub bneg |
574 | { |
575 | # (BINT or num_str) return BINT |
576 | # negate number or make a negated number from string |
577 | my ($self,$x,$a,$p,$r) = objectify(1,@_); |
578 | return $x if $x->modify('bneg'); |
579 | # for +0 dont negate (to have always normalized) |
580 | return $x if $x->is_zero(); |
581 | $x->{sign} =~ tr/+\-/-+/; # does nothing for NaN |
582 | # $x->round($a,$p,$r); # changing this makes $x - $y modify $y!! |
583 | $x; |
584 | } |
585 | |
586 | sub bcmp |
587 | { |
588 | # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort) |
589 | # (BINT or num_str, BINT or num_str) return cond_code |
590 | my ($self,$x,$y) = objectify(2,@_); |
591 | return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); |
592 | &cmp($x->{value},$y->{value},$x->{sign},$y->{sign}) <=> 0; |
593 | } |
594 | |
595 | sub bacmp |
596 | { |
597 | # Compares 2 values, ignoring their signs. |
598 | # Returns one of undef, <0, =0, >0. (suitable for sort) |
599 | # (BINT, BINT) return cond_code |
600 | my ($self,$x,$y) = objectify(2,@_); |
601 | return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); |
602 | acmp($x->{value},$y->{value}) <=> 0; |
603 | } |
604 | |
605 | sub badd |
606 | { |
607 | # add second arg (BINT or string) to first (BINT) (modifies first) |
608 | # return result as BINT |
609 | trace(@_); |
610 | my ($self,$x,$y,$a,$p,$r) = objectify(2,@_); |
611 | |
612 | return $x if $x->modify('badd'); |
613 | return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); |
614 | |
615 | # for round calls, make array |
616 | my @bn = ($a,$p,$r,$y); |
617 | # speed: no add for 0+y or x+0 |
618 | return $x->bnorm(@bn) if $y->is_zero(); # x+0 |
619 | if ($x->is_zero()) # 0+y |
620 | { |
621 | # make copy, clobbering up x |
622 | $x->{value} = [ @{$y->{value}} ]; |
623 | $x->{sign} = $y->{sign} || $nan; |
624 | return $x->round(@bn); |
625 | } |
626 | |
627 | # shortcuts |
628 | my $xv = $x->{value}; |
629 | my $yv = $y->{value}; |
630 | my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs |
631 | |
632 | if ($sx eq $sy) |
633 | { |
634 | add($xv,$yv); # if same sign, absolute add |
635 | $x->{sign} = $sx; |
636 | } |
637 | else |
638 | { |
639 | my $a = acmp ($yv,$xv); # absolute compare |
640 | if ($a > 0) |
641 | { |
642 | #print "swapped sub (a=$a)\n"; |
643 | &sub($yv,$xv,1); # absolute sub w/ swapped params |
644 | $x->{sign} = $sy; |
645 | } |
646 | elsif ($a == 0) |
647 | { |
648 | # speedup, if equal, set result to 0 |
649 | $x->{value} = [ 0 ]; |
650 | $x->{sign} = '+'; |
651 | } |
652 | else # a < 0 |
653 | { |
654 | #print "unswapped sub (a=$a)\n"; |
655 | &sub($xv, $yv); # absolute sub |
656 | $x->{sign} = $sx; |
a0d0e21e |
657 | } |
a0d0e21e |
658 | } |
58cde26e |
659 | return $x->round(@bn); |
660 | } |
661 | |
662 | sub bsub |
663 | { |
664 | # (BINT or num_str, BINT or num_str) return num_str |
665 | # subtract second arg from first, modify first |
666 | my ($self,$x,$y,$a,$p,$r) = objectify(2,@_); |
667 | |
668 | trace(@_); |
669 | return $x if $x->modify('bsub'); |
670 | $x->badd($y->bneg()); # badd does not leave internal zeros |
671 | $y->bneg(); # refix y, assumes no one reads $y in between |
672 | return $x->round($a,$p,$r,$y); |
673 | } |
674 | |
675 | sub binc |
676 | { |
677 | # increment arg by one |
678 | my ($self,$x,$a,$p,$r) = objectify(1,@_); |
679 | # my $x = shift; $x = $class->new($x) unless ref $x; my $self = ref($x); |
680 | trace(@_); |
681 | return $x if $x->modify('binc'); |
682 | $x->badd($self->_one())->round($a,$p,$r); |
683 | } |
684 | |
685 | sub bdec |
686 | { |
687 | # decrement arg by one |
688 | my ($self,$x,$a,$p,$r) = objectify(1,@_); |
689 | trace(@_); |
690 | return $x if $x->modify('bdec'); |
691 | $x->badd($self->_one('-'))->round($a,$p,$r); |
692 | } |
693 | |
694 | sub blcm |
695 | { |
696 | # (BINT or num_str, BINT or num_str) return BINT |
697 | # does not modify arguments, but returns new object |
698 | # Lowest Common Multiplicator |
699 | trace(@_); |
700 | |
701 | my ($self,@arg) = objectify(0,@_); |
702 | my $x = $self->new(shift @arg); |
703 | while (@arg) { $x = _lcm($x,shift @arg); } |
704 | $x; |
705 | } |
706 | |
707 | sub bgcd |
708 | { |
709 | # (BINT or num_str, BINT or num_str) return BINT |
710 | # does not modify arguments, but returns new object |
711 | # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff) |
712 | trace(@_); |
713 | |
714 | my ($self,@arg) = objectify(0,@_); |
715 | my $x = $self->new(shift @arg); |
716 | while (@arg) |
717 | { |
718 | #$x = _gcd($x,shift @arg); last if $x->is_one(); # new fast, but is slower |
719 | $x = _gcd_old($x,shift @arg); last if $x->is_one(); # old, slow, but faster |
720 | } |
721 | $x; |
722 | } |
723 | |
724 | sub bmod |
725 | { |
726 | # modulus |
727 | # (BINT or num_str, BINT or num_str) return BINT |
728 | my ($self,$x,$y) = objectify(2,@_); |
729 | |
730 | return $x if $x->modify('bmod'); |
731 | (&bdiv($self,$x,$y))[1]; |
732 | } |
733 | |
734 | sub bnot |
735 | { |
736 | # (num_str or BINT) return BINT |
737 | # represent ~x as twos-complement number |
738 | my ($self,$x) = objectify(1,@_); |
739 | return $x if $x->modify('bnot'); |
740 | $x->bneg(); $x->bdec(); # was: bsub(-1,$x);, time it someday |
741 | $x; |
742 | } |
743 | |
744 | sub is_zero |
745 | { |
746 | # return true if arg (BINT or num_str) is zero (array '+', '0') |
747 | #my ($self,$x) = objectify(1,@_); |
748 | #trace(@_); |
749 | my $x = shift; $x = $class->new($x) unless ref $x; |
750 | return (@{$x->{value}} == 1) && ($x->{sign} eq '+') |
751 | && ($x->{value}->[0] == 0); |
752 | } |
753 | |
754 | sub is_nan |
755 | { |
756 | # return true if arg (BINT or num_str) is NaN |
757 | #my ($self,$x) = objectify(1,@_); |
758 | #trace(@_); |
759 | my $x = shift; $x = $class->new($x) unless ref $x; |
760 | return ($x->{sign} eq $nan); |
761 | } |
762 | |
763 | sub is_inf |
764 | { |
765 | # return true if arg (BINT or num_str) is +-inf |
766 | #my ($self,$x) = objectify(1,@_); |
767 | #trace(@_); |
768 | my $x = shift; $x = $class->new($x) unless ref $x; |
769 | my $sign = shift || ''; |
770 | |
771 | return $x->{sign} =~ /^[+-]inf/ if $sign eq ''; |
772 | return $x->{sign} =~ /^[$sign]inf/; |
773 | } |
774 | |
775 | sub is_one |
776 | { |
777 | # return true if arg (BINT or num_str) is +1 (array '+', '1') |
778 | # or -1 if signis given |
779 | #my ($self,$x) = objectify(1,@_); |
780 | my $x = shift; $x = $class->new($x) unless ref $x; |
781 | my $sign = shift || '+'; #$_[2] || '+'; |
782 | return (@{$x->{value}} == 1) && ($x->{sign} eq $sign) |
783 | && ($x->{value}->[0] == 1); |
784 | } |
785 | |
786 | sub is_odd |
787 | { |
788 | # return true when arg (BINT or num_str) is odd, false for even |
789 | my $x = shift; $x = $class->new($x) unless ref $x; |
790 | #my ($self,$x) = objectify(1,@_); |
791 | return (($x->{sign} ne $nan) && ($x->{value}->[0] & 1)); |
792 | } |
793 | |
794 | sub is_even |
795 | { |
796 | # return true when arg (BINT or num_str) is even, false for odd |
797 | my $x = shift; $x = $class->new($x) unless ref $x; |
798 | #my ($self,$x) = objectify(1,@_); |
799 | return (($x->{sign} ne $nan) && (!($x->{value}->[0] & 1))); |
800 | } |
801 | |
802 | sub bmul |
803 | { |
804 | # multiply two numbers -- stolen from Knuth Vol 2 pg 233 |
805 | # (BINT or num_str, BINT or num_str) return BINT |
806 | my ($self,$x,$y,$a,$p,$r) = objectify(2,@_); |
807 | #print "$self bmul $x ",ref($x)," $y ",ref($y),"\n"; |
808 | trace(@_); |
809 | return $x if $x->modify('bmul'); |
810 | return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); |
811 | |
812 | mul($x,$y); # do actual math |
813 | return $x->round($a,$p,$r,$y); |
814 | } |
815 | |
816 | sub bdiv |
817 | { |
818 | # (dividend: BINT or num_str, divisor: BINT or num_str) return |
819 | # (BINT,BINT) (quo,rem) or BINT (only rem) |
820 | trace(@_); |
821 | my ($self,$x,$y,$a,$p,$r) = objectify(2,@_); |
822 | |
823 | return $x if $x->modify('bdiv'); |
824 | |
825 | # NaN? |
826 | return wantarray ? ($x->bnan(),bnan()) : $x->bnan() |
827 | if ($x->{sign} eq $nan || $y->{sign} eq $nan || $y->is_zero()); |
828 | |
829 | # 0 / something |
830 | return wantarray ? ($x,$self->bzero()) : $x if $x->is_zero(); |
831 | |
832 | # Is $x in the interval [0, $y) ? |
833 | my $cmp = acmp($x->{value},$y->{value}); |
834 | if (($cmp < 0) and ($x->{sign} eq $y->{sign})) |
835 | { |
836 | return $x->bzero() unless wantarray; |
837 | my $t = $x->copy(); # make copy first, because $x->bzero() clobbers $x |
838 | return ($x->bzero(),$t); |
839 | } |
840 | elsif ($cmp == 0) |
841 | { |
842 | # shortcut, both are the same, so set to +/- 1 |
843 | $x->_one( ($x->{sign} ne $y->{sign} ? '-' : '+') ); |
844 | return $x unless wantarray; |
845 | return ($x,$self->bzero()); |
846 | } |
847 | |
848 | # calc new sign and in case $y == +/- 1, return $x |
849 | $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+'); |
850 | # check for / +-1 (cant use $y->is_one due to '-' |
851 | if ((@{$y->{value}} == 1) && ($y->{value}->[0] == 1)) |
852 | { |
853 | return wantarray ? ($x,$self->bzero()) : $x; |
854 | } |
855 | |
856 | # call div here |
857 | my $rem = $self->bzero(); |
858 | $rem->{sign} = $y->{sign}; |
859 | ($x->{value},$rem->{value}) = div($x->{value},$y->{value}); |
860 | # do not leave rest "-0"; |
861 | $rem->{sign} = '+' if (@{$rem->{value}} == 1) && ($rem->{value}->[0] == 0); |
862 | if (($x->{sign} eq '-') and (!$rem->is_zero())) |
863 | { |
864 | $x->bdec(); |
865 | } |
866 | $x->round($a,$p,$r,$y); |
867 | if (wantarray) |
868 | { |
869 | $rem->round($a,$p,$r,$x,$y); |
870 | return ($x,$y-$rem) if $x->{sign} eq '-'; # was $x,$rem |
871 | return ($x,$rem); |
872 | } |
873 | return $x; |
874 | } |
875 | |
876 | sub bpow |
877 | { |
878 | # (BINT or num_str, BINT or num_str) return BINT |
879 | # compute power of two numbers -- stolen from Knuth Vol 2 pg 233 |
880 | # modifies first argument |
881 | #trace(@_); |
882 | my ($self,$x,$y,$a,$p,$r) = objectify(2,@_); |
883 | |
884 | return $x if $x->modify('bpow'); |
885 | |
886 | return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan; |
887 | return $x->_one() if $y->is_zero(); |
888 | return $x if $x->is_one() || $y->is_one(); |
889 | if ($x->{sign} eq '-' && @{$x->{value}} == 1 && $x->{value}->[0] == 1) |
890 | { |
891 | # if $x == -1 and odd/even y => +1/-1 |
892 | return $y->is_odd() ? $x : $x->_set(1); # $x->babs() would work to |
893 | # my Casio FX-5500L has here a bug, -1 ** 2 is -1, but -1 * -1 is 1 LOL |
894 | } |
895 | # shortcut for $x ** 2 |
896 | if ($y->{sign} eq '+' && @{$y->{value}} == 1 && $y->{value}->[0] == 2) |
897 | { |
898 | return $x->bmul($x)->bround($a,$p,$r); |
899 | } |
900 | # 1 ** -y => 1 / (1**y), so do test for negative $y after above's clause |
901 | return $x->bnan() if $y->{sign} eq '-'; |
902 | return $x if $x->is_zero(); # 0**y => 0 (if not y <= 0) |
903 | |
904 | # tels: 10**x is special (actually 100**x etc is special, too) but not here |
905 | #if ((@{$x->{value}} == 1) && ($x->{value}->[0] == 10)) |
906 | # { |
907 | # # 10**2 |
908 | # my $yi = int($y); my $yi5 = int($yi/5); |
909 | # $x->{value} = []; |
910 | # my $v = $x->{value}; |
911 | # if ($yi5 > 0) |
912 | # { |
913 | # # $x->{value}->[$yi5-1] = 0; # pre-padd array (no use) |
914 | # for (my $i = 0; $i < $yi5; $i++) |
915 | # { |
916 | # $v->[$i] = 0; |
917 | # } |
918 | # } |
919 | # push @{$v}, int( '1'.'0' x ($yi % 5)); |
920 | # if ($x->{sign} eq '-') |
921 | # { |
922 | # $x->{sign} = $y->is_odd() ? '-' : '+'; # -10**2 = 100, -10**3 = -1000 |
923 | # } |
924 | # return $x; |
925 | # } |
926 | |
927 | # based on the assumption that shifting in base 10 is fast, and that bpow() |
928 | # works faster if numbers are small: we count trailing zeros (this step is |
929 | # O(1)..O(N), but in case of O(N) we save much more time), stripping them |
930 | # out of the multiplication, and add $count * $y zeros afterwards: |
931 | # 300 ** 3 == 300*300*300 == 3*3*3 . '0' x 2 * 3 == 27 . '0' x 6 |
932 | my $zeros = $x->_trailing_zeros(); |
933 | if ($zeros > 0) |
934 | { |
935 | $x->brsft($zeros,10); # remove zeros |
936 | $x->bpow($y); # recursion (will not branch into here again) |
937 | $zeros = $y * $zeros; # real number of zeros to add |
938 | $x->blsft($zeros,10); |
939 | return $x; |
940 | } |
941 | |
942 | my $pow2 = $self->_one(); |
943 | my $y1 = $class->new($y); |
944 | my ($res); |
945 | while (!$y1->is_one()) |
946 | { |
947 | #print "bpow: p2: $pow2 x: $x y: $y1 r: $res\n"; |
948 | #print "len ",$x->length(),"\n"; |
949 | ($y1,$res)=&bdiv($y1,2); |
950 | if (!$res->is_zero()) { &bmul($pow2,$x); } |
951 | if (!$y1->is_zero()) { &bmul($x,$x); } |
952 | } |
953 | #print "bpow: e p2: $pow2 x: $x y: $y1 r: $res\n"; |
954 | &bmul($x,$pow2) if (!$pow2->is_one()); |
955 | #print "bpow: e p2: $pow2 x: $x y: $y1 r: $res\n"; |
956 | return $x->round($a,$p,$r); |
957 | } |
958 | |
959 | sub blsft |
960 | { |
961 | # (BINT or num_str, BINT or num_str) return BINT |
962 | # compute x << y, base n, y >= 0 |
963 | my ($self,$x,$y,$n) = objectify(2,@_); |
964 | |
965 | return $x if $x->modify('blsft'); |
966 | return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/); |
967 | |
968 | $n = 2 if !defined $n; return $x if $n == 0; |
969 | return $x->bnan() if $n < 0 || $y->{sign} eq '-'; |
970 | if ($n != 10) |
971 | { |
972 | $x->bmul( $self->bpow($n, $y) ); |
973 | } |
974 | else |
975 | { |
976 | # shortcut (faster) for shifting by 10) since we are in base 10eX |
977 | # multiples of 5: |
978 | my $src = scalar @{$x->{value}}; # source |
979 | my $len = $y->numify(); # shift-len as normal int |
980 | my $rem = $len % 5; # reminder to shift |
981 | my $dst = $src + int($len/5); # destination |
982 | |
983 | my $v = $x->{value}; # speed-up |
984 | my $vd; # further speedup |
985 | #print "src $src:",$v->[$src]||0," dst $dst:",$v->[$dst]||0," rem $rem\n"; |
986 | $v->[$src] = 0; # avoid first ||0 for speed |
987 | while ($src >= 0) |
988 | { |
989 | $vd = $v->[$src]; $vd = '00000'.$vd; |
990 | #print "s $src d $dst '$vd' "; |
991 | $vd = substr($vd,-5+$rem,5-$rem); |
992 | #print "'$vd' "; |
993 | $vd .= $src > 0 ? substr('00000'.$v->[$src-1],-5,$rem) : '0' x $rem; |
994 | #print "'$vd' "; |
995 | $vd = substr($vd,-5,5) if length($vd) > 5; |
996 | #print "'$vd'\n"; |
997 | $v->[$dst] = int($vd); |
998 | $dst--; $src--; |
999 | } |
1000 | # set lowest parts to 0 |
1001 | while ($dst >= 0) { $v->[$dst--] = 0; } |
1002 | # fix spurios last zero element |
1003 | splice @$v,-1 if $v->[-1] == 0; |
1004 | #print "elems: "; my $i = 0; |
1005 | #foreach (reverse @$v) { print "$i $_ "; $i++; } print "\n"; |
1006 | # old way: $x->bmul( $self->bpow($n, $y) ); |
1007 | } |
1008 | return $x; |
1009 | } |
1010 | |
1011 | sub brsft |
1012 | { |
1013 | # (BINT or num_str, BINT or num_str) return BINT |
1014 | # compute x >> y, base n, y >= 0 |
1015 | my ($self,$x,$y,$n) = objectify(2,@_); |
1016 | |
1017 | return $x if $x->modify('brsft'); |
1018 | return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/); |
1019 | |
1020 | $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-'; |
1021 | if ($n != 10) |
1022 | { |
1023 | scalar bdiv($x, $self->bpow($n, $y)); |
1024 | } |
1025 | else |
1026 | { |
1027 | # shortcut (faster) for shifting by 10) |
1028 | # multiples of 5: |
1029 | my $dst = 0; # destination |
1030 | my $src = $y->numify(); # as normal int |
1031 | my $rem = $src % 5; # reminder to shift |
1032 | $src = int($src / 5); # source |
1033 | my $len = scalar @{$x->{value}} - $src; # elems to go |
1034 | my $v = $x->{value}; # speed-up |
1035 | if ($rem == 0) |
1036 | { |
1037 | splice (@$v,0,$src); # even faster, 38.4 => 39.3 |
1038 | } |
1039 | else |
1040 | { |
1041 | my $vd; |
1042 | $v->[scalar @$v] = 0; # avoid || 0 test inside loop |
1043 | while ($dst < $len) |
1044 | { |
1045 | $vd = '00000'.$v->[$src]; |
1046 | #print "$dst $src '$vd' "; |
1047 | $vd = substr($vd,-5,5-$rem); |
1048 | #print "'$vd' "; |
1049 | $src++; |
1050 | $vd = substr('00000'.$v->[$src],-$rem,$rem) . $vd; |
1051 | #print "'$vd1' "; |
1052 | #print "'$vd'\n"; |
1053 | $vd = substr($vd,-5,5) if length($vd) > 5; |
1054 | $v->[$dst] = int($vd); |
1055 | $dst++; |
1056 | } |
1057 | splice (@$v,$dst) if $dst > 0; # kill left-over array elems |
1058 | pop @$v if $v->[-1] == 0; # kill last element |
1059 | } # else rem == 0 |
1060 | # old way: scalar bdiv($x, $self->bpow($n, $y)); |
1061 | } |
1062 | return $x; |
1063 | } |
1064 | |
1065 | sub band |
1066 | { |
1067 | #(BINT or num_str, BINT or num_str) return BINT |
1068 | # compute x & y |
1069 | trace(@_); |
1070 | my ($self,$x,$y) = objectify(2,@_); |
1071 | |
1072 | return $x if $x->modify('band'); |
1073 | |
1074 | return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/); |
1075 | return $x->bzero() if $y->is_zero(); |
1076 | my $r = $self->bzero(); my $m = new Math::BigInt 1; my ($xr,$yr); |
1077 | my $x10000 = new Math::BigInt (0x10000); |
1078 | my $y1 = copy(ref($x),$y); # make copy |
1079 | while (!$x->is_zero() && !$y1->is_zero()) |
1080 | { |
1081 | ($x, $xr) = bdiv($x, $x10000); |
1082 | ($y1, $yr) = bdiv($y1, $x10000); |
1083 | $r->badd( bmul( new Math::BigInt ( $xr->numify() & $yr->numify()), $m )); |
1084 | $m->bmul($x10000); |
1085 | } |
1086 | $x = $r; |
1087 | } |
1088 | |
1089 | sub bior |
1090 | { |
1091 | #(BINT or num_str, BINT or num_str) return BINT |
1092 | # compute x | y |
1093 | trace(@_); |
1094 | my ($self,$x,$y) = objectify(2,@_); |
1095 | |
1096 | return $x if $x->modify('bior'); |
1097 | |
1098 | return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/); |
1099 | return $x if $y->is_zero(); |
1100 | my $r = $self->bzero(); my $m = new Math::BigInt 1; my ($xr,$yr); |
1101 | my $x10000 = new Math::BigInt (0x10000); |
1102 | my $y1 = copy(ref($x),$y); # make copy |
1103 | while (!$x->is_zero() || !$y1->is_zero()) |
1104 | { |
1105 | ($x, $xr) = bdiv($x,$x10000); |
1106 | ($y1, $yr) = bdiv($y1,$x10000); |
1107 | $r->badd( bmul( new Math::BigInt ( $xr->numify() | $yr->numify()), $m )); |
1108 | $m->bmul($x10000); |
1109 | } |
1110 | $x = $r; |
1111 | } |
1112 | |
1113 | sub bxor |
1114 | { |
1115 | #(BINT or num_str, BINT or num_str) return BINT |
1116 | # compute x ^ y |
1117 | my ($self,$x,$y) = objectify(2,@_); |
1118 | |
1119 | return $x if $x->modify('bxor'); |
1120 | |
1121 | return $x->bnan() if ($x->{sign} eq $nan || $y->{sign} eq $nan); |
1122 | return $x if $y->is_zero(); |
1123 | return $x->bzero() if $x == $y; # shortcut |
1124 | my $r = $self->bzero(); my $m = new Math::BigInt 1; my ($xr,$yr); |
1125 | my $x10000 = new Math::BigInt (0x10000); |
1126 | my $y1 = copy(ref($x),$y); # make copy |
1127 | while (!$x->is_zero() || !$y1->is_zero()) |
1128 | { |
1129 | ($x, $xr) = bdiv($x, $x10000); |
1130 | ($y1, $yr) = bdiv($y1, $x10000); |
1131 | $r->badd( bmul( new Math::BigInt ( $xr->numify() ^ $yr->numify()), $m )); |
1132 | $m->bmul($x10000); |
1133 | } |
1134 | $x = $r; |
1135 | } |
1136 | |
1137 | sub length |
1138 | { |
1139 | my ($self,$x) = objectify(1,@_); |
1140 | |
1141 | return (_digits($x->{value}), 0) if wantarray; |
1142 | _digits($x->{value}); |
1143 | } |
1144 | |
1145 | sub digit |
1146 | { |
1147 | # return the nth digit, negative values count backward |
1148 | my $x = shift; |
1149 | my $n = shift || 0; |
1150 | |
1151 | my $len = $x->length(); |
1152 | |
1153 | $n = $len+$n if $n < 0; # -1 last, -2 second-to-last |
1154 | $n = abs($n); # if negatives are to big |
1155 | $len--; $n = $len if $n > $len; # n to big? |
1156 | |
1157 | my $elem = int($n / 5); # which array element |
1158 | my $digit = $n % 5; # which digit in this element |
1159 | $elem = '0000'.$x->{value}->[$elem]; # get element padded with 0's |
1160 | return substr($elem,-$digit-1,1); |
1161 | } |
1162 | |
1163 | sub _trailing_zeros |
1164 | { |
1165 | # return the amount of trailing zeros in $x |
1166 | my $x = shift; |
1167 | $x = $class->new($x) unless ref $x; |
1168 | |
1169 | return 0 if $x->is_zero() || $x->is_nan(); |
1170 | # check each array elem in _m for having 0 at end as long as elem == 0 |
1171 | # Upon finding a elem != 0, stop |
1172 | my $zeros = 0; my $elem; |
1173 | foreach my $e (@{$x->{value}}) |
1174 | { |
1175 | if ($e != 0) |
1176 | { |
1177 | $elem = "$e"; # preserve x |
1178 | $elem =~ s/.*?(0*$)/$1/; # strip anything not zero |
1179 | $zeros *= 5; # elems * 5 |
1180 | $zeros += CORE::length($elem); # count trailing zeros |
1181 | last; # early out |
1182 | } |
1183 | $zeros ++; # real else branch: 50% slower! |
1184 | } |
1185 | return $zeros; |
1186 | } |
1187 | |
1188 | sub bsqrt |
1189 | { |
1190 | my ($self,$x) = objectify(1,@_); |
1191 | |
1192 | return $x->bnan() if $x->{sign} =~ /\-|$nan/; # -x or NaN => NaN |
1193 | return $x->bzero() if $x->is_zero(); # 0 => 0 |
1194 | return $x if $x == 1; # 1 => 1 |
1195 | |
1196 | my $y = $x->copy(); # give us one more digit accur. |
1197 | my $l = int($x->length()/2); |
1198 | |
1199 | $x->bzero(); |
1200 | $x->binc(); # keep ref($x), but modify it |
1201 | $x *= 10 ** $l; |
1202 | |
1203 | # print "x: $y guess $x\n"; |
1204 | |
1205 | my $last = $self->bzero(); |
1206 | while ($last != $x) |
1207 | { |
1208 | $last = $x; |
1209 | $x += $y / $x; |
1210 | $x /= 2; |
1211 | } |
1212 | return $x; |
1213 | } |
1214 | |
1215 | sub exponent |
1216 | { |
1217 | # return a copy of the exponent (here always 0, NaN or 1 for $m == 0) |
1218 | my ($self,$x) = objectify(1,@_); |
1219 | |
1220 | return bnan() if $x->is_nan(); |
1221 | my $e = $class->bzero(); |
1222 | return $e->binc() if $x->is_zero(); |
1223 | $e += $x->_trailing_zeros(); |
1224 | return $e; |
1225 | } |
1226 | |
1227 | sub mantissa |
1228 | { |
1229 | # return a copy of the mantissa (here always $self) |
1230 | my ($self,$x) = objectify(1,@_); |
1231 | |
1232 | return bnan() if $x->is_nan(); |
1233 | my $m = $x->copy(); |
1234 | # that's inefficient |
1235 | my $zeros = $m->_trailing_zeros(); |
1236 | $m /= 10 ** $zeros if $zeros != 0; |
1237 | return $m; |
1238 | } |
1239 | |
1240 | sub parts |
1241 | { |
1242 | # return a copy of both the exponent and the mantissa (here 0 and self) |
1243 | my $self = shift; |
1244 | $self = $class->new($self) unless ref $self; |
1245 | |
1246 | return ($self->mantissa(),$self->exponent()); |
1247 | } |
1248 | |
1249 | ############################################################################## |
1250 | # rounding functions |
1251 | |
1252 | sub bfround |
1253 | { |
1254 | # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.' |
1255 | # $n == 0 => round to integer |
1256 | my $x = shift; $x = $class->new($x) unless ref $x; |
1257 | my ($scale,$mode) = $x->_scale_p($precision,$rnd_mode,@_); |
1258 | return $x if !defined $scale; # no-op |
1259 | |
1260 | # no-op for BigInts if $n <= 0 |
1261 | return $x if $scale <= 0; |
1262 | |
1263 | $x->bround( $x->length()-$scale, $mode); |
1264 | } |
1265 | |
1266 | sub _scan_for_nonzero |
1267 | { |
1268 | my $x = shift; |
1269 | my $pad = shift; |
1270 | |
1271 | my $len = $x->length(); |
1272 | return 0 if $len == 1; # '5' is trailed by invisible zeros |
1273 | my $follow = $pad - 1; |
1274 | return 0 if $follow > $len || $follow < 1; |
1275 | #print "checking $x $r\n"; |
1276 | # old, slow way checking string for non-zero characters |
1277 | my $r = substr ("$x",-$follow); |
1278 | return 1 if $r =~ /[^0]/; return 0; |
1279 | |
1280 | # faster way checking array contents; it is actually not faster (even in a |
1281 | # rounding-only-shoutout, so I leave the simpler code in) |
1282 | #my $rem = $follow % 5; my $div = $follow / 5; my $v = $x->{value}; |
1283 | # pad with zeros and extract |
1284 | #print "last part : ",'00000'.$v->[$div]," $rem = '"; |
1285 | #print substr('00000'.$v->[$div],-$rem,5),"'\n"; |
1286 | #my $r1 = substr ('00000'.$v->[$div],-$rem,5); |
1287 | #print "$r1\n"; |
1288 | #return 1 if $r1 =~ /[^0]/; |
1289 | # |
1290 | #for (my $j = $div-1; $j >= 0; $j --) |
1291 | # { |
1292 | # #print "part $v->[$j]\n"; |
1293 | # return 1 if $v->[$j] != 0; |
1294 | # } |
1295 | #return 0; |
1296 | } |
1297 | |
1298 | sub fround |
1299 | { |
1300 | # to make life easier for switch between MBF and MBI (autoload fxxx() |
1301 | # like MBF does for bxxx()?) |
1302 | my $x = shift; |
1303 | return $x->bround(@_); |
1304 | } |
1305 | |
1306 | sub bround |
1307 | { |
1308 | # accuracy: +$n preserve $n digits from left, |
1309 | # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF) |
1310 | # no-op for $n == 0 |
1311 | # and overwrite the rest with 0's, return normalized number |
1312 | # do not return $x->bnorm(), but $x |
1313 | my $x = shift; $x = $class->new($x) unless ref $x; |
1314 | my ($scale,$mode) = $x->_scale_a($accuracy,$rnd_mode,@_); |
1315 | return $x if !defined $scale; # no-op |
1316 | |
1317 | # print "MBI round: $x to $scale $mode\n"; |
1318 | # -scale means what? tom? hullo? -$scale needed by MBF round, but what for? |
1319 | return $x if $x->is_nan() || $x->is_zero() || $scale == 0; |
1320 | |
1321 | # we have fewer digits than we want to scale to |
1322 | my $len = $x->length(); |
1323 | # print "$len $scale\n"; |
1324 | return $x if $len < abs($scale); |
1325 | |
1326 | # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6 |
1327 | my ($pad,$digit_round,$digit_after); |
1328 | $pad = $len - $scale; |
1329 | $pad = abs($scale)+1 if $scale < 0; |
1330 | $digit_round = '0'; $digit_round = $x->digit($pad) if $pad < $len; |
1331 | $digit_after = '0'; $digit_after = $x->digit($pad-1) if $pad > 0; |
1332 | # print "r $x: pos:$pad l:$len s:$scale r:$digit_round a:$digit_after m: $mode\n"; |
1333 | |
1334 | # in case of 01234 we round down, for 6789 up, and only in case 5 we look |
1335 | # closer at the remaining digits of the original $x, remember decision |
1336 | my $round_up = 1; # default round up |
1337 | $round_up -- if |
1338 | ($mode eq 'trunc') || # trunc by round down |
1339 | ($digit_after =~ /[01234]/) || # round down anyway, |
1340 | # 6789 => round up |
1341 | ($digit_after eq '5') && # not 5000...0000 |
1342 | ($x->_scan_for_nonzero($pad) == 0) && |
1343 | ( |
1344 | ($mode eq 'even') && ($digit_round =~ /[24680]/) || |
1345 | ($mode eq 'odd') && ($digit_round =~ /[13579]/) || |
1346 | ($mode eq '+inf') && ($x->{sign} eq '-') || |
1347 | ($mode eq '-inf') && ($x->{sign} eq '+') || |
1348 | ($mode eq 'zero') # round down if zero, sign adjusted below |
1349 | ); |
1350 | # allow rounding one place left of mantissa |
1351 | #print "$pad $len $scale\n"; |
1352 | # this is triggering warnings, and buggy for $scale < 0 |
1353 | #if (-$scale != $len) |
1354 | { |
1355 | # split mantissa at $scale and then pad with zeros |
1356 | my $s5 = int($pad / 5); |
1357 | my $i = 0; |
1358 | while ($i < $s5) |
1359 | { |
1360 | $x->{value}->[$i++] = 0; # replace with 5 x 0 |
1361 | } |
1362 | $x->{value}->[$s5] = '00000'.$x->{value}->[$s5]; # pad with 0 |
1363 | my $rem = $pad % 5; # so much left over |
1364 | if ($rem > 0) |
1365 | { |
1366 | #print "remainder $rem\n"; |
1367 | #print "elem $x->{value}->[$s5]\n"; |
1368 | substr($x->{value}->[$s5],-$rem,$rem) = '0' x $rem; # stamp w/ '0' |
1369 | } |
1370 | $x->{value}->[$s5] = int ($x->{value}->[$s5]); # str '05' => int '5' |
1371 | } |
1372 | if ($round_up) # what gave test above? |
1373 | { |
1374 | $pad = $len if $scale < 0; # tlr: whack 0.51=>1.0 |
1375 | # modify $x in place, undef, undef to avoid rounding |
1376 | $x->badd( Math::BigInt->new($x->{sign}.'1'.'0'x$pad), |
1377 | undef,undef ); |
1378 | # str creation much faster than 10 ** something |
1379 | } |
1380 | $x; |
1381 | } |
1382 | |
1383 | sub bfloor |
1384 | { |
1385 | # return integer less or equal then number, since it is already integer, |
1386 | # always returns $self |
1387 | my ($self,$x,$a,$p,$r) = objectify(1,@_); |
1388 | |
1389 | # not needed: return $x if $x->modify('bfloor'); |
1390 | |
1391 | return $x->round($a,$p,$r); |
1392 | } |
1393 | |
1394 | sub bceil |
1395 | { |
1396 | # return integer greater or equal then number, since it is already integer, |
1397 | # always returns $self |
1398 | my ($self,$x,$a,$p,$r) = objectify(1,@_); |
1399 | |
1400 | # not needed: return $x if $x->modify('bceil'); |
1401 | |
1402 | return $x->round($a,$p,$r); |
1403 | } |
1404 | |
1405 | ############################################################################## |
1406 | # private stuff (internal use only) |
1407 | |
1408 | sub trace |
1409 | { |
1410 | # print out a number without using bstr (avoid deep recurse) for trace/debug |
1411 | return unless $trace; |
1412 | |
1413 | my ($package,$file,$line,$sub) = caller(1); |
1414 | print "'$sub' called from '$package' line $line:\n "; |
1415 | |
1416 | foreach my $x (@_) |
1417 | { |
1418 | if (!defined $x) |
1419 | { |
1420 | print "undef, "; next; |
1421 | } |
1422 | if (!ref($x)) |
1423 | { |
1424 | print "'$x' "; next; |
1425 | } |
1426 | next if (ref($x) ne "HASH"); |
1427 | print "$x->{sign} "; |
1428 | foreach (@{$x->{value}}) |
1429 | { |
1430 | print "$_ "; |
1431 | } |
1432 | print ", "; |
1433 | } |
1434 | print "\n"; |
1435 | } |
1436 | |
1437 | sub _set |
1438 | { |
1439 | # internal set routine to set X fast to an integer value < [+-]100000 |
1440 | my $self = shift; |
1441 | my $wanted = shift || 0; |
1442 | |
1443 | $self->{sign} = $nan, return if $wanted !~ /^[+-]?[0-9]+$/; |
1444 | $self->{sign} = '-'; $self->{sign} = '+' if $wanted >= 0; |
1445 | $self->{value} = [ abs($wanted) ]; |
1446 | return $self; |
1447 | } |
1448 | |
1449 | sub _one |
1450 | { |
1451 | # internal speedup, set argument to 1, or create a +/- 1 |
1452 | my $self = shift; |
1453 | my $x = $self->bzero(); $x->{value} = [ 1 ]; $x->{sign} = shift || '+'; $x; |
1454 | } |
1455 | |
1456 | sub _swap |
1457 | { |
1458 | # Overload will swap params if first one is no object ref so that the first |
1459 | # one is always an object ref. In this case, third param is true. |
1460 | # This routine is to overcome the effect of scalar,$object creating an object |
1461 | # of the class of this package, instead of the second param $object. This |
1462 | # happens inside overload, when the overload section of this package is |
1463 | # inherited by sub classes. |
1464 | # For overload cases (and this is used only there), we need to preserve the |
1465 | # args, hence the copy(). |
1466 | # You can override this method in a subclass, the overload section will call |
1467 | # $object->_swap() to make sure it arrives at the proper subclass, with some |
1468 | # exceptions like '+' and '-'. |
1469 | |
1470 | # object, (object|scalar) => preserve first and make copy |
1471 | # scalar, object => swapped, re-swap and create new from first |
1472 | # (using class of second object, not $class!!) |
1473 | my $self = shift; # for override in subclass |
1474 | #print "swap $self 0:$_[0] 1:$_[1] 2:$_[2]\n"; |
1475 | if ($_[2]) |
1476 | { |
1477 | my $c = ref ($_[0]) || $class; # fallback $class should not happen |
1478 | return ( $c->new($_[1]), $_[0] ); |
1479 | } |
1480 | else |
1481 | { |
1482 | return ( $_[0]->copy(), $_[1] ); |
1483 | } |
1484 | } |
1485 | |
1486 | sub objectify |
1487 | { |
1488 | # check for strings, if yes, return objects instead |
1489 | |
1490 | # the first argument is number of args objectify() should look at it will |
1491 | # return $count+1 elements, the first will be a classname. This is because |
1492 | # overloaded '""' calls bstr($object,undef,undef) and this would result in |
1493 | # useless objects beeing created and thrown away. So we cannot simple loop |
1494 | # over @_. If the given count is 0, all arguments will be used. |
1495 | |
1496 | # If the second arg is a ref, use it as class. |
1497 | # If not, try to use it as classname, unless undef, then use $class |
1498 | # (aka Math::BigInt). The latter shouldn't happen,though. |
1499 | |
1500 | # caller: gives us: |
1501 | # $x->badd(1); => ref x, scalar y |
1502 | # Class->badd(1,2); => classname x (scalar), scalar x, scalar y |
1503 | # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y |
1504 | # Math::BigInt::badd(1,2); => scalar x, scalar y |
1505 | # In the last case we check number of arguments to turn it silently into |
1506 | # $class,1,2. (We can not take '1' as class ;o) |
1507 | # badd($class,1) is not supported (it should, eventually, try to add undef) |
1508 | # currently it tries 'Math::BigInt' + 1, which will not work. |
1509 | |
1510 | trace(@_); |
1511 | my $count = abs(shift || 0); |
1512 | |
1513 | #print caller(),"\n"; |
1514 | |
1515 | my @a; # resulting array |
1516 | if (ref $_[0]) |
1517 | { |
1518 | # okay, got object as first |
1519 | $a[0] = ref $_[0]; |
1520 | } |
1521 | else |
1522 | { |
1523 | # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported) |
1524 | $a[0] = $class; |
1525 | #print "@_\n"; sleep(1); |
1526 | $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first? |
1527 | } |
1528 | #print caller(),"\n"; |
1529 | # print "Now in objectify, my class is today $a[0]\n"; |
1530 | my $k; |
1531 | if ($count == 0) |
1532 | { |
1533 | while (@_) |
1534 | { |
1535 | $k = shift; |
1536 | if (!ref($k)) |
1537 | { |
1538 | $k = $a[0]->new($k); |
1539 | } |
1540 | elsif (ref($k) ne $a[0]) |
1541 | { |
1542 | # foreign object, try to convert to integer |
1543 | $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k); |
e16b8f49 |
1544 | } |
58cde26e |
1545 | push @a,$k; |
1546 | } |
1547 | } |
1548 | else |
1549 | { |
1550 | while ($count > 0) |
1551 | { |
1552 | #print "$count\n"; |
1553 | $count--; |
1554 | $k = shift; |
1555 | if (!ref($k)) |
1556 | { |
1557 | $k = $a[0]->new($k); |
1558 | } |
1559 | elsif (ref($k) ne $a[0]) |
1560 | { |
1561 | # foreign object, try to convert to integer |
1562 | $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k); |
e16b8f49 |
1563 | } |
58cde26e |
1564 | push @a,$k; |
1565 | } |
1566 | push @a,@_; # return other params, too |
1567 | } |
1568 | #my $i = 0; |
1569 | #foreach (@a) |
1570 | # { |
1571 | # print "o $i $a[0]\n" if $i == 0; |
1572 | # print "o $i ",ref($_),"\n" if $i != 0; $i++; |
1573 | # } |
1574 | #print "objectify done: would return ",scalar @a," values\n"; |
1575 | #print caller(1),"\n" unless wantarray; |
1576 | die "$class objectify needs list context" unless wantarray; |
1577 | @a; |
1578 | } |
1579 | |
1580 | sub import |
1581 | { |
1582 | my $self = shift; |
1583 | #print "import $self @_\n"; |
1584 | for ( my $i = 0; $i < @_ ; $i++ ) |
1585 | { |
1586 | if ( $_[$i] eq ':constant' ) |
1587 | { |
1588 | # this rest causes overlord er load to step in |
1589 | overload::constant integer => sub { $self->new(shift) }; |
1590 | splice @_, $i, 1; last; |
1591 | } |
1592 | } |
1593 | # any non :constant stuff is handled by our parent, Exporter |
1594 | # even if @_ is empty, to give it a chance |
1595 | #$self->SUPER::import(@_); # does not work |
1596 | $self->export_to_level(1,$self,@_); # need this instead |
1597 | } |
1598 | |
1599 | sub _internal |
1600 | { |
1601 | # (ref to self, ref to string) return ref to num_array |
1602 | # Convert a number from string format to internal base 100000 format. |
1603 | # Assumes normalized value as input. |
1604 | my ($s,$d) = @_; |
1605 | my $il = CORE::length($$d)-1; |
1606 | # these leaves '00000' instead of int 0 and will be corrected after any op |
1607 | $s->{value} = [ reverse(unpack("a" . ($il%5+1) . ("a5" x ($il/5)), $$d)) ]; |
1608 | $s; |
1609 | } |
1610 | |
1611 | sub _strip_zeros |
1612 | { |
1613 | # internal normalization function that strips leading zeros from the array |
1614 | # args: ref to array |
1615 | #trace(@_); |
1616 | my $s = shift; |
1617 | |
1618 | my $cnt = scalar @$s; # get count of parts |
1619 | my $i = $cnt-1; |
1620 | #print "strip: cnt $cnt i $i\n"; |
1621 | # '0', '3', '4', '0', '0', |
1622 | # 0 1 2 3 4 |
1623 | # cnt = 5, i = 4 |
1624 | # i = 4 |
1625 | # i = 3 |
1626 | # => fcnt = cnt - i (5-2 => 3, cnt => 5-1 = 4, throw away from 4th pos) |
1627 | # >= 1: skip first part (this can be zero) |
1628 | while ($i > 0) { last if $s->[$i] != 0; $i--; } |
1629 | $i++; splice @$s,$i if ($i < $cnt); # $i cant be 0 |
1630 | return $s; |
1631 | } |
1632 | |
1633 | sub _from_hex |
1634 | { |
1635 | # convert a (ref to) big hex string to BigInt, return undef for error |
1636 | my $hs = shift; |
1637 | |
1638 | my $x = Math::BigInt->bzero(); |
1639 | return $x->bnan() if $$hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/; |
1640 | |
1641 | my $mul = Math::BigInt->bzero(); $mul++; |
1642 | my $x65536 = Math::BigInt->new(65536); |
1643 | |
1644 | my $len = CORE::length($$hs)-2; my $sign = '+'; |
1645 | if ($$hs =~ /^\-/) |
1646 | { |
1647 | $sign = '-'; $len--; |
1648 | } |
1649 | $len = int($len/4); # 4-digit parts, w/o '0x' |
1650 | my $val; my $i = -4; |
1651 | while ($len >= 0) |
1652 | { |
1653 | $val = substr($$hs,$i,4); |
1654 | $val =~ s/^[\-\+]?0x// if $len == 0; # for last part only because |
1655 | $val = hex($val); # hex does not like wrong chars |
1656 | # print "$val ",substr($$hs,$i,4),"\n"; |
1657 | $i -= 4; $len --; |
1658 | $x += $mul * $val if $val != 0; |
1659 | $mul *= $x65536 if $len >= 0; # skip last mul |
1660 | } |
1661 | $x->{sign} = $sign if !$x->is_zero(); |
1662 | return $x; |
1663 | } |
1664 | |
1665 | sub _from_bin |
1666 | { |
1667 | # convert a (ref to) big binary string to BigInt, return undef for error |
1668 | my $bs = shift; |
1669 | |
1670 | my $x = Math::BigInt->bzero(); |
1671 | return $x->bnan() if $$bs !~ /^[\-\+]?0b[01]+$/; |
1672 | |
1673 | my $mul = Math::BigInt->bzero(); $mul++; |
1674 | my $x256 = Math::BigInt->new(256); |
1675 | |
1676 | my $len = CORE::length($$bs)-2; my $sign = '+'; |
1677 | if ($$bs =~ /^\-/) |
1678 | { |
1679 | $sign = '-'; $len--; |
1680 | } |
1681 | $len = int($len/8); # 8-digit parts, w/o '0b' |
1682 | my $val; my $i = -8; |
1683 | while ($len >= 0) |
1684 | { |
1685 | $val = substr($$bs,$i,8); |
1686 | $val =~ s/^[\-\+]?0b// if $len == 0; # for last part only |
1687 | #$val = oct('0b'.$val); # does not work on Perl prior 5.6.0 |
1688 | $val = ('0' x (8-CORE::length($val))).$val if CORE::length($val) < 8; |
1689 | $val = ord(pack('B8',$val)); |
1690 | # print "$val ",substr($$bs,$i,16),"\n"; |
1691 | $i -= 8; $len --; |
1692 | $x += $mul * $val if $val != 0; |
1693 | $mul *= $x256 if $len >= 0; # skip last mul |
1694 | } |
1695 | $x->{sign} = $sign if !$x->is_zero(); |
1696 | return $x; |
1697 | } |
1698 | |
1699 | sub _split |
1700 | { |
1701 | # (ref to num_str) return num_str |
1702 | # internal, take apart a string and return the pieces |
1703 | my $x = shift; |
1704 | |
1705 | # pre-parse input |
1706 | $$x =~ s/^\s+//g; # strip white space at front |
1707 | $$x =~ s/\s+$//g; # strip white space at end |
1708 | #$$x =~ s/\s+//g; # strip white space (no longer) |
1709 | return if $$x eq ""; |
1710 | |
1711 | return _from_hex($x) if $$x =~ /^[\-\+]?0x/; # hex string |
1712 | return _from_bin($x) if $$x =~ /^[\-\+]?0b/; # binary string |
1713 | |
1714 | return if $$x !~ /^[\-\+]?\.?[0-9]/; |
1715 | |
1716 | $$x =~ s/(\d)_(\d)/$1$2/g; # strip underscores between digits |
1717 | $$x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3 |
1718 | |
1719 | # some possible inputs: |
1720 | # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2 |
1721 | # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 |
1722 | |
1723 | #print "input: '$$x' "; |
1724 | my ($m,$e) = split /[Ee]/,$$x; |
1725 | $e = '0' if !defined $e || $e eq ""; |
1726 | # print "m '$m' e '$e'\n"; |
1727 | # sign,value for exponent,mantint,mantfrac |
1728 | my ($es,$ev,$mis,$miv,$mfv); |
1729 | # valid exponent? |
1730 | if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros |
1731 | { |
1732 | $es = $1; $ev = $2; |
1733 | #print "'$m' '$e' e: $es $ev "; |
1734 | # valid mantissa? |
1735 | return if $m eq '.' || $m eq ''; |
1736 | my ($mi,$mf) = split /\./,$m; |
1737 | $mi = '0' if !defined $mi; |
1738 | $mi .= '0' if $mi =~ /^[\-\+]?$/; |
1739 | $mf = '0' if !defined $mf || $mf eq ''; |
1740 | if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros |
1741 | { |
1742 | $mis = $1||'+'; $miv = $2; |
1743 | #print "$mis $miv"; |
1744 | # valid, existing fraction part of mantissa? |
1745 | return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros |
1746 | $mfv = $1; |
1747 | #print " split: $mis $miv . $mfv E $es $ev\n"; |
1748 | return (\$mis,\$miv,\$mfv,\$es,\$ev); |
1749 | } |
1750 | } |
1751 | return; # NaN, not a number |
1752 | } |
1753 | |
1754 | sub _digits |
1755 | { |
1756 | # computer number of digits in bigint, minus the sign |
1757 | # int() because add/sub leaves sometimes strings (like '00005') instead of |
1758 | # int ('5') in this place, causing length to fail |
1759 | my $cx = shift; |
1760 | |
1761 | #print "len: ",(@$cx-1)*5+CORE::length(int($cx->[-1])),"\n"; |
1762 | return (@$cx-1)*5+CORE::length(int($cx->[-1])); |
1763 | } |
1764 | |
1765 | sub as_number |
1766 | { |
1767 | # an object might be asked to return itself as bigint on certain overloaded |
1768 | # operations, this does exactly this, so that sub classes can simple inherit |
1769 | # it or override with their own integer conversion routine |
1770 | my $self = shift; |
1771 | |
1772 | return $self->copy(); |
1773 | } |
1774 | |
1775 | ############################################################################## |
1776 | # internal calculation routines |
1777 | |
1778 | sub acmp |
1779 | { |
1780 | # internal absolute post-normalized compare (ignore signs) |
1781 | # ref to array, ref to array, return <0, 0, >0 |
1782 | # arrays must have at least on entry, this is not checked for |
1783 | |
1784 | my ($cx, $cy) = @_; |
1785 | |
1786 | #print "$cx $cy\n"; |
1787 | my ($i,$a,$x,$y,$k); |
1788 | # calculate length based on digits, not parts |
1789 | $x = _digits($cx); $y = _digits($cy); |
1790 | # print "length: ",($x-$y),"\n"; |
1791 | return $x-$y if ($x - $y); # if different in length |
1792 | #print "full compare\n"; |
1793 | $i = 0; $a = 0; |
1794 | # first way takes 5.49 sec instead of 4.87, but has the early out advantage |
1795 | # so grep is slightly faster, but more unflexible. hm. $_ instead if $k |
1796 | # yields 5.6 instead of 5.5 sec huh? |
1797 | # manual way (abort if unequal, good for early ne) |
1798 | my $j = scalar @$cx - 1; |
1799 | while ($j >= 0) |
1800 | { |
1801 | # print "$cx->[$j] $cy->[$j] $a",$cx->[$j]-$cy->[$j],"\n"; |
1802 | last if ($a = $cx->[$j] - $cy->[$j]); $j--; |
1803 | } |
1804 | return $a; |
1805 | # while it early aborts, it is even slower than the manual variant |
1806 | #grep { return $a if ($a = $_ - $cy->[$i++]); } @$cx; |
1807 | # grep way, go trough all (bad for early ne) |
1808 | #grep { $a = $_ - $cy->[$i++]; } @$cx; |
1809 | #return $a; |
1810 | } |
1811 | |
1812 | sub cmp |
1813 | { |
1814 | # post-normalized compare for internal use (honors signs) |
1815 | # ref to array, ref to array, return < 0, 0, >0 |
1816 | my ($cx,$cy,$sx,$sy) = @_; |
1817 | |
1818 | #return 0 if (is0($cx,$sx) && is0($cy,$sy)); |
1819 | |
1820 | if ($sx eq '+') |
1821 | { |
1822 | return 1 if $sy eq '-'; # 0 check handled above |
1823 | return acmp($cx,$cy); |
1824 | } |
1825 | else |
1826 | { |
1827 | # $sx eq '-' |
1828 | return -1 if ($sy eq '+'); |
1829 | return acmp($cy,$cx); |
1830 | } |
1831 | return 0; # equal |
1832 | } |
1833 | |
1834 | sub add |
1835 | { |
1836 | # (ref to int_num_array, ref to int_num_array) |
1837 | # routine to add two base 1e5 numbers |
1838 | # stolen from Knuth Vol 2 Algorithm A pg 231 |
1839 | # there are separate routines to add and sub as per Kunth pg 233 |
1840 | # This routine clobbers up array x, but not y. |
1841 | |
1842 | my ($x,$y) = @_; |
1843 | |
1844 | # for each in Y, add Y to X and carry. If after that, something is left in |
1845 | # X, foreach in X add carry to X and then return X, carry |
1846 | # Trades one "$j++" for having to shift arrays, $j could be made integer |
1847 | # but this would impose a limit to number-length to 2**32. |
1848 | my $i; my $car = 0; my $j = 0; |
1849 | for $i (@$y) |
1850 | { |
1851 | $x->[$j] -= $BASE |
1852 | if $car = (($x->[$j] += $i + $car) >= $BASE) ? 1 : 0; |
1853 | $j++; |
1854 | } |
1855 | while ($car != 0) |
1856 | { |
1857 | $x->[$j] -= $BASE if $car = (($x->[$j] += $car) >= $BASE) ? 1 : 0; $j++; |
1858 | } |
1859 | } |
1860 | |
1861 | sub sub |
1862 | { |
1863 | # (ref to int_num_array, ref to int_num_array) |
1864 | # subtract base 1e5 numbers -- stolen from Knuth Vol 2 pg 232, $x > $y |
1865 | # subtract Y from X (X is always greater/equal!) by modifiyng x in place |
1866 | my ($sx,$sy,$s) = @_; |
1867 | |
1868 | my $car = 0; my $i; my $j = 0; |
1869 | if (!$s) |
1870 | { |
1871 | #print "case 2\n"; |
1872 | for $i (@$sx) |
1873 | { |
1874 | last unless defined $sy->[$j] || $car; |
1875 | #print "x: $i y: $sy->[$j] c: $car\n"; |
1876 | $i += $BASE if $car = (($i -= ($sy->[$j] || 0) + $car) < 0); $j++; |
1877 | #print "x: $i y: $sy->[$j-1] c: $car\n"; |
1878 | } |
1879 | # might leave leading zeros, so fix that |
1880 | _strip_zeros($sx); |
1881 | return $sx; |
1882 | } |
1883 | else |
1884 | { |
1885 | #print "case 1 (swap)\n"; |
1886 | for $i (@$sx) |
1887 | { |
1888 | last unless defined $sy->[$j] || $car; |
1889 | #print "$sy->[$j] $i $car => $sx->[$j]\n"; |
1890 | $sy->[$j] += $BASE |
1891 | if $car = (($sy->[$j] = $i-($sy->[$j]||0) - $car) < 0); |
1892 | #print "$sy->[$j] $i $car => $sy->[$j]\n"; |
1893 | $j++; |
1894 | } |
1895 | # might leave leading zeros, so fix that |
1896 | _strip_zeros($sy); |
1897 | return $sy; |
1898 | } |
1899 | } |
1900 | |
1901 | sub mul |
1902 | { |
1903 | # (BINT, BINT) return nothing |
1904 | # multiply two numbers in internal representation |
1905 | # modifies first arg, second needs not be different from first |
1906 | my ($x,$y) = @_; |
1907 | |
1908 | $x->{sign} = $x->{sign} ne $y->{sign} ? '-' : '+'; |
1909 | my @prod = (); my ($prod,$car,$cty,$xi,$yi); |
1910 | my $xv = $x->{value}; |
1911 | my $yv = $y->{value}; |
1912 | # since multiplying $x with $x fails, make copy in this case |
1913 | $yv = [@$xv] if "$xv" eq "$yv"; |
1914 | for $xi (@$xv) |
1915 | { |
1916 | $car = 0; $cty = 0; |
1917 | for $yi (@$yv) |
1918 | { |
1919 | $prod = $xi * $yi + ($prod[$cty] || 0) + $car; |
1920 | $prod[$cty++] = |
1921 | $prod - ($car = int($prod * 1e-5)) * $BASE; # see USE_MUL |
1922 | } |
1923 | $prod[$cty] += $car if $car; # need really to check for 0? |
1924 | $xi = shift @prod; |
1925 | } |
1926 | push @$xv, @prod; |
1927 | _strip_zeros($x->{value}); |
1928 | # normalize (handled last to save check for $y->is_zero() |
1929 | $x->{sign} = '+' if @$xv == 1 && $xv->[0] == 0; # not is_zero due to '-' |
1930 | } |
1931 | |
1932 | sub div |
1933 | { |
1934 | # ref to array, ref to array, modify first array and return reminder if |
1935 | # in list context |
1936 | # does no longer handle sign |
1937 | my ($x,$yorg) = @_; |
1938 | my ($car,$bar,$prd,$dd,$xi,$yi,@q,$v2,$v1); |
1939 | |
1940 | my (@d,$tmp,$q,$u2,$u1,$u0); |
1941 | |
1942 | $car = $bar = $prd = 0; |
1943 | |
1944 | my $y = [ @$yorg ]; |
1945 | if (($dd = int($BASE/($y->[-1]+1))) != 1) |
1946 | { |
1947 | for $xi (@$x) |
1948 | { |
1949 | $xi = $xi * $dd + $car; |
1950 | $xi -= ($car = int($xi * $RBASE)) * $BASE; # see USE_MUL |
1951 | } |
1952 | push(@$x, $car); $car = 0; |
1953 | for $yi (@$y) |
1954 | { |
1955 | $yi = $yi * $dd + $car; |
1956 | $yi -= ($car = int($yi * $RBASE)) * $BASE; # see USE_MUL |
1957 | } |
1958 | } |
1959 | else |
1960 | { |
1961 | push(@$x, 0); |
1962 | } |
1963 | @q = (); ($v2,$v1) = @$y[-2,-1]; |
1964 | $v2 = 0 unless $v2; |
1965 | while ($#$x > $#$y) |
1966 | { |
1967 | ($u2,$u1,$u0) = @$x[-3..-1]; |
1968 | $u2 = 0 unless $u2; |
1969 | print "oups v1 is 0, u0: $u0 $y->[-2] $y->[-1] l ",scalar @$y,"\n" |
1970 | if $v1 == 0; |
1971 | $q = (($u0 == $v1) ? 99999 : int(($u0*$BASE+$u1)/$v1)); |
1972 | --$q while ($v2*$q > ($u0*1e5+$u1-$q*$v1)*$BASE+$u2); |
1973 | if ($q) |
1974 | { |
1975 | ($car, $bar) = (0,0); |
1976 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
1977 | { |
1978 | $prd = $q * $y->[$yi] + $car; |
1979 | $prd -= ($car = int($prd * $RBASE)) * $BASE; # see USE_MUL |
1980 | $x->[$xi] += 1e5 if ($bar = (($x->[$xi] -= $prd + $bar) < 0)); |
e16b8f49 |
1981 | } |
58cde26e |
1982 | if ($x->[-1] < $car + $bar) |
1983 | { |
1984 | $car = 0; --$q; |
1985 | for ($yi = 0, $xi = $#$x-$#$y-1; $yi <= $#$y; ++$yi,++$xi) |
1986 | { |
1987 | $x->[$xi] -= 1e5 |
1988 | if ($car = (($x->[$xi] += $y->[$yi] + $car) > $BASE)); |
1989 | } |
1990 | } |
1991 | } |
1992 | pop(@$x); unshift(@q, $q); |
e16b8f49 |
1993 | } |
58cde26e |
1994 | if (wantarray) |
1995 | { |
1996 | @d = (); |
1997 | if ($dd != 1) |
1998 | { |
1999 | $car = 0; |
2000 | for $xi (reverse @$x) |
2001 | { |
2002 | $prd = $car * $BASE + $xi; |
2003 | $car = $prd - ($tmp = int($prd / $dd)) * $dd; # see USE_MUL |
2004 | unshift(@d, $tmp); |
2005 | } |
2006 | } |
2007 | else |
2008 | { |
2009 | @d = @$x; |
2010 | } |
2011 | @$x = @q; |
2012 | _strip_zeros($x); |
2013 | _strip_zeros(\@d); |
2014 | return ($x,\@d); |
2015 | } |
2016 | @$x = @q; |
2017 | _strip_zeros($x); |
2018 | return $x; |
2019 | } |
2020 | |
2021 | sub _lcm |
2022 | { |
2023 | # (BINT or num_str, BINT or num_str) return BINT |
2024 | # does modify first argument |
2025 | # LCM |
2026 | |
2027 | my $x = shift; my $ty = shift; |
2028 | return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan); |
2029 | return $x * $ty / bgcd($x,$ty); |
2030 | } |
2031 | |
2032 | sub _gcd_old |
2033 | { |
2034 | # (BINT or num_str, BINT or num_str) return BINT |
2035 | # does modify first arg |
2036 | # GCD -- Euclids algorithm E, Knuth Vol 2 pg 296 |
2037 | trace(@_); |
2038 | |
2039 | my $x = shift; my $ty = $class->new(shift); # preserve y |
2040 | return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan); |
2041 | |
2042 | while (!$ty->is_zero()) |
2043 | { |
2044 | ($x, $ty) = ($ty,bmod($x,$ty)); |
2045 | } |
2046 | $x; |
2047 | } |
2048 | |
2049 | sub _gcd |
2050 | { |
2051 | # (BINT or num_str, BINT or num_str) return BINT |
2052 | # does not modify arguments |
2053 | # GCD -- Euclids algorithm, variant L (Lehmer), Knuth Vol 3 pg 347 ff |
2054 | # unfortunately, it is slower and also seems buggy (the A=0, B=1, C=1, D=0 |
2055 | # case..) |
2056 | trace(@_); |
2057 | |
2058 | my $u = $class->new(shift); my $v = $class->new(shift); # preserve u,v |
2059 | return $u->bnan() if ($u->{sign} eq $nan) || ($v->{sign} eq $nan); |
2060 | |
2061 | $u->babs(); $v->babs(); # Euclid is valid for |u| and |v| |
2062 | |
2063 | my ($U,$V,$A,$B,$C,$D,$T,$Q); # single precision variables |
2064 | my ($t); # multiprecision variables |
2065 | |
2066 | while ($v > $BASE) |
2067 | { |
2068 | #sleep 1; |
2069 | ($u,$v) = ($v,$u) if ($u < $v); # make sure that u >= v |
2070 | #print "gcd: $u $v\n"; |
2071 | # step L1, initialize |
2072 | $A = 1; $B = 0; $C = 0; $D = 1; |
2073 | $U = $u->{value}->[-1]; # leading digits of u |
2074 | $V = $v->{value}->[-1]; # leading digits of v |
2075 | |
2076 | # step L2, test quotient |
2077 | if (($V + $C != 0) && ($V + $D != 0)) # div by zero => go to L4 |
2078 | { |
2079 | $Q = int(($U + $A)/($V + $C)); # quotient |
2080 | #print "L1 A=$A B=$B C=$C D=$D U=$U V=$V Q=$Q\n"; |
2081 | # do L3? (div by zero => go to L4) |
2082 | while ($Q == int(($U + $B)/($V + $D))) |
2083 | { |
2084 | # step L3, emulate Euclid |
2085 | #print "L3a A=$A B=$B C=$C D=$D U=$U V=$V Q=$Q\n"; |
2086 | $T = $A - $Q*$C; $A = $C; $C = $T; |
2087 | $T = $B - $Q*$D; $B = $D; $D = $T; |
2088 | $T = $U - $Q*$V; $U = $V; $V = $T; |
2089 | last if ($V + $D == 0) || ($V + $C == 0); # div by zero => L4 |
2090 | $Q = int(($U + $A)/($V + $C)); # quotient for next test |
2091 | #print "L3b A=$A B=$B C=$C D=$D U=$U V=$V Q=$Q\n"; |
2092 | } |
2093 | } |
2094 | # step L4, multiprecision step |
2095 | # was if ($B == 0) |
2096 | # in case A = 0, B = 1, C = 0 and D = 1, this case would simple swap u & v |
2097 | # and loop endless. Not sure why this happens, Knuth does not make a |
2098 | # remark about this special case. bug? |
2099 | if (($B == 0) || (($A == 0) && ($C == 1) && ($D == 0))) |
2100 | { |
2101 | #print "L4b1: u=$u v=$v\n"; |
2102 | ($u,$v) = ($v,bmod($u,$v)); |
2103 | #$t = $u % $v; $u = $v->copy(); $v = $t; |
2104 | #print "L4b12: u=$u v=$v\n"; |
2105 | } |
2106 | else |
2107 | { |
2108 | #print "L4b: $u $v $A $B $C $D\n"; |
2109 | $t = $A*$u + $B*$v; $v *= $D; $v += $C*$u; $u = $t; |
2110 | #print "L4b2: $u $v\n"; |
2111 | } |
2112 | } # back to L1 |
e16b8f49 |
2113 | |
58cde26e |
2114 | return _gcd_old($u,$v) if $v != 1; # v too small |
2115 | return $v; # 1 |
2116 | } |
2117 | |
2118 | ############################################################################### |
2119 | # this method return 0 if the object can be modified, or 1 for not |
2120 | # We use a fast use constant statement here, to avoid costly calls. Subclasses |
2121 | # may override it with special code (f.i. Math::BigInt::Constant does so) |
2122 | |
2123 | use constant modify => 0; |
2124 | |
2125 | #sub modify |
2126 | # { |
2127 | # my $self = shift; |
2128 | # my $method = shift; |
2129 | # print "original $self modify by $method\n"; |
2130 | # return 0; # $self; |
2131 | # } |
e16b8f49 |
2132 | |
a0d0e21e |
2133 | 1; |
a5f75d66 |
2134 | __END__ |
2135 | |
2136 | =head1 NAME |
2137 | |
2138 | Math::BigInt - Arbitrary size integer math package |
2139 | |
2140 | =head1 SYNOPSIS |
2141 | |
2142 | use Math::BigInt; |
58cde26e |
2143 | |
2144 | # Number creation |
2145 | $x = Math::BigInt->new($str); # defaults to 0 |
2146 | $nan = Math::BigInt->bnan(); # create a NotANumber |
2147 | $zero = Math::BigInt->bzero();# create a "+0" |
2148 | |
2149 | # Testing |
2150 | $x->is_zero(); # return whether arg is zero or not |
2151 | $x->is_nan(); # return whether arg is NaN or not |
2152 | $x->is_one(); # return true if arg is +1 |
2153 | $x->is_one('-'); # return true if arg is -1 |
2154 | $x->is_odd(); # return true if odd, false for even |
2155 | $x->is_even(); # return true if even, false for odd |
2156 | $x->bcmp($y); # compare numbers (undef,<0,=0,>0) |
2157 | $x->bacmp($y); # compare absolutely (undef,<0,=0,>0) |
2158 | $x->sign(); # return the sign, either +,- or NaN |
2159 | $x->digit($n); # return the nth digit, counting from right |
2160 | $x->digit(-$n); # return the nth digit, counting from left |
2161 | |
2162 | # The following all modify their first argument: |
2163 | |
2164 | # set |
2165 | $x->bzero(); # set $x to 0 |
2166 | $x->bnan(); # set $x to NaN |
2167 | |
2168 | $x->bneg(); # negation |
2169 | $x->babs(); # absolute value |
2170 | $x->bnorm(); # normalize (no-op) |
2171 | $x->bnot(); # two's complement (bit wise not) |
2172 | $x->binc(); # increment x by 1 |
2173 | $x->bdec(); # decrement x by 1 |
2174 | |
2175 | $x->badd($y); # addition (add $y to $x) |
2176 | $x->bsub($y); # subtraction (subtract $y from $x) |
2177 | $x->bmul($y); # multiplication (multiply $x by $y) |
2178 | $x->bdiv($y); # divide, set $x to quotient |
2179 | # return (quo,rem) or quo if scalar |
2180 | |
2181 | $x->bmod($y); # modulus (x % y) |
2182 | $x->bpow($y); # power of arguments (x ** y) |
2183 | $x->blsft($y); # left shift |
2184 | $x->brsft($y); # right shift |
2185 | $x->blsft($y,$n); # left shift, by base $n (like 10) |
2186 | $x->brsft($y,$n); # right shift, by base $n (like 10) |
2187 | |
2188 | $x->band($y); # bitwise and |
2189 | $x->bior($y); # bitwise inclusive or |
2190 | $x->bxor($y); # bitwise exclusive or |
2191 | $x->bnot(); # bitwise not (two's complement) |
2192 | |
2193 | $x->bsqrt(); # calculate square-root |
2194 | |
2195 | $x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r |
2196 | $x->bround($N); # accuracy: preserve $N digits |
2197 | $x->bfround($N); # round to $Nth digit, no-op for BigInts |
2198 | |
2199 | # The following do not modify their arguments in BigInt, but do in BigFloat: |
2200 | $x->bfloor(); # return integer less or equal than $x |
2201 | $x->bceil(); # return integer greater or equal than $x |
2202 | |
2203 | # The following do not modify their arguments: |
2204 | |
2205 | bgcd(@values); # greatest common divisor |
2206 | blcm(@values); # lowest common multiplicator |
2207 | |
2208 | $x->bstr(); # normalized string |
2209 | $x->bsstr(); # normalized string in scientific notation |
2210 | $x->length(); # return number of digits in number |
2211 | ($x,$f) = $x->length(); # length of number and length of fraction part |
2212 | |
2213 | $x->exponent(); # return exponent as BigInt |
2214 | $x->mantissa(); # return mantissa as BigInt |
2215 | $x->parts(); # return (mantissa,exponent) as BigInt |
a5f75d66 |
2216 | |
2217 | =head1 DESCRIPTION |
2218 | |
58cde26e |
2219 | All operators (inlcuding basic math operations) are overloaded if you |
2220 | declare your big integers as |
a5f75d66 |
2221 | |
58cde26e |
2222 | $i = new Math::BigInt '123_456_789_123_456_789'; |
a5f75d66 |
2223 | |
58cde26e |
2224 | Operations with overloaded operators preserve the arguments which is |
2225 | exactly what you expect. |
a5f75d66 |
2226 | |
2227 | =over 2 |
2228 | |
2229 | =item Canonical notation |
2230 | |
58cde26e |
2231 | Big integer values are strings of the form C</^[+-]\d+$/> with leading |
a5f75d66 |
2232 | zeros suppressed. |
2233 | |
58cde26e |
2234 | '-0' canonical value '-0', normalized '0' |
2235 | ' -123_123_123' canonical value '-123123123' |
2236 | '1_23_456_7890' canonical value '1234567890' |
2237 | |
a5f75d66 |
2238 | =item Input |
2239 | |
58cde26e |
2240 | Input values to these routines may be either Math::BigInt objects or |
2241 | strings of the form C</^\s*[+-]?[\d]+\.?[\d]*E?[+-]?[\d]*$/>. |
2242 | |
2243 | You can include one underscore between any two digits. |
2244 | |
2245 | This means integer values like 1.01E2 or even 1000E-2 are also accepted. |
2246 | Non integer values result in NaN. |
2247 | |
2248 | Math::BigInt::new() defaults to 0, while Math::BigInt::new('') results |
2249 | in 'NaN'. |
2250 | |
2251 | bnorm() on a BigInt object is now effectively a no-op, since the numbers |
2252 | are always stored in normalized form. On a string, it creates a BigInt |
2253 | object. |
a5f75d66 |
2254 | |
2255 | =item Output |
2256 | |
58cde26e |
2257 | Output values are BigInt objects (normalized), except for bstr(), which |
2258 | returns a string in normalized form. |
2259 | Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>, |
2260 | C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>) |
2261 | return either undef, <0, 0 or >0 and are suited for sort. |
a5f75d66 |
2262 | |
2263 | =back |
2264 | |
58cde26e |
2265 | =head2 Rounding |
a5f75d66 |
2266 | |
58cde26e |
2267 | Only C<bround()> is defined for BigInts, for further details of rounding see |
2268 | L<Math::BigFloat>. |
2269 | |
2270 | =over 2 |
a5f75d66 |
2271 | |
58cde26e |
2272 | =item bfround ( +$scale ) rounds to the $scale'th place left from the '.' |
a5f75d66 |
2273 | |
58cde26e |
2274 | =item bround ( +$scale ) preserves accuracy to $scale sighnificant digits counted from the left and paddes the number with zeros |
2275 | |
2276 | =item bround ( -$scale ) preserves accuracy to $scale significant digits counted from the right and paddes the number with zeros. |
2277 | |
2278 | =back |
2279 | |
2280 | C<bfround()> does nothing in case of negative C<$scale>. Both C<bround()> and |
2281 | C<bfround()> are a no-ops for a scale of 0. |
2282 | |
2283 | All rounding functions take as a second parameter a rounding mode from one of |
2284 | the following: 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'. |
2285 | |
2286 | The default is 'even'. By using C<< Math::BigInt->round_mode($rnd_mode); >> |
2287 | you can get and set the default round mode for subsequent rounding. |
2288 | |
2289 | The second parameter to the round functions than overrides the default |
2290 | temporarily. |
2291 | |
2292 | =head2 Internals |
2293 | |
2294 | Actual math is done in an internal format consisting of an array of |
2295 | elements of base 100000 digits with the least significant digit first. |
2296 | The sign C</^[+-]$/> is stored separately. The string 'NaN' is used to |
2297 | represent the result when input arguments are not numbers, as well as |
2298 | the result of dividing by zero. |
2299 | |
2300 | You sould neither care nor depend on the internal represantation, it might |
2301 | change without notice. Use only method calls like C<< $x->sign(); >> instead |
2302 | relying on the internal hash keys like in C<< $x->{sign}; >>. |
2303 | |
2304 | =head2 mantissa(), exponent() and parts() |
2305 | |
2306 | C<mantissa()> and C<exponent()> return the said parts of the BigInt such |
2307 | that: |
2308 | |
2309 | $m = $x->mantissa(); |
2310 | $e = $x->exponent(); |
2311 | $y = $m * ( 10 ** $e ); |
2312 | print "ok\n" if $x == $y; |
2313 | |
2314 | C<($m,$e) = $x->parts()> is just a shortcut that gives you both of them in one |
2315 | go. Both the returned mantissa and exponent do have a sign. |
2316 | |
2317 | Currently, for BigInts C<$e> will be always 0, except for NaN where it will be |
2318 | NaN and for $x == 0, then it will be 1 (to be compatible with Math::BigFlaot's |
2319 | internal representation of a zero as C<0E1>). |
2320 | |
2321 | C<$m> will always be a copy of the original number. The relation between $e |
2322 | and $m might change in the future, but will be always equivalent in a |
2323 | numerical sense, e.g. $m might get minized. |
2324 | |
2325 | =head1 EXAMPLES |
2326 | |
2327 | use Math::BigInt qw(bstr bint); |
2328 | $x = bstr("1234") # string "1234" |
2329 | $x = "$x"; # same as bstr() |
2330 | $x = bneg("1234") # Bigint "-1234" |
2331 | $x = Math::BigInt->bneg("1234"); # Bigint "-1234" |
2332 | $x = Math::BigInt->babs("-12345"); # Bigint "12345" |
2333 | $x = Math::BigInt->bnorm("-0 00"); # BigInt "0" |
2334 | $x = bint(1) + bint(2); # BigInt "3" |
2335 | $x = bint(1) + "2"; # ditto (auto-BigIntify of "2") |
2336 | $x = bint(1); # BigInt "1" |
2337 | $x = $x + 5 / 2; # BigInt "3" |
2338 | $x = $x ** 3; # BigInt "27" |
2339 | $x *= 2; # BigInt "54" |
2340 | $x = new Math::BigInt; # BigInt "0" |
2341 | $x--; # BigInt "-1" |
2342 | $x = Math::BigInt->badd(4,5) # BigInt "9" |
2343 | $x = Math::BigInt::badd(4,5) # BigInt "9" |
2344 | print $x->bsstr(); # 9e+0 |
a5f75d66 |
2345 | |
b3ac6de7 |
2346 | =head1 Autocreating constants |
2347 | |
58cde26e |
2348 | After C<use Math::BigInt ':constant'> all the B<integer> decimal constants |
2349 | in the given scope are converted to C<Math::BigInt>. This conversion |
b3ac6de7 |
2350 | happens at compile time. |
2351 | |
2352 | In particular |
2353 | |
58cde26e |
2354 | perl -MMath::BigInt=:constant -e 'print 2**100,"\n"' |
2355 | |
2356 | prints the integer value of C<2**100>. Note that without conversion of |
2357 | constants the expression 2**100 will be calculated as floating point |
2358 | number. |
2359 | |
2360 | Please note that strings and floating point constants are not affected, |
2361 | so that |
2362 | |
2363 | use Math::BigInt qw/:constant/; |
2364 | |
2365 | $x = 1234567890123456789012345678901234567890 |
2366 | + 123456789123456789; |
2367 | $x = '1234567890123456789012345678901234567890' |
2368 | + '123456789123456789'; |
b3ac6de7 |
2369 | |
58cde26e |
2370 | do both not work. You need a explicit Math::BigInt->new() around one of them. |
2371 | |
2372 | =head1 PERFORMANCE |
2373 | |
2374 | Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x |
2375 | must be made in the second case. For long numbers, the copy can eat up to 20% |
2376 | of the work (in case of addition/subtraction, less for |
2377 | multiplication/division). If $y is very small compared to $x, the form |
2378 | $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes |
2379 | more time then the actual addition. |
2380 | |
2381 | With a technic called copy-on-write the cost of copying with overload could |
2382 | be minimized or even completely avoided. This is currently not implemented. |
2383 | |
2384 | The new version of this module is slower on new(), bstr() and numify(). Some |
2385 | operations may be slower for small numbers, but are significantly faster for |
2386 | big numbers. Other operations are now constant (O(1), like bneg(), babs() |
2387 | etc), instead of O(N) and thus nearly always take much less time. |
2388 | |
2389 | For more benchmark results see http://bloodgate.com/perl/benchmarks.html |
b3ac6de7 |
2390 | |
a5f75d66 |
2391 | =head1 BUGS |
2392 | |
58cde26e |
2393 | =over 2 |
2394 | |
2395 | =item :constant and eval() |
2396 | |
2397 | Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and |
2398 | C<eval()> in your code will crash with "Out of memory". This is probably an |
2399 | overload/exporter bug. You can workaround by not having C<eval()> |
2400 | and ':constant' at the same time or upgrade your Perl. |
2401 | |
2402 | =back |
2403 | |
2404 | =head1 CAVEATS |
2405 | |
2406 | Some things might not work as you expect them. Below is documented what is |
2407 | known to be troublesome: |
2408 | |
2409 | =over 1 |
2410 | |
2411 | =item stringify, bstr(), bsstr() and 'cmp' |
2412 | |
2413 | Both stringify and bstr() now drop the leading '+'. The old code would return |
2414 | '+3', the new returns '3'. This is to be consistent with Perl and to make |
2415 | cmp (especially with overloading) to work as you expect. It also solves |
2416 | problems with Test.pm, it's ok() uses 'eq' internally. |
2417 | |
2418 | Mark said, when asked about to drop the '+' altogether, or make only cmp work: |
2419 | |
2420 | I agree (with the first alternative), don't add the '+' on positive |
2421 | numbers. It's not as important anymore with the new internal |
2422 | form for numbers. It made doing things like abs and neg easier, |
2423 | but those have to be done differently now anyway. |
2424 | |
2425 | So, the following examples will now work all as expected: |
2426 | |
2427 | use Test; |
2428 | BEGIN { plan tests => 1 } |
2429 | use Math::BigInt; |
2430 | |
2431 | my $x = new Math::BigInt 3*3; |
2432 | my $y = new Math::BigInt 3*3; |
2433 | |
2434 | ok ($x,3*3); |
2435 | print "$x eq 9" if $x eq $y; |
2436 | print "$x eq 9" if $x eq '9'; |
2437 | print "$x eq 9" if $x eq 3*3; |
2438 | |
2439 | Additionally, the following still works: |
2440 | |
2441 | print "$x == 9" if $x == $y; |
2442 | print "$x == 9" if $x == 9; |
2443 | print "$x == 9" if $x == 3*3; |
2444 | |
2445 | There is now a C<bsstr()> method to get the string in scientific notation aka |
2446 | C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr() |
2447 | for comparisation, but Perl will represent some numbers as 100 and others |
2448 | as 1e+308. If in doubt, convert both arguments to Math::BigInt before doing eq: |
2449 | |
2450 | use Test; |
2451 | BEGIN { plan tests => 3 } |
2452 | use Math::BigInt; |
2453 | |
2454 | $x = Math::BigInt->new('1e56'); $y = 1e56; |
2455 | ok ($x,$y); # will fail |
2456 | ok ($x->bsstr(),$y); # okay |
2457 | $y = Math::BigInt->new($y); |
2458 | ok ($x,$y); # okay |
2459 | |
2460 | =item int() |
2461 | |
2462 | C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a |
2463 | Perl scalar: |
2464 | |
2465 | $x = Math::BigInt->new(123); |
2466 | $y = int($x); # BigInt 123 |
2467 | $x = Math::BigFloat->new(123.45); |
2468 | $y = int($x); # BigInt 123 |
2469 | |
2470 | In all Perl versions you can use C<as_number()> for the same effect: |
2471 | |
2472 | $x = Math::BigFloat->new(123.45); |
2473 | $y = $x->as_number(); # BigInt 123 |
2474 | |
2475 | This also works for other subclasses, like Math::String. |
2476 | |
2477 | =item bdiv |
2478 | |
2479 | The following will probably not do what you expect: |
2480 | |
2481 | print $c->bdiv(10000),"\n"; |
2482 | |
2483 | It prints both quotient and reminder since print calls C<bdiv()> in list |
2484 | context. Also, C<bdiv()> will modify $c, so be carefull. You probably want |
2485 | to use |
2486 | |
2487 | print $c / 10000,"\n"; |
2488 | print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c |
2489 | |
2490 | instead. |
2491 | |
2492 | The quotient is always the greatest integer less than or equal to the |
2493 | real-valued quotient of the two operands, and the remainder (when it is |
2494 | nonzero) always has the same sign as the second operand; so, for |
2495 | example, |
2496 | |
2497 | 1 / 4 => ( 0, 1) |
2498 | 1 / -4 => (-1,-3) |
2499 | -3 / 4 => (-1, 1) |
2500 | -3 / -4 => ( 0,-3) |
2501 | |
2502 | As a consequence, the behavior of the operator % agrees with the |
2503 | behavior of Perl's built-in % operator (as documented in the perlop |
2504 | manpage), and the equation |
2505 | |
2506 | $x == ($x / $y) * $y + ($x % $y) |
2507 | |
2508 | holds true for any $x and $y, which justifies calling the two return |
2509 | values of bdiv() the quotient and remainder. |
2510 | |
2511 | Perl's 'use integer;' changes the behaviour of % and / for scalars, but will |
2512 | not change BigInt's way to do things. This is because under 'use integer' Perl |
2513 | will do what the underlying C thinks is right and this is different for each |
2514 | system. If you need BigInt's behaving exactly like Perl's 'use integer', bug |
2515 | the author to implement it ;) |
2516 | |
2517 | =item Modifying and = |
2518 | |
2519 | Beware of: |
2520 | |
2521 | $x = Math::BigFloat->new(5); |
2522 | $y = $x; |
2523 | |
2524 | It will not do what you think, e.g. making a copy of $x. Instead it just makes |
2525 | a second reference to the B<same> object and stores it in $y. Thus anything |
2526 | that modifies $x will modify $y, and vice versa. |
2527 | |
2528 | $x->bmul(2); |
2529 | print "$x, $y\n"; # prints '10, 10' |
2530 | |
2531 | If you want a true copy of $x, use: |
2532 | |
2533 | $y = $x->copy(); |
2534 | |
2535 | See also the documentation in L<overload> regarding C<=>. |
2536 | |
2537 | =item bpow |
2538 | |
2539 | C<bpow()> (and the rounding functions) now modifies the first argument and |
2540 | return it, unlike the old code which left it alone and only returned the |
2541 | result. This is to be consistent with C<badd()> etc. The first three will |
2542 | modify $x, the last one won't: |
2543 | |
2544 | print bpow($x,$i),"\n"; # modify $x |
2545 | print $x->bpow($i),"\n"; # ditto |
2546 | print $x **= $i,"\n"; # the same |
2547 | print $x ** $i,"\n"; # leave $x alone |
2548 | |
2549 | The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though. |
2550 | |
2551 | =item Overloading -$x |
2552 | |
2553 | The following: |
2554 | |
2555 | $x = -$x; |
2556 | |
2557 | is slower than |
2558 | |
2559 | $x->bneg(); |
2560 | |
2561 | since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant |
2562 | needs to preserve $x since it does not know that it later will get overwritten. |
2563 | This makes a copy of $x and takes O(N). But $x->bneg() is O(1). |
2564 | |
2565 | With Copy-On-Write, this issue will be gone. Stay tuned... |
2566 | |
2567 | =item Mixing different object types |
2568 | |
2569 | In Perl you will get a floating point value if you do one of the following: |
2570 | |
2571 | $float = 5.0 + 2; |
2572 | $float = 2 + 5.0; |
2573 | $float = 5 / 2; |
2574 | |
2575 | With overloaded math, only the first two variants will result in a BigFloat: |
2576 | |
2577 | use Math::BigInt; |
2578 | use Math::BigFloat; |
2579 | |
2580 | $mbf = Math::BigFloat->new(5); |
2581 | $mbi2 = Math::BigInteger->new(5); |
2582 | $mbi = Math::BigInteger->new(2); |
2583 | |
2584 | # what actually gets called: |
2585 | $float = $mbf + $mbi; # $mbf->badd() |
2586 | $float = $mbf / $mbi; # $mbf->bdiv() |
2587 | $integer = $mbi + $mbf; # $mbi->badd() |
2588 | $integer = $mbi2 / $mbi; # $mbi2->bdiv() |
2589 | $integer = $mbi2 / $mbf; # $mbi2->bdiv() |
2590 | |
2591 | This is because math with overloaded operators follows the first (dominating) |
2592 | operand, this one's operation is called and returns thus the result. So, |
2593 | Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether |
2594 | the result should be a Math::BigFloat or the second operant is one. |
2595 | |
2596 | To get a Math::BigFloat you either need to call the operation manually, |
2597 | make sure the operands are already of the proper type or casted to that type |
2598 | via Math::BigFloat->new(): |
2599 | |
2600 | $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5 |
2601 | |
2602 | Beware of simple "casting" the entire expression, this would only convert |
2603 | the already computed result: |
2604 | |
2605 | $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong! |
2606 | |
2607 | Beware of the order of more complicated expressions like: |
2608 | |
2609 | $integer = ($mbi2 + $mbi) / $mbf; # int / float => int |
2610 | $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto |
2611 | |
2612 | If in doubt, break the expression into simpler terms, or cast all operands |
2613 | to the desired resulting type. |
2614 | |
2615 | Scalar values are a bit different, since: |
2616 | |
2617 | $float = 2 + $mbf; |
2618 | $float = $mbf + 2; |
2619 | |
2620 | will both result in the proper type due to the way the overloaded math works. |
2621 | |
2622 | This section also applies to other overloaded math packages, like Math::String. |
2623 | |
2624 | =item bsqrt() |
2625 | |
2626 | C<bsqrt()> works only good if the result is an big integer, e.g. the square |
2627 | root of 144 is 12, but from 12 the square root is 3, regardless of rounding |
2628 | mode. |
2629 | |
2630 | If you want a better approximation of the square root, then use: |
2631 | |
2632 | $x = Math::BigFloat->new(12); |
2633 | $Math::BigFloat::precision = 0; |
2634 | Math::BigFloat->round_mode('even'); |
2635 | print $x->copy->bsqrt(),"\n"; # 4 |
2636 | |
2637 | $Math::BigFloat::precision = 2; |
2638 | print $x->bsqrt(),"\n"; # 3.46 |
2639 | print $x->bsqrt(3),"\n"; # 3.464 |
2640 | |
2641 | =back |
2642 | |
2643 | =head1 LICENSE |
2644 | |
2645 | This program is free software; you may redistribute it and/or modify it under |
2646 | the same terms as Perl itself. |
a5f75d66 |
2647 | |
58cde26e |
2648 | =head1 AUTHORS |
a5f75d66 |
2649 | |
58cde26e |
2650 | Original code by Mark Biggar, overloaded interface by Ilya Zakharevich. |
2651 | Completely rewritten by Tels http://bloodgate.com in late 2000, 2001. |
a5f75d66 |
2652 | |
2653 | =cut |