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86530b38 AT |
1 | package Math::BigInt; |
2 | ||
3 | # | |
4 | # "Mike had an infinite amount to do and a negative amount of time in which | |
5 | # to do it." - Before and After | |
6 | # | |
7 | ||
8 | # The following hash values are used: | |
9 | # value: unsigned int with actual value (as a Math::BigInt::Calc or similiar) | |
10 | # sign : +,-,NaN,+inf,-inf | |
11 | # _a : accuracy | |
12 | # _p : precision | |
13 | # _f : flags, used by MBF to flag parts of a float as untouchable | |
14 | ||
15 | # Remember not to take shortcuts ala $xs = $x->{value}; $CALC->foo($xs); since | |
16 | # underlying lib might change the reference! | |
17 | ||
18 | my $class = "Math::BigInt"; | |
19 | require 5.005; | |
20 | ||
21 | # This is a patched v1.60, containing a fix for the "1234567890\n" bug | |
22 | $VERSION = '1.60'; | |
23 | use Exporter; | |
24 | @ISA = qw( Exporter ); | |
25 | @EXPORT_OK = qw( objectify _swap bgcd blcm); | |
26 | use vars qw/$round_mode $accuracy $precision $div_scale $rnd_mode/; | |
27 | use vars qw/$upgrade $downgrade/; | |
28 | use strict; | |
29 | ||
30 | # Inside overload, the first arg is always an object. If the original code had | |
31 | # it reversed (like $x = 2 * $y), then the third paramater indicates this | |
32 | # swapping. To make it work, we use a helper routine which not only reswaps the | |
33 | # params, but also makes a new object in this case. See _swap() for details, | |
34 | # especially the cases of operators with different classes. | |
35 | ||
36 | # For overloaded ops with only one argument we simple use $_[0]->copy() to | |
37 | # preserve the argument. | |
38 | ||
39 | # Thus inheritance of overload operators becomes possible and transparent for | |
40 | # our subclasses without the need to repeat the entire overload section there. | |
41 | ||
42 | use overload | |
43 | '=' => sub { $_[0]->copy(); }, | |
44 | ||
45 | # '+' and '-' do not use _swap, since it is a triffle slower. If you want to | |
46 | # override _swap (if ever), then override overload of '+' and '-', too! | |
47 | # for sub it is a bit tricky to keep b: b-a => -a+b | |
48 | '-' => sub { my $c = $_[0]->copy; $_[2] ? | |
49 | $c->bneg()->badd($_[1]) : | |
50 | $c->bsub( $_[1]) }, | |
51 | '+' => sub { $_[0]->copy()->badd($_[1]); }, | |
52 | ||
53 | # some shortcuts for speed (assumes that reversed order of arguments is routed | |
54 | # to normal '+' and we thus can always modify first arg. If this is changed, | |
55 | # this breaks and must be adjusted.) | |
56 | '+=' => sub { $_[0]->badd($_[1]); }, | |
57 | '-=' => sub { $_[0]->bsub($_[1]); }, | |
58 | '*=' => sub { $_[0]->bmul($_[1]); }, | |
59 | '/=' => sub { scalar $_[0]->bdiv($_[1]); }, | |
60 | '%=' => sub { $_[0]->bmod($_[1]); }, | |
61 | '^=' => sub { $_[0]->bxor($_[1]); }, | |
62 | '&=' => sub { $_[0]->band($_[1]); }, | |
63 | '|=' => sub { $_[0]->bior($_[1]); }, | |
64 | '**=' => sub { $_[0]->bpow($_[1]); }, | |
65 | ||
66 | # not supported by Perl yet | |
67 | '..' => \&_pointpoint, | |
68 | ||
69 | '<=>' => sub { $_[2] ? | |
70 | ref($_[0])->bcmp($_[1],$_[0]) : | |
71 | $_[0]->bcmp($_[1])}, | |
72 | 'cmp' => sub { | |
73 | $_[2] ? | |
74 | "$_[1]" cmp $_[0]->bstr() : | |
75 | $_[0]->bstr() cmp "$_[1]" }, | |
76 | ||
77 | 'log' => sub { $_[0]->copy()->blog(); }, | |
78 | 'int' => sub { $_[0]->copy(); }, | |
79 | 'neg' => sub { $_[0]->copy()->bneg(); }, | |
80 | 'abs' => sub { $_[0]->copy()->babs(); }, | |
81 | 'sqrt' => sub { $_[0]->copy()->bsqrt(); }, | |
82 | '~' => sub { $_[0]->copy()->bnot(); }, | |
83 | ||
84 | '*' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bmul($a[1]); }, | |
85 | '/' => sub { my @a = ref($_[0])->_swap(@_);scalar $a[0]->bdiv($a[1]);}, | |
86 | '%' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bmod($a[1]); }, | |
87 | '**' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bpow($a[1]); }, | |
88 | '<<' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->blsft($a[1]); }, | |
89 | '>>' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->brsft($a[1]); }, | |
90 | ||
91 | '&' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->band($a[1]); }, | |
92 | '|' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bior($a[1]); }, | |
93 | '^' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bxor($a[1]); }, | |
94 | ||
95 | # can modify arg of ++ and --, so avoid a new-copy for speed, but don't | |
96 | # use $_[0]->__one(), it modifies $_[0] to be 1! | |
97 | '++' => sub { $_[0]->binc() }, | |
98 | '--' => sub { $_[0]->bdec() }, | |
99 | ||
100 | # if overloaded, O(1) instead of O(N) and twice as fast for small numbers | |
101 | 'bool' => sub { | |
102 | # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/ | |
103 | # v5.6.1 dumps on that: return !$_[0]->is_zero() || undef; :-( | |
104 | my $t = !$_[0]->is_zero(); | |
105 | undef $t if $t == 0; | |
106 | $t; | |
107 | }, | |
108 | ||
109 | # the original qw() does not work with the TIESCALAR below, why? | |
110 | # Order of arguments unsignificant | |
111 | '""' => sub { $_[0]->bstr(); }, | |
112 | '0+' => sub { $_[0]->numify(); } | |
113 | ; | |
114 | ||
115 | ############################################################################## | |
116 | # global constants, flags and accessory | |
117 | ||
118 | use constant MB_NEVER_ROUND => 0x0001; | |
119 | ||
120 | my $NaNOK=1; # are NaNs ok? | |
121 | my $nan = 'NaN'; # constants for easier life | |
122 | ||
123 | my $CALC = 'Math::BigInt::Calc'; # module to do low level math | |
124 | my $IMPORT = 0; # did import() yet? | |
125 | ||
126 | $round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc' | |
127 | $accuracy = undef; | |
128 | $precision = undef; | |
129 | $div_scale = 40; | |
130 | ||
131 | $upgrade = undef; # default is no upgrade | |
132 | $downgrade = undef; # default is no downgrade | |
133 | ||
134 | ############################################################################## | |
135 | # the old code had $rnd_mode, so we need to support it, too | |
136 | ||
137 | $rnd_mode = 'even'; | |
138 | sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; } | |
139 | sub FETCH { return $round_mode; } | |
140 | sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); } | |
141 | ||
142 | BEGIN { tie $rnd_mode, 'Math::BigInt'; } | |
143 | ||
144 | ############################################################################## | |
145 | ||
146 | sub round_mode | |
147 | { | |
148 | no strict 'refs'; | |
149 | # make Class->round_mode() work | |
150 | my $self = shift; | |
151 | my $class = ref($self) || $self || __PACKAGE__; | |
152 | if (defined $_[0]) | |
153 | { | |
154 | my $m = shift; | |
155 | die "Unknown round mode $m" | |
156 | if $m !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/; | |
157 | return ${"${class}::round_mode"} = $m; | |
158 | } | |
159 | return ${"${class}::round_mode"}; | |
160 | } | |
161 | ||
162 | sub upgrade | |
163 | { | |
164 | no strict 'refs'; | |
165 | # make Class->upgrade() work | |
166 | my $self = shift; | |
167 | my $class = ref($self) || $self || __PACKAGE__; | |
168 | # need to set new value? | |
169 | if (@_ > 0) | |
170 | { | |
171 | my $u = shift; | |
172 | return ${"${class}::upgrade"} = $u; | |
173 | } | |
174 | return ${"${class}::upgrade"}; | |
175 | } | |
176 | ||
177 | sub downgrade | |
178 | { | |
179 | no strict 'refs'; | |
180 | # make Class->downgrade() work | |
181 | my $self = shift; | |
182 | my $class = ref($self) || $self || __PACKAGE__; | |
183 | # need to set new value? | |
184 | if (@_ > 0) | |
185 | { | |
186 | my $u = shift; | |
187 | return ${"${class}::downgrade"} = $u; | |
188 | } | |
189 | return ${"${class}::downgrade"}; | |
190 | } | |
191 | ||
192 | sub div_scale | |
193 | { | |
194 | no strict 'refs'; | |
195 | # make Class->round_mode() work | |
196 | my $self = shift; | |
197 | my $class = ref($self) || $self || __PACKAGE__; | |
198 | if (defined $_[0]) | |
199 | { | |
200 | die ('div_scale must be greater than zero') if $_[0] < 0; | |
201 | ${"${class}::div_scale"} = shift; | |
202 | } | |
203 | return ${"${class}::div_scale"}; | |
204 | } | |
205 | ||
206 | sub accuracy | |
207 | { | |
208 | # $x->accuracy($a); ref($x) $a | |
209 | # $x->accuracy(); ref($x) | |
210 | # Class->accuracy(); class | |
211 | # Class->accuracy($a); class $a | |
212 | ||
213 | my $x = shift; | |
214 | my $class = ref($x) || $x || __PACKAGE__; | |
215 | ||
216 | no strict 'refs'; | |
217 | # need to set new value? | |
218 | if (@_ > 0) | |
219 | { | |
220 | my $a = shift; | |
221 | die ('accuracy must not be zero') if defined $a && $a == 0; | |
222 | if (ref($x)) | |
223 | { | |
224 | # $object->accuracy() or fallback to global | |
225 | $x->bround($a) if defined $a; | |
226 | $x->{_a} = $a; # set/overwrite, even if not rounded | |
227 | $x->{_p} = undef; # clear P | |
228 | } | |
229 | else | |
230 | { | |
231 | # set global | |
232 | ${"${class}::accuracy"} = $a; | |
233 | ${"${class}::precision"} = undef; # clear P | |
234 | } | |
235 | return $a; # shortcut | |
236 | } | |
237 | ||
238 | my $r; | |
239 | # $object->accuracy() or fallback to global | |
240 | $r = $x->{_a} if ref($x); | |
241 | # but don't return global undef, when $x's accuracy is 0! | |
242 | $r = ${"${class}::accuracy"} if !defined $r; | |
243 | $r; | |
244 | } | |
245 | ||
246 | sub precision | |
247 | { | |
248 | # $x->precision($p); ref($x) $p | |
249 | # $x->precision(); ref($x) | |
250 | # Class->precision(); class | |
251 | # Class->precision($p); class $p | |
252 | ||
253 | my $x = shift; | |
254 | my $class = ref($x) || $x || __PACKAGE__; | |
255 | ||
256 | no strict 'refs'; | |
257 | # need to set new value? | |
258 | if (@_ > 0) | |
259 | { | |
260 | my $p = shift; | |
261 | if (ref($x)) | |
262 | { | |
263 | # $object->precision() or fallback to global | |
264 | $x->bfround($p) if defined $p; | |
265 | $x->{_p} = $p; # set/overwrite, even if not rounded | |
266 | $x->{_a} = undef; # clear A | |
267 | } | |
268 | else | |
269 | { | |
270 | # set global | |
271 | ${"${class}::precision"} = $p; | |
272 | ${"${class}::accuracy"} = undef; # clear A | |
273 | } | |
274 | return $p; # shortcut | |
275 | } | |
276 | ||
277 | my $r; | |
278 | # $object->precision() or fallback to global | |
279 | $r = $x->{_p} if ref($x); | |
280 | # but don't return global undef, when $x's precision is 0! | |
281 | $r = ${"${class}::precision"} if !defined $r; | |
282 | $r; | |
283 | } | |
284 | ||
285 | sub config | |
286 | { | |
287 | # return (later set?) configuration data as hash ref | |
288 | my $class = shift || 'Math::BigInt'; | |
289 | ||
290 | no strict 'refs'; | |
291 | my $lib = $CALC; | |
292 | my $cfg = { | |
293 | lib => $lib, | |
294 | lib_version => ${"${lib}::VERSION"}, | |
295 | class => $class, | |
296 | }; | |
297 | foreach ( | |
298 | qw/upgrade downgrade precision accuracy round_mode VERSION div_scale/) | |
299 | { | |
300 | $cfg->{lc($_)} = ${"${class}::$_"}; | |
301 | }; | |
302 | $cfg; | |
303 | } | |
304 | ||
305 | sub _scale_a | |
306 | { | |
307 | # select accuracy parameter based on precedence, | |
308 | # used by bround() and bfround(), may return undef for scale (means no op) | |
309 | my ($x,$s,$m,$scale,$mode) = @_; | |
310 | $scale = $x->{_a} if !defined $scale; | |
311 | $scale = $s if (!defined $scale); | |
312 | $mode = $m if !defined $mode; | |
313 | return ($scale,$mode); | |
314 | } | |
315 | ||
316 | sub _scale_p | |
317 | { | |
318 | # select precision parameter based on precedence, | |
319 | # used by bround() and bfround(), may return undef for scale (means no op) | |
320 | my ($x,$s,$m,$scale,$mode) = @_; | |
321 | $scale = $x->{_p} if !defined $scale; | |
322 | $scale = $s if (!defined $scale); | |
323 | $mode = $m if !defined $mode; | |
324 | return ($scale,$mode); | |
325 | } | |
326 | ||
327 | ############################################################################## | |
328 | # constructors | |
329 | ||
330 | sub copy | |
331 | { | |
332 | my ($c,$x); | |
333 | if (@_ > 1) | |
334 | { | |
335 | # if two arguments, the first one is the class to "swallow" subclasses | |
336 | ($c,$x) = @_; | |
337 | } | |
338 | else | |
339 | { | |
340 | $x = shift; | |
341 | $c = ref($x); | |
342 | } | |
343 | return unless ref($x); # only for objects | |
344 | ||
345 | my $self = {}; bless $self,$c; | |
346 | my $r; | |
347 | foreach my $k (keys %$x) | |
348 | { | |
349 | if ($k eq 'value') | |
350 | { | |
351 | $self->{value} = $CALC->_copy($x->{value}); next; | |
352 | } | |
353 | if (!($r = ref($x->{$k}))) | |
354 | { | |
355 | $self->{$k} = $x->{$k}; next; | |
356 | } | |
357 | if ($r eq 'SCALAR') | |
358 | { | |
359 | $self->{$k} = \${$x->{$k}}; | |
360 | } | |
361 | elsif ($r eq 'ARRAY') | |
362 | { | |
363 | $self->{$k} = [ @{$x->{$k}} ]; | |
364 | } | |
365 | elsif ($r eq 'HASH') | |
366 | { | |
367 | # only one level deep! | |
368 | foreach my $h (keys %{$x->{$k}}) | |
369 | { | |
370 | $self->{$k}->{$h} = $x->{$k}->{$h}; | |
371 | } | |
372 | } | |
373 | else # normal ref | |
374 | { | |
375 | my $xk = $x->{$k}; | |
376 | if ($xk->can('copy')) | |
377 | { | |
378 | $self->{$k} = $xk->copy(); | |
379 | } | |
380 | else | |
381 | { | |
382 | $self->{$k} = $xk->new($xk); | |
383 | } | |
384 | } | |
385 | } | |
386 | $self; | |
387 | } | |
388 | ||
389 | sub new | |
390 | { | |
391 | # create a new BigInt object from a string or another BigInt object. | |
392 | # see hash keys documented at top | |
393 | ||
394 | # the argument could be an object, so avoid ||, && etc on it, this would | |
395 | # cause costly overloaded code to be called. The only allowed ops are | |
396 | # ref() and defined. | |
397 | ||
398 | my ($class,$wanted,$a,$p,$r) = @_; | |
399 | ||
400 | # avoid numify-calls by not using || on $wanted! | |
401 | return $class->bzero($a,$p) if !defined $wanted; # default to 0 | |
402 | return $class->copy($wanted,$a,$p,$r) | |
403 | if ref($wanted) && $wanted->isa($class); # MBI or subclass | |
404 | ||
405 | $class->import() if $IMPORT == 0; # make require work | |
406 | ||
407 | my $self = bless {}, $class; | |
408 | ||
409 | # shortcut for "normal" numbers | |
410 | if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*\z/)) | |
411 | { | |
412 | $self->{sign} = $1 || '+'; | |
413 | my $ref = \$wanted; | |
414 | if ($wanted =~ /^[+-]/) | |
415 | { | |
416 | # remove sign without touching wanted | |
417 | my $t = $wanted; $t =~ s/^[+-]//; $ref = \$t; | |
418 | } | |
419 | $self->{value} = $CALC->_new($ref); | |
420 | no strict 'refs'; | |
421 | if ( (defined $a) || (defined $p) | |
422 | || (defined ${"${class}::precision"}) | |
423 | || (defined ${"${class}::accuracy"}) | |
424 | ) | |
425 | { | |
426 | $self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p); | |
427 | } | |
428 | return $self; | |
429 | } | |
430 | ||
431 | # handle '+inf', '-inf' first | |
432 | if ($wanted =~ /^[+-]?inf$/) | |
433 | { | |
434 | $self->{value} = $CALC->_zero(); | |
435 | $self->{sign} = $wanted; $self->{sign} = '+inf' if $self->{sign} eq 'inf'; | |
436 | return $self; | |
437 | } | |
438 | # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign | |
439 | my ($mis,$miv,$mfv,$es,$ev) = _split(\$wanted); | |
440 | if (!ref $mis) | |
441 | { | |
442 | die "$wanted is not a number initialized to $class" if !$NaNOK; | |
443 | #print "NaN 1\n"; | |
444 | $self->{value} = $CALC->_zero(); | |
445 | $self->{sign} = $nan; | |
446 | return $self; | |
447 | } | |
448 | if (!ref $miv) | |
449 | { | |
450 | # _from_hex or _from_bin | |
451 | $self->{value} = $mis->{value}; | |
452 | $self->{sign} = $mis->{sign}; | |
453 | return $self; # throw away $mis | |
454 | } | |
455 | # make integer from mantissa by adjusting exp, then convert to bigint | |
456 | $self->{sign} = $$mis; # store sign | |
457 | $self->{value} = $CALC->_zero(); # for all the NaN cases | |
458 | my $e = int("$$es$$ev"); # exponent (avoid recursion) | |
459 | if ($e > 0) | |
460 | { | |
461 | my $diff = $e - CORE::length($$mfv); | |
462 | if ($diff < 0) # Not integer | |
463 | { | |
464 | #print "NOI 1\n"; | |
465 | return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade; | |
466 | $self->{sign} = $nan; | |
467 | } | |
468 | else # diff >= 0 | |
469 | { | |
470 | # adjust fraction and add it to value | |
471 | # print "diff > 0 $$miv\n"; | |
472 | $$miv = $$miv . ($$mfv . '0' x $diff); | |
473 | } | |
474 | } | |
475 | else | |
476 | { | |
477 | if ($$mfv ne '') # e <= 0 | |
478 | { | |
479 | # fraction and negative/zero E => NOI | |
480 | #print "NOI 2 \$\$mfv '$$mfv'\n"; | |
481 | return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade; | |
482 | $self->{sign} = $nan; | |
483 | } | |
484 | elsif ($e < 0) | |
485 | { | |
486 | # xE-y, and empty mfv | |
487 | #print "xE-y\n"; | |
488 | $e = abs($e); | |
489 | if ($$miv !~ s/0{$e}$//) # can strip so many zero's? | |
490 | { | |
491 | #print "NOI 3\n"; | |
492 | return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade; | |
493 | $self->{sign} = $nan; | |
494 | } | |
495 | } | |
496 | } | |
497 | $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0 | |
498 | $self->{value} = $CALC->_new($miv) if $self->{sign} =~ /^[+-]$/; | |
499 | # if any of the globals is set, use them to round and store them inside $self | |
500 | # do not round for new($x,undef,undef) since that is used by MBF to signal | |
501 | # no rounding | |
502 | $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p; | |
503 | $self; | |
504 | } | |
505 | ||
506 | sub bnan | |
507 | { | |
508 | # create a bigint 'NaN', if given a BigInt, set it to 'NaN' | |
509 | my $self = shift; | |
510 | $self = $class if !defined $self; | |
511 | if (!ref($self)) | |
512 | { | |
513 | my $c = $self; $self = {}; bless $self, $c; | |
514 | } | |
515 | $self->import() if $IMPORT == 0; # make require work | |
516 | return if $self->modify('bnan'); | |
517 | my $c = ref($self); | |
518 | if ($self->can('_bnan')) | |
519 | { | |
520 | # use subclass to initialize | |
521 | $self->_bnan(); | |
522 | } | |
523 | else | |
524 | { | |
525 | # otherwise do our own thing | |
526 | $self->{value} = $CALC->_zero(); | |
527 | } | |
528 | $self->{sign} = $nan; | |
529 | delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly | |
530 | return $self; | |
531 | } | |
532 | ||
533 | sub binf | |
534 | { | |
535 | # create a bigint '+-inf', if given a BigInt, set it to '+-inf' | |
536 | # the sign is either '+', or if given, used from there | |
537 | my $self = shift; | |
538 | my $sign = shift; $sign = '+' if !defined $sign || $sign !~ /^-(inf)?$/; | |
539 | $self = $class if !defined $self; | |
540 | if (!ref($self)) | |
541 | { | |
542 | my $c = $self; $self = {}; bless $self, $c; | |
543 | } | |
544 | $self->import() if $IMPORT == 0; # make require work | |
545 | return if $self->modify('binf'); | |
546 | my $c = ref($self); | |
547 | if ($self->can('_binf')) | |
548 | { | |
549 | # use subclass to initialize | |
550 | $self->_binf(); | |
551 | } | |
552 | else | |
553 | { | |
554 | # otherwise do our own thing | |
555 | $self->{value} = $CALC->_zero(); | |
556 | } | |
557 | $sign = $sign . 'inf' if $sign !~ /inf$/; # - => -inf | |
558 | $self->{sign} = $sign; | |
559 | ($self->{_a},$self->{_p}) = @_; # take over requested rounding | |
560 | return $self; | |
561 | } | |
562 | ||
563 | sub bzero | |
564 | { | |
565 | # create a bigint '+0', if given a BigInt, set it to 0 | |
566 | my $self = shift; | |
567 | $self = $class if !defined $self; | |
568 | ||
569 | if (!ref($self)) | |
570 | { | |
571 | my $c = $self; $self = {}; bless $self, $c; | |
572 | } | |
573 | $self->import() if $IMPORT == 0; # make require work | |
574 | return if $self->modify('bzero'); | |
575 | ||
576 | if ($self->can('_bzero')) | |
577 | { | |
578 | # use subclass to initialize | |
579 | $self->_bzero(); | |
580 | } | |
581 | else | |
582 | { | |
583 | # otherwise do our own thing | |
584 | $self->{value} = $CALC->_zero(); | |
585 | } | |
586 | $self->{sign} = '+'; | |
587 | if (@_ > 0) | |
588 | { | |
589 | if (@_ > 3) | |
590 | { | |
591 | # call like: $x->bzero($a,$p,$r,$y); | |
592 | ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_); | |
593 | } | |
594 | else | |
595 | { | |
596 | $self->{_a} = $_[0] | |
597 | if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a})); | |
598 | $self->{_p} = $_[1] | |
599 | if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p})); | |
600 | } | |
601 | } | |
602 | $self; | |
603 | } | |
604 | ||
605 | sub bone | |
606 | { | |
607 | # create a bigint '+1' (or -1 if given sign '-'), | |
608 | # if given a BigInt, set it to +1 or -1, respecively | |
609 | my $self = shift; | |
610 | my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-'; | |
611 | $self = $class if !defined $self; | |
612 | ||
613 | if (!ref($self)) | |
614 | { | |
615 | my $c = $self; $self = {}; bless $self, $c; | |
616 | } | |
617 | $self->import() if $IMPORT == 0; # make require work | |
618 | return if $self->modify('bone'); | |
619 | ||
620 | if ($self->can('_bone')) | |
621 | { | |
622 | # use subclass to initialize | |
623 | $self->_bone(); | |
624 | } | |
625 | else | |
626 | { | |
627 | # otherwise do our own thing | |
628 | $self->{value} = $CALC->_one(); | |
629 | } | |
630 | $self->{sign} = $sign; | |
631 | if (@_ > 0) | |
632 | { | |
633 | if (@_ > 3) | |
634 | { | |
635 | # call like: $x->bone($sign,$a,$p,$r,$y); | |
636 | ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_); | |
637 | } | |
638 | else | |
639 | { | |
640 | $self->{_a} = $_[0] | |
641 | if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a})); | |
642 | $self->{_p} = $_[1] | |
643 | if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p})); | |
644 | } | |
645 | } | |
646 | $self; | |
647 | } | |
648 | ||
649 | ############################################################################## | |
650 | # string conversation | |
651 | ||
652 | sub bsstr | |
653 | { | |
654 | # (ref to BFLOAT or num_str ) return num_str | |
655 | # Convert number from internal format to scientific string format. | |
656 | # internal format is always normalized (no leading zeros, "-0E0" => "+0E0") | |
657 | my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x); | |
658 | # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
659 | ||
660 | if ($x->{sign} !~ /^[+-]$/) | |
661 | { | |
662 | return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN | |
663 | return 'inf'; # +inf | |
664 | } | |
665 | my ($m,$e) = $x->parts(); | |
666 | # e can only be positive | |
667 | my $sign = 'e+'; | |
668 | # MBF: my $s = $e->{sign}; $s = '' if $s eq '-'; my $sep = 'e'.$s; | |
669 | return $m->bstr().$sign.$e->bstr(); | |
670 | } | |
671 | ||
672 | sub bstr | |
673 | { | |
674 | # make a string from bigint object | |
675 | my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x); | |
676 | # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
677 | ||
678 | if ($x->{sign} !~ /^[+-]$/) | |
679 | { | |
680 | return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN | |
681 | return 'inf'; # +inf | |
682 | } | |
683 | my $es = ''; $es = $x->{sign} if $x->{sign} eq '-'; | |
684 | return $es.${$CALC->_str($x->{value})}; | |
685 | } | |
686 | ||
687 | sub numify | |
688 | { | |
689 | # Make a "normal" scalar from a BigInt object | |
690 | my $x = shift; $x = $class->new($x) unless ref $x; | |
691 | return $x->{sign} if $x->{sign} !~ /^[+-]$/; | |
692 | my $num = $CALC->_num($x->{value}); | |
693 | return -$num if $x->{sign} eq '-'; | |
694 | $num; | |
695 | } | |
696 | ||
697 | ############################################################################## | |
698 | # public stuff (usually prefixed with "b") | |
699 | ||
700 | sub sign | |
701 | { | |
702 | # return the sign of the number: +/-/-inf/+inf/NaN | |
703 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
704 | ||
705 | $x->{sign}; | |
706 | } | |
707 | ||
708 | sub _find_round_parameters | |
709 | { | |
710 | # After any operation or when calling round(), the result is rounded by | |
711 | # regarding the A & P from arguments, local parameters, or globals. | |
712 | ||
713 | # This procedure finds the round parameters, but it is for speed reasons | |
714 | # duplicated in round. Otherwise, it is tested by the testsuite and used | |
715 | # by fdiv(). | |
716 | ||
717 | my ($self,$a,$p,$r,@args) = @_; | |
718 | # $a accuracy, if given by caller | |
719 | # $p precision, if given by caller | |
720 | # $r round_mode, if given by caller | |
721 | # @args all 'other' arguments (0 for unary, 1 for binary ops) | |
722 | ||
723 | # leave bigfloat parts alone | |
724 | return ($self) if exists $self->{_f} && $self->{_f} & MB_NEVER_ROUND != 0; | |
725 | ||
726 | my $c = ref($self); # find out class of argument(s) | |
727 | no strict 'refs'; | |
728 | ||
729 | # now pick $a or $p, but only if we have got "arguments" | |
730 | if (!defined $a) | |
731 | { | |
732 | foreach ($self,@args) | |
733 | { | |
734 | # take the defined one, or if both defined, the one that is smaller | |
735 | $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a); | |
736 | } | |
737 | } | |
738 | if (!defined $p) | |
739 | { | |
740 | # even if $a is defined, take $p, to signal error for both defined | |
741 | foreach ($self,@args) | |
742 | { | |
743 | # take the defined one, or if both defined, the one that is bigger | |
744 | # -2 > -3, and 3 > 2 | |
745 | $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p); | |
746 | } | |
747 | } | |
748 | # if still none defined, use globals (#2) | |
749 | $a = ${"$c\::accuracy"} unless defined $a; | |
750 | $p = ${"$c\::precision"} unless defined $p; | |
751 | ||
752 | # no rounding today? | |
753 | return ($self) unless defined $a || defined $p; # early out | |
754 | ||
755 | # set A and set P is an fatal error | |
756 | return ($self->bnan()) if defined $a && defined $p; | |
757 | ||
758 | $r = ${"$c\::round_mode"} unless defined $r; | |
759 | die "Unknown round mode '$r'" if $r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/; | |
760 | ||
761 | return ($self,$a,$p,$r); | |
762 | } | |
763 | ||
764 | sub round | |
765 | { | |
766 | # Round $self according to given parameters, or given second argument's | |
767 | # parameters or global defaults | |
768 | ||
769 | # for speed reasons, _find_round_parameters is embeded here: | |
770 | ||
771 | my ($self,$a,$p,$r,@args) = @_; | |
772 | # $a accuracy, if given by caller | |
773 | # $p precision, if given by caller | |
774 | # $r round_mode, if given by caller | |
775 | # @args all 'other' arguments (0 for unary, 1 for binary ops) | |
776 | ||
777 | # leave bigfloat parts alone | |
778 | return ($self) if exists $self->{_f} && $self->{_f} & MB_NEVER_ROUND != 0; | |
779 | ||
780 | my $c = ref($self); # find out class of argument(s) | |
781 | no strict 'refs'; | |
782 | ||
783 | # now pick $a or $p, but only if we have got "arguments" | |
784 | if (!defined $a) | |
785 | { | |
786 | foreach ($self,@args) | |
787 | { | |
788 | # take the defined one, or if both defined, the one that is smaller | |
789 | $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a); | |
790 | } | |
791 | } | |
792 | if (!defined $p) | |
793 | { | |
794 | # even if $a is defined, take $p, to signal error for both defined | |
795 | foreach ($self,@args) | |
796 | { | |
797 | # take the defined one, or if both defined, the one that is bigger | |
798 | # -2 > -3, and 3 > 2 | |
799 | $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p); | |
800 | } | |
801 | } | |
802 | # if still none defined, use globals (#2) | |
803 | $a = ${"$c\::accuracy"} unless defined $a; | |
804 | $p = ${"$c\::precision"} unless defined $p; | |
805 | ||
806 | # no rounding today? | |
807 | return $self unless defined $a || defined $p; # early out | |
808 | ||
809 | # set A and set P is an fatal error | |
810 | return $self->bnan() if defined $a && defined $p; | |
811 | ||
812 | $r = ${"$c\::round_mode"} unless defined $r; | |
813 | die "Unknown round mode '$r'" if $r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/; | |
814 | ||
815 | # now round, by calling either fround or ffround: | |
816 | if (defined $a) | |
817 | { | |
818 | $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a; | |
819 | } | |
820 | else # both can't be undefined due to early out | |
821 | { | |
822 | $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p; | |
823 | } | |
824 | $self->bnorm(); # after round, normalize | |
825 | } | |
826 | ||
827 | sub bnorm | |
828 | { | |
829 | # (numstr or BINT) return BINT | |
830 | # Normalize number -- no-op here | |
831 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
832 | $x; | |
833 | } | |
834 | ||
835 | sub babs | |
836 | { | |
837 | # (BINT or num_str) return BINT | |
838 | # make number absolute, or return absolute BINT from string | |
839 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
840 | ||
841 | return $x if $x->modify('babs'); | |
842 | # post-normalized abs for internal use (does nothing for NaN) | |
843 | $x->{sign} =~ s/^-/+/; | |
844 | $x; | |
845 | } | |
846 | ||
847 | sub bneg | |
848 | { | |
849 | # (BINT or num_str) return BINT | |
850 | # negate number or make a negated number from string | |
851 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
852 | ||
853 | return $x if $x->modify('bneg'); | |
854 | ||
855 | # for +0 dont negate (to have always normalized) | |
856 | $x->{sign} =~ tr/+-/-+/ if !$x->is_zero(); # does nothing for NaN | |
857 | $x; | |
858 | } | |
859 | ||
860 | sub bcmp | |
861 | { | |
862 | # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort) | |
863 | # (BINT or num_str, BINT or num_str) return cond_code | |
864 | ||
865 | # set up parameters | |
866 | my ($self,$x,$y) = (ref($_[0]),@_); | |
867 | ||
868 | # objectify is costly, so avoid it | |
869 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
870 | { | |
871 | ($self,$x,$y) = objectify(2,@_); | |
872 | } | |
873 | ||
874 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/)) | |
875 | { | |
876 | # handle +-inf and NaN | |
877 | return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); | |
878 | return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/; | |
879 | return +1 if $x->{sign} eq '+inf'; | |
880 | return -1 if $x->{sign} eq '-inf'; | |
881 | return -1 if $y->{sign} eq '+inf'; | |
882 | return +1; | |
883 | } | |
884 | # check sign for speed first | |
885 | return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y | |
886 | return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0 | |
887 | ||
888 | # have same sign, so compare absolute values. Don't make tests for zero here | |
889 | # because it's actually slower than testin in Calc (especially w/ Pari et al) | |
890 | ||
891 | # post-normalized compare for internal use (honors signs) | |
892 | if ($x->{sign} eq '+') | |
893 | { | |
894 | # $x and $y both > 0 | |
895 | return $CALC->_acmp($x->{value},$y->{value}); | |
896 | } | |
897 | ||
898 | # $x && $y both < 0 | |
899 | $CALC->_acmp($y->{value},$x->{value}); # swaped (lib returns 0,1,-1) | |
900 | } | |
901 | ||
902 | sub bacmp | |
903 | { | |
904 | # Compares 2 values, ignoring their signs. | |
905 | # Returns one of undef, <0, =0, >0. (suitable for sort) | |
906 | # (BINT, BINT) return cond_code | |
907 | ||
908 | # set up parameters | |
909 | my ($self,$x,$y) = (ref($_[0]),@_); | |
910 | # objectify is costly, so avoid it | |
911 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
912 | { | |
913 | ($self,$x,$y) = objectify(2,@_); | |
914 | } | |
915 | ||
916 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/)) | |
917 | { | |
918 | # handle +-inf and NaN | |
919 | return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); | |
920 | return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/; | |
921 | return +1; # inf is always bigger | |
922 | } | |
923 | $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1 | |
924 | } | |
925 | ||
926 | sub badd | |
927 | { | |
928 | # add second arg (BINT or string) to first (BINT) (modifies first) | |
929 | # return result as BINT | |
930 | ||
931 | # set up parameters | |
932 | my ($self,$x,$y,@r) = (ref($_[0]),@_); | |
933 | # objectify is costly, so avoid it | |
934 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
935 | { | |
936 | ($self,$x,$y,@r) = objectify(2,@_); | |
937 | } | |
938 | ||
939 | return $x if $x->modify('badd'); | |
940 | return $upgrade->badd($x,$y,@r) if defined $upgrade && | |
941 | ((!$x->isa($self)) || (!$y->isa($self))); | |
942 | ||
943 | $r[3] = $y; # no push! | |
944 | # inf and NaN handling | |
945 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/)) | |
946 | { | |
947 | # NaN first | |
948 | return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); | |
949 | # inf handling | |
950 | if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/)) | |
951 | { | |
952 | # +inf++inf or -inf+-inf => same, rest is NaN | |
953 | return $x if $x->{sign} eq $y->{sign}; | |
954 | return $x->bnan(); | |
955 | } | |
956 | # +-inf + something => +inf | |
957 | # something +-inf => +-inf | |
958 | $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/; | |
959 | return $x; | |
960 | } | |
961 | ||
962 | my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs | |
963 | ||
964 | if ($sx eq $sy) | |
965 | { | |
966 | $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add | |
967 | $x->{sign} = $sx; | |
968 | } | |
969 | else | |
970 | { | |
971 | my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare | |
972 | if ($a > 0) | |
973 | { | |
974 | #print "swapped sub (a=$a)\n"; | |
975 | $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap | |
976 | $x->{sign} = $sy; | |
977 | } | |
978 | elsif ($a == 0) | |
979 | { | |
980 | # speedup, if equal, set result to 0 | |
981 | #print "equal sub, result = 0\n"; | |
982 | $x->{value} = $CALC->_zero(); | |
983 | $x->{sign} = '+'; | |
984 | } | |
985 | else # a < 0 | |
986 | { | |
987 | #print "unswapped sub (a=$a)\n"; | |
988 | $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub | |
989 | $x->{sign} = $sx; | |
990 | } | |
991 | } | |
992 | $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; | |
993 | $x; | |
994 | } | |
995 | ||
996 | sub bsub | |
997 | { | |
998 | # (BINT or num_str, BINT or num_str) return num_str | |
999 | # subtract second arg from first, modify first | |
1000 | ||
1001 | # set up parameters | |
1002 | my ($self,$x,$y,@r) = (ref($_[0]),@_); | |
1003 | # objectify is costly, so avoid it | |
1004 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1005 | { | |
1006 | ($self,$x,$y,@r) = objectify(2,@_); | |
1007 | } | |
1008 | ||
1009 | return $x if $x->modify('bsub'); | |
1010 | ||
1011 | # upgrade done by badd(): | |
1012 | # return $upgrade->badd($x,$y,@r) if defined $upgrade && | |
1013 | # ((!$x->isa($self)) || (!$y->isa($self))); | |
1014 | ||
1015 | if ($y->is_zero()) | |
1016 | { | |
1017 | $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; | |
1018 | return $x; | |
1019 | } | |
1020 | ||
1021 | $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN | |
1022 | $x->badd($y,@r); # badd does not leave internal zeros | |
1023 | $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN) | |
1024 | $x; # already rounded by badd() or no round necc. | |
1025 | } | |
1026 | ||
1027 | sub binc | |
1028 | { | |
1029 | # increment arg by one | |
1030 | my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); | |
1031 | return $x if $x->modify('binc'); | |
1032 | ||
1033 | if ($x->{sign} eq '+') | |
1034 | { | |
1035 | $x->{value} = $CALC->_inc($x->{value}); | |
1036 | $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; | |
1037 | return $x; | |
1038 | } | |
1039 | elsif ($x->{sign} eq '-') | |
1040 | { | |
1041 | $x->{value} = $CALC->_dec($x->{value}); | |
1042 | $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0 | |
1043 | $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; | |
1044 | return $x; | |
1045 | } | |
1046 | # inf, nan handling etc | |
1047 | $x->badd($self->__one(),$a,$p,$r); # badd does round | |
1048 | } | |
1049 | ||
1050 | sub bdec | |
1051 | { | |
1052 | # decrement arg by one | |
1053 | my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); | |
1054 | return $x if $x->modify('bdec'); | |
1055 | ||
1056 | my $zero = $CALC->_is_zero($x->{value}) && $x->{sign} eq '+'; | |
1057 | # <= 0 | |
1058 | if (($x->{sign} eq '-') || $zero) | |
1059 | { | |
1060 | $x->{value} = $CALC->_inc($x->{value}); | |
1061 | $x->{sign} = '-' if $zero; # 0 => 1 => -1 | |
1062 | $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0 | |
1063 | $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; | |
1064 | return $x; | |
1065 | } | |
1066 | # > 0 | |
1067 | elsif ($x->{sign} eq '+') | |
1068 | { | |
1069 | $x->{value} = $CALC->_dec($x->{value}); | |
1070 | $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; | |
1071 | return $x; | |
1072 | } | |
1073 | # inf, nan handling etc | |
1074 | $x->badd($self->__one('-'),$a,$p,$r); # badd does round | |
1075 | } | |
1076 | ||
1077 | sub blog | |
1078 | { | |
1079 | # not implemented yet | |
1080 | my ($self,$x,$base,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); | |
1081 | ||
1082 | return $upgrade->blog($x,$base,$a,$p,$r) if defined $upgrade; | |
1083 | ||
1084 | return $x->bnan(); | |
1085 | } | |
1086 | ||
1087 | sub blcm | |
1088 | { | |
1089 | # (BINT or num_str, BINT or num_str) return BINT | |
1090 | # does not modify arguments, but returns new object | |
1091 | # Lowest Common Multiplicator | |
1092 | ||
1093 | my $y = shift; my ($x); | |
1094 | if (ref($y)) | |
1095 | { | |
1096 | $x = $y->copy(); | |
1097 | } | |
1098 | else | |
1099 | { | |
1100 | $x = $class->new($y); | |
1101 | } | |
1102 | while (@_) { $x = __lcm($x,shift); } | |
1103 | $x; | |
1104 | } | |
1105 | ||
1106 | sub bgcd | |
1107 | { | |
1108 | # (BINT or num_str, BINT or num_str) return BINT | |
1109 | # does not modify arguments, but returns new object | |
1110 | # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff) | |
1111 | ||
1112 | my $y = shift; | |
1113 | $y = __PACKAGE__->new($y) if !ref($y); | |
1114 | my $self = ref($y); | |
1115 | my $x = $y->copy(); # keep arguments | |
1116 | if ($CALC->can('_gcd')) | |
1117 | { | |
1118 | while (@_) | |
1119 | { | |
1120 | $y = shift; $y = $self->new($y) if !ref($y); | |
1121 | next if $y->is_zero(); | |
1122 | return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN? | |
1123 | $x->{value} = $CALC->_gcd($x->{value},$y->{value}); last if $x->is_one(); | |
1124 | } | |
1125 | } | |
1126 | else | |
1127 | { | |
1128 | while (@_) | |
1129 | { | |
1130 | $y = shift; $y = $self->new($y) if !ref($y); | |
1131 | $x = __gcd($x,$y->copy()); last if $x->is_one(); # _gcd handles NaN | |
1132 | } | |
1133 | } | |
1134 | $x->babs(); | |
1135 | } | |
1136 | ||
1137 | sub bnot | |
1138 | { | |
1139 | # (num_str or BINT) return BINT | |
1140 | # represent ~x as twos-complement number | |
1141 | # we don't need $self, so undef instead of ref($_[0]) make it slightly faster | |
1142 | my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_); | |
1143 | ||
1144 | return $x if $x->modify('bnot'); | |
1145 | $x->bneg()->bdec(); # bdec already does round | |
1146 | } | |
1147 | ||
1148 | # is_foo test routines | |
1149 | ||
1150 | sub is_zero | |
1151 | { | |
1152 | # return true if arg (BINT or num_str) is zero (array '+', '0') | |
1153 | # we don't need $self, so undef instead of ref($_[0]) make it slightly faster | |
1154 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); | |
1155 | ||
1156 | return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't | |
1157 | $CALC->_is_zero($x->{value}); | |
1158 | } | |
1159 | ||
1160 | sub is_nan | |
1161 | { | |
1162 | # return true if arg (BINT or num_str) is NaN | |
1163 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
1164 | ||
1165 | return 1 if $x->{sign} eq $nan; | |
1166 | 0; | |
1167 | } | |
1168 | ||
1169 | sub is_inf | |
1170 | { | |
1171 | # return true if arg (BINT or num_str) is +-inf | |
1172 | my ($self,$x,$sign) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); | |
1173 | ||
1174 | $sign = '' if !defined $sign; | |
1175 | return 1 if $sign eq $x->{sign}; # match ("+inf" eq "+inf") | |
1176 | return 0 if $sign !~ /^([+-]|)$/; | |
1177 | ||
1178 | if ($sign eq '') | |
1179 | { | |
1180 | return 1 if ($x->{sign} =~ /^[+-]inf$/); | |
1181 | return 0; | |
1182 | } | |
1183 | $sign = quotemeta($sign.'inf'); | |
1184 | return 1 if ($x->{sign} =~ /^$sign$/); | |
1185 | 0; | |
1186 | } | |
1187 | ||
1188 | sub is_one | |
1189 | { | |
1190 | # return true if arg (BINT or num_str) is +1 | |
1191 | # or -1 if sign is given | |
1192 | # we don't need $self, so undef instead of ref($_[0]) make it slightly faster | |
1193 | my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_); | |
1194 | ||
1195 | $sign = '' if !defined $sign; $sign = '+' if $sign ne '-'; | |
1196 | ||
1197 | return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either | |
1198 | $CALC->_is_one($x->{value}); | |
1199 | } | |
1200 | ||
1201 | sub is_odd | |
1202 | { | |
1203 | # return true when arg (BINT or num_str) is odd, false for even | |
1204 | # we don't need $self, so undef instead of ref($_[0]) make it slightly faster | |
1205 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); | |
1206 | ||
1207 | return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't | |
1208 | $CALC->_is_odd($x->{value}); | |
1209 | } | |
1210 | ||
1211 | sub is_even | |
1212 | { | |
1213 | # return true when arg (BINT or num_str) is even, false for odd | |
1214 | # we don't need $self, so undef instead of ref($_[0]) make it slightly faster | |
1215 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); | |
1216 | ||
1217 | return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't | |
1218 | $CALC->_is_even($x->{value}); | |
1219 | } | |
1220 | ||
1221 | sub is_positive | |
1222 | { | |
1223 | # return true when arg (BINT or num_str) is positive (>= 0) | |
1224 | # we don't need $self, so undef instead of ref($_[0]) make it slightly faster | |
1225 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); | |
1226 | ||
1227 | return 1 if $x->{sign} =~ /^\+/; | |
1228 | 0; | |
1229 | } | |
1230 | ||
1231 | sub is_negative | |
1232 | { | |
1233 | # return true when arg (BINT or num_str) is negative (< 0) | |
1234 | # we don't need $self, so undef instead of ref($_[0]) make it slightly faster | |
1235 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); | |
1236 | ||
1237 | return 1 if ($x->{sign} =~ /^-/); | |
1238 | 0; | |
1239 | } | |
1240 | ||
1241 | sub is_int | |
1242 | { | |
1243 | # return true when arg (BINT or num_str) is an integer | |
1244 | # always true for BigInt, but different for Floats | |
1245 | # we don't need $self, so undef instead of ref($_[0]) make it slightly faster | |
1246 | my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_); | |
1247 | ||
1248 | $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't | |
1249 | } | |
1250 | ||
1251 | ############################################################################### | |
1252 | ||
1253 | sub bmul | |
1254 | { | |
1255 | # multiply two numbers -- stolen from Knuth Vol 2 pg 233 | |
1256 | # (BINT or num_str, BINT or num_str) return BINT | |
1257 | ||
1258 | # set up parameters | |
1259 | my ($self,$x,$y,@r) = (ref($_[0]),@_); | |
1260 | # objectify is costly, so avoid it | |
1261 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1262 | { | |
1263 | ($self,$x,$y,@r) = objectify(2,@_); | |
1264 | } | |
1265 | ||
1266 | return $x if $x->modify('bmul'); | |
1267 | ||
1268 | return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan)); | |
1269 | ||
1270 | # inf handling | |
1271 | if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/)) | |
1272 | { | |
1273 | return $x->bnan() if $x->is_zero() || $y->is_zero(); | |
1274 | # result will always be +-inf: | |
1275 | # +inf * +/+inf => +inf, -inf * -/-inf => +inf | |
1276 | # +inf * -/-inf => -inf, -inf * +/+inf => -inf | |
1277 | return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/); | |
1278 | return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/); | |
1279 | return $x->binf('-'); | |
1280 | } | |
1281 | ||
1282 | return $upgrade->bmul($x,$y,@r) | |
1283 | if defined $upgrade && $y->isa($upgrade); | |
1284 | ||
1285 | $r[3] = $y; # no push here | |
1286 | ||
1287 | $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => + | |
1288 | ||
1289 | $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math | |
1290 | $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0 | |
1291 | ||
1292 | $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; | |
1293 | $x; | |
1294 | } | |
1295 | ||
1296 | sub _div_inf | |
1297 | { | |
1298 | # helper function that handles +-inf cases for bdiv()/bmod() to reuse code | |
1299 | my ($self,$x,$y) = @_; | |
1300 | ||
1301 | # NaN if x == NaN or y == NaN or x==y==0 | |
1302 | return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan() | |
1303 | if (($x->is_nan() || $y->is_nan()) || | |
1304 | ($x->is_zero() && $y->is_zero())); | |
1305 | ||
1306 | # +-inf / +-inf == NaN, reminder also NaN | |
1307 | if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/)) | |
1308 | { | |
1309 | return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan(); | |
1310 | } | |
1311 | # x / +-inf => 0, remainder x (works even if x == 0) | |
1312 | if ($y->{sign} =~ /^[+-]inf$/) | |
1313 | { | |
1314 | my $t = $x->copy(); # bzero clobbers up $x | |
1315 | return wantarray ? ($x->bzero(),$t) : $x->bzero() | |
1316 | } | |
1317 | ||
1318 | # 5 / 0 => +inf, -6 / 0 => -inf | |
1319 | # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf | |
1320 | # exception: -8 / 0 has remainder -8, not 8 | |
1321 | # exception: -inf / 0 has remainder -inf, not inf | |
1322 | if ($y->is_zero()) | |
1323 | { | |
1324 | # +-inf / 0 => special case for -inf | |
1325 | return wantarray ? ($x,$x->copy()) : $x if $x->is_inf(); | |
1326 | if (!$x->is_zero() && !$x->is_inf()) | |
1327 | { | |
1328 | my $t = $x->copy(); # binf clobbers up $x | |
1329 | return wantarray ? | |
1330 | ($x->binf($x->{sign}),$t) : $x->binf($x->{sign}) | |
1331 | } | |
1332 | } | |
1333 | ||
1334 | # last case: +-inf / ordinary number | |
1335 | my $sign = '+inf'; | |
1336 | $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign}; | |
1337 | $x->{sign} = $sign; | |
1338 | return wantarray ? ($x,$self->bzero()) : $x; | |
1339 | } | |
1340 | ||
1341 | sub bdiv | |
1342 | { | |
1343 | # (dividend: BINT or num_str, divisor: BINT or num_str) return | |
1344 | # (BINT,BINT) (quo,rem) or BINT (only rem) | |
1345 | ||
1346 | # set up parameters | |
1347 | my ($self,$x,$y,@r) = (ref($_[0]),@_); | |
1348 | # objectify is costly, so avoid it | |
1349 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1350 | { | |
1351 | ($self,$x,$y,@r) = objectify(2,@_); | |
1352 | } | |
1353 | ||
1354 | return $x if $x->modify('bdiv'); | |
1355 | ||
1356 | return $self->_div_inf($x,$y) | |
1357 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero()); | |
1358 | ||
1359 | return $upgrade->bdiv($upgrade->new($x),$y,@r) | |
1360 | if defined $upgrade && !$y->isa($self); | |
1361 | ||
1362 | $r[3] = $y; # no push! | |
1363 | ||
1364 | # 0 / something | |
1365 | return | |
1366 | wantarray ? ($x->round(@r),$self->bzero(@r)):$x->round(@r) if $x->is_zero(); | |
1367 | ||
1368 | # Is $x in the interval [0, $y) (aka $x <= $y) ? | |
1369 | my $cmp = $CALC->_acmp($x->{value},$y->{value}); | |
1370 | if (($cmp < 0) and (($x->{sign} eq $y->{sign}) or !wantarray)) | |
1371 | { | |
1372 | return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r) | |
1373 | if defined $upgrade; | |
1374 | ||
1375 | return $x->bzero()->round(@r) unless wantarray; | |
1376 | my $t = $x->copy(); # make copy first, because $x->bzero() clobbers $x | |
1377 | return ($x->bzero()->round(@r),$t); | |
1378 | } | |
1379 | elsif ($cmp == 0) | |
1380 | { | |
1381 | # shortcut, both are the same, so set to +/- 1 | |
1382 | $x->__one( ($x->{sign} ne $y->{sign} ? '-' : '+') ); | |
1383 | return $x unless wantarray; | |
1384 | return ($x->round(@r),$self->bzero(@r)); | |
1385 | } | |
1386 | return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r) | |
1387 | if defined $upgrade; | |
1388 | ||
1389 | # calc new sign and in case $y == +/- 1, return $x | |
1390 | my $xsign = $x->{sign}; # keep | |
1391 | $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+'); | |
1392 | # check for / +-1 (cant use $y->is_one due to '-' | |
1393 | if ($CALC->_is_one($y->{value})) | |
1394 | { | |
1395 | return wantarray ? ($x->round(@r),$self->bzero(@r)) : $x->round(@r); | |
1396 | } | |
1397 | ||
1398 | if (wantarray) | |
1399 | { | |
1400 | my $rem = $self->bzero(); | |
1401 | ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value}); | |
1402 | $x->{sign} = '+' if $CALC->_is_zero($x->{value}); | |
1403 | $rem->{_a} = $x->{_a}; | |
1404 | $rem->{_p} = $x->{_p}; | |
1405 | $x->round(@r); | |
1406 | if (! $CALC->_is_zero($rem->{value})) | |
1407 | { | |
1408 | $rem->{sign} = $y->{sign}; | |
1409 | $rem = $y-$rem if $xsign ne $y->{sign}; # one of them '-' | |
1410 | } | |
1411 | else | |
1412 | { | |
1413 | $rem->{sign} = '+'; # dont leave -0 | |
1414 | } | |
1415 | return ($x,$rem->round(@r)); | |
1416 | } | |
1417 | ||
1418 | $x->{value} = $CALC->_div($x->{value},$y->{value}); | |
1419 | $x->{sign} = '+' if $CALC->_is_zero($x->{value}); | |
1420 | ||
1421 | $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; | |
1422 | $x; | |
1423 | } | |
1424 | ||
1425 | ############################################################################### | |
1426 | # modulus functions | |
1427 | ||
1428 | sub bmod | |
1429 | { | |
1430 | # modulus (or remainder) | |
1431 | # (BINT or num_str, BINT or num_str) return BINT | |
1432 | ||
1433 | # set up parameters | |
1434 | my ($self,$x,$y,@r) = (ref($_[0]),@_); | |
1435 | # objectify is costly, so avoid it | |
1436 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1437 | { | |
1438 | ($self,$x,$y,@r) = objectify(2,@_); | |
1439 | } | |
1440 | ||
1441 | return $x if $x->modify('bmod'); | |
1442 | $r[3] = $y; # no push! | |
1443 | if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero()) | |
1444 | { | |
1445 | my ($d,$r) = $self->_div_inf($x,$y); | |
1446 | $x->{sign} = $r->{sign}; | |
1447 | $x->{value} = $r->{value}; | |
1448 | return $x->round(@r); | |
1449 | } | |
1450 | ||
1451 | if ($CALC->can('_mod')) | |
1452 | { | |
1453 | # calc new sign and in case $y == +/- 1, return $x | |
1454 | $x->{value} = $CALC->_mod($x->{value},$y->{value}); | |
1455 | if (!$CALC->_is_zero($x->{value})) | |
1456 | { | |
1457 | my $xsign = $x->{sign}; | |
1458 | $x->{sign} = $y->{sign}; | |
1459 | if ($xsign ne $y->{sign}) | |
1460 | { | |
1461 | my $t = $CALC->_copy($x->{value}); # copy $x | |
1462 | $x->{value} = $CALC->_copy($y->{value}); # copy $y to $x | |
1463 | $x->{value} = $CALC->_sub($y->{value},$t,1); # $y-$x | |
1464 | } | |
1465 | } | |
1466 | else | |
1467 | { | |
1468 | $x->{sign} = '+'; # dont leave -0 | |
1469 | } | |
1470 | $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; | |
1471 | return $x; | |
1472 | } | |
1473 | my ($t,$rem) = $self->bdiv($x->copy(),$y,@r); # slow way (also rounds) | |
1474 | # modify in place | |
1475 | foreach (qw/value sign _a _p/) | |
1476 | { | |
1477 | $x->{$_} = $rem->{$_}; | |
1478 | } | |
1479 | $x; | |
1480 | } | |
1481 | ||
1482 | sub bmodinv | |
1483 | { | |
1484 | # modular inverse. given a number which is (hopefully) relatively | |
1485 | # prime to the modulus, calculate its inverse using Euclid's | |
1486 | # alogrithm. if the number is not relatively prime to the modulus | |
1487 | # (i.e. their gcd is not one) then NaN is returned. | |
1488 | ||
1489 | # set up parameters | |
1490 | my ($self,$x,$y,@r) = (ref($_[0]),@_); | |
1491 | # objectify is costly, so avoid it | |
1492 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1493 | { | |
1494 | ($self,$x,$y,@r) = objectify(2,@_); | |
1495 | } | |
1496 | ||
1497 | return $x if $x->modify('bmodinv'); | |
1498 | ||
1499 | return $x->bnan() | |
1500 | if ($y->{sign} ne '+' # -, NaN, +inf, -inf | |
1501 | || $x->is_zero() # or num == 0 | |
1502 | || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf | |
1503 | ); | |
1504 | ||
1505 | # put least residue into $x if $x was negative, and thus make it positive | |
1506 | $x->bmod($y) if $x->{sign} eq '-'; | |
1507 | ||
1508 | if ($CALC->can('_modinv')) | |
1509 | { | |
1510 | $x->{value} = $CALC->_modinv($x->{value},$y->{value}); | |
1511 | $x->bnan() if !defined $x->{value} ; # in case there was none | |
1512 | return $x; | |
1513 | } | |
1514 | ||
1515 | my ($u, $u1) = ($self->bzero(), $self->bone()); | |
1516 | my ($a, $b) = ($y->copy(), $x->copy()); | |
1517 | ||
1518 | # first step need always be done since $num (and thus $b) is never 0 | |
1519 | # Note that the loop is aligned so that the check occurs between #2 and #1 | |
1520 | # thus saving us one step #2 at the loop end. Typical loop count is 1. Even | |
1521 | # a case with 28 loops still gains about 3% with this layout. | |
1522 | my $q; | |
1523 | ($a, $q, $b) = ($b, $a->bdiv($b)); # step #1 | |
1524 | # Euclid's Algorithm | |
1525 | while (!$b->is_zero()) | |
1526 | { | |
1527 | ($u, $u1) = ($u1, $u->bsub($u1->copy()->bmul($q))); # step #2 | |
1528 | ($a, $q, $b) = ($b, $a->bdiv($b)); # step #1 again | |
1529 | } | |
1530 | ||
1531 | # if the gcd is not 1, then return NaN! It would be pointless to | |
1532 | # have called bgcd to check this first, because we would then be performing | |
1533 | # the same Euclidean Algorithm *twice* | |
1534 | return $x->bnan() unless $a->is_one(); | |
1535 | ||
1536 | $u1->bmod($y); | |
1537 | $x->{value} = $u1->{value}; | |
1538 | $x->{sign} = $u1->{sign}; | |
1539 | $x; | |
1540 | } | |
1541 | ||
1542 | sub bmodpow | |
1543 | { | |
1544 | # takes a very large number to a very large exponent in a given very | |
1545 | # large modulus, quickly, thanks to binary exponentation. supports | |
1546 | # negative exponents. | |
1547 | my ($self,$num,$exp,$mod,@r) = objectify(3,@_); | |
1548 | ||
1549 | return $num if $num->modify('bmodpow'); | |
1550 | ||
1551 | # check modulus for valid values | |
1552 | return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf | |
1553 | || $mod->is_zero()); | |
1554 | ||
1555 | # check exponent for valid values | |
1556 | if ($exp->{sign} =~ /\w/) | |
1557 | { | |
1558 | # i.e., if it's NaN, +inf, or -inf... | |
1559 | return $num->bnan(); | |
1560 | } | |
1561 | ||
1562 | $num->bmodinv ($mod) if ($exp->{sign} eq '-'); | |
1563 | ||
1564 | # check num for valid values (also NaN if there was no inverse but $exp < 0) | |
1565 | return $num->bnan() if $num->{sign} !~ /^[+-]$/; | |
1566 | ||
1567 | if ($CALC->can('_modpow')) | |
1568 | { | |
1569 | # $mod is positive, sign on $exp is ignored, result also positive | |
1570 | $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value}); | |
1571 | return $num; | |
1572 | } | |
1573 | ||
1574 | # in the trivial case, | |
1575 | return $num->bzero(@r) if $mod->is_one(); | |
1576 | return $num->bone('+',@r) if $num->is_zero() or $num->is_one(); | |
1577 | ||
1578 | # $num->bmod($mod); # if $x is large, make it smaller first | |
1579 | my $acc = $num->copy(); # but this is not really faster... | |
1580 | ||
1581 | $num->bone(); # keep ref to $num | |
1582 | ||
1583 | my $expbin = $exp->as_bin(); $expbin =~ s/^[-]?0b//; # ignore sign and prefix | |
1584 | my $len = length($expbin); | |
1585 | while (--$len >= 0) | |
1586 | { | |
1587 | if( substr($expbin,$len,1) eq '1') | |
1588 | { | |
1589 | $num->bmul($acc)->bmod($mod); | |
1590 | } | |
1591 | $acc->bmul($acc)->bmod($mod); | |
1592 | } | |
1593 | ||
1594 | $num; | |
1595 | } | |
1596 | ||
1597 | ############################################################################### | |
1598 | ||
1599 | sub bfac | |
1600 | { | |
1601 | # (BINT or num_str, BINT or num_str) return BINT | |
1602 | # compute factorial numbers | |
1603 | # modifies first argument | |
1604 | my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); | |
1605 | ||
1606 | return $x if $x->modify('bfac'); | |
1607 | ||
1608 | return $x->bnan() if $x->{sign} ne '+'; # inf, NnN, <0 etc => NaN | |
1609 | return $x->bone('+',@r) if $x->is_zero() || $x->is_one(); # 0 or 1 => 1 | |
1610 | ||
1611 | if ($CALC->can('_fac')) | |
1612 | { | |
1613 | $x->{value} = $CALC->_fac($x->{value}); | |
1614 | return $x->round(@r); | |
1615 | } | |
1616 | ||
1617 | my $n = $x->copy(); | |
1618 | $x->bone(); | |
1619 | # seems we need not to temp. clear A/P of $x since the result is the same | |
1620 | my $f = $self->new(2); | |
1621 | while ($f->bacmp($n) < 0) | |
1622 | { | |
1623 | $x->bmul($f); $f->binc(); | |
1624 | } | |
1625 | $x->bmul($f,@r); # last step and also round | |
1626 | } | |
1627 | ||
1628 | sub bpow | |
1629 | { | |
1630 | # (BINT or num_str, BINT or num_str) return BINT | |
1631 | # compute power of two numbers -- stolen from Knuth Vol 2 pg 233 | |
1632 | # modifies first argument | |
1633 | ||
1634 | # set up parameters | |
1635 | my ($self,$x,$y,@r) = (ref($_[0]),@_); | |
1636 | # objectify is costly, so avoid it | |
1637 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1638 | { | |
1639 | ($self,$x,$y,@r) = objectify(2,@_); | |
1640 | } | |
1641 | ||
1642 | return $x if $x->modify('bpow'); | |
1643 | ||
1644 | return $upgrade->bpow($upgrade->new($x),$y,@r) | |
1645 | if defined $upgrade && !$y->isa($self); | |
1646 | ||
1647 | $r[3] = $y; # no push! | |
1648 | return $x if $x->{sign} =~ /^[+-]inf$/; # -inf/+inf ** x | |
1649 | return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan; | |
1650 | return $x->bone('+',@r) if $y->is_zero(); | |
1651 | return $x->round(@r) if $x->is_one() || $y->is_one(); | |
1652 | if ($x->{sign} eq '-' && $CALC->_is_one($x->{value})) | |
1653 | { | |
1654 | # if $x == -1 and odd/even y => +1/-1 | |
1655 | return $y->is_odd() ? $x->round(@r) : $x->babs()->round(@r); | |
1656 | # my Casio FX-5500L has a bug here: -1 ** 2 is -1, but -1 * -1 is 1; | |
1657 | } | |
1658 | # 1 ** -y => 1 / (1 ** |y|) | |
1659 | # so do test for negative $y after above's clause | |
1660 | return $x->bnan() if $y->{sign} eq '-'; | |
1661 | return $x->round(@r) if $x->is_zero(); # 0**y => 0 (if not y <= 0) | |
1662 | ||
1663 | if ($CALC->can('_pow')) | |
1664 | { | |
1665 | $x->{value} = $CALC->_pow($x->{value},$y->{value}); | |
1666 | $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; | |
1667 | return $x; | |
1668 | } | |
1669 | ||
1670 | # based on the assumption that shifting in base 10 is fast, and that mul | |
1671 | # works faster if numbers are small: we count trailing zeros (this step is | |
1672 | # O(1)..O(N), but in case of O(N) we save much more time due to this), | |
1673 | # stripping them out of the multiplication, and add $count * $y zeros | |
1674 | # afterwards like this: | |
1675 | # 300 ** 3 == 300*300*300 == 3*3*3 . '0' x 2 * 3 == 27 . '0' x 6 | |
1676 | # creates deep recursion since brsft/blsft use bpow sometimes. | |
1677 | # my $zeros = $x->_trailing_zeros(); | |
1678 | # if ($zeros > 0) | |
1679 | # { | |
1680 | # $x->brsft($zeros,10); # remove zeros | |
1681 | # $x->bpow($y); # recursion (will not branch into here again) | |
1682 | # $zeros = $y * $zeros; # real number of zeros to add | |
1683 | # $x->blsft($zeros,10); | |
1684 | # return $x->round(@r); | |
1685 | # } | |
1686 | ||
1687 | my $pow2 = $self->__one(); | |
1688 | my $y_bin = $y->as_bin(); $y_bin =~ s/^0b//; | |
1689 | my $len = length($y_bin); | |
1690 | while (--$len > 0) | |
1691 | { | |
1692 | $pow2->bmul($x) if substr($y_bin,$len,1) eq '1'; # is odd? | |
1693 | $x->bmul($x); | |
1694 | } | |
1695 | $x->bmul($pow2); | |
1696 | $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0; | |
1697 | $x; | |
1698 | } | |
1699 | ||
1700 | sub blsft | |
1701 | { | |
1702 | # (BINT or num_str, BINT or num_str) return BINT | |
1703 | # compute x << y, base n, y >= 0 | |
1704 | ||
1705 | # set up parameters | |
1706 | my ($self,$x,$y,$n,@r) = (ref($_[0]),@_); | |
1707 | # objectify is costly, so avoid it | |
1708 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1709 | { | |
1710 | ($self,$x,$y,$n,@r) = objectify(2,@_); | |
1711 | } | |
1712 | ||
1713 | return $x if $x->modify('blsft'); | |
1714 | return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/); | |
1715 | return $x->round(@r) if $y->is_zero(); | |
1716 | ||
1717 | $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-'; | |
1718 | ||
1719 | my $t; $t = $CALC->_lsft($x->{value},$y->{value},$n) if $CALC->can('_lsft'); | |
1720 | if (defined $t) | |
1721 | { | |
1722 | $x->{value} = $t; return $x->round(@r); | |
1723 | } | |
1724 | # fallback | |
1725 | return $x->bmul( $self->bpow($n, $y, @r), @r ); | |
1726 | } | |
1727 | ||
1728 | sub brsft | |
1729 | { | |
1730 | # (BINT or num_str, BINT or num_str) return BINT | |
1731 | # compute x >> y, base n, y >= 0 | |
1732 | ||
1733 | # set up parameters | |
1734 | my ($self,$x,$y,$n,@r) = (ref($_[0]),@_); | |
1735 | # objectify is costly, so avoid it | |
1736 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1737 | { | |
1738 | ($self,$x,$y,$n,@r) = objectify(2,@_); | |
1739 | } | |
1740 | ||
1741 | return $x if $x->modify('brsft'); | |
1742 | return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/); | |
1743 | return $x->round(@r) if $y->is_zero(); | |
1744 | return $x->bzero(@r) if $x->is_zero(); # 0 => 0 | |
1745 | ||
1746 | $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-'; | |
1747 | ||
1748 | # this only works for negative numbers when shifting in base 2 | |
1749 | if (($x->{sign} eq '-') && ($n == 2)) | |
1750 | { | |
1751 | return $x->round(@r) if $x->is_one('-'); # -1 => -1 | |
1752 | if (!$y->is_one()) | |
1753 | { | |
1754 | # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al | |
1755 | # but perhaps there is a better emulation for two's complement shift... | |
1756 | # if $y != 1, we must simulate it by doing: | |
1757 | # convert to bin, flip all bits, shift, and be done | |
1758 | $x->binc(); # -3 => -2 | |
1759 | my $bin = $x->as_bin(); | |
1760 | $bin =~ s/^-0b//; # strip '-0b' prefix | |
1761 | $bin =~ tr/10/01/; # flip bits | |
1762 | # now shift | |
1763 | if (CORE::length($bin) <= $y) | |
1764 | { | |
1765 | $bin = '0'; # shifting to far right creates -1 | |
1766 | # 0, because later increment makes | |
1767 | # that 1, attached '-' makes it '-1' | |
1768 | # because -1 >> x == -1 ! | |
1769 | } | |
1770 | else | |
1771 | { | |
1772 | $bin =~ s/.{$y}$//; # cut off at the right side | |
1773 | $bin = '1' . $bin; # extend left side by one dummy '1' | |
1774 | $bin =~ tr/10/01/; # flip bits back | |
1775 | } | |
1776 | my $res = $self->new('0b'.$bin); # add prefix and convert back | |
1777 | $res->binc(); # remember to increment | |
1778 | $x->{value} = $res->{value}; # take over value | |
1779 | return $x->round(@r); # we are done now, magic, isn't? | |
1780 | } | |
1781 | $x->bdec(); # n == 2, but $y == 1: this fixes it | |
1782 | } | |
1783 | ||
1784 | my $t; $t = $CALC->_rsft($x->{value},$y->{value},$n) if $CALC->can('_rsft'); | |
1785 | if (defined $t) | |
1786 | { | |
1787 | $x->{value} = $t; | |
1788 | return $x->round(@r); | |
1789 | } | |
1790 | # fallback | |
1791 | $x->bdiv($self->bpow($n,$y, @r), @r); | |
1792 | $x; | |
1793 | } | |
1794 | ||
1795 | sub band | |
1796 | { | |
1797 | #(BINT or num_str, BINT or num_str) return BINT | |
1798 | # compute x & y | |
1799 | ||
1800 | # set up parameters | |
1801 | my ($self,$x,$y,@r) = (ref($_[0]),@_); | |
1802 | # objectify is costly, so avoid it | |
1803 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1804 | { | |
1805 | ($self,$x,$y,@r) = objectify(2,@_); | |
1806 | } | |
1807 | ||
1808 | return $x if $x->modify('band'); | |
1809 | ||
1810 | $r[3] = $y; # no push! | |
1811 | local $Math::BigInt::upgrade = undef; | |
1812 | ||
1813 | return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/); | |
1814 | return $x->bzero(@r) if $y->is_zero() || $x->is_zero(); | |
1815 | ||
1816 | my $sign = 0; # sign of result | |
1817 | $sign = 1 if ($x->{sign} eq '-') && ($y->{sign} eq '-'); | |
1818 | my $sx = 1; $sx = -1 if $x->{sign} eq '-'; | |
1819 | my $sy = 1; $sy = -1 if $y->{sign} eq '-'; | |
1820 | ||
1821 | if ($CALC->can('_and') && $sx == 1 && $sy == 1) | |
1822 | { | |
1823 | $x->{value} = $CALC->_and($x->{value},$y->{value}); | |
1824 | return $x->round(@r); | |
1825 | } | |
1826 | ||
1827 | my $m = $self->bone(); my ($xr,$yr); | |
1828 | my $x10000 = $self->new (0x1000); | |
1829 | my $y1 = copy(ref($x),$y); # make copy | |
1830 | $y1->babs(); # and positive | |
1831 | my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place! | |
1832 | use integer; # need this for negative bools | |
1833 | while (!$x1->is_zero() && !$y1->is_zero()) | |
1834 | { | |
1835 | ($x1, $xr) = bdiv($x1, $x10000); | |
1836 | ($y1, $yr) = bdiv($y1, $x10000); | |
1837 | # make both op's numbers! | |
1838 | $x->badd( bmul( $class->new( | |
1839 | abs($sx*int($xr->numify()) & $sy*int($yr->numify()))), | |
1840 | $m)); | |
1841 | $m->bmul($x10000); | |
1842 | } | |
1843 | $x->bneg() if $sign; | |
1844 | $x->round(@r); | |
1845 | } | |
1846 | ||
1847 | sub bior | |
1848 | { | |
1849 | #(BINT or num_str, BINT or num_str) return BINT | |
1850 | # compute x | y | |
1851 | ||
1852 | # set up parameters | |
1853 | my ($self,$x,$y,@r) = (ref($_[0]),@_); | |
1854 | # objectify is costly, so avoid it | |
1855 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1856 | { | |
1857 | ($self,$x,$y,@r) = objectify(2,@_); | |
1858 | } | |
1859 | ||
1860 | return $x if $x->modify('bior'); | |
1861 | $r[3] = $y; # no push! | |
1862 | ||
1863 | local $Math::BigInt::upgrade = undef; | |
1864 | ||
1865 | return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/); | |
1866 | return $x->round(@r) if $y->is_zero(); | |
1867 | ||
1868 | my $sign = 0; # sign of result | |
1869 | $sign = 1 if ($x->{sign} eq '-') || ($y->{sign} eq '-'); | |
1870 | my $sx = 1; $sx = -1 if $x->{sign} eq '-'; | |
1871 | my $sy = 1; $sy = -1 if $y->{sign} eq '-'; | |
1872 | ||
1873 | # don't use lib for negative values | |
1874 | if ($CALC->can('_or') && $sx == 1 && $sy == 1) | |
1875 | { | |
1876 | $x->{value} = $CALC->_or($x->{value},$y->{value}); | |
1877 | return $x->round(@r); | |
1878 | } | |
1879 | ||
1880 | my $m = $self->bone(); my ($xr,$yr); | |
1881 | my $x10000 = $self->new(0x10000); | |
1882 | my $y1 = copy(ref($x),$y); # make copy | |
1883 | $y1->babs(); # and positive | |
1884 | my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place! | |
1885 | use integer; # need this for negative bools | |
1886 | while (!$x1->is_zero() || !$y1->is_zero()) | |
1887 | { | |
1888 | ($x1, $xr) = bdiv($x1,$x10000); | |
1889 | ($y1, $yr) = bdiv($y1,$x10000); | |
1890 | # make both op's numbers! | |
1891 | $x->badd( bmul( $class->new( | |
1892 | abs($sx*int($xr->numify()) | $sy*int($yr->numify()))), | |
1893 | $m)); | |
1894 | $m->bmul($x10000); | |
1895 | } | |
1896 | $x->bneg() if $sign; | |
1897 | $x->round(@r); | |
1898 | } | |
1899 | ||
1900 | sub bxor | |
1901 | { | |
1902 | #(BINT or num_str, BINT or num_str) return BINT | |
1903 | # compute x ^ y | |
1904 | ||
1905 | # set up parameters | |
1906 | my ($self,$x,$y,@r) = (ref($_[0]),@_); | |
1907 | # objectify is costly, so avoid it | |
1908 | if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) | |
1909 | { | |
1910 | ($self,$x,$y,@r) = objectify(2,@_); | |
1911 | } | |
1912 | ||
1913 | return $x if $x->modify('bxor'); | |
1914 | $r[3] = $y; # no push! | |
1915 | ||
1916 | local $Math::BigInt::upgrade = undef; | |
1917 | ||
1918 | return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/); | |
1919 | return $x->round(@r) if $y->is_zero(); | |
1920 | ||
1921 | my $sign = 0; # sign of result | |
1922 | $sign = 1 if $x->{sign} ne $y->{sign}; | |
1923 | my $sx = 1; $sx = -1 if $x->{sign} eq '-'; | |
1924 | my $sy = 1; $sy = -1 if $y->{sign} eq '-'; | |
1925 | ||
1926 | # don't use lib for negative values | |
1927 | if ($CALC->can('_xor') && $sx == 1 && $sy == 1) | |
1928 | { | |
1929 | $x->{value} = $CALC->_xor($x->{value},$y->{value}); | |
1930 | return $x->round(@r); | |
1931 | } | |
1932 | ||
1933 | my $m = $self->bone(); my ($xr,$yr); | |
1934 | my $x10000 = $self->new(0x10000); | |
1935 | my $y1 = copy(ref($x),$y); # make copy | |
1936 | $y1->babs(); # and positive | |
1937 | my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place! | |
1938 | use integer; # need this for negative bools | |
1939 | while (!$x1->is_zero() || !$y1->is_zero()) | |
1940 | { | |
1941 | ($x1, $xr) = bdiv($x1, $x10000); | |
1942 | ($y1, $yr) = bdiv($y1, $x10000); | |
1943 | # make both op's numbers! | |
1944 | $x->badd( bmul( $class->new( | |
1945 | abs($sx*int($xr->numify()) ^ $sy*int($yr->numify()))), | |
1946 | $m)); | |
1947 | $m->bmul($x10000); | |
1948 | } | |
1949 | $x->bneg() if $sign; | |
1950 | $x->round(@r); | |
1951 | } | |
1952 | ||
1953 | sub length | |
1954 | { | |
1955 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
1956 | ||
1957 | my $e = $CALC->_len($x->{value}); | |
1958 | return wantarray ? ($e,0) : $e; | |
1959 | } | |
1960 | ||
1961 | sub digit | |
1962 | { | |
1963 | # return the nth decimal digit, negative values count backward, 0 is right | |
1964 | my ($self,$x,$n) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); | |
1965 | ||
1966 | $CALC->_digit($x->{value},$n||0); | |
1967 | } | |
1968 | ||
1969 | sub _trailing_zeros | |
1970 | { | |
1971 | # return the amount of trailing zeros in $x | |
1972 | my $x = shift; | |
1973 | $x = $class->new($x) unless ref $x; | |
1974 | ||
1975 | return 0 if $x->is_zero() || $x->is_odd() || $x->{sign} !~ /^[+-]$/; | |
1976 | ||
1977 | return $CALC->_zeros($x->{value}) if $CALC->can('_zeros'); | |
1978 | ||
1979 | # if not: since we do not know underlying internal representation: | |
1980 | my $es = "$x"; $es =~ /([0]*)$/; | |
1981 | return 0 if !defined $1; # no zeros | |
1982 | CORE::length("$1"); # as string, not as +0! | |
1983 | } | |
1984 | ||
1985 | sub bsqrt | |
1986 | { | |
1987 | my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); | |
1988 | ||
1989 | return $x if $x->modify('bsqrt'); | |
1990 | ||
1991 | return $x->bnan() if $x->{sign} ne '+'; # -x or inf or NaN => NaN | |
1992 | return $x->bzero(@r) if $x->is_zero(); # 0 => 0 | |
1993 | return $x->round(@r) if $x->is_one(); # 1 => 1 | |
1994 | ||
1995 | return $upgrade->bsqrt($x,@r) if defined $upgrade; | |
1996 | ||
1997 | if ($CALC->can('_sqrt')) | |
1998 | { | |
1999 | $x->{value} = $CALC->_sqrt($x->{value}); | |
2000 | return $x->round(@r); | |
2001 | } | |
2002 | ||
2003 | return $x->bone('+',@r) if $x < 4; # 2,3 => 1 | |
2004 | my $y = $x->copy(); | |
2005 | my $l = int($x->length()/2); | |
2006 | ||
2007 | $x->bone(); # keep ref($x), but modify it | |
2008 | $x->blsft($l,10); | |
2009 | ||
2010 | my $last = $self->bzero(); | |
2011 | my $two = $self->new(2); | |
2012 | my $lastlast = $x+$two; | |
2013 | while ($last != $x && $lastlast != $x) | |
2014 | { | |
2015 | $lastlast = $last; $last = $x; | |
2016 | $x += $y / $x; | |
2017 | $x /= $two; | |
2018 | } | |
2019 | $x-- if $x * $x > $y; # overshot? | |
2020 | $x->round(@r); | |
2021 | } | |
2022 | ||
2023 | sub exponent | |
2024 | { | |
2025 | # return a copy of the exponent (here always 0, NaN or 1 for $m == 0) | |
2026 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
2027 | ||
2028 | if ($x->{sign} !~ /^[+-]$/) | |
2029 | { | |
2030 | my $s = $x->{sign}; $s =~ s/^[+-]//; | |
2031 | return $self->new($s); # -inf,+inf => inf | |
2032 | } | |
2033 | my $e = $class->bzero(); | |
2034 | return $e->binc() if $x->is_zero(); | |
2035 | $e += $x->_trailing_zeros(); | |
2036 | $e; | |
2037 | } | |
2038 | ||
2039 | sub mantissa | |
2040 | { | |
2041 | # return the mantissa (compatible to Math::BigFloat, e.g. reduced) | |
2042 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
2043 | ||
2044 | if ($x->{sign} !~ /^[+-]$/) | |
2045 | { | |
2046 | return $self->new($x->{sign}); # keep + or - sign | |
2047 | } | |
2048 | my $m = $x->copy(); | |
2049 | # that's inefficient | |
2050 | my $zeros = $m->_trailing_zeros(); | |
2051 | $m->brsft($zeros,10) if $zeros != 0; | |
2052 | $m; | |
2053 | } | |
2054 | ||
2055 | sub parts | |
2056 | { | |
2057 | # return a copy of both the exponent and the mantissa | |
2058 | my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_); | |
2059 | ||
2060 | return ($x->mantissa(),$x->exponent()); | |
2061 | } | |
2062 | ||
2063 | ############################################################################## | |
2064 | # rounding functions | |
2065 | ||
2066 | sub bfround | |
2067 | { | |
2068 | # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.' | |
2069 | # $n == 0 || $n == 1 => round to integer | |
2070 | my $x = shift; $x = $class->new($x) unless ref $x; | |
2071 | my ($scale,$mode) = $x->_scale_p($x->precision(),$x->round_mode(),@_); | |
2072 | return $x if !defined $scale; # no-op | |
2073 | return $x if $x->modify('bfround'); | |
2074 | ||
2075 | # no-op for BigInts if $n <= 0 | |
2076 | if ($scale <= 0) | |
2077 | { | |
2078 | $x->{_a} = undef; # clear an eventual set A | |
2079 | $x->{_p} = $scale; return $x; | |
2080 | } | |
2081 | ||
2082 | $x->bround( $x->length()-$scale, $mode); | |
2083 | $x->{_a} = undef; # bround sets {_a} | |
2084 | $x->{_p} = $scale; # so correct it | |
2085 | $x; | |
2086 | } | |
2087 | ||
2088 | sub _scan_for_nonzero | |
2089 | { | |
2090 | my $x = shift; | |
2091 | my $pad = shift; | |
2092 | my $xs = shift; | |
2093 | ||
2094 | my $len = $x->length(); | |
2095 | return 0 if $len == 1; # '5' is trailed by invisible zeros | |
2096 | my $follow = $pad - 1; | |
2097 | return 0 if $follow > $len || $follow < 1; | |
2098 | ||
2099 | # since we do not know underlying represention of $x, use decimal string | |
2100 | #my $r = substr ($$xs,-$follow); | |
2101 | my $r = substr ("$x",-$follow); | |
2102 | return 1 if $r =~ /[^0]/; | |
2103 | 0; | |
2104 | } | |
2105 | ||
2106 | sub fround | |
2107 | { | |
2108 | # to make life easier for switch between MBF and MBI (autoload fxxx() | |
2109 | # like MBF does for bxxx()?) | |
2110 | my $x = shift; | |
2111 | return $x->bround(@_); | |
2112 | } | |
2113 | ||
2114 | sub bround | |
2115 | { | |
2116 | # accuracy: +$n preserve $n digits from left, | |
2117 | # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF) | |
2118 | # no-op for $n == 0 | |
2119 | # and overwrite the rest with 0's, return normalized number | |
2120 | # do not return $x->bnorm(), but $x | |
2121 | ||
2122 | my $x = shift; $x = $class->new($x) unless ref $x; | |
2123 | my ($scale,$mode) = $x->_scale_a($x->accuracy(),$x->round_mode(),@_); | |
2124 | return $x if !defined $scale; # no-op | |
2125 | return $x if $x->modify('bround'); | |
2126 | ||
2127 | if ($x->is_zero() || $scale == 0) | |
2128 | { | |
2129 | $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2 | |
2130 | return $x; | |
2131 | } | |
2132 | return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN | |
2133 | ||
2134 | # we have fewer digits than we want to scale to | |
2135 | my $len = $x->length(); | |
2136 | # scale < 0, but > -len (not >=!) | |
2137 | if (($scale < 0 && $scale < -$len-1) || ($scale >= $len)) | |
2138 | { | |
2139 | $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2 | |
2140 | return $x; | |
2141 | } | |
2142 | ||
2143 | # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6 | |
2144 | my ($pad,$digit_round,$digit_after); | |
2145 | $pad = $len - $scale; | |
2146 | $pad = abs($scale-1) if $scale < 0; | |
2147 | ||
2148 | # do not use digit(), it is costly for binary => decimal | |
2149 | ||
2150 | my $xs = $CALC->_str($x->{value}); | |
2151 | my $pl = -$pad-1; | |
2152 | ||
2153 | # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4 | |
2154 | # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3 | |
2155 | $digit_round = '0'; $digit_round = substr($$xs,$pl,1) if $pad <= $len; | |
2156 | $pl++; $pl ++ if $pad >= $len; | |
2157 | $digit_after = '0'; $digit_after = substr($$xs,$pl,1) if $pad > 0; | |
2158 | ||
2159 | # in case of 01234 we round down, for 6789 up, and only in case 5 we look | |
2160 | # closer at the remaining digits of the original $x, remember decision | |
2161 | my $round_up = 1; # default round up | |
2162 | $round_up -- if | |
2163 | ($mode eq 'trunc') || # trunc by round down | |
2164 | ($digit_after =~ /[01234]/) || # round down anyway, | |
2165 | # 6789 => round up | |
2166 | ($digit_after eq '5') && # not 5000...0000 | |
2167 | ($x->_scan_for_nonzero($pad,$xs) == 0) && | |
2168 | ( | |
2169 | ($mode eq 'even') && ($digit_round =~ /[24680]/) || | |
2170 | ($mode eq 'odd') && ($digit_round =~ /[13579]/) || | |
2171 | ($mode eq '+inf') && ($x->{sign} eq '-') || | |
2172 | ($mode eq '-inf') && ($x->{sign} eq '+') || | |
2173 | ($mode eq 'zero') # round down if zero, sign adjusted below | |
2174 | ); | |
2175 | my $put_back = 0; # not yet modified | |
2176 | ||
2177 | if (($pad > 0) && ($pad <= $len)) | |
2178 | { | |
2179 | substr($$xs,-$pad,$pad) = '0' x $pad; | |
2180 | $put_back = 1; | |
2181 | } | |
2182 | elsif ($pad > $len) | |
2183 | { | |
2184 | $x->bzero(); # round to '0' | |
2185 | } | |
2186 | ||
2187 | if ($round_up) # what gave test above? | |
2188 | { | |
2189 | $put_back = 1; | |
2190 | $pad = $len, $$xs = '0'x$pad if $scale < 0; # tlr: whack 0.51=>1.0 | |
2191 | ||
2192 | # we modify directly the string variant instead of creating a number and | |
2193 | # adding it, since that is faster (we already have the string) | |
2194 | my $c = 0; $pad ++; # for $pad == $len case | |
2195 | while ($pad <= $len) | |
2196 | { | |
2197 | $c = substr($$xs,-$pad,1) + 1; $c = '0' if $c eq '10'; | |
2198 | substr($$xs,-$pad,1) = $c; $pad++; | |
2199 | last if $c != 0; # no overflow => early out | |
2200 | } | |
2201 | $$xs = '1'.$$xs if $c == 0; | |
2202 | ||
2203 | } | |
2204 | $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back in if needed | |
2205 | ||
2206 | $x->{_a} = $scale if $scale >= 0; | |
2207 | if ($scale < 0) | |
2208 | { | |
2209 | $x->{_a} = $len+$scale; | |
2210 | $x->{_a} = 0 if $scale < -$len; | |
2211 | } | |
2212 | $x; | |
2213 | } | |
2214 | ||
2215 | sub bfloor | |
2216 | { | |
2217 | # return integer less or equal then number, since it is already integer, | |
2218 | # always returns $self | |
2219 | my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); | |
2220 | ||
2221 | $x->round(@r); | |
2222 | } | |
2223 | ||
2224 | sub bceil | |
2225 | { | |
2226 | # return integer greater or equal then number, since it is already integer, | |
2227 | # always returns $self | |
2228 | my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_); | |
2229 | ||
2230 | $x->round(@r); | |
2231 | } | |
2232 | ||
2233 | ############################################################################## | |
2234 | # private stuff (internal use only) | |
2235 | ||
2236 | sub __one | |
2237 | { | |
2238 | # internal speedup, set argument to 1, or create a +/- 1 | |
2239 | my $self = shift; | |
2240 | my $x = $self->bone(); # $x->{value} = $CALC->_one(); | |
2241 | $x->{sign} = shift || '+'; | |
2242 | $x; | |
2243 | } | |
2244 | ||
2245 | sub _swap | |
2246 | { | |
2247 | # Overload will swap params if first one is no object ref so that the first | |
2248 | # one is always an object ref. In this case, third param is true. | |
2249 | # This routine is to overcome the effect of scalar,$object creating an object | |
2250 | # of the class of this package, instead of the second param $object. This | |
2251 | # happens inside overload, when the overload section of this package is | |
2252 | # inherited by sub classes. | |
2253 | # For overload cases (and this is used only there), we need to preserve the | |
2254 | # args, hence the copy(). | |
2255 | # You can override this method in a subclass, the overload section will call | |
2256 | # $object->_swap() to make sure it arrives at the proper subclass, with some | |
2257 | # exceptions like '+' and '-'. To make '+' and '-' work, you also need to | |
2258 | # specify your own overload for them. | |
2259 | ||
2260 | # object, (object|scalar) => preserve first and make copy | |
2261 | # scalar, object => swapped, re-swap and create new from first | |
2262 | # (using class of second object, not $class!!) | |
2263 | my $self = shift; # for override in subclass | |
2264 | if ($_[2]) | |
2265 | { | |
2266 | my $c = ref ($_[0]) || $class; # fallback $class should not happen | |
2267 | return ( $c->new($_[1]), $_[0] ); | |
2268 | } | |
2269 | return ( $_[0]->copy(), $_[1] ); | |
2270 | } | |
2271 | ||
2272 | sub objectify | |
2273 | { | |
2274 | # check for strings, if yes, return objects instead | |
2275 | ||
2276 | # the first argument is number of args objectify() should look at it will | |
2277 | # return $count+1 elements, the first will be a classname. This is because | |
2278 | # overloaded '""' calls bstr($object,undef,undef) and this would result in | |
2279 | # useless objects beeing created and thrown away. So we cannot simple loop | |
2280 | # over @_. If the given count is 0, all arguments will be used. | |
2281 | ||
2282 | # If the second arg is a ref, use it as class. | |
2283 | # If not, try to use it as classname, unless undef, then use $class | |
2284 | # (aka Math::BigInt). The latter shouldn't happen,though. | |
2285 | ||
2286 | # caller: gives us: | |
2287 | # $x->badd(1); => ref x, scalar y | |
2288 | # Class->badd(1,2); => classname x (scalar), scalar x, scalar y | |
2289 | # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y | |
2290 | # Math::BigInt::badd(1,2); => scalar x, scalar y | |
2291 | # In the last case we check number of arguments to turn it silently into | |
2292 | # $class,1,2. (We can not take '1' as class ;o) | |
2293 | # badd($class,1) is not supported (it should, eventually, try to add undef) | |
2294 | # currently it tries 'Math::BigInt' + 1, which will not work. | |
2295 | ||
2296 | # some shortcut for the common cases | |
2297 | # $x->unary_op(); | |
2298 | return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]); | |
2299 | ||
2300 | my $count = abs(shift || 0); | |
2301 | ||
2302 | my (@a,$k,$d); # resulting array, temp, and downgrade | |
2303 | if (ref $_[0]) | |
2304 | { | |
2305 | # okay, got object as first | |
2306 | $a[0] = ref $_[0]; | |
2307 | } | |
2308 | else | |
2309 | { | |
2310 | # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported) | |
2311 | $a[0] = $class; | |
2312 | $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first? | |
2313 | } | |
2314 | ||
2315 | no strict 'refs'; | |
2316 | # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats | |
2317 | if (defined ${"$a[0]::downgrade"}) | |
2318 | { | |
2319 | $d = ${"$a[0]::downgrade"}; | |
2320 | ${"$a[0]::downgrade"} = undef; | |
2321 | } | |
2322 | ||
2323 | my $up = ${"$a[0]::upgrade"}; | |
2324 | # print "Now in objectify, my class is today $a[0]\n"; | |
2325 | if ($count == 0) | |
2326 | { | |
2327 | while (@_) | |
2328 | { | |
2329 | $k = shift; | |
2330 | if (!ref($k)) | |
2331 | { | |
2332 | $k = $a[0]->new($k); | |
2333 | } | |
2334 | elsif (!defined $up && ref($k) ne $a[0]) | |
2335 | { | |
2336 | # foreign object, try to convert to integer | |
2337 | $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k); | |
2338 | } | |
2339 | push @a,$k; | |
2340 | } | |
2341 | } | |
2342 | else | |
2343 | { | |
2344 | while ($count > 0) | |
2345 | { | |
2346 | $count--; | |
2347 | $k = shift; | |
2348 | if (!ref($k)) | |
2349 | { | |
2350 | $k = $a[0]->new($k); | |
2351 | } | |
2352 | elsif (!defined $up && ref($k) ne $a[0]) | |
2353 | { | |
2354 | # foreign object, try to convert to integer | |
2355 | $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k); | |
2356 | } | |
2357 | push @a,$k; | |
2358 | } | |
2359 | push @a,@_; # return other params, too | |
2360 | } | |
2361 | die "$class objectify needs list context" unless wantarray; | |
2362 | ${"$a[0]::downgrade"} = $d; | |
2363 | @a; | |
2364 | } | |
2365 | ||
2366 | sub import | |
2367 | { | |
2368 | my $self = shift; | |
2369 | ||
2370 | $IMPORT++; | |
2371 | my @a; my $l = scalar @_; | |
2372 | for ( my $i = 0; $i < $l ; $i++ ) | |
2373 | { | |
2374 | if ($_[$i] eq ':constant') | |
2375 | { | |
2376 | # this causes overlord er load to step in | |
2377 | overload::constant integer => sub { $self->new(shift) }; | |
2378 | overload::constant binary => sub { $self->new(shift) }; | |
2379 | } | |
2380 | elsif ($_[$i] eq 'upgrade') | |
2381 | { | |
2382 | # this causes upgrading | |
2383 | $upgrade = $_[$i+1]; # or undef to disable | |
2384 | $i++; | |
2385 | } | |
2386 | elsif ($_[$i] =~ /^lib$/i) | |
2387 | { | |
2388 | # this causes a different low lib to take care... | |
2389 | $CALC = $_[$i+1] || ''; | |
2390 | $i++; | |
2391 | } | |
2392 | else | |
2393 | { | |
2394 | push @a, $_[$i]; | |
2395 | } | |
2396 | } | |
2397 | # any non :constant stuff is handled by our parent, Exporter | |
2398 | # even if @_ is empty, to give it a chance | |
2399 | $self->SUPER::import(@a); # need it for subclasses | |
2400 | $self->export_to_level(1,$self,@a); # need it for MBF | |
2401 | ||
2402 | # try to load core math lib | |
2403 | my @c = split /\s*,\s*/,$CALC; | |
2404 | push @c,'Calc'; # if all fail, try this | |
2405 | $CALC = ''; # signal error | |
2406 | foreach my $lib (@c) | |
2407 | { | |
2408 | next if ($lib || '') eq ''; | |
2409 | $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i; | |
2410 | $lib =~ s/\.pm$//; | |
2411 | if ($] < 5.006) | |
2412 | { | |
2413 | # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is | |
2414 | # used in the same script, or eval inside import(). | |
2415 | my @parts = split /::/, $lib; # Math::BigInt => Math BigInt | |
2416 | my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm | |
2417 | require File::Spec; | |
2418 | $file = File::Spec->catfile (@parts, $file); | |
2419 | eval { require "$file"; $lib->import( @c ); } | |
2420 | } | |
2421 | else | |
2422 | { | |
2423 | eval "use $lib qw/@c/;"; | |
2424 | } | |
2425 | $CALC = $lib, last if $@ eq ''; # no error in loading lib? | |
2426 | } | |
2427 | die "Couldn't load any math lib, not even the default" if $CALC eq ''; | |
2428 | } | |
2429 | ||
2430 | sub __from_hex | |
2431 | { | |
2432 | # convert a (ref to) big hex string to BigInt, return undef for error | |
2433 | my $hs = shift; | |
2434 | ||
2435 | my $x = Math::BigInt->bzero(); | |
2436 | ||
2437 | # strip underscores | |
2438 | $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g; | |
2439 | $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g; | |
2440 | ||
2441 | return $x->bnan() if $$hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/; | |
2442 | ||
2443 | my $sign = '+'; $sign = '-' if ($$hs =~ /^-/); | |
2444 | ||
2445 | $$hs =~ s/^[+-]//; # strip sign | |
2446 | if ($CALC->can('_from_hex')) | |
2447 | { | |
2448 | $x->{value} = $CALC->_from_hex($hs); | |
2449 | } | |
2450 | else | |
2451 | { | |
2452 | # fallback to pure perl | |
2453 | my $mul = Math::BigInt->bzero(); $mul++; | |
2454 | my $x65536 = Math::BigInt->new(65536); | |
2455 | my $len = CORE::length($$hs)-2; | |
2456 | $len = int($len/4); # 4-digit parts, w/o '0x' | |
2457 | my $val; my $i = -4; | |
2458 | while ($len >= 0) | |
2459 | { | |
2460 | $val = substr($$hs,$i,4); | |
2461 | $val =~ s/^[+-]?0x// if $len == 0; # for last part only because | |
2462 | $val = hex($val); # hex does not like wrong chars | |
2463 | $i -= 4; $len --; | |
2464 | $x += $mul * $val if $val != 0; | |
2465 | $mul *= $x65536 if $len >= 0; # skip last mul | |
2466 | } | |
2467 | } | |
2468 | $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0' | |
2469 | $x; | |
2470 | } | |
2471 | ||
2472 | sub __from_bin | |
2473 | { | |
2474 | # convert a (ref to) big binary string to BigInt, return undef for error | |
2475 | my $bs = shift; | |
2476 | ||
2477 | my $x = Math::BigInt->bzero(); | |
2478 | # strip underscores | |
2479 | $$bs =~ s/([01])_([01])/$1$2/g; | |
2480 | $$bs =~ s/([01])_([01])/$1$2/g; | |
2481 | return $x->bnan() if $$bs !~ /^[+-]?0b[01]+$/; | |
2482 | ||
2483 | my $sign = '+'; $sign = '-' if ($$bs =~ /^\-/); | |
2484 | $$bs =~ s/^[+-]//; # strip sign | |
2485 | if ($CALC->can('_from_bin')) | |
2486 | { | |
2487 | $x->{value} = $CALC->_from_bin($bs); | |
2488 | } | |
2489 | else | |
2490 | { | |
2491 | my $mul = Math::BigInt->bzero(); $mul++; | |
2492 | my $x256 = Math::BigInt->new(256); | |
2493 | my $len = CORE::length($$bs)-2; | |
2494 | $len = int($len/8); # 8-digit parts, w/o '0b' | |
2495 | my $val; my $i = -8; | |
2496 | while ($len >= 0) | |
2497 | { | |
2498 | $val = substr($$bs,$i,8); | |
2499 | $val =~ s/^[+-]?0b// if $len == 0; # for last part only | |
2500 | #$val = oct('0b'.$val); # does not work on Perl prior to 5.6.0 | |
2501 | # slower: | |
2502 | # $val = ('0' x (8-CORE::length($val))).$val if CORE::length($val) < 8; | |
2503 | $val = ord(pack('B8',substr('00000000'.$val,-8,8))); | |
2504 | $i -= 8; $len --; | |
2505 | $x += $mul * $val if $val != 0; | |
2506 | $mul *= $x256 if $len >= 0; # skip last mul | |
2507 | } | |
2508 | } | |
2509 | $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0' | |
2510 | $x; | |
2511 | } | |
2512 | ||
2513 | sub _split | |
2514 | { | |
2515 | # (ref to num_str) return num_str | |
2516 | # internal, take apart a string and return the pieces | |
2517 | # strip leading/trailing whitespace, leading zeros, underscore and reject | |
2518 | # invalid input | |
2519 | my $x = shift; | |
2520 | ||
2521 | # strip white space at front, also extranous leading zeros | |
2522 | $$x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2' | |
2523 | $$x =~ s/^\s+//; # but this will | |
2524 | $$x =~ s/\s+$//g; # strip white space at end | |
2525 | ||
2526 | # shortcut, if nothing to split, return early | |
2527 | if ($$x =~ /^[+-]?\d+\z/) | |
2528 | { | |
2529 | $$x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+'; | |
2530 | return (\$sign, $x, \'', \'', \0); | |
2531 | } | |
2532 | ||
2533 | # invalid starting char? | |
2534 | return if $$x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/; | |
2535 | ||
2536 | return __from_hex($x) if $$x =~ /^[\-\+]?0x/; # hex string | |
2537 | return __from_bin($x) if $$x =~ /^[\-\+]?0b/; # binary string | |
2538 | ||
2539 | # strip underscores between digits | |
2540 | $$x =~ s/(\d)_(\d)/$1$2/g; | |
2541 | $$x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3 | |
2542 | ||
2543 | # some possible inputs: | |
2544 | # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2 | |
2545 | # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 | |
2546 | ||
2547 | return if $$x =~ /[Ee].*[Ee]/; # more than one E => error | |
2548 | ||
2549 | my ($m,$e) = split /[Ee]/,$$x; | |
2550 | $e = '0' if !defined $e || $e eq ""; | |
2551 | # sign,value for exponent,mantint,mantfrac | |
2552 | my ($es,$ev,$mis,$miv,$mfv); | |
2553 | # valid exponent? | |
2554 | if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros | |
2555 | { | |
2556 | $es = $1; $ev = $2; | |
2557 | # valid mantissa? | |
2558 | return if $m eq '.' || $m eq ''; | |
2559 | my ($mi,$mf,$last) = split /\./,$m; | |
2560 | return if defined $last; # last defined => 1.2.3 or others | |
2561 | $mi = '0' if !defined $mi; | |
2562 | $mi .= '0' if $mi =~ /^[\-\+]?$/; | |
2563 | $mf = '0' if !defined $mf || $mf eq ''; | |
2564 | if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros | |
2565 | { | |
2566 | $mis = $1||'+'; $miv = $2; | |
2567 | return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros | |
2568 | $mfv = $1; | |
2569 | return (\$mis,\$miv,\$mfv,\$es,\$ev); | |
2570 | } | |
2571 | } | |
2572 | return; # NaN, not a number | |
2573 | } | |
2574 | ||
2575 | sub as_number | |
2576 | { | |
2577 | # an object might be asked to return itself as bigint on certain overloaded | |
2578 | # operations, this does exactly this, so that sub classes can simple inherit | |
2579 | # it or override with their own integer conversion routine | |
2580 | my $self = shift; | |
2581 | ||
2582 | $self->copy(); | |
2583 | } | |
2584 | ||
2585 | sub as_hex | |
2586 | { | |
2587 | # return as hex string, with prefixed 0x | |
2588 | my $x = shift; $x = $class->new($x) if !ref($x); | |
2589 | ||
2590 | return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc | |
2591 | return '0x0' if $x->is_zero(); | |
2592 | ||
2593 | my $es = ''; my $s = ''; | |
2594 | $s = $x->{sign} if $x->{sign} eq '-'; | |
2595 | if ($CALC->can('_as_hex')) | |
2596 | { | |
2597 | $es = ${$CALC->_as_hex($x->{value})}; | |
2598 | } | |
2599 | else | |
2600 | { | |
2601 | my $x1 = $x->copy()->babs(); my ($xr,$x10000,$h); | |
2602 | if ($] >= 5.006) | |
2603 | { | |
2604 | $x10000 = Math::BigInt->new (0x10000); $h = 'h4'; | |
2605 | } | |
2606 | else | |
2607 | { | |
2608 | $x10000 = Math::BigInt->new (0x1000); $h = 'h3'; | |
2609 | } | |
2610 | while (!$x1->is_zero()) | |
2611 | { | |
2612 | ($x1, $xr) = bdiv($x1,$x10000); | |
2613 | $es .= unpack($h,pack('v',$xr->numify())); | |
2614 | } | |
2615 | $es = reverse $es; | |
2616 | $es =~ s/^[0]+//; # strip leading zeros | |
2617 | $s .= '0x'; | |
2618 | } | |
2619 | $s . $es; | |
2620 | } | |
2621 | ||
2622 | sub as_bin | |
2623 | { | |
2624 | # return as binary string, with prefixed 0b | |
2625 | my $x = shift; $x = $class->new($x) if !ref($x); | |
2626 | ||
2627 | return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc | |
2628 | return '0b0' if $x->is_zero(); | |
2629 | ||
2630 | my $es = ''; my $s = ''; | |
2631 | $s = $x->{sign} if $x->{sign} eq '-'; | |
2632 | if ($CALC->can('_as_bin')) | |
2633 | { | |
2634 | $es = ${$CALC->_as_bin($x->{value})}; | |
2635 | } | |
2636 | else | |
2637 | { | |
2638 | my $x1 = $x->copy()->babs(); my ($xr,$x10000,$b); | |
2639 | if ($] >= 5.006) | |
2640 | { | |
2641 | $x10000 = Math::BigInt->new (0x10000); $b = 'b16'; | |
2642 | } | |
2643 | else | |
2644 | { | |
2645 | $x10000 = Math::BigInt->new (0x1000); $b = 'b12'; | |
2646 | } | |
2647 | while (!$x1->is_zero()) | |
2648 | { | |
2649 | ($x1, $xr) = bdiv($x1,$x10000); | |
2650 | $es .= unpack($b,pack('v',$xr->numify())); | |
2651 | } | |
2652 | $es = reverse $es; | |
2653 | $es =~ s/^[0]+//; # strip leading zeros | |
2654 | $s .= '0b'; | |
2655 | } | |
2656 | $s . $es; | |
2657 | } | |
2658 | ||
2659 | ############################################################################## | |
2660 | # internal calculation routines (others are in Math::BigInt::Calc etc) | |
2661 | ||
2662 | sub __lcm | |
2663 | { | |
2664 | # (BINT or num_str, BINT or num_str) return BINT | |
2665 | # does modify first argument | |
2666 | # LCM | |
2667 | ||
2668 | my $x = shift; my $ty = shift; | |
2669 | return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan); | |
2670 | return $x * $ty / bgcd($x,$ty); | |
2671 | } | |
2672 | ||
2673 | sub __gcd | |
2674 | { | |
2675 | # (BINT or num_str, BINT or num_str) return BINT | |
2676 | # does modify both arguments | |
2677 | # GCD -- Euclids algorithm E, Knuth Vol 2 pg 296 | |
2678 | my ($x,$ty) = @_; | |
2679 | ||
2680 | return $x->bnan() if $x->{sign} !~ /^[+-]$/ || $ty->{sign} !~ /^[+-]$/; | |
2681 | ||
2682 | while (!$ty->is_zero()) | |
2683 | { | |
2684 | ($x, $ty) = ($ty,bmod($x,$ty)); | |
2685 | } | |
2686 | $x; | |
2687 | } | |
2688 | ||
2689 | ############################################################################### | |
2690 | # this method return 0 if the object can be modified, or 1 for not | |
2691 | # We use a fast use constant statement here, to avoid costly calls. Subclasses | |
2692 | # may override it with special code (f.i. Math::BigInt::Constant does so) | |
2693 | ||
2694 | sub modify () { 0; } | |
2695 | ||
2696 | 1; | |
2697 | __END__ | |
2698 | ||
2699 | =head1 NAME | |
2700 | ||
2701 | Math::BigInt - Arbitrary size integer math package | |
2702 | ||
2703 | =head1 SYNOPSIS | |
2704 | ||
2705 | use Math::BigInt; | |
2706 | ||
2707 | # Number creation | |
2708 | $x = Math::BigInt->new($str); # defaults to 0 | |
2709 | $nan = Math::BigInt->bnan(); # create a NotANumber | |
2710 | $zero = Math::BigInt->bzero(); # create a +0 | |
2711 | $inf = Math::BigInt->binf(); # create a +inf | |
2712 | $inf = Math::BigInt->binf('-'); # create a -inf | |
2713 | $one = Math::BigInt->bone(); # create a +1 | |
2714 | $one = Math::BigInt->bone('-'); # create a -1 | |
2715 | ||
2716 | # Testing | |
2717 | $x->is_zero(); # true if arg is +0 | |
2718 | $x->is_nan(); # true if arg is NaN | |
2719 | $x->is_one(); # true if arg is +1 | |
2720 | $x->is_one('-'); # true if arg is -1 | |
2721 | $x->is_odd(); # true if odd, false for even | |
2722 | $x->is_even(); # true if even, false for odd | |
2723 | $x->is_positive(); # true if >= 0 | |
2724 | $x->is_negative(); # true if < 0 | |
2725 | $x->is_inf(sign); # true if +inf, or -inf (sign is default '+') | |
2726 | $x->is_int(); # true if $x is an integer (not a float) | |
2727 | ||
2728 | $x->bcmp($y); # compare numbers (undef,<0,=0,>0) | |
2729 | $x->bacmp($y); # compare absolutely (undef,<0,=0,>0) | |
2730 | $x->sign(); # return the sign, either +,- or NaN | |
2731 | $x->digit($n); # return the nth digit, counting from right | |
2732 | $x->digit(-$n); # return the nth digit, counting from left | |
2733 | ||
2734 | # The following all modify their first argument: | |
2735 | ||
2736 | # set | |
2737 | $x->bzero(); # set $x to 0 | |
2738 | $x->bnan(); # set $x to NaN | |
2739 | $x->bone(); # set $x to +1 | |
2740 | $x->bone('-'); # set $x to -1 | |
2741 | $x->binf(); # set $x to inf | |
2742 | $x->binf('-'); # set $x to -inf | |
2743 | ||
2744 | $x->bneg(); # negation | |
2745 | $x->babs(); # absolute value | |
2746 | $x->bnorm(); # normalize (no-op) | |
2747 | $x->bnot(); # two's complement (bit wise not) | |
2748 | $x->binc(); # increment x by 1 | |
2749 | $x->bdec(); # decrement x by 1 | |
2750 | ||
2751 | $x->badd($y); # addition (add $y to $x) | |
2752 | $x->bsub($y); # subtraction (subtract $y from $x) | |
2753 | $x->bmul($y); # multiplication (multiply $x by $y) | |
2754 | $x->bdiv($y); # divide, set $x to quotient | |
2755 | # return (quo,rem) or quo if scalar | |
2756 | ||
2757 | $x->bmod($y); # modulus (x % y) | |
2758 | $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod)) | |
2759 | $x->bmodinv($mod); # the inverse of $x in the given modulus $mod | |
2760 | ||
2761 | $x->bpow($y); # power of arguments (x ** y) | |
2762 | $x->blsft($y); # left shift | |
2763 | $x->brsft($y); # right shift | |
2764 | $x->blsft($y,$n); # left shift, by base $n (like 10) | |
2765 | $x->brsft($y,$n); # right shift, by base $n (like 10) | |
2766 | ||
2767 | $x->band($y); # bitwise and | |
2768 | $x->bior($y); # bitwise inclusive or | |
2769 | $x->bxor($y); # bitwise exclusive or | |
2770 | $x->bnot(); # bitwise not (two's complement) | |
2771 | ||
2772 | $x->bsqrt(); # calculate square-root | |
2773 | $x->bfac(); # factorial of $x (1*2*3*4*..$x) | |
2774 | ||
2775 | $x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r | |
2776 | $x->bround($N); # accuracy: preserve $N digits | |
2777 | $x->bfround($N); # round to $Nth digit, no-op for BigInts | |
2778 | ||
2779 | # The following do not modify their arguments in BigInt, but do in BigFloat: | |
2780 | $x->bfloor(); # return integer less or equal than $x | |
2781 | $x->bceil(); # return integer greater or equal than $x | |
2782 | ||
2783 | # The following do not modify their arguments: | |
2784 | ||
2785 | bgcd(@values); # greatest common divisor (no OO style) | |
2786 | blcm(@values); # lowest common multiplicator (no OO style) | |
2787 | ||
2788 | $x->length(); # return number of digits in number | |
2789 | ($x,$f) = $x->length(); # length of number and length of fraction part, | |
2790 | # latter is always 0 digits long for BigInt's | |
2791 | ||
2792 | $x->exponent(); # return exponent as BigInt | |
2793 | $x->mantissa(); # return (signed) mantissa as BigInt | |
2794 | $x->parts(); # return (mantissa,exponent) as BigInt | |
2795 | $x->copy(); # make a true copy of $x (unlike $y = $x;) | |
2796 | $x->as_number(); # return as BigInt (in BigInt: same as copy()) | |
2797 | ||
2798 | # conversation to string | |
2799 | $x->bstr(); # normalized string | |
2800 | $x->bsstr(); # normalized string in scientific notation | |
2801 | $x->as_hex(); # as signed hexadecimal string with prefixed 0x | |
2802 | $x->as_bin(); # as signed binary string with prefixed 0b | |
2803 | ||
2804 | Math::BigInt->config(); # return hash containing configuration/version | |
2805 | ||
2806 | # precision and accuracy (see section about rounding for more) | |
2807 | $x->precision(); # return P of $x (or global, if P of $x undef) | |
2808 | $x->precision($n); # set P of $x to $n | |
2809 | $x->accuracy(); # return A of $x (or global, if A of $x undef) | |
2810 | $x->accuracy($n); # set A $x to $n | |
2811 | ||
2812 | Math::BigInt->precision(); # get/set global P for all BigInt objects | |
2813 | Math::BigInt->accuracy(); # get/set global A for all BigInt objects | |
2814 | ||
2815 | =head1 DESCRIPTION | |
2816 | ||
2817 | All operators (inlcuding basic math operations) are overloaded if you | |
2818 | declare your big integers as | |
2819 | ||
2820 | $i = new Math::BigInt '123_456_789_123_456_789'; | |
2821 | ||
2822 | Operations with overloaded operators preserve the arguments which is | |
2823 | exactly what you expect. | |
2824 | ||
2825 | =over 2 | |
2826 | ||
2827 | =item Canonical notation | |
2828 | ||
2829 | Big integer values are strings of the form C</^[+-]\d+$/> with leading | |
2830 | zeros suppressed. | |
2831 | ||
2832 | '-0' canonical value '-0', normalized '0' | |
2833 | ' -123_123_123' canonical value '-123123123' | |
2834 | '1_23_456_7890' canonical value '1234567890' | |
2835 | ||
2836 | =item Input | |
2837 | ||
2838 | Input values to these routines may be either Math::BigInt objects or | |
2839 | strings of the form C</^[+-]?[\d]+\.?[\d]*E?[+-]?[\d]*$/>. | |
2840 | ||
2841 | You can include one underscore between any two digits. The input string may | |
2842 | have leading and trailing whitespace, which will be ignored. In later | |
2843 | versions, a more strict (no whitespace at all) or more lax (whitespace | |
2844 | allowed everywhere) input checking will also be possible. | |
2845 | ||
2846 | This means integer values like 1.01E2 or even 1000E-2 are also accepted. | |
2847 | Non integer values result in NaN. | |
2848 | ||
2849 | Math::BigInt::new() defaults to 0, while Math::BigInt::new('') results | |
2850 | in 'NaN'. | |
2851 | ||
2852 | bnorm() on a BigInt object is now effectively a no-op, since the numbers | |
2853 | are always stored in normalized form. On a string, it creates a BigInt | |
2854 | object. | |
2855 | ||
2856 | =item Output | |
2857 | ||
2858 | Output values are BigInt objects (normalized), except for bstr(), which | |
2859 | returns a string in normalized form. | |
2860 | Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>, | |
2861 | C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>) | |
2862 | return either undef, <0, 0 or >0 and are suited for sort. | |
2863 | ||
2864 | =back | |
2865 | ||
2866 | =head1 METHODS | |
2867 | ||
2868 | Each of the methods below accepts three additional parameters. These arguments | |
2869 | $A, $P and $R are accuracy, precision and round_mode. Please see more in the | |
2870 | section about ACCURACY and ROUNDIND. | |
2871 | ||
2872 | =head2 config | |
2873 | ||
2874 | use Data::Dumper; | |
2875 | ||
2876 | print Dumper ( Math::BigInt->config() ); | |
2877 | ||
2878 | Returns a hash containing the configuration, e.g. the version number, lib | |
2879 | loaded etc. | |
2880 | ||
2881 | =head2 accuracy | |
2882 | ||
2883 | $x->accuracy(5); # local for $x | |
2884 | $class->accuracy(5); # global for all members of $class | |
2885 | ||
2886 | Set or get the global or local accuracy, aka how many significant digits the | |
2887 | results have. Please see the section about L<ACCURACY AND PRECISION> for | |
2888 | further details. | |
2889 | ||
2890 | Value must be greater than zero. Pass an undef value to disable it: | |
2891 | ||
2892 | $x->accuracy(undef); | |
2893 | Math::BigInt->accuracy(undef); | |
2894 | ||
2895 | Returns the current accuracy. For C<$x->accuracy()> it will return either the | |
2896 | local accuracy, or if not defined, the global. This means the return value | |
2897 | represents the accuracy that will be in effect for $x: | |
2898 | ||
2899 | $y = Math::BigInt->new(1234567); # unrounded | |
2900 | print Math::BigInt->accuracy(4),"\n"; # set 4, print 4 | |
2901 | $x = Math::BigInt->new(123456); # will be automatically rounded | |
2902 | print "$x $y\n"; # '123500 1234567' | |
2903 | print $x->accuracy(),"\n"; # will be 4 | |
2904 | print $y->accuracy(),"\n"; # also 4, since global is 4 | |
2905 | print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5 | |
2906 | print $x->accuracy(),"\n"; # still 4 | |
2907 | print $y->accuracy(),"\n"; # 5, since global is 5 | |
2908 | ||
2909 | =head2 brsft | |
2910 | ||
2911 | $x->brsft($y,$n); | |
2912 | ||
2913 | Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and | |
2914 | 2, but others work, too. | |
2915 | ||
2916 | Right shifting usually amounts to dividing $x by $n ** $y and truncating the | |
2917 | result: | |
2918 | ||
2919 | ||
2920 | $x = Math::BigInt->new(10); | |
2921 | $x->brsft(1); # same as $x >> 1: 5 | |
2922 | $x = Math::BigInt->new(1234); | |
2923 | $x->brsft(2,10); # result 12 | |
2924 | ||
2925 | There is one exception, and that is base 2 with negative $x: | |
2926 | ||
2927 | ||
2928 | $x = Math::BigInt->new(-5); | |
2929 | print $x->brsft(1); | |
2930 | ||
2931 | This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the | |
2932 | result). | |
2933 | ||
2934 | =head2 new | |
2935 | ||
2936 | $x = Math::BigInt->new($str,$A,$P,$R); | |
2937 | ||
2938 | Creates a new BigInt object from a string or another BigInt object. The | |
2939 | input is accepted as decimal, hex (with leading '0x') or binary (with leading | |
2940 | '0b'). | |
2941 | ||
2942 | =head2 bnan | |
2943 | ||
2944 | $x = Math::BigInt->bnan(); | |
2945 | ||
2946 | Creates a new BigInt object representing NaN (Not A Number). | |
2947 | If used on an object, it will set it to NaN: | |
2948 | ||
2949 | $x->bnan(); | |
2950 | ||
2951 | =head2 bzero | |
2952 | ||
2953 | $x = Math::BigInt->bzero(); | |
2954 | ||
2955 | Creates a new BigInt object representing zero. | |
2956 | If used on an object, it will set it to zero: | |
2957 | ||
2958 | $x->bzero(); | |
2959 | ||
2960 | =head2 binf | |
2961 | ||
2962 | $x = Math::BigInt->binf($sign); | |
2963 | ||
2964 | Creates a new BigInt object representing infinity. The optional argument is | |
2965 | either '-' or '+', indicating whether you want infinity or minus infinity. | |
2966 | If used on an object, it will set it to infinity: | |
2967 | ||
2968 | $x->binf(); | |
2969 | $x->binf('-'); | |
2970 | ||
2971 | =head2 bone | |
2972 | ||
2973 | $x = Math::BigInt->binf($sign); | |
2974 | ||
2975 | Creates a new BigInt object representing one. The optional argument is | |
2976 | either '-' or '+', indicating whether you want one or minus one. | |
2977 | If used on an object, it will set it to one: | |
2978 | ||
2979 | $x->bone(); # +1 | |
2980 | $x->bone('-'); # -1 | |
2981 | ||
2982 | =head2 is_one()/is_zero()/is_nan()/is_inf() | |
2983 | ||
2984 | ||
2985 | $x->is_zero(); # true if arg is +0 | |
2986 | $x->is_nan(); # true if arg is NaN | |
2987 | $x->is_one(); # true if arg is +1 | |
2988 | $x->is_one('-'); # true if arg is -1 | |
2989 | $x->is_inf(); # true if +inf | |
2990 | $x->is_inf('-'); # true if -inf (sign is default '+') | |
2991 | ||
2992 | These methods all test the BigInt for beeing one specific value and return | |
2993 | true or false depending on the input. These are faster than doing something | |
2994 | like: | |
2995 | ||
2996 | if ($x == 0) | |
2997 | ||
2998 | =head2 is_positive()/is_negative() | |
2999 | ||
3000 | $x->is_positive(); # true if >= 0 | |
3001 | $x->is_negative(); # true if < 0 | |
3002 | ||
3003 | The methods return true if the argument is positive or negative, respectively. | |
3004 | C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and | |
3005 | C<-inf> is negative. A C<zero> is positive. | |
3006 | ||
3007 | These methods are only testing the sign, and not the value. | |
3008 | ||
3009 | =head2 is_odd()/is_even()/is_int() | |
3010 | ||
3011 | $x->is_odd(); # true if odd, false for even | |
3012 | $x->is_even(); # true if even, false for odd | |
3013 | $x->is_int(); # true if $x is an integer | |
3014 | ||
3015 | The return true when the argument satisfies the condition. C<NaN>, C<+inf>, | |
3016 | C<-inf> are not integers and are neither odd nor even. | |
3017 | ||
3018 | =head2 bcmp | |
3019 | ||
3020 | $x->bcmp($y); | |
3021 | ||
3022 | Compares $x with $y and takes the sign into account. | |
3023 | Returns -1, 0, 1 or undef. | |
3024 | ||
3025 | =head2 bacmp | |
3026 | ||
3027 | $x->bacmp($y); | |
3028 | ||
3029 | Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef. | |
3030 | ||
3031 | =head2 sign | |
3032 | ||
3033 | $x->sign(); | |
3034 | ||
3035 | Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN. | |
3036 | ||
3037 | =head2 bcmp | |
3038 | ||
3039 | $x->digit($n); # return the nth digit, counting from right | |
3040 | ||
3041 | =head2 bneg | |
3042 | ||
3043 | $x->bneg(); | |
3044 | ||
3045 | Negate the number, e.g. change the sign between '+' and '-', or between '+inf' | |
3046 | and '-inf', respectively. Does nothing for NaN or zero. | |
3047 | ||
3048 | =head2 babs | |
3049 | ||
3050 | $x->babs(); | |
3051 | ||
3052 | Set the number to it's absolute value, e.g. change the sign from '-' to '+' | |
3053 | and from '-inf' to '+inf', respectively. Does nothing for NaN or positive | |
3054 | numbers. | |
3055 | ||
3056 | =head2 bnorm | |
3057 | ||
3058 | $x->bnorm(); # normalize (no-op) | |
3059 | ||
3060 | =head2 bnot | |
3061 | ||
3062 | $x->bnot(); # two's complement (bit wise not) | |
3063 | ||
3064 | =head2 binc | |
3065 | ||
3066 | $x->binc(); # increment x by 1 | |
3067 | ||
3068 | =head2 bdec | |
3069 | ||
3070 | $x->bdec(); # decrement x by 1 | |
3071 | ||
3072 | =head2 badd | |
3073 | ||
3074 | $x->badd($y); # addition (add $y to $x) | |
3075 | ||
3076 | =head2 bsub | |
3077 | ||
3078 | $x->bsub($y); # subtraction (subtract $y from $x) | |
3079 | ||
3080 | =head2 bmul | |
3081 | ||
3082 | $x->bmul($y); # multiplication (multiply $x by $y) | |
3083 | ||
3084 | =head2 bdiv | |
3085 | ||
3086 | $x->bdiv($y); # divide, set $x to quotient | |
3087 | # return (quo,rem) or quo if scalar | |
3088 | ||
3089 | =head2 bmod | |
3090 | ||
3091 | $x->bmod($y); # modulus (x % y) | |
3092 | ||
3093 | =head2 bmodinv | |
3094 | ||
3095 | $num->bmodinv($mod); # modular inverse | |
3096 | ||
3097 | Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is | |
3098 | returned unless C<$num> is relatively prime to C<$mod>, i.e. unless | |
3099 | C<bgcd($num, $mod)==1>. | |
3100 | ||
3101 | =head2 bmodpow | |
3102 | ||
3103 | $num->bmodpow($exp,$mod); # modular exponentation ($num**$exp % $mod) | |
3104 | ||
3105 | Returns the value of C<$num> taken to the power C<$exp> in the modulus | |
3106 | C<$mod> using binary exponentation. C<bmodpow> is far superior to | |
3107 | writing | |
3108 | ||
3109 | $num ** $exp % $mod | |
3110 | ||
3111 | because C<bmodpow> is much faster--it reduces internal variables into | |
3112 | the modulus whenever possible, so it operates on smaller numbers. | |
3113 | ||
3114 | C<bmodpow> also supports negative exponents. | |
3115 | ||
3116 | bmodpow($num, -1, $mod) | |
3117 | ||
3118 | is exactly equivalent to | |
3119 | ||
3120 | bmodinv($num, $mod) | |
3121 | ||
3122 | =head2 bpow | |
3123 | ||
3124 | $x->bpow($y); # power of arguments (x ** y) | |
3125 | ||
3126 | =head2 blsft | |
3127 | ||
3128 | $x->blsft($y); # left shift | |
3129 | $x->blsft($y,$n); # left shift, by base $n (like 10) | |
3130 | ||
3131 | =head2 brsft | |
3132 | ||
3133 | $x->brsft($y); # right shift | |
3134 | $x->brsft($y,$n); # right shift, by base $n (like 10) | |
3135 | ||
3136 | =head2 band | |
3137 | ||
3138 | $x->band($y); # bitwise and | |
3139 | ||
3140 | =head2 bior | |
3141 | ||
3142 | $x->bior($y); # bitwise inclusive or | |
3143 | ||
3144 | =head2 bxor | |
3145 | ||
3146 | $x->bxor($y); # bitwise exclusive or | |
3147 | ||
3148 | =head2 bnot | |
3149 | ||
3150 | $x->bnot(); # bitwise not (two's complement) | |
3151 | ||
3152 | =head2 bsqrt | |
3153 | ||
3154 | $x->bsqrt(); # calculate square-root | |
3155 | ||
3156 | =head2 bfac | |
3157 | ||
3158 | $x->bfac(); # factorial of $x (1*2*3*4*..$x) | |
3159 | ||
3160 | =head2 round | |
3161 | ||
3162 | $x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r | |
3163 | ||
3164 | =head2 bround | |
3165 | ||
3166 | $x->bround($N); # accuracy: preserve $N digits | |
3167 | ||
3168 | =head2 bfround | |
3169 | ||
3170 | $x->bfround($N); # round to $Nth digit, no-op for BigInts | |
3171 | ||
3172 | =head2 bfloor | |
3173 | ||
3174 | $x->bfloor(); | |
3175 | ||
3176 | Set $x to the integer less or equal than $x. This is a no-op in BigInt, but | |
3177 | does change $x in BigFloat. | |
3178 | ||
3179 | =head2 bceil | |
3180 | ||
3181 | $x->bceil(); | |
3182 | ||
3183 | Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but | |
3184 | does change $x in BigFloat. | |
3185 | ||
3186 | =head2 bgcd | |
3187 | ||
3188 | bgcd(@values); # greatest common divisor (no OO style) | |
3189 | ||
3190 | =head2 blcm | |
3191 | ||
3192 | blcm(@values); # lowest common multiplicator (no OO style) | |
3193 | ||
3194 | head2 length | |
3195 | ||
3196 | $x->length(); | |
3197 | ($xl,$fl) = $x->length(); | |
3198 | ||
3199 | Returns the number of digits in the decimal representation of the number. | |
3200 | In list context, returns the length of the integer and fraction part. For | |
3201 | BigInt's, the length of the fraction part will always be 0. | |
3202 | ||
3203 | =head2 exponent | |
3204 | ||
3205 | $x->exponent(); | |
3206 | ||
3207 | Return the exponent of $x as BigInt. | |
3208 | ||
3209 | =head2 mantissa | |
3210 | ||
3211 | $x->mantissa(); | |
3212 | ||
3213 | Return the signed mantissa of $x as BigInt. | |
3214 | ||
3215 | =head2 parts | |
3216 | ||
3217 | $x->parts(); # return (mantissa,exponent) as BigInt | |
3218 | ||
3219 | =head2 copy | |
3220 | ||
3221 | $x->copy(); # make a true copy of $x (unlike $y = $x;) | |
3222 | ||
3223 | =head2 as_number | |
3224 | ||
3225 | $x->as_number(); # return as BigInt (in BigInt: same as copy()) | |
3226 | ||
3227 | =head2 bsrt | |
3228 | ||
3229 | $x->bstr(); # normalized string | |
3230 | ||
3231 | =head2 bsstr | |
3232 | ||
3233 | $x->bsstr(); # normalized string in scientific notation | |
3234 | ||
3235 | =head2 as_hex | |
3236 | ||
3237 | $x->as_hex(); # as signed hexadecimal string with prefixed 0x | |
3238 | ||
3239 | =head2 as_bin | |
3240 | ||
3241 | $x->as_bin(); # as signed binary string with prefixed 0b | |
3242 | ||
3243 | =head1 ACCURACY and PRECISION | |
3244 | ||
3245 | Since version v1.33, Math::BigInt and Math::BigFloat have full support for | |
3246 | accuracy and precision based rounding, both automatically after every | |
3247 | operation as well as manually. | |
3248 | ||
3249 | This section describes the accuracy/precision handling in Math::Big* as it | |
3250 | used to be and as it is now, complete with an explanation of all terms and | |
3251 | abbreviations. | |
3252 | ||
3253 | Not yet implemented things (but with correct description) are marked with '!', | |
3254 | things that need to be answered are marked with '?'. | |
3255 | ||
3256 | In the next paragraph follows a short description of terms used here (because | |
3257 | these may differ from terms used by others people or documentation). | |
3258 | ||
3259 | During the rest of this document, the shortcuts A (for accuracy), P (for | |
3260 | precision), F (fallback) and R (rounding mode) will be used. | |
3261 | ||
3262 | =head2 Precision P | |
3263 | ||
3264 | A fixed number of digits before (positive) or after (negative) | |
3265 | the decimal point. For example, 123.45 has a precision of -2. 0 means an | |
3266 | integer like 123 (or 120). A precision of 2 means two digits to the left | |
3267 | of the decimal point are zero, so 123 with P = 1 becomes 120. Note that | |
3268 | numbers with zeros before the decimal point may have different precisions, | |
3269 | because 1200 can have p = 0, 1 or 2 (depending on what the inital value | |
3270 | was). It could also have p < 0, when the digits after the decimal point | |
3271 | are zero. | |
3272 | ||
3273 | The string output (of floating point numbers) will be padded with zeros: | |
3274 | ||
3275 | Initial value P A Result String | |
3276 | ------------------------------------------------------------ | |
3277 | 1234.01 -3 1000 1000 | |
3278 | 1234 -2 1200 1200 | |
3279 | 1234.5 -1 1230 1230 | |
3280 | 1234.001 1 1234 1234.0 | |
3281 | 1234.01 0 1234 1234 | |
3282 | 1234.01 2 1234.01 1234.01 | |
3283 | 1234.01 5 1234.01 1234.01000 | |
3284 | ||
3285 | For BigInts, no padding occurs. | |
3286 | ||
3287 | =head2 Accuracy A | |
3288 | ||
3289 | Number of significant digits. Leading zeros are not counted. A | |
3290 | number may have an accuracy greater than the non-zero digits | |
3291 | when there are zeros in it or trailing zeros. For example, 123.456 has | |
3292 | A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3. | |
3293 | ||
3294 | The string output (of floating point numbers) will be padded with zeros: | |
3295 | ||
3296 | Initial value P A Result String | |
3297 | ------------------------------------------------------------ | |
3298 | 1234.01 3 1230 1230 | |
3299 | 1234.01 6 1234.01 1234.01 | |
3300 | 1234.1 8 1234.1 1234.1000 | |
3301 | ||
3302 | For BigInts, no padding occurs. | |
3303 | ||
3304 | =head2 Fallback F | |
3305 | ||
3306 | When both A and P are undefined, this is used as a fallback accuracy when | |
3307 | dividing numbers. | |
3308 | ||
3309 | =head2 Rounding mode R | |
3310 | ||
3311 | When rounding a number, different 'styles' or 'kinds' | |
3312 | of rounding are possible. (Note that random rounding, as in | |
3313 | Math::Round, is not implemented.) | |
3314 | ||
3315 | =over 2 | |
3316 | ||
3317 | =item 'trunc' | |
3318 | ||
3319 | truncation invariably removes all digits following the | |
3320 | rounding place, replacing them with zeros. Thus, 987.65 rounded | |
3321 | to tens (P=1) becomes 980, and rounded to the fourth sigdig | |
3322 | becomes 987.6 (A=4). 123.456 rounded to the second place after the | |
3323 | decimal point (P=-2) becomes 123.46. | |
3324 | ||
3325 | All other implemented styles of rounding attempt to round to the | |
3326 | "nearest digit." If the digit D immediately to the right of the | |
3327 | rounding place (skipping the decimal point) is greater than 5, the | |
3328 | number is incremented at the rounding place (possibly causing a | |
3329 | cascade of incrementation): e.g. when rounding to units, 0.9 rounds | |
3330 | to 1, and -19.9 rounds to -20. If D < 5, the number is similarly | |
3331 | truncated at the rounding place: e.g. when rounding to units, 0.4 | |
3332 | rounds to 0, and -19.4 rounds to -19. | |
3333 | ||
3334 | However the results of other styles of rounding differ if the | |
3335 | digit immediately to the right of the rounding place (skipping the | |
3336 | decimal point) is 5 and if there are no digits, or no digits other | |
3337 | than 0, after that 5. In such cases: | |
3338 | ||
3339 | =item 'even' | |
3340 | ||
3341 | rounds the digit at the rounding place to 0, 2, 4, 6, or 8 | |
3342 | if it is not already. E.g., when rounding to the first sigdig, 0.45 | |
3343 | becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5. | |
3344 | ||
3345 | =item 'odd' | |
3346 | ||
3347 | rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if | |
3348 | it is not already. E.g., when rounding to the first sigdig, 0.45 | |
3349 | becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6. | |
3350 | ||
3351 | =item '+inf' | |
3352 | ||
3353 | round to plus infinity, i.e. always round up. E.g., when | |
3354 | rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5, | |
3355 | and 0.4501 also becomes 0.5. | |
3356 | ||
3357 | =item '-inf' | |
3358 | ||
3359 | round to minus infinity, i.e. always round down. E.g., when | |
3360 | rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6, | |
3361 | but 0.4501 becomes 0.5. | |
3362 | ||
3363 | =item 'zero' | |
3364 | ||
3365 | round to zero, i.e. positive numbers down, negative ones up. | |
3366 | E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55 | |
3367 | becomes -0.5, but 0.4501 becomes 0.5. | |
3368 | ||
3369 | =back | |
3370 | ||
3371 | The handling of A & P in MBI/MBF (the old core code shipped with Perl | |
3372 | versions <= 5.7.2) is like this: | |
3373 | ||
3374 | =over 2 | |
3375 | ||
3376 | =item Precision | |
3377 | ||
3378 | * ffround($p) is able to round to $p number of digits after the decimal | |
3379 | point | |
3380 | * otherwise P is unused | |
3381 | ||
3382 | =item Accuracy (significant digits) | |
3383 | ||
3384 | * fround($a) rounds to $a significant digits | |
3385 | * only fdiv() and fsqrt() take A as (optional) paramater | |
3386 | + other operations simply create the same number (fneg etc), or more (fmul) | |
3387 | of digits | |
3388 | + rounding/truncating is only done when explicitly calling one of fround | |
3389 | or ffround, and never for BigInt (not implemented) | |
3390 | * fsqrt() simply hands its accuracy argument over to fdiv. | |
3391 | * the documentation and the comment in the code indicate two different ways | |
3392 | on how fdiv() determines the maximum number of digits it should calculate, | |
3393 | and the actual code does yet another thing | |
3394 | POD: | |
3395 | max($Math::BigFloat::div_scale,length(dividend)+length(divisor)) | |
3396 | Comment: | |
3397 | result has at most max(scale, length(dividend), length(divisor)) digits | |
3398 | Actual code: | |
3399 | scale = max(scale, length(dividend)-1,length(divisor)-1); | |
3400 | scale += length(divisior) - length(dividend); | |
3401 | So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3). | |
3402 | Actually, the 'difference' added to the scale is calculated from the | |
3403 | number of "significant digits" in dividend and divisor, which is derived | |
3404 | by looking at the length of the mantissa. Which is wrong, since it includes | |
3405 | the + sign (oups) and actually gets 2 for '+100' and 4 for '+101'. Oups | |
3406 | again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange | |
3407 | assumption that 124 has 3 significant digits, while 120/7 will get you | |
3408 | '17', not '17.1' since 120 is thought to have 2 significant digits. | |
3409 | The rounding after the division then uses the remainder and $y to determine | |
3410 | wether it must round up or down. | |
3411 | ? I have no idea which is the right way. That's why I used a slightly more | |
3412 | ? simple scheme and tweaked the few failing testcases to match it. | |
3413 | ||
3414 | =back | |
3415 | ||
3416 | This is how it works now: | |
3417 | ||
3418 | =over 2 | |
3419 | ||
3420 | =item Setting/Accessing | |
3421 | ||
3422 | * You can set the A global via Math::BigInt->accuracy() or | |
3423 | Math::BigFloat->accuracy() or whatever class you are using. | |
3424 | * You can also set P globally by using Math::SomeClass->precision() likewise. | |
3425 | * Globals are classwide, and not inherited by subclasses. | |
3426 | * to undefine A, use Math::SomeCLass->accuracy(undef); | |
3427 | * to undefine P, use Math::SomeClass->precision(undef); | |
3428 | * Setting Math::SomeClass->accuracy() clears automatically | |
3429 | Math::SomeClass->precision(), and vice versa. | |
3430 | * To be valid, A must be > 0, P can have any value. | |
3431 | * If P is negative, this means round to the P'th place to the right of the | |
3432 | decimal point; positive values mean to the left of the decimal point. | |
3433 | P of 0 means round to integer. | |
3434 | * to find out the current global A, take Math::SomeClass->accuracy() | |
3435 | * to find out the current global P, take Math::SomeClass->precision() | |
3436 | * use $x->accuracy() respective $x->precision() for the local setting of $x. | |
3437 | * Please note that $x->accuracy() respecive $x->precision() fall back to the | |
3438 | defined globals, when $x's A or P is not set. | |
3439 | ||
3440 | =item Creating numbers | |
3441 | ||
3442 | * When you create a number, you can give it's desired A or P via: | |
3443 | $x = Math::BigInt->new($number,$A,$P); | |
3444 | * Only one of A or P can be defined, otherwise the result is NaN | |
3445 | * If no A or P is give ($x = Math::BigInt->new($number) form), then the | |
3446 | globals (if set) will be used. Thus changing the global defaults later on | |
3447 | will not change the A or P of previously created numbers (i.e., A and P of | |
3448 | $x will be what was in effect when $x was created) | |
3449 | * If given undef for A and P, B<no> rounding will occur, and the globals will | |
3450 | B<not> be used. This is used by subclasses to create numbers without | |
3451 | suffering rounding in the parent. Thus a subclass is able to have it's own | |
3452 | globals enforced upon creation of a number by using | |
3453 | $x = Math::BigInt->new($number,undef,undef): | |
3454 | ||
3455 | use Math::Bigint::SomeSubclass; | |
3456 | use Math::BigInt; | |
3457 | ||
3458 | Math::BigInt->accuracy(2); | |
3459 | Math::BigInt::SomeSubClass->accuracy(3); | |
3460 | $x = Math::BigInt::SomeSubClass->new(1234); | |
3461 | ||
3462 | $x is now 1230, and not 1200. A subclass might choose to implement | |
3463 | this otherwise, e.g. falling back to the parent's A and P. | |
3464 | ||
3465 | =item Usage | |
3466 | ||
3467 | * If A or P are enabled/defined, they are used to round the result of each | |
3468 | operation according to the rules below | |
3469 | * Negative P is ignored in Math::BigInt, since BigInts never have digits | |
3470 | after the decimal point | |
3471 | * Math::BigFloat uses Math::BigInts internally, but setting A or P inside | |
3472 | Math::BigInt as globals should not tamper with the parts of a BigFloat. | |
3473 | Thus a flag is used to mark all Math::BigFloat numbers as 'never round' | |
3474 | ||
3475 | =item Precedence | |
3476 | ||
3477 | * It only makes sense that a number has only one of A or P at a time. | |
3478 | Since you can set/get both A and P, there is a rule that will practically | |
3479 | enforce only A or P to be in effect at a time, even if both are set. | |
3480 | This is called precedence. | |
3481 | * If two objects are involved in an operation, and one of them has A in | |
3482 | effect, and the other P, this results in an error (NaN). | |
3483 | * A takes precendence over P (Hint: A comes before P). If A is defined, it | |
3484 | is used, otherwise P is used. If neither of them is defined, nothing is | |
3485 | used, i.e. the result will have as many digits as it can (with an | |
3486 | exception for fdiv/fsqrt) and will not be rounded. | |
3487 | * There is another setting for fdiv() (and thus for fsqrt()). If neither of | |
3488 | A or P is defined, fdiv() will use a fallback (F) of $div_scale digits. | |
3489 | If either the dividend's or the divisor's mantissa has more digits than | |
3490 | the value of F, the higher value will be used instead of F. | |
3491 | This is to limit the digits (A) of the result (just consider what would | |
3492 | happen with unlimited A and P in the case of 1/3 :-) | |
3493 | * fdiv will calculate (at least) 4 more digits than required (determined by | |
3494 | A, P or F), and, if F is not used, round the result | |
3495 | (this will still fail in the case of a result like 0.12345000000001 with A | |
3496 | or P of 5, but this can not be helped - or can it?) | |
3497 | * Thus you can have the math done by on Math::Big* class in three modes: | |
3498 | + never round (this is the default): | |
3499 | This is done by setting A and P to undef. No math operation | |
3500 | will round the result, with fdiv() and fsqrt() as exceptions to guard | |
3501 | against overflows. You must explicitely call bround(), bfround() or | |
3502 | round() (the latter with parameters). | |
3503 | Note: Once you have rounded a number, the settings will 'stick' on it | |
3504 | and 'infect' all other numbers engaged in math operations with it, since | |
3505 | local settings have the highest precedence. So, to get SaferRound[tm], | |
3506 | use a copy() before rounding like this: | |
3507 | ||
3508 | $x = Math::BigFloat->new(12.34); | |
3509 | $y = Math::BigFloat->new(98.76); | |
3510 | $z = $x * $y; # 1218.6984 | |
3511 | print $x->copy()->fround(3); # 12.3 (but A is now 3!) | |
3512 | $z = $x * $y; # still 1218.6984, without | |
3513 | # copy would have been 1210! | |
3514 | ||
3515 | + round after each op: | |
3516 | After each single operation (except for testing like is_zero()), the | |
3517 | method round() is called and the result is rounded appropriately. By | |
3518 | setting proper values for A and P, you can have all-the-same-A or | |
3519 | all-the-same-P modes. For example, Math::Currency might set A to undef, | |
3520 | and P to -2, globally. | |
3521 | ||
3522 | ?Maybe an extra option that forbids local A & P settings would be in order, | |
3523 | ?so that intermediate rounding does not 'poison' further math? | |
3524 | ||
3525 | =item Overriding globals | |
3526 | ||
3527 | * you will be able to give A, P and R as an argument to all the calculation | |
3528 | routines; the second parameter is A, the third one is P, and the fourth is | |
3529 | R (shift right by one for binary operations like badd). P is used only if | |
3530 | the first parameter (A) is undefined. These three parameters override the | |
3531 | globals in the order detailed as follows, i.e. the first defined value | |
3532 | wins: | |
3533 | (local: per object, global: global default, parameter: argument to sub) | |
3534 | + parameter A | |
3535 | + parameter P | |
3536 | + local A (if defined on both of the operands: smaller one is taken) | |
3537 | + local P (if defined on both of the operands: bigger one is taken) | |
3538 | + global A | |
3539 | + global P | |
3540 | + global F | |
3541 | * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two | |
3542 | arguments (A and P) instead of one | |
3543 | ||
3544 | =item Local settings | |
3545 | ||
3546 | * You can set A and P locally by using $x->accuracy() and $x->precision() | |
3547 | and thus force different A and P for different objects/numbers. | |
3548 | * Setting A or P this way immediately rounds $x to the new value. | |
3549 | * $x->accuracy() clears $x->precision(), and vice versa. | |
3550 | ||
3551 | =item Rounding | |
3552 | ||
3553 | * the rounding routines will use the respective global or local settings. | |
3554 | fround()/bround() is for accuracy rounding, while ffround()/bfround() | |
3555 | is for precision | |
3556 | * the two rounding functions take as the second parameter one of the | |
3557 | following rounding modes (R): | |
3558 | 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' | |
3559 | * you can set and get the global R by using Math::SomeClass->round_mode() | |
3560 | or by setting $Math::SomeClass::round_mode | |
3561 | * after each operation, $result->round() is called, and the result may | |
3562 | eventually be rounded (that is, if A or P were set either locally, | |
3563 | globally or as parameter to the operation) | |
3564 | * to manually round a number, call $x->round($A,$P,$round_mode); | |
3565 | this will round the number by using the appropriate rounding function | |
3566 | and then normalize it. | |
3567 | * rounding modifies the local settings of the number: | |
3568 | ||
3569 | $x = Math::BigFloat->new(123.456); | |
3570 | $x->accuracy(5); | |
3571 | $x->bround(4); | |
3572 | ||
3573 | Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy() | |
3574 | will be 4 from now on. | |
3575 | ||
3576 | =item Default values | |
3577 | ||
3578 | * R: 'even' | |
3579 | * F: 40 | |
3580 | * A: undef | |
3581 | * P: undef | |
3582 | ||
3583 | =item Remarks | |
3584 | ||
3585 | * The defaults are set up so that the new code gives the same results as | |
3586 | the old code (except in a few cases on fdiv): | |
3587 | + Both A and P are undefined and thus will not be used for rounding | |
3588 | after each operation. | |
3589 | + round() is thus a no-op, unless given extra parameters A and P | |
3590 | ||
3591 | =back | |
3592 | ||
3593 | =head1 INTERNALS | |
3594 | ||
3595 | The actual numbers are stored as unsigned big integers (with seperate sign). | |
3596 | You should neither care about nor depend on the internal representation; it | |
3597 | might change without notice. Use only method calls like C<< $x->sign(); >> | |
3598 | instead relying on the internal hash keys like in C<< $x->{sign}; >>. | |
3599 | ||
3600 | =head2 MATH LIBRARY | |
3601 | ||
3602 | Math with the numbers is done (by default) by a module called | |
3603 | Math::BigInt::Calc. This is equivalent to saying: | |
3604 | ||
3605 | use Math::BigInt lib => 'Calc'; | |
3606 | ||
3607 | You can change this by using: | |
3608 | ||
3609 | use Math::BigInt lib => 'BitVect'; | |
3610 | ||
3611 | The following would first try to find Math::BigInt::Foo, then | |
3612 | Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc: | |
3613 | ||
3614 | use Math::BigInt lib => 'Foo,Math::BigInt::Bar'; | |
3615 | ||
3616 | Calc.pm uses as internal format an array of elements of some decimal base | |
3617 | (usually 1e5 or 1e7) with the least significant digit first, while BitVect.pm | |
3618 | uses a bit vector of base 2, most significant bit first. Other modules might | |
3619 | use even different means of representing the numbers. See the respective | |
3620 | module documentation for further details. | |
3621 | ||
3622 | =head2 SIGN | |
3623 | ||
3624 | The sign is either '+', '-', 'NaN', '+inf' or '-inf' and stored seperately. | |
3625 | ||
3626 | A sign of 'NaN' is used to represent the result when input arguments are not | |
3627 | numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively | |
3628 | minus infinity. You will get '+inf' when dividing a positive number by 0, and | |
3629 | '-inf' when dividing any negative number by 0. | |
3630 | ||
3631 | =head2 mantissa(), exponent() and parts() | |
3632 | ||
3633 | C<mantissa()> and C<exponent()> return the said parts of the BigInt such | |
3634 | that: | |
3635 | ||
3636 | $m = $x->mantissa(); | |
3637 | $e = $x->exponent(); | |
3638 | $y = $m * ( 10 ** $e ); | |
3639 | print "ok\n" if $x == $y; | |
3640 | ||
3641 | C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them | |
3642 | in one go. Both the returned mantissa and exponent have a sign. | |
3643 | ||
3644 | Currently, for BigInts C<$e> will be always 0, except for NaN, +inf and -inf, | |
3645 | where it will be NaN; and for $x == 0, where it will be 1 | |
3646 | (to be compatible with Math::BigFloat's internal representation of a zero as | |
3647 | C<0E1>). | |
3648 | ||
3649 | C<$m> will always be a copy of the original number. The relation between $e | |
3650 | and $m might change in the future, but will always be equivalent in a | |
3651 | numerical sense, e.g. $m might get minimized. | |
3652 | ||
3653 | =head1 EXAMPLES | |
3654 | ||
3655 | use Math::BigInt; | |
3656 | ||
3657 | sub bint { Math::BigInt->new(shift); } | |
3658 | ||
3659 | $x = Math::BigInt->bstr("1234") # string "1234" | |
3660 | $x = "$x"; # same as bstr() | |
3661 | $x = Math::BigInt->bneg("1234"); # Bigint "-1234" | |
3662 | $x = Math::BigInt->babs("-12345"); # Bigint "12345" | |
3663 | $x = Math::BigInt->bnorm("-0 00"); # BigInt "0" | |
3664 | $x = bint(1) + bint(2); # BigInt "3" | |
3665 | $x = bint(1) + "2"; # ditto (auto-BigIntify of "2") | |
3666 | $x = bint(1); # BigInt "1" | |
3667 | $x = $x + 5 / 2; # BigInt "3" | |
3668 | $x = $x ** 3; # BigInt "27" | |
3669 | $x *= 2; # BigInt "54" | |
3670 | $x = Math::BigInt->new(0); # BigInt "0" | |
3671 | $x--; # BigInt "-1" | |
3672 | $x = Math::BigInt->badd(4,5) # BigInt "9" | |
3673 | print $x->bsstr(); # 9e+0 | |
3674 | ||
3675 | Examples for rounding: | |
3676 | ||
3677 | use Math::BigFloat; | |
3678 | use Test; | |
3679 | ||
3680 | $x = Math::BigFloat->new(123.4567); | |
3681 | $y = Math::BigFloat->new(123.456789); | |
3682 | Math::BigFloat->accuracy(4); # no more A than 4 | |
3683 | ||
3684 | ok ($x->copy()->fround(),123.4); # even rounding | |
3685 | print $x->copy()->fround(),"\n"; # 123.4 | |
3686 | Math::BigFloat->round_mode('odd'); # round to odd | |
3687 | print $x->copy()->fround(),"\n"; # 123.5 | |
3688 | Math::BigFloat->accuracy(5); # no more A than 5 | |
3689 | Math::BigFloat->round_mode('odd'); # round to odd | |
3690 | print $x->copy()->fround(),"\n"; # 123.46 | |
3691 | $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4 | |
3692 | print "$y, ",$y->accuracy(),"\n"; # 123.4, 4 | |
3693 | ||
3694 | Math::BigFloat->accuracy(undef); # A not important now | |
3695 | Math::BigFloat->precision(2); # P important | |
3696 | print $x->copy()->bnorm(),"\n"; # 123.46 | |
3697 | print $x->copy()->fround(),"\n"; # 123.46 | |
3698 | ||
3699 | Examples for converting: | |
3700 | ||
3701 | my $x = Math::BigInt->new('0b1'.'01' x 123); | |
3702 | print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n"; | |
3703 | ||
3704 | =head1 Autocreating constants | |
3705 | ||
3706 | After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal | |
3707 | and binary constants in the given scope are converted to C<Math::BigInt>. | |
3708 | This conversion happens at compile time. | |
3709 | ||
3710 | In particular, | |
3711 | ||
3712 | perl -MMath::BigInt=:constant -e 'print 2**100,"\n"' | |
3713 | ||
3714 | prints the integer value of C<2**100>. Note that without conversion of | |
3715 | constants the expression 2**100 will be calculated as perl scalar. | |
3716 | ||
3717 | Please note that strings and floating point constants are not affected, | |
3718 | so that | |
3719 | ||
3720 | use Math::BigInt qw/:constant/; | |
3721 | ||
3722 | $x = 1234567890123456789012345678901234567890 | |
3723 | + 123456789123456789; | |
3724 | $y = '1234567890123456789012345678901234567890' | |
3725 | + '123456789123456789'; | |
3726 | ||
3727 | do not work. You need an explicit Math::BigInt->new() around one of the | |
3728 | operands. You should also quote large constants to protect loss of precision: | |
3729 | ||
3730 | use Math::Bigint; | |
3731 | ||
3732 | $x = Math::BigInt->new('1234567889123456789123456789123456789'); | |
3733 | ||
3734 | Without the quotes Perl would convert the large number to a floating point | |
3735 | constant at compile time and then hand the result to BigInt, which results in | |
3736 | an truncated result or a NaN. | |
3737 | ||
3738 | This also applies to integers that look like floating point constants: | |
3739 | ||
3740 | use Math::BigInt ':constant'; | |
3741 | ||
3742 | print ref(123e2),"\n"; | |
3743 | print ref(123.2e2),"\n"; | |
3744 | ||
3745 | will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat> | |
3746 | to get this to work. | |
3747 | ||
3748 | =head1 PERFORMANCE | |
3749 | ||
3750 | Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x | |
3751 | must be made in the second case. For long numbers, the copy can eat up to 20% | |
3752 | of the work (in the case of addition/subtraction, less for | |
3753 | multiplication/division). If $y is very small compared to $x, the form | |
3754 | $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes | |
3755 | more time then the actual addition. | |
3756 | ||
3757 | With a technique called copy-on-write, the cost of copying with overload could | |
3758 | be minimized or even completely avoided. A test implementation of COW did show | |
3759 | performance gains for overloaded math, but introduced a performance loss due | |
3760 | to a constant overhead for all other operatons. | |
3761 | ||
3762 | The rewritten version of this module is slower on certain operations, like | |
3763 | new(), bstr() and numify(). The reason are that it does now more work and | |
3764 | handles more cases. The time spent in these operations is usually gained in | |
3765 | the other operations so that programs on the average should get faster. If | |
3766 | they don't, please contect the author. | |
3767 | ||
3768 | Some operations may be slower for small numbers, but are significantly faster | |
3769 | for big numbers. Other operations are now constant (O(1), like bneg(), babs() | |
3770 | etc), instead of O(N) and thus nearly always take much less time. These | |
3771 | optimizations were done on purpose. | |
3772 | ||
3773 | If you find the Calc module to slow, try to install any of the replacement | |
3774 | modules and see if they help you. | |
3775 | ||
3776 | =head2 Alternative math libraries | |
3777 | ||
3778 | You can use an alternative library to drive Math::BigInt via: | |
3779 | ||
3780 | use Math::BigInt lib => 'Module'; | |
3781 | ||
3782 | See L<MATH LIBRARY> for more information. | |
3783 | ||
3784 | For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>. | |
3785 | ||
3786 | =head2 SUBCLASSING | |
3787 | ||
3788 | =head1 Subclassing Math::BigInt | |
3789 | ||
3790 | The basic design of Math::BigInt allows simple subclasses with very little | |
3791 | work, as long as a few simple rules are followed: | |
3792 | ||
3793 | =over 2 | |
3794 | ||
3795 | =item * | |
3796 | ||
3797 | The public API must remain consistent, i.e. if a sub-class is overloading | |
3798 | addition, the sub-class must use the same name, in this case badd(). The | |
3799 | reason for this is that Math::BigInt is optimized to call the object methods | |
3800 | directly. | |
3801 | ||
3802 | =item * | |
3803 | ||
3804 | The private object hash keys like C<$x->{sign}> may not be changed, but | |
3805 | additional keys can be added, like C<$x->{_custom}>. | |
3806 | ||
3807 | =item * | |
3808 | ||
3809 | Accessor functions are available for all existing object hash keys and should | |
3810 | be used instead of directly accessing the internal hash keys. The reason for | |
3811 | this is that Math::BigInt itself has a pluggable interface which permits it | |
3812 | to support different storage methods. | |
3813 | ||
3814 | =back | |
3815 | ||
3816 | More complex sub-classes may have to replicate more of the logic internal of | |
3817 | Math::BigInt if they need to change more basic behaviors. A subclass that | |
3818 | needs to merely change the output only needs to overload C<bstr()>. | |
3819 | ||
3820 | All other object methods and overloaded functions can be directly inherited | |
3821 | from the parent class. | |
3822 | ||
3823 | At the very minimum, any subclass will need to provide it's own C<new()> and can | |
3824 | store additional hash keys in the object. There are also some package globals | |
3825 | that must be defined, e.g.: | |
3826 | ||
3827 | # Globals | |
3828 | $accuracy = undef; | |
3829 | $precision = -2; # round to 2 decimal places | |
3830 | $round_mode = 'even'; | |
3831 | $div_scale = 40; | |
3832 | ||
3833 | Additionally, you might want to provide the following two globals to allow | |
3834 | auto-upgrading and auto-downgrading to work correctly: | |
3835 | ||
3836 | $upgrade = undef; | |
3837 | $downgrade = undef; | |
3838 | ||
3839 | This allows Math::BigInt to correctly retrieve package globals from the | |
3840 | subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or | |
3841 | t/Math/BigFloat/SubClass.pm completely functional subclass examples. | |
3842 | ||
3843 | Don't forget to | |
3844 | ||
3845 | use overload; | |
3846 | ||
3847 | in your subclass to automatically inherit the overloading from the parent. If | |
3848 | you like, you can change part of the overloading, look at Math::String for an | |
3849 | example. | |
3850 | ||
3851 | =head1 UPGRADING | |
3852 | ||
3853 | When used like this: | |
3854 | ||
3855 | use Math::BigInt upgrade => 'Foo::Bar'; | |
3856 | ||
3857 | certain operations will 'upgrade' their calculation and thus the result to | |
3858 | the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat: | |
3859 | ||
3860 | use Math::BigInt upgrade => 'Math::BigFloat'; | |
3861 | ||
3862 | As a shortcut, you can use the module C<bignum>: | |
3863 | ||
3864 | use bignum; | |
3865 | ||
3866 | Also good for oneliners: | |
3867 | ||
3868 | perl -Mbignum -le 'print 2 ** 255' | |
3869 | ||
3870 | This makes it possible to mix arguments of different classes (as in 2.5 + 2) | |
3871 | as well es preserve accuracy (as in sqrt(3)). | |
3872 | ||
3873 | Beware: This feature is not fully implemented yet. | |
3874 | ||
3875 | =head2 Auto-upgrade | |
3876 | ||
3877 | The following methods upgrade themselves unconditionally; that is if upgrade | |
3878 | is in effect, they will always hand up their work: | |
3879 | ||
3880 | =over 2 | |
3881 | ||
3882 | =item bsqrt() | |
3883 | ||
3884 | =item div() | |
3885 | ||
3886 | =item blog() | |
3887 | ||
3888 | =back | |
3889 | ||
3890 | Beware: This list is not complete. | |
3891 | ||
3892 | All other methods upgrade themselves only when one (or all) of their | |
3893 | arguments are of the class mentioned in $upgrade (This might change in later | |
3894 | versions to a more sophisticated scheme): | |
3895 | ||
3896 | =head1 BUGS | |
3897 | ||
3898 | =over 2 | |
3899 | ||
3900 | =item Out of Memory! | |
3901 | ||
3902 | Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and | |
3903 | C<eval()> in your code will crash with "Out of memory". This is probably an | |
3904 | overload/exporter bug. You can workaround by not having C<eval()> | |
3905 | and ':constant' at the same time or upgrade your Perl to a newer version. | |
3906 | ||
3907 | =item Fails to load Calc on Perl prior 5.6.0 | |
3908 | ||
3909 | Since eval(' use ...') can not be used in conjunction with ':constant', BigInt | |
3910 | will fall back to eval { require ... } when loading the math lib on Perls | |
3911 | prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on | |
3912 | filesystems using a different seperator. | |
3913 | ||
3914 | =back | |
3915 | ||
3916 | =head1 CAVEATS | |
3917 | ||
3918 | Some things might not work as you expect them. Below is documented what is | |
3919 | known to be troublesome: | |
3920 | ||
3921 | =over 1 | |
3922 | ||
3923 | =item stringify, bstr(), bsstr() and 'cmp' | |
3924 | ||
3925 | Both stringify and bstr() now drop the leading '+'. The old code would return | |
3926 | '+3', the new returns '3'. This is to be consistent with Perl and to make | |
3927 | cmp (especially with overloading) to work as you expect. It also solves | |
3928 | problems with Test.pm, it's ok() uses 'eq' internally. | |
3929 | ||
3930 | Mark said, when asked about to drop the '+' altogether, or make only cmp work: | |
3931 | ||
3932 | I agree (with the first alternative), don't add the '+' on positive | |
3933 | numbers. It's not as important anymore with the new internal | |
3934 | form for numbers. It made doing things like abs and neg easier, | |
3935 | but those have to be done differently now anyway. | |
3936 | ||
3937 | So, the following examples will now work all as expected: | |
3938 | ||
3939 | use Test; | |
3940 | BEGIN { plan tests => 1 } | |
3941 | use Math::BigInt; | |
3942 | ||
3943 | my $x = new Math::BigInt 3*3; | |
3944 | my $y = new Math::BigInt 3*3; | |
3945 | ||
3946 | ok ($x,3*3); | |
3947 | print "$x eq 9" if $x eq $y; | |
3948 | print "$x eq 9" if $x eq '9'; | |
3949 | print "$x eq 9" if $x eq 3*3; | |
3950 | ||
3951 | Additionally, the following still works: | |
3952 | ||
3953 | print "$x == 9" if $x == $y; | |
3954 | print "$x == 9" if $x == 9; | |
3955 | print "$x == 9" if $x == 3*3; | |
3956 | ||
3957 | There is now a C<bsstr()> method to get the string in scientific notation aka | |
3958 | C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr() | |
3959 | for comparisation, but Perl will represent some numbers as 100 and others | |
3960 | as 1e+308. If in doubt, convert both arguments to Math::BigInt before doing eq: | |
3961 | ||
3962 | use Test; | |
3963 | BEGIN { plan tests => 3 } | |
3964 | use Math::BigInt; | |
3965 | ||
3966 | $x = Math::BigInt->new('1e56'); $y = 1e56; | |
3967 | ok ($x,$y); # will fail | |
3968 | ok ($x->bsstr(),$y); # okay | |
3969 | $y = Math::BigInt->new($y); | |
3970 | ok ($x,$y); # okay | |
3971 | ||
3972 | Alternatively, simple use <=> for comparisations, that will get it always | |
3973 | right. There is not yet a way to get a number automatically represented as | |
3974 | a string that matches exactly the way Perl represents it. | |
3975 | ||
3976 | =item int() | |
3977 | ||
3978 | C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a | |
3979 | Perl scalar: | |
3980 | ||
3981 | $x = Math::BigInt->new(123); | |
3982 | $y = int($x); # BigInt 123 | |
3983 | $x = Math::BigFloat->new(123.45); | |
3984 | $y = int($x); # BigInt 123 | |
3985 | ||
3986 | In all Perl versions you can use C<as_number()> for the same effect: | |
3987 | ||
3988 | $x = Math::BigFloat->new(123.45); | |
3989 | $y = $x->as_number(); # BigInt 123 | |
3990 | ||
3991 | This also works for other subclasses, like Math::String. | |
3992 | ||
3993 | It is yet unlcear whether overloaded int() should return a scalar or a BigInt. | |
3994 | ||
3995 | =item length | |
3996 | ||
3997 | The following will probably not do what you expect: | |
3998 | ||
3999 | $c = Math::BigInt->new(123); | |
4000 | print $c->length(),"\n"; # prints 30 | |
4001 | ||
4002 | It prints both the number of digits in the number and in the fraction part | |
4003 | since print calls C<length()> in list context. Use something like: | |
4004 | ||
4005 | print scalar $c->length(),"\n"; # prints 3 | |
4006 | ||
4007 | =item bdiv | |
4008 | ||
4009 | The following will probably not do what you expect: | |
4010 | ||
4011 | print $c->bdiv(10000),"\n"; | |
4012 | ||
4013 | It prints both quotient and remainder since print calls C<bdiv()> in list | |
4014 | context. Also, C<bdiv()> will modify $c, so be carefull. You probably want | |
4015 | to use | |
4016 | ||
4017 | print $c / 10000,"\n"; | |
4018 | print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c | |
4019 | ||
4020 | instead. | |
4021 | ||
4022 | The quotient is always the greatest integer less than or equal to the | |
4023 | real-valued quotient of the two operands, and the remainder (when it is | |
4024 | nonzero) always has the same sign as the second operand; so, for | |
4025 | example, | |
4026 | ||
4027 | 1 / 4 => ( 0, 1) | |
4028 | 1 / -4 => (-1,-3) | |
4029 | -3 / 4 => (-1, 1) | |
4030 | -3 / -4 => ( 0,-3) | |
4031 | -11 / 2 => (-5,1) | |
4032 | 11 /-2 => (-5,-1) | |
4033 | ||
4034 | As a consequence, the behavior of the operator % agrees with the | |
4035 | behavior of Perl's built-in % operator (as documented in the perlop | |
4036 | manpage), and the equation | |
4037 | ||
4038 | $x == ($x / $y) * $y + ($x % $y) | |
4039 | ||
4040 | holds true for any $x and $y, which justifies calling the two return | |
4041 | values of bdiv() the quotient and remainder. The only exception to this rule | |
4042 | are when $y == 0 and $x is negative, then the remainder will also be | |
4043 | negative. See below under "infinity handling" for the reasoning behing this. | |
4044 | ||
4045 | Perl's 'use integer;' changes the behaviour of % and / for scalars, but will | |
4046 | not change BigInt's way to do things. This is because under 'use integer' Perl | |
4047 | will do what the underlying C thinks is right and this is different for each | |
4048 | system. If you need BigInt's behaving exactly like Perl's 'use integer', bug | |
4049 | the author to implement it ;) | |
4050 | ||
4051 | =item infinity handling | |
4052 | ||
4053 | Here are some examples that explain the reasons why certain results occur while | |
4054 | handling infinity: | |
4055 | ||
4056 | The following table shows the result of the division and the remainder, so that | |
4057 | the equation above holds true. Some "ordinary" cases are strewn in to show more | |
4058 | clearly the reasoning: | |
4059 | ||
4060 | A / B = C, R so that C * B + R = A | |
4061 | ========================================================= | |
4062 | 5 / 8 = 0, 5 0 * 8 + 5 = 5 | |
4063 | 0 / 8 = 0, 0 0 * 8 + 0 = 0 | |
4064 | 0 / inf = 0, 0 0 * inf + 0 = 0 | |
4065 | 0 /-inf = 0, 0 0 * -inf + 0 = 0 | |
4066 | 5 / inf = 0, 5 0 * inf + 5 = 5 | |
4067 | 5 /-inf = 0, 5 0 * -inf + 5 = 5 | |
4068 | -5/ inf = 0, -5 0 * inf + -5 = -5 | |
4069 | -5/-inf = 0, -5 0 * -inf + -5 = -5 | |
4070 | inf/ 5 = inf, 0 inf * 5 + 0 = inf | |
4071 | -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf | |
4072 | inf/ -5 = -inf, 0 -inf * -5 + 0 = inf | |
4073 | -inf/ -5 = inf, 0 inf * -5 + 0 = -inf | |
4074 | 5/ 5 = 1, 0 1 * 5 + 0 = 5 | |
4075 | -5/ -5 = 1, 0 1 * -5 + 0 = -5 | |
4076 | inf/ inf = 1, 0 1 * inf + 0 = inf | |
4077 | -inf/-inf = 1, 0 1 * -inf + 0 = -inf | |
4078 | inf/-inf = -1, 0 -1 * -inf + 0 = inf | |
4079 | -inf/ inf = -1, 0 1 * -inf + 0 = -inf | |
4080 | 8/ 0 = inf, 8 inf * 0 + 8 = 8 | |
4081 | inf/ 0 = inf, inf inf * 0 + inf = inf | |
4082 | 0/ 0 = NaN | |
4083 | ||
4084 | These cases below violate the "remainder has the sign of the second of the two | |
4085 | arguments", since they wouldn't match up otherwise. | |
4086 | ||
4087 | A / B = C, R so that C * B + R = A | |
4088 | ======================================================== | |
4089 | -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf | |
4090 | -8/ 0 = -inf, -8 -inf * 0 + 8 = -8 | |
4091 | ||
4092 | =item Modifying and = | |
4093 | ||
4094 | Beware of: | |
4095 | ||
4096 | $x = Math::BigFloat->new(5); | |
4097 | $y = $x; | |
4098 | ||
4099 | It will not do what you think, e.g. making a copy of $x. Instead it just makes | |
4100 | a second reference to the B<same> object and stores it in $y. Thus anything | |
4101 | that modifies $x (except overloaded operators) will modify $y, and vice versa. | |
4102 | Or in other words, C<=> is only safe if you modify your BigInts only via | |
4103 | overloaded math. As soon as you use a method call it breaks: | |
4104 | ||
4105 | $x->bmul(2); | |
4106 | print "$x, $y\n"; # prints '10, 10' | |
4107 | ||
4108 | If you want a true copy of $x, use: | |
4109 | ||
4110 | $y = $x->copy(); | |
4111 | ||
4112 | You can also chain the calls like this, this will make first a copy and then | |
4113 | multiply it by 2: | |
4114 | ||
4115 | $y = $x->copy()->bmul(2); | |
4116 | ||
4117 | See also the documentation for overload.pm regarding C<=>. | |
4118 | ||
4119 | =item bpow | |
4120 | ||
4121 | C<bpow()> (and the rounding functions) now modifies the first argument and | |
4122 | returns it, unlike the old code which left it alone and only returned the | |
4123 | result. This is to be consistent with C<badd()> etc. The first three will | |
4124 | modify $x, the last one won't: | |
4125 | ||
4126 | print bpow($x,$i),"\n"; # modify $x | |
4127 | print $x->bpow($i),"\n"; # ditto | |
4128 | print $x **= $i,"\n"; # the same | |
4129 | print $x ** $i,"\n"; # leave $x alone | |
4130 | ||
4131 | The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though. | |
4132 | ||
4133 | =item Overloading -$x | |
4134 | ||
4135 | The following: | |
4136 | ||
4137 | $x = -$x; | |
4138 | ||
4139 | is slower than | |
4140 | ||
4141 | $x->bneg(); | |
4142 | ||
4143 | since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant | |
4144 | needs to preserve $x since it does not know that it later will get overwritten. | |
4145 | This makes a copy of $x and takes O(N), but $x->bneg() is O(1). | |
4146 | ||
4147 | With Copy-On-Write, this issue would be gone, but C-o-W is not implemented | |
4148 | since it is slower for all other things. | |
4149 | ||
4150 | =item Mixing different object types | |
4151 | ||
4152 | In Perl you will get a floating point value if you do one of the following: | |
4153 | ||
4154 | $float = 5.0 + 2; | |
4155 | $float = 2 + 5.0; | |
4156 | $float = 5 / 2; | |
4157 | ||
4158 | With overloaded math, only the first two variants will result in a BigFloat: | |
4159 | ||
4160 | use Math::BigInt; | |
4161 | use Math::BigFloat; | |
4162 | ||
4163 | $mbf = Math::BigFloat->new(5); | |
4164 | $mbi2 = Math::BigInteger->new(5); | |
4165 | $mbi = Math::BigInteger->new(2); | |
4166 | ||
4167 | # what actually gets called: | |
4168 | $float = $mbf + $mbi; # $mbf->badd() | |
4169 | $float = $mbf / $mbi; # $mbf->bdiv() | |
4170 | $integer = $mbi + $mbf; # $mbi->badd() | |
4171 | $integer = $mbi2 / $mbi; # $mbi2->bdiv() | |
4172 | $integer = $mbi2 / $mbf; # $mbi2->bdiv() | |
4173 | ||
4174 | This is because math with overloaded operators follows the first (dominating) | |
4175 | operand, and the operation of that is called and returns thus the result. So, | |
4176 | Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether | |
4177 | the result should be a Math::BigFloat or the second operant is one. | |
4178 | ||
4179 | To get a Math::BigFloat you either need to call the operation manually, | |
4180 | make sure the operands are already of the proper type or casted to that type | |
4181 | via Math::BigFloat->new(): | |
4182 | ||
4183 | $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5 | |
4184 | ||
4185 | Beware of simple "casting" the entire expression, this would only convert | |
4186 | the already computed result: | |
4187 | ||
4188 | $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong! | |
4189 | ||
4190 | Beware also of the order of more complicated expressions like: | |
4191 | ||
4192 | $integer = ($mbi2 + $mbi) / $mbf; # int / float => int | |
4193 | $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto | |
4194 | ||
4195 | If in doubt, break the expression into simpler terms, or cast all operands | |
4196 | to the desired resulting type. | |
4197 | ||
4198 | Scalar values are a bit different, since: | |
4199 | ||
4200 | $float = 2 + $mbf; | |
4201 | $float = $mbf + 2; | |
4202 | ||
4203 | will both result in the proper type due to the way the overloaded math works. | |
4204 | ||
4205 | This section also applies to other overloaded math packages, like Math::String. | |
4206 | ||
4207 | One solution to you problem might be L<autoupgrading|upgrading>. | |
4208 | ||
4209 | =item bsqrt() | |
4210 | ||
4211 | C<bsqrt()> works only good if the result is a big integer, e.g. the square | |
4212 | root of 144 is 12, but from 12 the square root is 3, regardless of rounding | |
4213 | mode. | |
4214 | ||
4215 | If you want a better approximation of the square root, then use: | |
4216 | ||
4217 | $x = Math::BigFloat->new(12); | |
4218 | Math::BigFloat->precision(0); | |
4219 | Math::BigFloat->round_mode('even'); | |
4220 | print $x->copy->bsqrt(),"\n"; # 4 | |
4221 | ||
4222 | Math::BigFloat->precision(2); | |
4223 | print $x->bsqrt(),"\n"; # 3.46 | |
4224 | print $x->bsqrt(3),"\n"; # 3.464 | |
4225 | ||
4226 | =item brsft() | |
4227 | ||
4228 | For negative numbers in base see also L<brsft|brsft>. | |
4229 | ||
4230 | =back | |
4231 | ||
4232 | =head1 LICENSE | |
4233 | ||
4234 | This program is free software; you may redistribute it and/or modify it under | |
4235 | the same terms as Perl itself. | |
4236 | ||
4237 | =head1 SEE ALSO | |
4238 | ||
4239 | L<Math::BigFloat> and L<Math::Big> as well as L<Math::BigInt::BitVect>, | |
4240 | L<Math::BigInt::Pari> and L<Math::BigInt::GMP>. | |
4241 | ||
4242 | The package at | |
4243 | L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains | |
4244 | more documentation including a full version history, testcases, empty | |
4245 | subclass files and benchmarks. | |
4246 | ||
4247 | =head1 AUTHORS | |
4248 | ||
4249 | Original code by Mark Biggar, overloaded interface by Ilya Zakharevich. | |
4250 | Completely rewritten by Tels http://bloodgate.com in late 2000, 2001. | |
4251 | ||
4252 | =cut |