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Initial commit of OpenSPARC T2 design and verification files.
[OpenSPARC-T2-DV] / tools / perl-5.8.0 / lib / 5.8.0 / Math / BigInt.pm
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AT		 1package 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
18my $class = "Math::BigInt";
19require 5.005;
20
21# This is a patched v1.60, containing a fix for the "1234567890\n" bug
22$VERSION = '1.60';
23use Exporter;
24@ISA = qw( Exporter );
25@EXPORT_OK = qw( objectify _swap bgcd blcm);
26use vars qw/$round_mode $accuracy $precision $div_scale $rnd_mode/;
27use vars qw/$upgrade $downgrade/;
28use 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
42use 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
118use constant MB_NEVER_ROUND => 0x0001;
119
120my $NaNOK=1; # are NaNs ok?
121my $nan = 'NaN'; # constants for easier life
122
123my $CALC = 'Math::BigInt::Calc'; # module to do low level math
124my $IMPORT = 0; # did import() yet?
125
126$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';
138sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; }
139sub FETCH { return $round_mode; }
140sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
141
142BEGIN { tie $rnd_mode, 'Math::BigInt'; }
143
144##############################################################################
145
146sub 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
162sub 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
177sub 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
192sub 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
206sub 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
246sub 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
285sub 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 $cfg;
303 }
304
305sub _scale_a
306 {
307 # select accuracy parameter based on precedence,
308 # used by bround() and bfround(), may return undef for scale (means no op)
309 my ($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,$mode);
314 }
315
316sub _scale_p
317 {
318 # select precision parameter based on precedence,
319 # used by bround() and bfround(), may return undef for scale (means no op)
320 my ($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,$mode);
325 }
326
327##############################################################################
328# constructors
329
330sub copy
331 {
332 my ($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 $self;
387 }
388
389sub new
390 {
391 # create a new BigInt object from a string or another BigInt object.
392 # see hash keys documented at top
393
394 # the argument could be an object, so avoid ||, && etc on it, this would
395 # cause costly overloaded code to be called. The only allowed ops are
396 # ref() and defined.
397
398 my ($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
506sub 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
533sub 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
563sub 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
605sub 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
652sub 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
672sub 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
687sub 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 $num;
695 }
696
697##############################################################################
698# public stuff (usually prefixed with "b")
699
700sub sign
701 {
702 # return the sign of the number: +/-/-inf/+inf/NaN
703 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
704
705 $x->{sign};
706 }
707
708sub _find_round_parameters
709 {
710 # After any operation or when calling round(), the result is rounded by
711 # regarding the A & P from arguments, local parameters, or globals.
712
713 # This procedure finds the round parameters, but it is for speed reasons
714 # duplicated in round. Otherwise, it is tested by the testsuite and used
715 # by fdiv().
716
717 my ($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
764sub 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 $self->bnorm(); # after round, normalize
825 }
826
827sub bnorm
828 {
829 # (numstr or BINT) return BINT
830 # Normalize number -- no-op here
831 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
832 $x;
833 }
834
835sub babs
836 {
837 # (BINT or num_str) return BINT
838 # make number absolute, or return absolute BINT from string
839 my ($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 $x;
845 }
846
847sub bneg
848 {
849 # (BINT or num_str) return BINT
850 # negate number or make a negated number from string
851 my ($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 $x;
858 }
859
860sub bcmp
861 {
862 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
863 # (BINT or num_str, BINT or num_str) return cond_code
864
865 # set up parameters
866 my ($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},$x->{value}); # swaped (lib returns 0,1,-1)
900 }
901
902sub bacmp
903 {
904 # Compares 2 values, ignoring their signs.
905 # Returns one of undef, <0, =0, >0. (suitable for sort)
906 # (BINT, BINT) return cond_code
907
908 # set up parameters
909 my ($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},$y->{value}); # lib does only 0,1,-1
924 }
925
926sub badd
927 {
928 # add second arg (BINT or string) to first (BINT) (modifies first)
929 # return result as BINT
930
931 # set up parameters
932 my ($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 $x;
994 }
995
996sub bsub
997 {
998 # (BINT or num_str, BINT or num_str) return num_str
999 # subtract second arg from first, modify first
1000
1001 # set up parameters
1002 my ($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 $x; # already rounded by badd() or no round necc.
1025 }
1026
1027sub binc
1028 {
1029 # increment arg by one
1030 my ($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,$r); # badd does round
1048 }
1049
1050sub bdec
1051 {
1052 # decrement arg by one
1053 my ($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,$r); # badd does round
1075 }
1076
1077sub blog
1078 {
1079 # not implemented yet
1080 my ($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
1087sub 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
1106sub 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
1137sub 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
1150sub 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
1160sub 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 $nan;
1166 0;
1167 }
1168
1169sub is_inf
1170 {
1171 # return true if arg (BINT or num_str) is +-inf
1172 my ($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
1188sub 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
1201sub 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
1211sub 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
1221sub 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
1231sub 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
1241sub 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
1253sub 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 $x;
1294 }
1295
1296sub _div_inf
1297 {
1298 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1299 my ($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()) : $x;
1339 }
1340
1341sub bdiv
1342 {
1343 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1344 # (BINT,BINT) (quo,rem) or BINT (only rem)
1345
1346 # set up parameters
1347 my ($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 $x;
1423 }
1424
1425###############################################################################
1426# modulus functions
1427
1428sub bmod
1429 {
1430 # modulus (or remainder)
1431 # (BINT or num_str, BINT or num_str) return BINT
1432
1433 # set up parameters
1434 my ($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 $x;
1480 }
1481
1482sub bmodinv
1483 {
1484 # modular inverse. given a number which is (hopefully) relatively
1485 # prime to the modulus, calculate its inverse using Euclid's
1486 # alogrithm. if the number is not relatively prime to the modulus
1487 # (i.e. their gcd is not one) then NaN is returned.
1488
1489 # set up parameters
1490 my ($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 $x;
1540 }
1541
1542sub bmodpow
1543 {
1544 # takes a very large number to a very large exponent in a given very
1545 # large modulus, quickly, thanks to binary exponentation. supports
1546 # negative exponents.
1547 my ($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 $num;
1595 }
1596
1597###############################################################################
1598
1599sub bfac
1600 {
1601 # (BINT or num_str, BINT or num_str) return BINT
1602 # compute factorial numbers
1603 # modifies first argument
1604 my ($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
1628sub 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 $x;
1698 }
1699
1700sub blsft
1701 {
1702 # (BINT or num_str, BINT or num_str) return BINT
1703 # compute x << y, base n, y >= 0
1704
1705 # set up parameters
1706 my ($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, $y, @r), @r );
1726 }
1727
1728sub brsft
1729 {
1730 # (BINT or num_str, BINT or num_str) return BINT
1731 # compute x >> y, base n, y >= 0
1732
1733 # set up parameters
1734 my ($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 $x;
1793 }
1794
1795sub band
1796 {
1797 #(BINT or num_str, BINT or num_str) return BINT
1798 # compute x & y
1799
1800 # set up parameters
1801 my ($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 $x->round(@r);
1845 }
1846
1847sub bior
1848 {
1849 #(BINT or num_str, BINT or num_str) return BINT
1850 # compute x | y
1851
1852 # set up parameters
1853 my ($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 $x->round(@r);
1898 }
1899
1900sub bxor
1901 {
1902 #(BINT or num_str, BINT or num_str) return BINT
1903 # compute x ^ y
1904
1905 # set up parameters
1906 my ($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 $x->round(@r);
1951 }
1952
1953sub length
1954 {
1955 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1956
1957 my $e = $CALC->_len($x->{value});
1958 return wantarray ? ($e,0) : $e;
1959 }
1960
1961sub digit
1962 {
1963 # return the nth decimal digit, negative values count backward, 0 is right
1964 my ($self,$x,$n) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1965
1966 $CALC->_digit($x->{value},$n||0);
1967 }
1968
1969sub _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("$1"); # as string, not as +0!
1983 }
1984
1985sub bsqrt
1986 {
1987 my ($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
2023sub 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 $e;
2037 }
2038
2039sub mantissa
2040 {
2041 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
2042 my ($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 $m;
2053 }
2054
2055sub parts
2056 {
2057 # return a copy of both the exponent and the mantissa
2058 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2059
2060 return ($x->mantissa(),$x->exponent());
2061 }
2062
2063##############################################################################
2064# rounding functions
2065
2066sub 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
2088sub _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 ($$xs,-$follow);
2101 my $r = substr ("$x",-$follow);
2102 return 1 if $r =~ /[^0]/;
2103 0;
2104 }
2105
2106sub fround
2107 {
2108 # to make life easier for switch between MBF and MBI (autoload fxxx()
2109 # like MBF does for bxxx()?)
2110 my $x = shift;
2111 return $x->bround(@_);
2112 }
2113
2114sub bround
2115 {
2116 # accuracy: +$n preserve $n digits from left,
2117 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
2118 # no-op for $n == 0
2119 # and overwrite the rest with 0's, return normalized number
2120 # do not return $x->bnorm(), but $x
2121
2122 my $x = shift; $x = $class->new($x) unless ref $x;
2123 my ($scale,$mode) = $x->_scale_a($x->accuracy(),$x->round_mode(),@_);
2124 return $x if !defined $scale; # no-op
2125 return $x if $x->modify('bround');
2126
2127 if ($x->is_zero() || $scale == 0)
2128 {
2129 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2130 return $x;
2131 }
2132 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2133
2134 # we have fewer digits than we want to scale to
2135 my $len = $x->length();
2136 # scale < 0, but > -len (not >=!)
2137 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2138 {
2139 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2140 return $x;
2141 }
2142
2143 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2144 my ($pad,$digit_round,$digit_after);
2145 $pad = $len - $scale;
2146 $pad = abs($scale-1) if $scale < 0;
2147
2148 # do not use digit(), it is costly for binary => decimal
2149
2150 my $xs = $CALC->_str($x->{value});
2151 my $pl = -$pad-1;
2152
2153 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2154 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2155 $digit_round = '0'; $digit_round = substr($$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
2215sub 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
2224sub 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
2236sub __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
2245sub _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
2272sub 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
2366sub 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
2430sub __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
2472sub __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($$bs,$i,8);
2499 $val =~ s/^[+-]?0b// if $len == 0; # for last part only
2500 #$val = oct('0b'.$val); # does not work on Perl prior to 5.6.0
2501 # slower:
2502 # $val = ('0' x (8-CORE::length($val))).$val if CORE::length($val) < 8;
2503 $val = ord(pack('B8',substr('00000000'.$val,-8,8)));
2504 $i -= 8; $len --;
2505 $x += $mul * $val if $val != 0;
2506 $mul *= $x256 if $len >= 0; # skip last mul
2507 }
2508 }
2509 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2510 $x;
2511 }
2512
2513sub _split
2514 {
2515 # (ref to num_str) return num_str
2516 # internal, take apart a string and return the pieces
2517 # strip leading/trailing whitespace, leading zeros, underscore and reject
2518 # invalid input
2519 my $x = shift;
2520
2521 # strip white space at front, also extranous leading zeros
2522 $$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
2575sub 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
2585sub 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
2622sub 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
2662sub __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,$ty);
2671 }
2672
2673sub __gcd
2674 {
2675 # (BINT or num_str, BINT or num_str) return BINT
2676 # does modify both arguments
2677 # GCD -- Euclids algorithm E, Knuth Vol 2 pg 296
2678 my ($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
2694sub modify () { 0; }
2695
26961;
2697__END__
2698
2699=head1 NAME
2700
2701Math::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
2817All operators (inlcuding basic math operations) are overloaded if you
2818declare your big integers as
2819
2820 $i = new Math::BigInt '123_456_789_123_456_789';
2821
2822Operations with overloaded operators preserve the arguments which is
2823exactly what you expect.
2824
2825=over 2
2826
2827=item Canonical notation
2828
2829Big integer values are strings of the form C</^[+-]\d+$/> with leading
2830zeros 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
2838Input values to these routines may be either Math::BigInt objects or
2839strings of the form C</^[+-]?[\d]+\.?[\d]*E?[+-]?[\d]*$/>.
2840
2841You can include one underscore between any two digits. The input string may
2842have leading and trailing whitespace, which will be ignored. In later
2843versions, a more strict (no whitespace at all) or more lax (whitespace
2844allowed everywhere) input checking will also be possible.
2845
2846This means integer values like 1.01E2 or even 1000E-2 are also accepted.
2847Non integer values result in NaN.
2848
2849Math::BigInt::new() defaults to 0, while Math::BigInt::new('') results
2850in 'NaN'.
2851
2852bnorm() on a BigInt object is now effectively a no-op, since the numbers
2853are always stored in normalized form. On a string, it creates a BigInt
2854object.
2855
2856=item Output
2857
2858Output values are BigInt objects (normalized), except for bstr(), which
2859returns a string in normalized form.
2860Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
2861C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
2862return either undef, <0, 0 or >0 and are suited for sort.
2863
2864=back
2865
2866=head1 METHODS
2867
2868Each of the methods below accepts three additional parameters. These arguments
2869$A, $P and $R are accuracy, precision and round_mode. Please see more in the
2870section about ACCURACY and ROUNDIND.
2871
2872=head2 config
2873
2874 use Data::Dumper;
2875
2876 print Dumper ( Math::BigInt->config() );
2877
2878Returns a hash containing the configuration, e.g. the version number, lib
2879loaded etc.
2880
2881=head2 accuracy
2882
2883 $x->accuracy(5); # local for $x
2884 $class->accuracy(5); # global for all members of $class
2885
2886Set or get the global or local accuracy, aka how many significant digits the
2887results have. Please see the section about L<ACCURACY AND PRECISION> for
2888further details.
2889
2890Value must be greater than zero. Pass an undef value to disable it:
2891
2892 $x->accuracy(undef);
2893 Math::BigInt->accuracy(undef);
2894
2895Returns the current accuracy. For C<$x->accuracy()> it will return either the
2896local accuracy, or if not defined, the global. This means the return value
2897represents 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
2913Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
29142, but others work, too.
2915
2916Right shifting usually amounts to dividing $x by $n ** $y and truncating the
2917result:
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
2925There 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
2931This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
2932result).
2933
2934=head2 new
2935
2936 $x = Math::BigInt->new($str,$A,$P,$R);
2937
2938Creates a new BigInt object from a string or another BigInt object. The
2939input 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
2946Creates a new BigInt object representing NaN (Not A Number).
2947If 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
2955Creates a new BigInt object representing zero.
2956If 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
2964Creates a new BigInt object representing infinity. The optional argument is
2965either '-' or '+', indicating whether you want infinity or minus infinity.
2966If 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
2975Creates a new BigInt object representing one. The optional argument is
2976either '-' or '+', indicating whether you want one or minus one.
2977If 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 $x->is_inf('-'); # true if -inf (sign is default '+')
2991
2992These methods all test the BigInt for beeing one specific value and return
2993true or false depending on the input. These are faster than doing something
2994like:
2995
2996 if ($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
3003The methods return true if the argument is positive or negative, respectively.
3004C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
3005C<-inf> is negative. A C<zero> is positive.
3006
3007These 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
3015The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
3016C<-inf> are not integers and are neither odd nor even.
3017
3018=head2 bcmp
3019
3020 $x->bcmp($y);
3021
3022Compares $x with $y and takes the sign into account.
3023Returns -1, 0, 1 or undef.
3024
3025=head2 bacmp
3026
3027 $x->bacmp($y);
3028
3029Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
3030
3031=head2 sign
3032
3033 $x->sign();
3034
3035Return 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
3045Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
3046and '-inf', respectively. Does nothing for NaN or zero.
3047
3048=head2 babs
3049
3050 $x->babs();
3051
3052Set the number to it's absolute value, e.g. change the sign from '-' to '+'
3053and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
3054numbers.
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
3097Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
3098returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
3099C<bgcd($num, $mod)==1>.
3100
3101=head2 bmodpow
3102
3103 $num->bmodpow($exp,$mod); # modular exponentation ($num**$exp % $mod)
3104
3105Returns the value of C<$num> taken to the power C<$exp> in the modulus
3106C<$mod> using binary exponentation. C<bmodpow> is far superior to
3107writing
3108
3109 $num ** $exp % $mod
3110
3111because C<bmodpow> is much faster--it reduces internal variables into
3112the modulus whenever possible, so it operates on smaller numbers.
3113
3114C<bmodpow> also supports negative exponents.
3115
3116 bmodpow($num, -1, $mod)
3117
3118is exactly equivalent to
3119
3120 bmodinv($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
3176Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3177does change $x in BigFloat.
3178
3179=head2 bceil
3180
3181 $x->bceil();
3182
3183Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3184does 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
3194head2 length
3195
3196 $x->length();
3197 ($xl,$fl) = $x->length();
3198
3199Returns the number of digits in the decimal representation of the number.
3200In list context, returns the length of the integer and fraction part. For
3201BigInt's, the length of the fraction part will always be 0.
3202
3203=head2 exponent
3204
3205 $x->exponent();
3206
3207Return the exponent of $x as BigInt.
3208
3209=head2 mantissa
3210
3211 $x->mantissa();
3212
3213Return 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
3245Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3246accuracy and precision based rounding, both automatically after every
3247operation as well as manually.
3248
3249This section describes the accuracy/precision handling in Math::Big* as it
3250used to be and as it is now, complete with an explanation of all terms and
3251abbreviations.
3252
3253Not yet implemented things (but with correct description) are marked with '!',
3254things that need to be answered are marked with '?'.
3255
3256In the next paragraph follows a short description of terms used here (because
3257these may differ from terms used by others people or documentation).
3258
3259During the rest of this document, the shortcuts A (for accuracy), P (for
3260precision), F (fallback) and R (rounding mode) will be used.
3261
3262=head2 Precision P
3263
3264A fixed number of digits before (positive) or after (negative)
3265the decimal point. For example, 123.45 has a precision of -2. 0 means an
3266integer like 123 (or 120). A precision of 2 means two digits to the left
3267of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3268numbers with zeros before the decimal point may have different precisions,
3269because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3270was). It could also have p < 0, when the digits after the decimal point
3271are zero.
3272
3273The 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
3285For BigInts, no padding occurs.
3286
3287=head2 Accuracy A
3288
3289Number of significant digits. Leading zeros are not counted. A
3290number may have an accuracy greater than the non-zero digits
3291when there are zeros in it or trailing zeros. For example, 123.456 has
3292A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3293
3294The 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
3302For BigInts, no padding occurs.
3303
3304=head2 Fallback F
3305
3306When both A and P are undefined, this is used as a fallback accuracy when
3307dividing numbers.
3308
3309=head2 Rounding mode R
3310
3311When rounding a number, different 'styles' or 'kinds'
3312of rounding are possible. (Note that random rounding, as in
3313Math::Round, is not implemented.)
3314
3315=over 2
3316
3317=item 'trunc'
3318
3319truncation invariably removes all digits following the
3320rounding place, replacing them with zeros. Thus, 987.65 rounded
3321to tens (P=1) becomes 980, and rounded to the fourth sigdig
3322becomes 987.6 (A=4). 123.456 rounded to the second place after the
3323decimal point (P=-2) becomes 123.46.
3324
3325All other implemented styles of rounding attempt to round to the
3326"nearest digit." If the digit D immediately to the right of the
3327rounding place (skipping the decimal point) is greater than 5, the
3328number is incremented at the rounding place (possibly causing a
3329cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3330to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3331truncated at the rounding place: e.g. when rounding to units, 0.4
3332rounds to 0, and -19.4 rounds to -19.
3333
3334However the results of other styles of rounding differ if the
3335digit immediately to the right of the rounding place (skipping the
3336decimal point) is 5 and if there are no digits, or no digits other
3337than 0, after that 5. In such cases:
3338
3339=item 'even'
3340
3341rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3342if it is not already. E.g., when rounding to the first sigdig, 0.45
3343becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3344
3345=item 'odd'
3346
3347rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3348it is not already. E.g., when rounding to the first sigdig, 0.45
3349becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3350
3351=item '+inf'
3352
3353round to plus infinity, i.e. always round up. E.g., when
3354rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3355and 0.4501 also becomes 0.5.
3356
3357=item '-inf'
3358
3359round to minus infinity, i.e. always round down. E.g., when
3360rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3361but 0.4501 becomes 0.5.
3362
3363=item 'zero'
3364
3365round to zero, i.e. positive numbers down, negative ones up.
3366E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3367becomes -0.5, but 0.4501 becomes 0.5.
3368
3369=back
3370
3371The handling of A & P in MBI/MBF (the old core code shipped with Perl
3372versions <= 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
3416This 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
3595The actual numbers are stored as unsigned big integers (with seperate sign).
3596You should neither care about nor depend on the internal representation; it
3597might change without notice. Use only method calls like C<< $x->sign(); >>
3598instead relying on the internal hash keys like in C<< $x->{sign}; >>.
3599
3600=head2 MATH LIBRARY
3601
3602Math with the numbers is done (by default) by a module called
3603Math::BigInt::Calc. This is equivalent to saying:
3604
3605 use Math::BigInt lib => 'Calc';
3606
3607You can change this by using:
3608
3609 use Math::BigInt lib => 'BitVect';
3610
3611The following would first try to find Math::BigInt::Foo, then
3612Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
3613
3614 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
3615
3616Calc.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
3618uses a bit vector of base 2, most significant bit first. Other modules might
3619use even different means of representing the numbers. See the respective
3620module documentation for further details.
3621
3622=head2 SIGN
3623
3624The sign is either '+', '-', 'NaN', '+inf' or '-inf' and stored seperately.
3625
3626A sign of 'NaN' is used to represent the result when input arguments are not
3627numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
3628minus 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
3633C<mantissa()> and C<exponent()> return the said parts of the BigInt such
3634that:
3635
3636 $m = $x->mantissa();
3637 $e = $x->exponent();
3638 $y = $m * ( 10 ** $e );
3639 print "ok\n" if $x == $y;
3640
3641C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
3642in one go. Both the returned mantissa and exponent have a sign.
3643
3644Currently, for BigInts C<$e> will be always 0, except for NaN, +inf and -inf,
3645where it will be NaN; and for $x == 0, where it will be 1
3646(to be compatible with Math::BigFloat's internal representation of a zero as
3647C<0E1>).
3648
3649C<$m> will always be a copy of the original number. The relation between $e
3650and $m might change in the future, but will always be equivalent in a
3651numerical 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
3675Examples 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
3699Examples 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
3706After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
3707and binary constants in the given scope are converted to C<Math::BigInt>.
3708This conversion happens at compile time.
3709
3710In particular,
3711
3712 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
3713
3714prints the integer value of C<2**100>. Note that without conversion of
3715constants the expression 2**100 will be calculated as perl scalar.
3716
3717Please note that strings and floating point constants are not affected,
3718so that
3719
3720 use Math::BigInt qw/:constant/;
3721
3722 $x = 1234567890123456789012345678901234567890
3723 + 123456789123456789;
3724 $y = '1234567890123456789012345678901234567890'
3725 + '123456789123456789';
3726
3727do not work. You need an explicit Math::BigInt->new() around one of the
3728operands. You should also quote large constants to protect loss of precision:
3729
3730 use Math::Bigint;
3731
3732 $x = Math::BigInt->new('1234567889123456789123456789123456789');
3733
3734Without the quotes Perl would convert the large number to a floating point
3735constant at compile time and then hand the result to BigInt, which results in
3736an truncated result or a NaN.
3737
3738This 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
3745will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
3746to get this to work.
3747
3748=head1 PERFORMANCE
3749
3750Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
3751must be made in the second case. For long numbers, the copy can eat up to 20%
3752of the work (in the case of addition/subtraction, less for
3753multiplication/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 $x takes
3755more time then the actual addition.
3756
3757With a technique called copy-on-write, the cost of copying with overload could
3758be minimized or even completely avoided. A test implementation of COW did show
3759performance gains for overloaded math, but introduced a performance loss due
3760to a constant overhead for all other operatons.
3761
3762The rewritten version of this module is slower on certain operations, like
3763new(), bstr() and numify(). The reason are that it does now more work and
3764handles more cases. The time spent in these operations is usually gained in
3765the other operations so that programs on the average should get faster. If
3766they don't, please contect the author.
3767
3768Some operations may be slower for small numbers, but are significantly faster
3769for big numbers. Other operations are now constant (O(1), like bneg(), babs()
3770etc), instead of O(N) and thus nearly always take much less time. These
3771optimizations were done on purpose.
3772
3773If you find the Calc module to slow, try to install any of the replacement
3774modules and see if they help you.
3775
3776=head2 Alternative math libraries
3777
3778You can use an alternative library to drive Math::BigInt via:
3779
3780 use Math::BigInt lib => 'Module';
3781
3782See L<MATH LIBRARY> for more information.
3783
3784For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
3785
3786=head2 SUBCLASSING
3787
3788=head1 Subclassing Math::BigInt
3789
3790The basic design of Math::BigInt allows simple subclasses with very little
3791work, as long as a few simple rules are followed:
3792
3793=over 2
3794
3795=item *
3796
3797The public API must remain consistent, i.e. if a sub-class is overloading
3798addition, the sub-class must use the same name, in this case badd(). The
3799reason for this is that Math::BigInt is optimized to call the object methods
3800directly.
3801
3802=item *
3803
3804The private object hash keys like C<$x->{sign}> may not be changed, but
3805additional keys can be added, like C<$x->{_custom}>.
3806
3807=item *
3808
3809Accessor functions are available for all existing object hash keys and should
3810be used instead of directly accessing the internal hash keys. The reason for
3811this is that Math::BigInt itself has a pluggable interface which permits it
3812to support different storage methods.
3813
3814=back
3815
3816More complex sub-classes may have to replicate more of the logic internal of
3817Math::BigInt if they need to change more basic behaviors. A subclass that
3818needs to merely change the output only needs to overload C<bstr()>.
3819
3820All other object methods and overloaded functions can be directly inherited
3821from the parent class.
3822
3823At the very minimum, any subclass will need to provide it's own C<new()> and can
3824store additional hash keys in the object. There are also some package globals
3825that 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
3833Additionally, you might want to provide the following two globals to allow
3834auto-upgrading and auto-downgrading to work correctly:
3835
3836 $upgrade = undef;
3837 $downgrade = undef;
3838
3839This allows Math::BigInt to correctly retrieve package globals from the
3840subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
3841t/Math/BigFloat/SubClass.pm completely functional subclass examples.
3842
3843Don't forget to
3844
3845 use overload;
3846
3847in your subclass to automatically inherit the overloading from the parent. If
3848you like, you can change part of the overloading, look at Math::String for an
3849example.
3850
3851=head1 UPGRADING
3852
3853When used like this:
3854
3855 use Math::BigInt upgrade => 'Foo::Bar';
3856
3857certain operations will 'upgrade' their calculation and thus the result to
3858the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
3859
3860 use Math::BigInt upgrade => 'Math::BigFloat';
3861
3862As a shortcut, you can use the module C<bignum>:
3863
3864 use bignum;
3865
3866Also good for oneliners:
3867
3868 perl -Mbignum -le 'print 2 ** 255'
3869
3870This makes it possible to mix arguments of different classes (as in 2.5 + 2)
3871as well es preserve accuracy (as in sqrt(3)).
3872
3873Beware: This feature is not fully implemented yet.
3874
3875=head2 Auto-upgrade
3876
3877The following methods upgrade themselves unconditionally; that is if upgrade
3878is 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
3890Beware: This list is not complete.
3891
3892All other methods upgrade themselves only when one (or all) of their
3893arguments are of the class mentioned in $upgrade (This might change in later
3894versions to a more sophisticated scheme):
3895
3896=head1 BUGS
3897
3898=over 2
3899
3900=item Out of Memory!
3901
3902Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
3903C<eval()> in your code will crash with "Out of memory". This is probably an
3904overload/exporter bug. You can workaround by not having C<eval()>
3905and ':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
3909Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
3910will fall back to eval { require ... } when loading the math lib on Perls
3911prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
3912filesystems using a different seperator.
3913
3914=back
3915
3916=head1 CAVEATS
3917
3918Some things might not work as you expect them. Below is documented what is
3919known to be troublesome:
3920
3921=over 1
3922
3923=item stringify, bstr(), bsstr() and 'cmp'
3924
3925Both 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
3927cmp (especially with overloading) to work as you expect. It also solves
3928problems with Test.pm, it's ok() uses 'eq' internally.
3929
3930Mark 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
3937So, 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 $x eq 3*3;
3950
3951Additionally, the following still works:
3952
3953 print "$x == 9" if $x == $y;
3954 print "$x == 9" if $x == 9;
3955 print "$x == 9" if $x == 3*3;
3956
3957There is now a C<bsstr()> method to get the string in scientific notation aka
3958C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
3959for comparisation, but Perl will represent some numbers as 100 and others
3960as 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
3972Alternatively, simple use <=> for comparisations, that will get it always
3973right. There is not yet a way to get a number automatically represented as
3974a string that matches exactly the way Perl represents it.
3975
3976=item int()
3977
3978C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
3979Perl 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
3986In 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
3991This also works for other subclasses, like Math::String.
3992
3993It is yet unlcear whether overloaded int() should return a scalar or a BigInt.
3994
3995=item length
3996
3997The following will probably not do what you expect:
3998
3999 $c = Math::BigInt->new(123);
4000 print $c->length(),"\n"; # prints 30
4001
4002It prints both the number of digits in the number and in the fraction part
4003since print calls C<length()> in list context. Use something like:
4004
4005 print scalar $c->length(),"\n"; # prints 3
4006
4007=item bdiv
4008
4009The following will probably not do what you expect:
4010
4011 print $c->bdiv(10000),"\n";
4012
4013It prints both quotient and remainder since print calls C<bdiv()> in list
4014context. Also, C<bdiv()> will modify $c, so be carefull. You probably want
4015to use
4016
4017 print $c / 10000,"\n";
4018 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
4019
4020instead.
4021
4022The quotient is always the greatest integer less than or equal to the
4023real-valued quotient of the two operands, and the remainder (when it is
4024nonzero) always has the same sign as the second operand; so, for
4025example,
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
4034As a consequence, the behavior of the operator % agrees with the
4035behavior of Perl's built-in % operator (as documented in the perlop
4036manpage), and the equation
4037
4038 $x == ($x / $y) * $y + ($x % $y)
4039
4040holds true for any $x and $y, which justifies calling the two return
4041values of bdiv() the quotient and remainder. The only exception to this rule
4042are when $y == 0 and $x is negative, then the remainder will also be
4043negative. See below under "infinity handling" for the reasoning behing this.
4044
4045Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
4046not change BigInt's way to do things. This is because under 'use integer' Perl
4047will do what the underlying C thinks is right and this is different for each
4048system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
4049the author to implement it ;)
4050
4051=item infinity handling
4052
4053Here are some examples that explain the reasons why certain results occur while
4054handling infinity:
4055
4056The following table shows the result of the division and the remainder, so that
4057the equation above holds true. Some "ordinary" cases are strewn in to show more
4058clearly 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
4084These cases below violate the "remainder has the sign of the second of the two
4085arguments", 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
4094Beware of:
4095
4096 $x = Math::BigFloat->new(5);
4097 $y = $x;
4098
4099It will not do what you think, e.g. making a copy of $x. Instead it just makes
4100a second reference to the B<same> object and stores it in $y. Thus anything
4101that modifies $x (except overloaded operators) will modify $y, and vice versa.
4102Or in other words, C<=> is only safe if you modify your BigInts only via
4103overloaded 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
4108If you want a true copy of $x, use:
4109
4110 $y = $x->copy();
4111
4112You can also chain the calls like this, this will make first a copy and then
4113multiply it by 2:
4114
4115 $y = $x->copy()->bmul(2);
4116
4117See also the documentation for overload.pm regarding C<=>.
4118
4119=item bpow
4120
4121C<bpow()> (and the rounding functions) now modifies the first argument and
4122returns it, unlike the old code which left it alone and only returned the
4123result. This is to be consistent with C<badd()> etc. The first three will
4124modify $x, the last one won't:
4125
4126 print bpow($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 $x alone
4130
4131The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4132
4133=item Overloading -$x
4134
4135The following:
4136
4137 $x = -$x;
4138
4139is slower than
4140
4141 $x->bneg();
4142
4143since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4144needs to preserve $x since it does not know that it later will get overwritten.
4145This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4146
4147With Copy-On-Write, this issue would be gone, but C-o-W is not implemented
4148since it is slower for all other things.
4149
4150=item Mixing different object types
4151
4152In 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
4158With 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
4174This is because math with overloaded operators follows the first (dominating)
4175operand, and the operation of that is called and returns thus the result. So,
4176Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4177the result should be a Math::BigFloat or the second operant is one.
4178
4179To get a Math::BigFloat you either need to call the operation manually,
4180make sure the operands are already of the proper type or casted to that type
4181via Math::BigFloat->new():
4182
4183 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4184
4185Beware of simple "casting" the entire expression, this would only convert
4186the already computed result:
4187
4188 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4189
4190Beware 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
4195If in doubt, break the expression into simpler terms, or cast all operands
4196to the desired resulting type.
4197
4198Scalar values are a bit different, since:
4199
4200 $float = 2 + $mbf;
4201 $float = $mbf + 2;
4202
4203will both result in the proper type due to the way the overloaded math works.
4204
4205This section also applies to other overloaded math packages, like Math::String.
4206
4207One solution to you problem might be L<autoupgrading|upgrading>.
4208
4209=item bsqrt()
4210
4211C<bsqrt()> works only good if the result is a big integer, e.g. the square
4212root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4213mode.
4214
4215If 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
4228For negative numbers in base see also L<brsft|brsft>.
4229
4230=back
4231
4232=head1 LICENSE
4233
4234This program is free software; you may redistribute it and/or modify it under
4235the same terms as Perl itself.
4236
4237=head1 SEE ALSO
4238
4239L<Math::BigFloat> and L<Math::Big> as well as L<Math::BigInt::BitVect>,
4240L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4241
4242The package at
4243L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4244more documentation including a full version history, testcases, empty
4245subclass files and benchmarks.
4246
4247=head1 AUTHORS
4248
4249Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4250Completely rewritten by Tels http://bloodgate.com in late 2000, 2001.
4251
4252=cut
Full OpenSPARC design & verification tarball including full HDL.