source: trunk/essentials/dev-lang/perl/lib/Math/BigInt.pm@ 3951

Last change on this file since 3951 was 3181, checked in by bird, 19 years ago

perl 5.8.8
File size: 130.4 KB
Line 
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 $class = "Math::BigInt";
19require 5.005;
20
21$VERSION = '1.77';
22
23@ISA = qw(Exporter);
24@EXPORT_OK = qw(objectify bgcd blcm);
25
26# _trap_inf and _trap_nan are internal and should never be accessed from the
27# outside
28use vars qw/$round_mode $accuracy $precision $div_scale $rnd_mode
29 $upgrade $downgrade $_trap_nan $_trap_inf/;
30use strict;
31
32# Inside overload, the first arg is always an object. If the original code had
33# it reversed (like $x = 2 * $y), then the third paramater is true.
34# In some cases (like add, $x = $x + 2 is the same as $x = 2 + $x) this makes
35# no difference, but in some cases it does.
36
37# For overloaded ops with only one argument we simple use $_[0]->copy() to
38# preserve the argument.
39
40# Thus inheritance of overload operators becomes possible and transparent for
41# our subclasses without the need to repeat the entire overload section there.
42
43use overload
44'=' => sub { $_[0]->copy(); },
45
46# some shortcuts for speed (assumes that reversed order of arguments is routed
47# to normal '+' and we thus can always modify first arg. If this is changed,
48# this breaks and must be adjusted.)
49'+=' => sub { $_[0]->badd($_[1]); },
50'-=' => sub { $_[0]->bsub($_[1]); },
51'*=' => sub { $_[0]->bmul($_[1]); },
52'/=' => sub { scalar $_[0]->bdiv($_[1]); },
53'%=' => sub { $_[0]->bmod($_[1]); },
54'^=' => sub { $_[0]->bxor($_[1]); },
55'&=' => sub { $_[0]->band($_[1]); },
56'|=' => sub { $_[0]->bior($_[1]); },
57
58'**=' => sub { $_[0]->bpow($_[1]); },
59'<<=' => sub { $_[0]->blsft($_[1]); },
60'>>=' => sub { $_[0]->brsft($_[1]); },
61
62# not supported by Perl yet
63'..' => \&_pointpoint,
64
65# we might need '==' and '!=' to get things like "NaN == NaN" right
66'<=>' => sub { $_[2] ?
67 ref($_[0])->bcmp($_[1],$_[0]) :
68 $_[0]->bcmp($_[1]); },
69'cmp' => sub {
70 $_[2] ?
71 "$_[1]" cmp $_[0]->bstr() :
72 $_[0]->bstr() cmp "$_[1]" },
73
74# make cos()/sin()/exp() "work" with BigInt's or subclasses
75'cos' => sub { cos($_[0]->numify()) },
76'sin' => sub { sin($_[0]->numify()) },
77'exp' => sub { exp($_[0]->numify()) },
78'atan2' => sub { $_[2] ?
79 atan2($_[1],$_[0]->numify()) :
80 atan2($_[0]->numify(),$_[1]) },
81
82# are not yet overloadable
83#'hex' => sub { print "hex"; $_[0]; },
84#'oct' => sub { print "oct"; $_[0]; },
85
86'log' => sub { $_[0]->copy()->blog($_[1]); },
87'int' => sub { $_[0]->copy(); },
88'neg' => sub { $_[0]->copy()->bneg(); },
89'abs' => sub { $_[0]->copy()->babs(); },
90'sqrt' => sub { $_[0]->copy()->bsqrt(); },
91'~' => sub { $_[0]->copy()->bnot(); },
92
93# for subtract it's a bit tricky to not modify b: b-a => -a+b
94'-' => sub { my $c = $_[0]->copy; $_[2] ?
95 $c->bneg()->badd( $_[1]) :
96 $c->bsub( $_[1]) },
97'+' => sub { $_[0]->copy()->badd($_[1]); },
98'*' => sub { $_[0]->copy()->bmul($_[1]); },
99
100'/' => sub {
101 $_[2] ? ref($_[0])->new($_[1])->bdiv($_[0]) : $_[0]->copy->bdiv($_[1]);
102 },
103'%' => sub {
104 $_[2] ? ref($_[0])->new($_[1])->bmod($_[0]) : $_[0]->copy->bmod($_[1]);
105 },
106'**' => sub {
107 $_[2] ? ref($_[0])->new($_[1])->bpow($_[0]) : $_[0]->copy->bpow($_[1]);
108 },
109'<<' => sub {
110 $_[2] ? ref($_[0])->new($_[1])->blsft($_[0]) : $_[0]->copy->blsft($_[1]);
111 },
112'>>' => sub {
113 $_[2] ? ref($_[0])->new($_[1])->brsft($_[0]) : $_[0]->copy->brsft($_[1]);
114 },
115'&' => sub {
116 $_[2] ? ref($_[0])->new($_[1])->band($_[0]) : $_[0]->copy->band($_[1]);
117 },
118'|' => sub {
119 $_[2] ? ref($_[0])->new($_[1])->bior($_[0]) : $_[0]->copy->bior($_[1]);
120 },
121'^' => sub {
122 $_[2] ? ref($_[0])->new($_[1])->bxor($_[0]) : $_[0]->copy->bxor($_[1]);
123 },
124
125# can modify arg of ++ and --, so avoid a copy() for speed, but don't
126# use $_[0]->bone(), it would modify $_[0] to be 1!
127'++' => sub { $_[0]->binc() },
128'--' => sub { $_[0]->bdec() },
129
130# if overloaded, O(1) instead of O(N) and twice as fast for small numbers
131'bool' => sub {
132 # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/
133 # v5.6.1 dumps on this: return !$_[0]->is_zero() || undef; :-(
134 my $t = undef;
135 $t = 1 if !$_[0]->is_zero();
136 $t;
137 },
138
139# the original qw() does not work with the TIESCALAR below, why?
140# Order of arguments unsignificant
141'""' => sub { $_[0]->bstr(); },
142'0+' => sub { $_[0]->numify(); }
143;
144
145##############################################################################
146# global constants, flags and accessory
147
148# These vars are public, but their direct usage is not recommended, use the
149# accessor methods instead
150
151$round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
152$accuracy = undef;
153$precision = undef;
154$div_scale = 40;
155
156$upgrade = undef; # default is no upgrade
157$downgrade = undef; # default is no downgrade
158
159# These are internally, and not to be used from the outside at all
160
161$_trap_nan = 0; # are NaNs ok? set w/ config()
162$_trap_inf = 0; # are infs ok? set w/ config()
163my $nan = 'NaN'; # constants for easier life
164
165my $CALC = 'Math::BigInt::FastCalc'; # module to do the low level math
166 # default is FastCalc.pm
167my $IMPORT = 0; # was import() called yet?
168 # used to make require work
169my %WARN; # warn only once for low-level libs
170my %CAN; # cache for $CALC->can(...)
171my %CALLBACKS; # callbacks to notify on lib loads
172my $EMU_LIB = 'Math/BigInt/CalcEmu.pm'; # emulate low-level math
173
174##############################################################################
175# the old code had $rnd_mode, so we need to support it, too
176
177$rnd_mode = 'even';
178sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; }
179sub FETCH { return $round_mode; }
180sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
181
182BEGIN
183 {
184 # tie to enable $rnd_mode to work transparently
185 tie $rnd_mode, 'Math::BigInt';
186
187 # set up some handy alias names
188 *as_int = \&as_number;
189 *is_pos = \&is_positive;
190 *is_neg = \&is_negative;
191 }
192
193##############################################################################
194
195sub round_mode
196 {
197 no strict 'refs';
198 # make Class->round_mode() work
199 my $self = shift;
200 my $class = ref($self) || $self || __PACKAGE__;
201 if (defined $_[0])
202 {
203 my $m = shift;
204 if ($m !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
205 {
206 require Carp; Carp::croak ("Unknown round mode '$m'");
207 }
208 return ${"${class}::round_mode"} = $m;
209 }
210 ${"${class}::round_mode"};
211 }
212
213sub upgrade
214 {
215 no strict 'refs';
216 # make Class->upgrade() work
217 my $self = shift;
218 my $class = ref($self) || $self || __PACKAGE__;
219 # need to set new value?
220 if (@_ > 0)
221 {
222 return ${"${class}::upgrade"} = $_[0];
223 }
224 ${"${class}::upgrade"};
225 }
226
227sub downgrade
228 {
229 no strict 'refs';
230 # make Class->downgrade() work
231 my $self = shift;
232 my $class = ref($self) || $self || __PACKAGE__;
233 # need to set new value?
234 if (@_ > 0)
235 {
236 return ${"${class}::downgrade"} = $_[0];
237 }
238 ${"${class}::downgrade"};
239 }
240
241sub div_scale
242 {
243 no strict 'refs';
244 # make Class->div_scale() work
245 my $self = shift;
246 my $class = ref($self) || $self || __PACKAGE__;
247 if (defined $_[0])
248 {
249 if ($_[0] < 0)
250 {
251 require Carp; Carp::croak ('div_scale must be greater than zero');
252 }
253 ${"${class}::div_scale"} = $_[0];
254 }
255 ${"${class}::div_scale"};
256 }
257
258sub accuracy
259 {
260 # $x->accuracy($a); ref($x) $a
261 # $x->accuracy(); ref($x)
262 # Class->accuracy(); class
263 # Class->accuracy($a); class $a
264
265 my $x = shift;
266 my $class = ref($x) || $x || __PACKAGE__;
267
268 no strict 'refs';
269 # need to set new value?
270 if (@_ > 0)
271 {
272 my $a = shift;
273 # convert objects to scalars to avoid deep recursion. If object doesn't
274 # have numify(), then hopefully it will have overloading for int() and
275 # boolean test without wandering into a deep recursion path...
276 $a = $a->numify() if ref($a) && $a->can('numify');
277
278 if (defined $a)
279 {
280 # also croak on non-numerical
281 if (!$a || $a <= 0)
282 {
283 require Carp;
284 Carp::croak ('Argument to accuracy must be greater than zero');
285 }
286 if (int($a) != $a)
287 {
288 require Carp; Carp::croak ('Argument to accuracy must be an integer');
289 }
290 }
291 if (ref($x))
292 {
293 # $object->accuracy() or fallback to global
294 $x->bround($a) if $a; # not for undef, 0
295 $x->{_a} = $a; # set/overwrite, even if not rounded
296 delete $x->{_p}; # clear P
297 $a = ${"${class}::accuracy"} unless defined $a; # proper return value
298 }
299 else
300 {
301 ${"${class}::accuracy"} = $a; # set global A
302 ${"${class}::precision"} = undef; # clear global P
303 }
304 return $a; # shortcut
305 }
306
307 my $a;
308 # $object->accuracy() or fallback to global
309 $a = $x->{_a} if ref($x);
310 # but don't return global undef, when $x's accuracy is 0!
311 $a = ${"${class}::accuracy"} if !defined $a;
312 $a;
313 }
314
315sub precision
316 {
317 # $x->precision($p); ref($x) $p
318 # $x->precision(); ref($x)
319 # Class->precision(); class
320 # Class->precision($p); class $p
321
322 my $x = shift;
323 my $class = ref($x) || $x || __PACKAGE__;
324
325 no strict 'refs';
326 if (@_ > 0)
327 {
328 my $p = shift;
329 # convert objects to scalars to avoid deep recursion. If object doesn't
330 # have numify(), then hopefully it will have overloading for int() and
331 # boolean test without wandering into a deep recursion path...
332 $p = $p->numify() if ref($p) && $p->can('numify');
333 if ((defined $p) && (int($p) != $p))
334 {
335 require Carp; Carp::croak ('Argument to precision must be an integer');
336 }
337 if (ref($x))
338 {
339 # $object->precision() or fallback to global
340 $x->bfround($p) if $p; # not for undef, 0
341 $x->{_p} = $p; # set/overwrite, even if not rounded
342 delete $x->{_a}; # clear A
343 $p = ${"${class}::precision"} unless defined $p; # proper return value
344 }
345 else
346 {
347 ${"${class}::precision"} = $p; # set global P
348 ${"${class}::accuracy"} = undef; # clear global A
349 }
350 return $p; # shortcut
351 }
352
353 my $p;
354 # $object->precision() or fallback to global
355 $p = $x->{_p} if ref($x);
356 # but don't return global undef, when $x's precision is 0!
357 $p = ${"${class}::precision"} if !defined $p;
358 $p;
359 }
360
361sub config
362 {
363 # return (or set) configuration data as hash ref
364 my $class = shift || 'Math::BigInt';
365
366 no strict 'refs';
367 if (@_ > 0)
368 {
369 # try to set given options as arguments from hash
370
371 my $args = $_[0];
372 if (ref($args) ne 'HASH')
373 {
374 $args = { @_ };
375 }
376 # these values can be "set"
377 my $set_args = {};
378 foreach my $key (
379 qw/trap_inf trap_nan
380 upgrade downgrade precision accuracy round_mode div_scale/
381 )
382 {
383 $set_args->{$key} = $args->{$key} if exists $args->{$key};
384 delete $args->{$key};
385 }
386 if (keys %$args > 0)
387 {
388 require Carp;
389 Carp::croak ("Illegal key(s) '",
390 join("','",keys %$args),"' passed to $class\->config()");
391 }
392 foreach my $key (keys %$set_args)
393 {
394 if ($key =~ /^trap_(inf|nan)\z/)
395 {
396 ${"${class}::_trap_$1"} = ($set_args->{"trap_$1"} ? 1 : 0);
397 next;
398 }
399 # use a call instead of just setting the $variable to check argument
400 $class->$key($set_args->{$key});
401 }
402 }
403
404 # now return actual configuration
405
406 my $cfg = {
407 lib => $CALC,
408 lib_version => ${"${CALC}::VERSION"},
409 class => $class,
410 trap_nan => ${"${class}::_trap_nan"},
411 trap_inf => ${"${class}::_trap_inf"},
412 version => ${"${class}::VERSION"},
413 };
414 foreach my $key (qw/
415 upgrade downgrade precision accuracy round_mode div_scale
416 /)
417 {
418 $cfg->{$key} = ${"${class}::$key"};
419 };
420 $cfg;
421 }
422
423sub _scale_a
424 {
425 # select accuracy parameter based on precedence,
426 # used by bround() and bfround(), may return undef for scale (means no op)
427 my ($x,$scale,$mode) = @_;
428
429 $scale = $x->{_a} unless defined $scale;
430
431 no strict 'refs';
432 my $class = ref($x);
433
434 $scale = ${ $class . '::accuracy' } unless defined $scale;
435 $mode = ${ $class . '::round_mode' } unless defined $mode;
436
437 ($scale,$mode);
438 }
439
440sub _scale_p
441 {
442 # select precision parameter based on precedence,
443 # used by bround() and bfround(), may return undef for scale (means no op)
444 my ($x,$scale,$mode) = @_;
445
446 $scale = $x->{_p} unless defined $scale;
447
448 no strict 'refs';
449 my $class = ref($x);
450
451 $scale = ${ $class . '::precision' } unless defined $scale;
452 $mode = ${ $class . '::round_mode' } unless defined $mode;
453
454 ($scale,$mode);
455 }
456
457##############################################################################
458# constructors
459
460sub copy
461 {
462 my ($c,$x);
463 if (@_ > 1)
464 {
465 # if two arguments, the first one is the class to "swallow" subclasses
466 ($c,$x) = @_;
467 }
468 else
469 {
470 $x = shift;
471 $c = ref($x);
472 }
473 return unless ref($x); # only for objects
474
475 my $self = bless {}, $c;
476
477 $self->{sign} = $x->{sign};
478 $self->{value} = $CALC->_copy($x->{value});
479 $self->{_a} = $x->{_a} if defined $x->{_a};
480 $self->{_p} = $x->{_p} if defined $x->{_p};
481 $self;
482 }
483
484sub new
485 {
486 # create a new BigInt object from a string or another BigInt object.
487 # see hash keys documented at top
488
489 # the argument could be an object, so avoid ||, && etc on it, this would
490 # cause costly overloaded code to be called. The only allowed ops are
491 # ref() and defined.
492
493 my ($class,$wanted,$a,$p,$r) = @_;
494
495 # avoid numify-calls by not using || on $wanted!
496 return $class->bzero($a,$p) if !defined $wanted; # default to 0
497 return $class->copy($wanted,$a,$p,$r)
498 if ref($wanted) && $wanted->isa($class); # MBI or subclass
499
500 $class->import() if $IMPORT == 0; # make require work
501
502 my $self = bless {}, $class;
503
504 # shortcut for "normal" numbers
505 if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*\z/))
506 {
507 $self->{sign} = $1 || '+';
508
509 if ($wanted =~ /^[+-]/)
510 {
511 # remove sign without touching wanted to make it work with constants
512 my $t = $wanted; $t =~ s/^[+-]//;
513 $self->{value} = $CALC->_new($t);
514 }
515 else
516 {
517 $self->{value} = $CALC->_new($wanted);
518 }
519 no strict 'refs';
520 if ( (defined $a) || (defined $p)
521 || (defined ${"${class}::precision"})
522 || (defined ${"${class}::accuracy"})
523 )
524 {
525 $self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p);
526 }
527 return $self;
528 }
529
530 # handle '+inf', '-inf' first
531 if ($wanted =~ /^[+-]?inf\z/)
532 {
533 $self->{sign} = $wanted; # set a default sign for bstr()
534 return $self->binf($wanted);
535 }
536 # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign
537 my ($mis,$miv,$mfv,$es,$ev) = _split($wanted);
538 if (!ref $mis)
539 {
540 if ($_trap_nan)
541 {
542 require Carp; Carp::croak("$wanted is not a number in $class");
543 }
544 $self->{value} = $CALC->_zero();
545 $self->{sign} = $nan;
546 return $self;
547 }
548 if (!ref $miv)
549 {
550 # _from_hex or _from_bin
551 $self->{value} = $mis->{value};
552 $self->{sign} = $mis->{sign};
553 return $self; # throw away $mis
554 }
555 # make integer from mantissa by adjusting exp, then convert to bigint
556 $self->{sign} = $$mis; # store sign
557 $self->{value} = $CALC->_zero(); # for all the NaN cases
558 my $e = int("$$es$$ev"); # exponent (avoid recursion)
559 if ($e > 0)
560 {
561 my $diff = $e - CORE::length($$mfv);
562 if ($diff < 0) # Not integer
563 {
564 if ($_trap_nan)
565 {
566 require Carp; Carp::croak("$wanted not an integer in $class");
567 }
568 #print "NOI 1\n";
569 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
570 $self->{sign} = $nan;
571 }
572 else # diff >= 0
573 {
574 # adjust fraction and add it to value
575 #print "diff > 0 $$miv\n";
576 $$miv = $$miv . ($$mfv . '0' x $diff);
577 }
578 }
579 else
580 {
581 if ($$mfv ne '') # e <= 0
582 {
583 # fraction and negative/zero E => NOI
584 if ($_trap_nan)
585 {
586 require Carp; Carp::croak("$wanted not an integer in $class");
587 }
588 #print "NOI 2 \$\$mfv '$$mfv'\n";
589 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
590 $self->{sign} = $nan;
591 }
592 elsif ($e < 0)
593 {
594 # xE-y, and empty mfv
595 #print "xE-y\n";
596 $e = abs($e);
597 if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
598 {
599 if ($_trap_nan)
600 {
601 require Carp; Carp::croak("$wanted not an integer in $class");
602 }
603 #print "NOI 3\n";
604 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
605 $self->{sign} = $nan;
606 }
607 }
608 }
609 $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
610 $self->{value} = $CALC->_new($$miv) if $self->{sign} =~ /^[+-]$/;
611 # if any of the globals is set, use them to round and store them inside $self
612 # do not round for new($x,undef,undef) since that is used by MBF to signal
613 # no rounding
614 $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p;
615 $self;
616 }
617
618sub bnan
619 {
620 # create a bigint 'NaN', if given a BigInt, set it to 'NaN'
621 my $self = shift;
622 $self = $class if !defined $self;
623 if (!ref($self))
624 {
625 my $c = $self; $self = {}; bless $self, $c;
626 }
627 no strict 'refs';
628 if (${"${class}::_trap_nan"})
629 {
630 require Carp;
631 Carp::croak ("Tried to set $self to NaN in $class\::bnan()");
632 }
633 $self->import() if $IMPORT == 0; # make require work
634 return if $self->modify('bnan');
635 if ($self->can('_bnan'))
636 {
637 # use subclass to initialize
638 $self->_bnan();
639 }
640 else
641 {
642 # otherwise do our own thing
643 $self->{value} = $CALC->_zero();
644 }
645 $self->{sign} = $nan;
646 delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly
647 $self;
648 }
649
650sub binf
651 {
652 # create a bigint '+-inf', if given a BigInt, set it to '+-inf'
653 # the sign is either '+', or if given, used from there
654 my $self = shift;
655 my $sign = shift; $sign = '+' if !defined $sign || $sign !~ /^-(inf)?$/;
656 $self = $class if !defined $self;
657 if (!ref($self))
658 {
659 my $c = $self; $self = {}; bless $self, $c;
660 }
661 no strict 'refs';
662 if (${"${class}::_trap_inf"})
663 {
664 require Carp;
665 Carp::croak ("Tried to set $self to +-inf in $class\::binf()");
666 }
667 $self->import() if $IMPORT == 0; # make require work
668 return if $self->modify('binf');
669 if ($self->can('_binf'))
670 {
671 # use subclass to initialize
672 $self->_binf();
673 }
674 else
675 {
676 # otherwise do our own thing
677 $self->{value} = $CALC->_zero();
678 }
679 $sign = $sign . 'inf' if $sign !~ /inf$/; # - => -inf
680 $self->{sign} = $sign;
681 ($self->{_a},$self->{_p}) = @_; # take over requested rounding
682 $self;
683 }
684
685sub bzero
686 {
687 # create a bigint '+0', if given a BigInt, set it to 0
688 my $self = shift;
689 $self = __PACKAGE__ if !defined $self;
690
691 if (!ref($self))
692 {
693 my $c = $self; $self = {}; bless $self, $c;
694 }
695 $self->import() if $IMPORT == 0; # make require work
696 return if $self->modify('bzero');
697
698 if ($self->can('_bzero'))
699 {
700 # use subclass to initialize
701 $self->_bzero();
702 }
703 else
704 {
705 # otherwise do our own thing
706 $self->{value} = $CALC->_zero();
707 }
708 $self->{sign} = '+';
709 if (@_ > 0)
710 {
711 if (@_ > 3)
712 {
713 # call like: $x->bzero($a,$p,$r,$y);
714 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
715 }
716 else
717 {
718 $self->{_a} = $_[0]
719 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
720 $self->{_p} = $_[1]
721 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
722 }
723 }
724 $self;
725 }
726
727sub bone
728 {
729 # create a bigint '+1' (or -1 if given sign '-'),
730 # if given a BigInt, set it to +1 or -1, respecively
731 my $self = shift;
732 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
733 $self = $class if !defined $self;
734
735 if (!ref($self))
736 {
737 my $c = $self; $self = {}; bless $self, $c;
738 }
739 $self->import() if $IMPORT == 0; # make require work
740 return if $self->modify('bone');
741
742 if ($self->can('_bone'))
743 {
744 # use subclass to initialize
745 $self->_bone();
746 }
747 else
748 {
749 # otherwise do our own thing
750 $self->{value} = $CALC->_one();
751 }
752 $self->{sign} = $sign;
753 if (@_ > 0)
754 {
755 if (@_ > 3)
756 {
757 # call like: $x->bone($sign,$a,$p,$r,$y);
758 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
759 }
760 else
761 {
762 # call like: $x->bone($sign,$a,$p,$r);
763 $self->{_a} = $_[0]
764 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
765 $self->{_p} = $_[1]
766 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
767 }
768 }
769 $self;
770 }
771
772##############################################################################
773# string conversation
774
775sub bsstr
776 {
777 # (ref to BFLOAT or num_str ) return num_str
778 # Convert number from internal format to scientific string format.
779 # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
780 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
781
782 if ($x->{sign} !~ /^[+-]$/)
783 {
784 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
785 return 'inf'; # +inf
786 }
787 my ($m,$e) = $x->parts();
788 #$m->bstr() . 'e+' . $e->bstr(); # e can only be positive in BigInt
789 # 'e+' because E can only be positive in BigInt
790 $m->bstr() . 'e+' . $CALC->_str($e->{value});
791 }
792
793sub bstr
794 {
795 # make a string from bigint object
796 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
797
798 if ($x->{sign} !~ /^[+-]$/)
799 {
800 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
801 return 'inf'; # +inf
802 }
803 my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
804 $es.$CALC->_str($x->{value});
805 }
806
807sub numify
808 {
809 # Make a "normal" scalar from a BigInt object
810 my $x = shift; $x = $class->new($x) unless ref $x;
811
812 return $x->bstr() if $x->{sign} !~ /^[+-]$/;
813 my $num = $CALC->_num($x->{value});
814 return -$num if $x->{sign} eq '-';
815 $num;
816 }
817
818##############################################################################
819# public stuff (usually prefixed with "b")
820
821sub sign
822 {
823 # return the sign of the number: +/-/-inf/+inf/NaN
824 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
825
826 $x->{sign};
827 }
828
829sub _find_round_parameters
830 {
831 # After any operation or when calling round(), the result is rounded by
832 # regarding the A & P from arguments, local parameters, or globals.
833
834 # !!!!!!! If you change this, remember to change round(), too! !!!!!!!!!!
835
836 # This procedure finds the round parameters, but it is for speed reasons
837 # duplicated in round. Otherwise, it is tested by the testsuite and used
838 # by fdiv().
839
840 # returns ($self) or ($self,$a,$p,$r) - sets $self to NaN of both A and P
841 # were requested/defined (locally or globally or both)
842
843 my ($self,$a,$p,$r,@args) = @_;
844 # $a accuracy, if given by caller
845 # $p precision, if given by caller
846 # $r round_mode, if given by caller
847 # @args all 'other' arguments (0 for unary, 1 for binary ops)
848
849 my $c = ref($self); # find out class of argument(s)
850 no strict 'refs';
851
852 # now pick $a or $p, but only if we have got "arguments"
853 if (!defined $a)
854 {
855 foreach ($self,@args)
856 {
857 # take the defined one, or if both defined, the one that is smaller
858 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
859 }
860 }
861 if (!defined $p)
862 {
863 # even if $a is defined, take $p, to signal error for both defined
864 foreach ($self,@args)
865 {
866 # take the defined one, or if both defined, the one that is bigger
867 # -2 > -3, and 3 > 2
868 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
869 }
870 }
871 # if still none defined, use globals (#2)
872 $a = ${"$c\::accuracy"} unless defined $a;
873 $p = ${"$c\::precision"} unless defined $p;
874
875 # A == 0 is useless, so undef it to signal no rounding
876 $a = undef if defined $a && $a == 0;
877
878 # no rounding today?
879 return ($self) unless defined $a || defined $p; # early out
880
881 # set A and set P is an fatal error
882 return ($self->bnan()) if defined $a && defined $p; # error
883
884 $r = ${"$c\::round_mode"} unless defined $r;
885 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
886 {
887 require Carp; Carp::croak ("Unknown round mode '$r'");
888 }
889
890 ($self,$a,$p,$r);
891 }
892
893sub round
894 {
895 # Round $self according to given parameters, or given second argument's
896 # parameters or global defaults
897
898 # for speed reasons, _find_round_parameters is embeded here:
899
900 my ($self,$a,$p,$r,@args) = @_;
901 # $a accuracy, if given by caller
902 # $p precision, if given by caller
903 # $r round_mode, if given by caller
904 # @args all 'other' arguments (0 for unary, 1 for binary ops)
905
906 my $c = ref($self); # find out class of argument(s)
907 no strict 'refs';
908
909 # now pick $a or $p, but only if we have got "arguments"
910 if (!defined $a)
911 {
912 foreach ($self,@args)
913 {
914 # take the defined one, or if both defined, the one that is smaller
915 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
916 }
917 }
918 if (!defined $p)
919 {
920 # even if $a is defined, take $p, to signal error for both defined
921 foreach ($self,@args)
922 {
923 # take the defined one, or if both defined, the one that is bigger
924 # -2 > -3, and 3 > 2
925 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
926 }
927 }
928 # if still none defined, use globals (#2)
929 $a = ${"$c\::accuracy"} unless defined $a;
930 $p = ${"$c\::precision"} unless defined $p;
931
932 # A == 0 is useless, so undef it to signal no rounding
933 $a = undef if defined $a && $a == 0;
934
935 # no rounding today?
936 return $self unless defined $a || defined $p; # early out
937
938 # set A and set P is an fatal error
939 return $self->bnan() if defined $a && defined $p;
940
941 $r = ${"$c\::round_mode"} unless defined $r;
942 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
943 {
944 require Carp; Carp::croak ("Unknown round mode '$r'");
945 }
946
947 # now round, by calling either fround or ffround:
948 if (defined $a)
949 {
950 $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a;
951 }
952 else # both can't be undefined due to early out
953 {
954 $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p;
955 }
956 # bround() or bfround() already callled bnorm() if necc.
957 $self;
958 }
959
960sub bnorm
961 {
962 # (numstr or BINT) return BINT
963 # Normalize number -- no-op here
964 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
965 $x;
966 }
967
968sub babs
969 {
970 # (BINT or num_str) return BINT
971 # make number absolute, or return absolute BINT from string
972 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
973
974 return $x if $x->modify('babs');
975 # post-normalized abs for internal use (does nothing for NaN)
976 $x->{sign} =~ s/^-/+/;
977 $x;
978 }
979
980sub bneg
981 {
982 # (BINT or num_str) return BINT
983 # negate number or make a negated number from string
984 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
985
986 return $x if $x->modify('bneg');
987
988 # for +0 dont negate (to have always normalized +0). Does nothing for 'NaN'
989 $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $CALC->_is_zero($x->{value}));
990 $x;
991 }
992
993sub bcmp
994 {
995 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
996 # (BINT or num_str, BINT or num_str) return cond_code
997
998 # set up parameters
999 my ($self,$x,$y) = (ref($_[0]),@_);
1000
1001 # objectify is costly, so avoid it
1002 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1003 {
1004 ($self,$x,$y) = objectify(2,@_);
1005 }
1006
1007 return $upgrade->bcmp($x,$y) if defined $upgrade &&
1008 ((!$x->isa($self)) || (!$y->isa($self)));
1009
1010 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1011 {
1012 # handle +-inf and NaN
1013 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1014 return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
1015 return +1 if $x->{sign} eq '+inf';
1016 return -1 if $x->{sign} eq '-inf';
1017 return -1 if $y->{sign} eq '+inf';
1018 return +1;
1019 }
1020 # check sign for speed first
1021 return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
1022 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
1023
1024 # have same sign, so compare absolute values. Don't make tests for zero here
1025 # because it's actually slower than testin in Calc (especially w/ Pari et al)
1026
1027 # post-normalized compare for internal use (honors signs)
1028 if ($x->{sign} eq '+')
1029 {
1030 # $x and $y both > 0
1031 return $CALC->_acmp($x->{value},$y->{value});
1032 }
1033
1034 # $x && $y both < 0
1035 $CALC->_acmp($y->{value},$x->{value}); # swaped acmp (lib returns 0,1,-1)
1036 }
1037
1038sub bacmp
1039 {
1040 # Compares 2 values, ignoring their signs.
1041 # Returns one of undef, <0, =0, >0. (suitable for sort)
1042 # (BINT, BINT) return cond_code
1043
1044 # set up parameters
1045 my ($self,$x,$y) = (ref($_[0]),@_);
1046 # objectify is costly, so avoid it
1047 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1048 {
1049 ($self,$x,$y) = objectify(2,@_);
1050 }
1051
1052 return $upgrade->bacmp($x,$y) if defined $upgrade &&
1053 ((!$x->isa($self)) || (!$y->isa($self)));
1054
1055 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1056 {
1057 # handle +-inf and NaN
1058 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1059 return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
1060 return 1 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} !~ /^[+-]inf$/;
1061 return -1;
1062 }
1063 $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
1064 }
1065
1066sub badd
1067 {
1068 # add second arg (BINT or string) to first (BINT) (modifies first)
1069 # return result as BINT
1070
1071 # set up parameters
1072 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1073 # objectify is costly, so avoid it
1074 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1075 {
1076 ($self,$x,$y,@r) = objectify(2,@_);
1077 }
1078
1079 return $x if $x->modify('badd');
1080 return $upgrade->badd($upgrade->new($x),$upgrade->new($y),@r) if defined $upgrade &&
1081 ((!$x->isa($self)) || (!$y->isa($self)));
1082
1083 $r[3] = $y; # no push!
1084 # inf and NaN handling
1085 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1086 {
1087 # NaN first
1088 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1089 # inf handling
1090 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1091 {
1092 # +inf++inf or -inf+-inf => same, rest is NaN
1093 return $x if $x->{sign} eq $y->{sign};
1094 return $x->bnan();
1095 }
1096 # +-inf + something => +inf
1097 # something +-inf => +-inf
1098 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
1099 return $x;
1100 }
1101
1102 my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
1103
1104 if ($sx eq $sy)
1105 {
1106 $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
1107 }
1108 else
1109 {
1110 my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
1111 if ($a > 0)
1112 {
1113 $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
1114 $x->{sign} = $sy;
1115 }
1116 elsif ($a == 0)
1117 {
1118 # speedup, if equal, set result to 0
1119 $x->{value} = $CALC->_zero();
1120 $x->{sign} = '+';
1121 }
1122 else # a < 0
1123 {
1124 $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
1125 }
1126 }
1127 $x->round(@r);
1128 }
1129
1130sub bsub
1131 {
1132 # (BINT or num_str, BINT or num_str) return BINT
1133 # subtract second arg from first, modify first
1134
1135 # set up parameters
1136 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1137 # objectify is costly, so avoid it
1138 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1139 {
1140 ($self,$x,$y,@r) = objectify(2,@_);
1141 }
1142
1143 return $x if $x->modify('bsub');
1144
1145 return $upgrade->new($x)->bsub($upgrade->new($y),@r) if defined $upgrade &&
1146 ((!$x->isa($self)) || (!$y->isa($self)));
1147
1148 return $x->round(@r) if $y->is_zero();
1149
1150 # To correctly handle the lone special case $x->bsub($x), we note the sign
1151 # of $x, then flip the sign from $y, and if the sign of $x did change, too,
1152 # then we caught the special case:
1153 my $xsign = $x->{sign};
1154 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
1155 if ($xsign ne $x->{sign})
1156 {
1157 # special case of $x->bsub($x) results in 0
1158 return $x->bzero(@r) if $xsign =~ /^[+-]$/;
1159 return $x->bnan(); # NaN, -inf, +inf
1160 }
1161 $x->badd($y,@r); # badd does not leave internal zeros
1162 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
1163 $x; # already rounded by badd() or no round necc.
1164 }
1165
1166sub binc
1167 {
1168 # increment arg by one
1169 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1170 return $x if $x->modify('binc');
1171
1172 if ($x->{sign} eq '+')
1173 {
1174 $x->{value} = $CALC->_inc($x->{value});
1175 return $x->round($a,$p,$r);
1176 }
1177 elsif ($x->{sign} eq '-')
1178 {
1179 $x->{value} = $CALC->_dec($x->{value});
1180 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1181 return $x->round($a,$p,$r);
1182 }
1183 # inf, nan handling etc
1184 $x->badd($self->bone(),$a,$p,$r); # badd does round
1185 }
1186
1187sub bdec
1188 {
1189 # decrement arg by one
1190 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1191 return $x if $x->modify('bdec');
1192
1193 if ($x->{sign} eq '-')
1194 {
1195 # x already < 0
1196 $x->{value} = $CALC->_inc($x->{value});
1197 }
1198 else
1199 {
1200 return $x->badd($self->bone('-'),@r) unless $x->{sign} eq '+'; # inf or NaN
1201 # >= 0
1202 if ($CALC->_is_zero($x->{value}))
1203 {
1204 # == 0
1205 $x->{value} = $CALC->_one(); $x->{sign} = '-'; # 0 => -1
1206 }
1207 else
1208 {
1209 # > 0
1210 $x->{value} = $CALC->_dec($x->{value});
1211 }
1212 }
1213 $x->round(@r);
1214 }
1215
1216sub blog
1217 {
1218 # calculate $x = $a ** $base + $b and return $a (e.g. the log() to base
1219 # $base of $x)
1220
1221 # set up parameters
1222 my ($self,$x,$base,@r) = (undef,@_);
1223 # objectify is costly, so avoid it
1224 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1225 {
1226 ($self,$x,$base,@r) = objectify(1,ref($x),@_);
1227 }
1228
1229 return $x if $x->modify('blog');
1230
1231 # inf, -inf, NaN, <0 => NaN
1232 return $x->bnan()
1233 if $x->{sign} ne '+' || (defined $base && $base->{sign} ne '+');
1234
1235 return $upgrade->blog($upgrade->new($x),$base,@r) if
1236 defined $upgrade;
1237
1238 my ($rc,$exact) = $CALC->_log_int($x->{value},$base->{value});
1239 return $x->bnan() unless defined $rc; # not possible to take log?
1240 $x->{value} = $rc;
1241 $x->round(@r);
1242 }
1243
1244sub blcm
1245 {
1246 # (BINT or num_str, BINT or num_str) return BINT
1247 # does not modify arguments, but returns new object
1248 # Lowest Common Multiplicator
1249
1250 my $y = shift; my ($x);
1251 if (ref($y))
1252 {
1253 $x = $y->copy();
1254 }
1255 else
1256 {
1257 $x = $class->new($y);
1258 }
1259 my $self = ref($x);
1260 while (@_)
1261 {
1262 my $y = shift; $y = $self->new($y) if !ref ($y);
1263 $x = __lcm($x,$y);
1264 }
1265 $x;
1266 }
1267
1268sub bgcd
1269 {
1270 # (BINT or num_str, BINT or num_str) return BINT
1271 # does not modify arguments, but returns new object
1272 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1273
1274 my $y = shift;
1275 $y = $class->new($y) if !ref($y);
1276 my $self = ref($y);
1277 my $x = $y->copy()->babs(); # keep arguments
1278 return $x->bnan() if $x->{sign} !~ /^[+-]$/; # x NaN?
1279
1280 while (@_)
1281 {
1282 $y = shift; $y = $self->new($y) if !ref($y);
1283 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1284 $x->{value} = $CALC->_gcd($x->{value},$y->{value});
1285 last if $CALC->_is_one($x->{value});
1286 }
1287 $x;
1288 }
1289
1290sub bnot
1291 {
1292 # (num_str or BINT) return BINT
1293 # represent ~x as twos-complement number
1294 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1295 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1296
1297 return $x if $x->modify('bnot');
1298 $x->binc()->bneg(); # binc already does round
1299 }
1300
1301##############################################################################
1302# is_foo test routines
1303# we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1304
1305sub is_zero
1306 {
1307 # return true if arg (BINT or num_str) is zero (array '+', '0')
1308 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1309
1310 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1311 $CALC->_is_zero($x->{value});
1312 }
1313
1314sub is_nan
1315 {
1316 # return true if arg (BINT or num_str) is NaN
1317 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1318
1319 $x->{sign} eq $nan ? 1 : 0;
1320 }
1321
1322sub is_inf
1323 {
1324 # return true if arg (BINT or num_str) is +-inf
1325 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1326
1327 if (defined $sign)
1328 {
1329 $sign = '[+-]inf' if $sign eq ''; # +- doesn't matter, only that's inf
1330 $sign = "[$1]inf" if $sign =~ /^([+-])(inf)?$/; # extract '+' or '-'
1331 return $x->{sign} =~ /^$sign$/ ? 1 : 0;
1332 }
1333 $x->{sign} =~ /^[+-]inf$/ ? 1 : 0; # only +-inf is infinity
1334 }
1335
1336sub is_one
1337 {
1338 # return true if arg (BINT or num_str) is +1, or -1 if sign is given
1339 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1340
1341 $sign = '+' if !defined $sign || $sign ne '-';
1342
1343 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1344 $CALC->_is_one($x->{value});
1345 }
1346
1347sub is_odd
1348 {
1349 # return true when arg (BINT or num_str) is odd, false for even
1350 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1351
1352 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1353 $CALC->_is_odd($x->{value});
1354 }
1355
1356sub is_even
1357 {
1358 # return true when arg (BINT or num_str) is even, false for odd
1359 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1360
1361 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1362 $CALC->_is_even($x->{value});
1363 }
1364
1365sub is_positive
1366 {
1367 # return true when arg (BINT or num_str) is positive (>= 0)
1368 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1369
1370 return 1 if $x->{sign} eq '+inf'; # +inf is positive
1371
1372 # 0+ is neither positive nor negative
1373 ($x->{sign} eq '+' && !$x->is_zero()) ? 1 : 0;
1374 }
1375
1376sub is_negative
1377 {
1378 # return true when arg (BINT or num_str) is negative (< 0)
1379 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1380
1381 $x->{sign} =~ /^-/ ? 1 : 0; # -inf is negative, but NaN is not
1382 }
1383
1384sub is_int
1385 {
1386 # return true when arg (BINT or num_str) is an integer
1387 # always true for BigInt, but different for BigFloats
1388 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1389
1390 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1391 }
1392
1393###############################################################################
1394
1395sub bmul
1396 {
1397 # multiply two numbers -- stolen from Knuth Vol 2 pg 233
1398 # (BINT or num_str, BINT or num_str) return BINT
1399
1400 # set up parameters
1401 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1402 # objectify is costly, so avoid it
1403 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1404 {
1405 ($self,$x,$y,@r) = objectify(2,@_);
1406 }
1407
1408 return $x if $x->modify('bmul');
1409
1410 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1411
1412 # inf handling
1413 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1414 {
1415 return $x->bnan() if $x->is_zero() || $y->is_zero();
1416 # result will always be +-inf:
1417 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1418 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1419 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1420 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1421 return $x->binf('-');
1422 }
1423
1424 return $upgrade->bmul($x,$upgrade->new($y),@r)
1425 if defined $upgrade && !$y->isa($self);
1426
1427 $r[3] = $y; # no push here
1428
1429 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1430
1431 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1432 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1433
1434 $x->round(@r);
1435 }
1436
1437sub _div_inf
1438 {
1439 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1440 my ($self,$x,$y) = @_;
1441
1442 # NaN if x == NaN or y == NaN or x==y==0
1443 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1444 if (($x->is_nan() || $y->is_nan()) ||
1445 ($x->is_zero() && $y->is_zero()));
1446
1447 # +-inf / +-inf == NaN, reminder also NaN
1448 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1449 {
1450 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1451 }
1452 # x / +-inf => 0, remainder x (works even if x == 0)
1453 if ($y->{sign} =~ /^[+-]inf$/)
1454 {
1455 my $t = $x->copy(); # bzero clobbers up $x
1456 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1457 }
1458
1459 # 5 / 0 => +inf, -6 / 0 => -inf
1460 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1461 # exception: -8 / 0 has remainder -8, not 8
1462 # exception: -inf / 0 has remainder -inf, not inf
1463 if ($y->is_zero())
1464 {
1465 # +-inf / 0 => special case for -inf
1466 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1467 if (!$x->is_zero() && !$x->is_inf())
1468 {
1469 my $t = $x->copy(); # binf clobbers up $x
1470 return wantarray ?
1471 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1472 }
1473 }
1474
1475 # last case: +-inf / ordinary number
1476 my $sign = '+inf';
1477 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1478 $x->{sign} = $sign;
1479 return wantarray ? ($x,$self->bzero()) : $x;
1480 }
1481
1482sub bdiv
1483 {
1484 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1485 # (BINT,BINT) (quo,rem) or BINT (only rem)
1486
1487 # set up parameters
1488 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1489 # objectify is costly, so avoid it
1490 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1491 {
1492 ($self,$x,$y,@r) = objectify(2,@_);
1493 }
1494
1495 return $x if $x->modify('bdiv');
1496
1497 return $self->_div_inf($x,$y)
1498 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1499
1500 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1501 if defined $upgrade;
1502
1503 $r[3] = $y; # no push!
1504
1505 # calc new sign and in case $y == +/- 1, return $x
1506 my $xsign = $x->{sign}; # keep
1507 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1508
1509 if (wantarray)
1510 {
1511 my $rem = $self->bzero();
1512 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1513 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1514 $rem->{_a} = $x->{_a};
1515 $rem->{_p} = $x->{_p};
1516 $x->round(@r);
1517 if (! $CALC->_is_zero($rem->{value}))
1518 {
1519 $rem->{sign} = $y->{sign};
1520 $rem = $y->copy()->bsub($rem) if $xsign ne $y->{sign}; # one of them '-'
1521 }
1522 else
1523 {
1524 $rem->{sign} = '+'; # dont leave -0
1525 }
1526 $rem->round(@r);
1527 return ($x,$rem);
1528 }
1529
1530 $x->{value} = $CALC->_div($x->{value},$y->{value});
1531 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1532
1533 $x->round(@r);
1534 }
1535
1536###############################################################################
1537# modulus functions
1538
1539sub bmod
1540 {
1541 # modulus (or remainder)
1542 # (BINT or num_str, BINT or num_str) return BINT
1543
1544 # set up parameters
1545 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1546 # objectify is costly, so avoid it
1547 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1548 {
1549 ($self,$x,$y,@r) = objectify(2,@_);
1550 }
1551
1552 return $x if $x->modify('bmod');
1553 $r[3] = $y; # no push!
1554 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1555 {
1556 my ($d,$r) = $self->_div_inf($x,$y);
1557 $x->{sign} = $r->{sign};
1558 $x->{value} = $r->{value};
1559 return $x->round(@r);
1560 }
1561
1562 # calc new sign and in case $y == +/- 1, return $x
1563 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1564 if (!$CALC->_is_zero($x->{value}))
1565 {
1566 $x->{value} = $CALC->_sub($y->{value},$x->{value},1) # $y-$x
1567 if ($x->{sign} ne $y->{sign});
1568 $x->{sign} = $y->{sign};
1569 }
1570 else
1571 {
1572 $x->{sign} = '+'; # dont leave -0
1573 }
1574 $x->round(@r);
1575 }
1576
1577sub bmodinv
1578 {
1579 # Modular inverse. given a number which is (hopefully) relatively
1580 # prime to the modulus, calculate its inverse using Euclid's
1581 # alogrithm. If the number is not relatively prime to the modulus
1582 # (i.e. their gcd is not one) then NaN is returned.
1583
1584 # set up parameters
1585 my ($self,$x,$y,@r) = (undef,@_);
1586 # objectify is costly, so avoid it
1587 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1588 {
1589 ($self,$x,$y,@r) = objectify(2,@_);
1590 }
1591
1592 return $x if $x->modify('bmodinv');
1593
1594 return $x->bnan()
1595 if ($y->{sign} ne '+' # -, NaN, +inf, -inf
1596 || $x->is_zero() # or num == 0
1597 || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1598 );
1599
1600 # put least residue into $x if $x was negative, and thus make it positive
1601 $x->bmod($y) if $x->{sign} eq '-';
1602
1603 my $sign;
1604 ($x->{value},$sign) = $CALC->_modinv($x->{value},$y->{value});
1605 return $x->bnan() if !defined $x->{value}; # in case no GCD found
1606 return $x if !defined $sign; # already real result
1607 $x->{sign} = $sign; # flip/flop see below
1608 $x->bmod($y); # calc real result
1609 $x;
1610 }
1611
1612sub bmodpow
1613 {
1614 # takes a very large number to a very large exponent in a given very
1615 # large modulus, quickly, thanks to binary exponentation. supports
1616 # negative exponents.
1617 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1618
1619 return $num if $num->modify('bmodpow');
1620
1621 # check modulus for valid values
1622 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1623 || $mod->is_zero());
1624
1625 # check exponent for valid values
1626 if ($exp->{sign} =~ /\w/)
1627 {
1628 # i.e., if it's NaN, +inf, or -inf...
1629 return $num->bnan();
1630 }
1631
1632 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
1633
1634 # check num for valid values (also NaN if there was no inverse but $exp < 0)
1635 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1636
1637 # $mod is positive, sign on $exp is ignored, result also positive
1638 $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
1639 $num;
1640 }
1641
1642###############################################################################
1643
1644sub bfac
1645 {
1646 # (BINT or num_str, BINT or num_str) return BINT
1647 # compute factorial number from $x, modify $x in place
1648 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1649
1650 return $x if $x->modify('bfac') || $x->{sign} eq '+inf'; # inf => inf
1651 return $x->bnan() if $x->{sign} ne '+'; # NaN, <0 etc => NaN
1652
1653 $x->{value} = $CALC->_fac($x->{value});
1654 $x->round(@r);
1655 }
1656
1657sub bpow
1658 {
1659 # (BINT or num_str, BINT or num_str) return BINT
1660 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1661 # modifies first argument
1662
1663 # set up parameters
1664 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1665 # objectify is costly, so avoid it
1666 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1667 {
1668 ($self,$x,$y,@r) = objectify(2,@_);
1669 }
1670
1671 return $x if $x->modify('bpow');
1672
1673 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1674
1675 # inf handling
1676 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1677 {
1678 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1679 {
1680 # +-inf ** +-inf
1681 return $x->bnan();
1682 }
1683 # +-inf ** Y
1684 if ($x->{sign} =~ /^[+-]inf/)
1685 {
1686 # +inf ** 0 => NaN
1687 return $x->bnan() if $y->is_zero();
1688 # -inf ** -1 => 1/inf => 0
1689 return $x->bzero() if $y->is_one('-') && $x->is_negative();
1690
1691 # +inf ** Y => inf
1692 return $x if $x->{sign} eq '+inf';
1693
1694 # -inf ** Y => -inf if Y is odd
1695 return $x if $y->is_odd();
1696 return $x->babs();
1697 }
1698 # X ** +-inf
1699
1700 # 1 ** +inf => 1
1701 return $x if $x->is_one();
1702
1703 # 0 ** inf => 0
1704 return $x if $x->is_zero() && $y->{sign} =~ /^[+]/;
1705
1706 # 0 ** -inf => inf
1707 return $x->binf() if $x->is_zero();
1708
1709 # -1 ** -inf => NaN
1710 return $x->bnan() if $x->is_one('-') && $y->{sign} =~ /^[-]/;
1711
1712 # -X ** -inf => 0
1713 return $x->bzero() if $x->{sign} eq '-' && $y->{sign} =~ /^[-]/;
1714
1715 # -1 ** inf => NaN
1716 return $x->bnan() if $x->{sign} eq '-';
1717
1718 # X ** inf => inf
1719 return $x->binf() if $y->{sign} =~ /^[+]/;
1720 # X ** -inf => 0
1721 return $x->bzero();
1722 }
1723
1724 return $upgrade->bpow($upgrade->new($x),$y,@r)
1725 if defined $upgrade && !$y->isa($self);
1726
1727 $r[3] = $y; # no push!
1728
1729 # cases 0 ** Y, X ** 0, X ** 1, 1 ** Y are handled by Calc or Emu
1730
1731 my $new_sign = '+';
1732 $new_sign = $y->is_odd() ? '-' : '+' if ($x->{sign} ne '+');
1733
1734 # 0 ** -7 => ( 1 / (0 ** 7)) => 1 / 0 => +inf
1735 return $x->binf()
1736 if $y->{sign} eq '-' && $x->{sign} eq '+' && $CALC->_is_zero($x->{value});
1737 # 1 ** -y => 1 / (1 ** |y|)
1738 # so do test for negative $y after above's clause
1739 return $x->bnan() if $y->{sign} eq '-' && !$CALC->_is_one($x->{value});
1740
1741 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1742 $x->{sign} = $new_sign;
1743 $x->{sign} = '+' if $CALC->_is_zero($y->{value});
1744 $x->round(@r);
1745 }
1746
1747sub blsft
1748 {
1749 # (BINT or num_str, BINT or num_str) return BINT
1750 # compute x << y, base n, y >= 0
1751
1752 # set up parameters
1753 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1754 # objectify is costly, so avoid it
1755 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1756 {
1757 ($self,$x,$y,$n,@r) = objectify(2,@_);
1758 }
1759
1760 return $x if $x->modify('blsft');
1761 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1762 return $x->round(@r) if $y->is_zero();
1763
1764 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1765
1766 $x->{value} = $CALC->_lsft($x->{value},$y->{value},$n);
1767 $x->round(@r);
1768 }
1769
1770sub brsft
1771 {
1772 # (BINT or num_str, BINT or num_str) return BINT
1773 # compute x >> y, base n, y >= 0
1774
1775 # set up parameters
1776 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1777 # objectify is costly, so avoid it
1778 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1779 {
1780 ($self,$x,$y,$n,@r) = objectify(2,@_);
1781 }
1782
1783 return $x if $x->modify('brsft');
1784 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1785 return $x->round(@r) if $y->is_zero();
1786 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
1787
1788 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1789
1790 # this only works for negative numbers when shifting in base 2
1791 if (($x->{sign} eq '-') && ($n == 2))
1792 {
1793 return $x->round(@r) if $x->is_one('-'); # -1 => -1
1794 if (!$y->is_one())
1795 {
1796 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1797 # but perhaps there is a better emulation for two's complement shift...
1798 # if $y != 1, we must simulate it by doing:
1799 # convert to bin, flip all bits, shift, and be done
1800 $x->binc(); # -3 => -2
1801 my $bin = $x->as_bin();
1802 $bin =~ s/^-0b//; # strip '-0b' prefix
1803 $bin =~ tr/10/01/; # flip bits
1804 # now shift
1805 if (CORE::length($bin) <= $y)
1806 {
1807 $bin = '0'; # shifting to far right creates -1
1808 # 0, because later increment makes
1809 # that 1, attached '-' makes it '-1'
1810 # because -1 >> x == -1 !
1811 }
1812 else
1813 {
1814 $bin =~ s/.{$y}$//; # cut off at the right side
1815 $bin = '1' . $bin; # extend left side by one dummy '1'
1816 $bin =~ tr/10/01/; # flip bits back
1817 }
1818 my $res = $self->new('0b'.$bin); # add prefix and convert back
1819 $res->binc(); # remember to increment
1820 $x->{value} = $res->{value}; # take over value
1821 return $x->round(@r); # we are done now, magic, isn't?
1822 }
1823 # x < 0, n == 2, y == 1
1824 $x->bdec(); # n == 2, but $y == 1: this fixes it
1825 }
1826
1827 $x->{value} = $CALC->_rsft($x->{value},$y->{value},$n);
1828 $x->round(@r);
1829 }
1830
1831sub band
1832 {
1833 #(BINT or num_str, BINT or num_str) return BINT
1834 # compute x & y
1835
1836 # set up parameters
1837 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1838 # objectify is costly, so avoid it
1839 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1840 {
1841 ($self,$x,$y,@r) = objectify(2,@_);
1842 }
1843
1844 return $x if $x->modify('band');
1845
1846 $r[3] = $y; # no push!
1847
1848 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1849
1850 my $sx = $x->{sign} eq '+' ? 1 : -1;
1851 my $sy = $y->{sign} eq '+' ? 1 : -1;
1852
1853 if ($sx == 1 && $sy == 1)
1854 {
1855 $x->{value} = $CALC->_and($x->{value},$y->{value});
1856 return $x->round(@r);
1857 }
1858
1859 if ($CAN{signed_and})
1860 {
1861 $x->{value} = $CALC->_signed_and($x->{value},$y->{value},$sx,$sy);
1862 return $x->round(@r);
1863 }
1864
1865 require $EMU_LIB;
1866 __emu_band($self,$x,$y,$sx,$sy,@r);
1867 }
1868
1869sub bior
1870 {
1871 #(BINT or num_str, BINT or num_str) return BINT
1872 # compute x | y
1873
1874 # set up parameters
1875 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1876 # objectify is costly, so avoid it
1877 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1878 {
1879 ($self,$x,$y,@r) = objectify(2,@_);
1880 }
1881
1882 return $x if $x->modify('bior');
1883 $r[3] = $y; # no push!
1884
1885 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1886
1887 my $sx = $x->{sign} eq '+' ? 1 : -1;
1888 my $sy = $y->{sign} eq '+' ? 1 : -1;
1889
1890 # the sign of X follows the sign of X, e.g. sign of Y irrelevant for bior()
1891
1892 # don't use lib for negative values
1893 if ($sx == 1 && $sy == 1)
1894 {
1895 $x->{value} = $CALC->_or($x->{value},$y->{value});
1896 return $x->round(@r);
1897 }
1898
1899 # if lib can do negative values, let it handle this
1900 if ($CAN{signed_or})
1901 {
1902 $x->{value} = $CALC->_signed_or($x->{value},$y->{value},$sx,$sy);
1903 return $x->round(@r);
1904 }
1905
1906 require $EMU_LIB;
1907 __emu_bior($self,$x,$y,$sx,$sy,@r);
1908 }
1909
1910sub bxor
1911 {
1912 #(BINT or num_str, BINT or num_str) return BINT
1913 # compute x ^ y
1914
1915 # set up parameters
1916 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1917 # objectify is costly, so avoid it
1918 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1919 {
1920 ($self,$x,$y,@r) = objectify(2,@_);
1921 }
1922
1923 return $x if $x->modify('bxor');
1924 $r[3] = $y; # no push!
1925
1926 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1927
1928 my $sx = $x->{sign} eq '+' ? 1 : -1;
1929 my $sy = $y->{sign} eq '+' ? 1 : -1;
1930
1931 # don't use lib for negative values
1932 if ($sx == 1 && $sy == 1)
1933 {
1934 $x->{value} = $CALC->_xor($x->{value},$y->{value});
1935 return $x->round(@r);
1936 }
1937
1938 # if lib can do negative values, let it handle this
1939 if ($CAN{signed_xor})
1940 {
1941 $x->{value} = $CALC->_signed_xor($x->{value},$y->{value},$sx,$sy);
1942 return $x->round(@r);
1943 }
1944
1945 require $EMU_LIB;
1946 __emu_bxor($self,$x,$y,$sx,$sy,@r);
1947 }
1948
1949sub length
1950 {
1951 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1952
1953 my $e = $CALC->_len($x->{value});
1954 wantarray ? ($e,0) : $e;
1955 }
1956
1957sub digit
1958 {
1959 # return the nth decimal digit, negative values count backward, 0 is right
1960 my ($self,$x,$n) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1961
1962 $n = $n->numify() if ref($n);
1963 $CALC->_digit($x->{value},$n||0);
1964 }
1965
1966sub _trailing_zeros
1967 {
1968 # return the amount of trailing zeros in $x (as scalar)
1969 my $x = shift;
1970 $x = $class->new($x) unless ref $x;
1971
1972 return 0 if $x->{sign} !~ /^[+-]$/; # NaN, inf, -inf etc
1973
1974 $CALC->_zeros($x->{value}); # must handle odd values, 0 etc
1975 }
1976
1977sub bsqrt
1978 {
1979 # calculate square root of $x
1980 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1981
1982 return $x if $x->modify('bsqrt');
1983
1984 return $x->bnan() if $x->{sign} !~ /^\+/; # -x or -inf or NaN => NaN
1985 return $x if $x->{sign} eq '+inf'; # sqrt(+inf) == inf
1986
1987 return $upgrade->bsqrt($x,@r) if defined $upgrade;
1988
1989 $x->{value} = $CALC->_sqrt($x->{value});
1990 $x->round(@r);
1991 }
1992
1993sub broot
1994 {
1995 # calculate $y'th root of $x
1996
1997 # set up parameters
1998 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1999
2000 $y = $self->new(2) unless defined $y;
2001
2002 # objectify is costly, so avoid it
2003 if ((!ref($x)) || (ref($x) ne ref($y)))
2004 {
2005 ($self,$x,$y,@r) = objectify(2,$self || $class,@_);
2006 }
2007
2008 return $x if $x->modify('broot');
2009
2010 # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
2011 return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
2012 $y->{sign} !~ /^\+$/;
2013
2014 return $x->round(@r)
2015 if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
2016
2017 return $upgrade->new($x)->broot($upgrade->new($y),@r) if defined $upgrade;
2018
2019 $x->{value} = $CALC->_root($x->{value},$y->{value});
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/^[+-]//; # NaN, -inf,+inf => NaN or inf
2031 return $self->new($s);
2032 }
2033 return $self->bone() if $x->is_zero();
2034
2035 $self->new($x->_trailing_zeros());
2036 }
2037
2038sub mantissa
2039 {
2040 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
2041 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2042
2043 if ($x->{sign} !~ /^[+-]$/)
2044 {
2045 # for NaN, +inf, -inf: keep the sign
2046 return $self->new($x->{sign});
2047 }
2048 my $m = $x->copy(); delete $m->{_p}; delete $m->{_a};
2049 # that's a bit 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]) ? (undef,$_[0]) : objectify(1,@_);
2059
2060 ($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; my $self = ref($x) || $x; $x = $self->new($x) unless ref $x;
2071
2072 my ($scale,$mode) = $x->_scale_p(@_);
2073
2074 return $x if !defined $scale || $x->modify('bfround'); # no-op
2075
2076 # no-op for BigInts if $n <= 0
2077 $x->bround( $x->length()-$scale, $mode) if $scale > 0;
2078
2079 delete $x->{_a}; # delete to save memory
2080 $x->{_p} = $scale; # store new _p
2081 $x;
2082 }
2083
2084sub _scan_for_nonzero
2085 {
2086 # internal, used by bround() to scan for non-zeros after a '5'
2087 my ($x,$pad,$xs,$len) = @_;
2088
2089 return 0 if $len == 1; # "5" is trailed by invisible zeros
2090 my $follow = $pad - 1;
2091 return 0 if $follow > $len || $follow < 1;
2092
2093 # use the string form to check whether only '0's follow or not
2094 substr ($xs,-$follow) =~ /[^0]/ ? 1 : 0;
2095 }
2096
2097sub fround
2098 {
2099 # Exists to make life easier for switch between MBF and MBI (should we
2100 # autoload fxxx() like MBF does for bxxx()?)
2101 my $x = shift; $x = $class->new($x) unless ref $x;
2102 $x->bround(@_);
2103 }
2104
2105sub bround
2106 {
2107 # accuracy: +$n preserve $n digits from left,
2108 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
2109 # no-op for $n == 0
2110 # and overwrite the rest with 0's, return normalized number
2111 # do not return $x->bnorm(), but $x
2112
2113 my $x = shift; $x = $class->new($x) unless ref $x;
2114 my ($scale,$mode) = $x->_scale_a(@_);
2115 return $x if !defined $scale || $x->modify('bround'); # no-op
2116
2117 if ($x->is_zero() || $scale == 0)
2118 {
2119 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2120 return $x;
2121 }
2122 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2123
2124 # we have fewer digits than we want to scale to
2125 my $len = $x->length();
2126 # convert $scale to a scalar in case it is an object (put's a limit on the
2127 # number length, but this would already limited by memory constraints), makes
2128 # it faster
2129 $scale = $scale->numify() if ref ($scale);
2130
2131 # scale < 0, but > -len (not >=!)
2132 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2133 {
2134 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2135 return $x;
2136 }
2137
2138 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2139 my ($pad,$digit_round,$digit_after);
2140 $pad = $len - $scale;
2141 $pad = abs($scale-1) if $scale < 0;
2142
2143 # do not use digit(), it is very costly for binary => decimal
2144 # getting the entire string is also costly, but we need to do it only once
2145 my $xs = $CALC->_str($x->{value});
2146 my $pl = -$pad-1;
2147
2148 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2149 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2150 $digit_round = '0'; $digit_round = substr($xs,$pl,1) if $pad <= $len;
2151 $pl++; $pl ++ if $pad >= $len;
2152 $digit_after = '0'; $digit_after = substr($xs,$pl,1) if $pad > 0;
2153
2154 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2155 # closer at the remaining digits of the original $x, remember decision
2156 my $round_up = 1; # default round up
2157 $round_up -- if
2158 ($mode eq 'trunc') || # trunc by round down
2159 ($digit_after =~ /[01234]/) || # round down anyway,
2160 # 6789 => round up
2161 ($digit_after eq '5') && # not 5000...0000
2162 ($x->_scan_for_nonzero($pad,$xs,$len) == 0) &&
2163 (
2164 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2165 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2166 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2167 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2168 ($mode eq 'zero') # round down if zero, sign adjusted below
2169 );
2170 my $put_back = 0; # not yet modified
2171
2172 if (($pad > 0) && ($pad <= $len))
2173 {
2174 substr($xs,-$pad,$pad) = '0' x $pad; # replace with '00...'
2175 $put_back = 1; # need to put back
2176 }
2177 elsif ($pad > $len)
2178 {
2179 $x->bzero(); # round to '0'
2180 }
2181
2182 if ($round_up) # what gave test above?
2183 {
2184 $put_back = 1; # need to put back
2185 $pad = $len, $xs = '0' x $pad if $scale < 0; # tlr: whack 0.51=>1.0
2186
2187 # we modify directly the string variant instead of creating a number and
2188 # adding it, since that is faster (we already have the string)
2189 my $c = 0; $pad ++; # for $pad == $len case
2190 while ($pad <= $len)
2191 {
2192 $c = substr($xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2193 substr($xs,-$pad,1) = $c; $pad++;
2194 last if $c != 0; # no overflow => early out
2195 }
2196 $xs = '1'.$xs if $c == 0;
2197
2198 }
2199 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back, if needed
2200
2201 $x->{_a} = $scale if $scale >= 0;
2202 if ($scale < 0)
2203 {
2204 $x->{_a} = $len+$scale;
2205 $x->{_a} = 0 if $scale < -$len;
2206 }
2207 $x;
2208 }
2209
2210sub bfloor
2211 {
2212 # return integer less or equal then number; no-op since it's already integer
2213 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2214
2215 $x->round(@r);
2216 }
2217
2218sub bceil
2219 {
2220 # return integer greater or equal then number; no-op since it's already int
2221 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2222
2223 $x->round(@r);
2224 }
2225
2226sub as_number
2227 {
2228 # An object might be asked to return itself as bigint on certain overloaded
2229 # operations, this does exactly this, so that sub classes can simple inherit
2230 # it or override with their own integer conversion routine.
2231 $_[0]->copy();
2232 }
2233
2234sub as_hex
2235 {
2236 # return as hex string, with prefixed 0x
2237 my $x = shift; $x = $class->new($x) if !ref($x);
2238
2239 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2240
2241 my $s = '';
2242 $s = $x->{sign} if $x->{sign} eq '-';
2243 $s . $CALC->_as_hex($x->{value});
2244 }
2245
2246sub as_bin
2247 {
2248 # return as binary string, with prefixed 0b
2249 my $x = shift; $x = $class->new($x) if !ref($x);
2250
2251 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2252
2253 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2254 return $s . $CALC->_as_bin($x->{value});
2255 }
2256
2257##############################################################################
2258# private stuff (internal use only)
2259
2260sub objectify
2261 {
2262 # check for strings, if yes, return objects instead
2263
2264 # the first argument is number of args objectify() should look at it will
2265 # return $count+1 elements, the first will be a classname. This is because
2266 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2267 # useless objects beeing created and thrown away. So we cannot simple loop
2268 # over @_. If the given count is 0, all arguments will be used.
2269
2270 # If the second arg is a ref, use it as class.
2271 # If not, try to use it as classname, unless undef, then use $class
2272 # (aka Math::BigInt). The latter shouldn't happen,though.
2273
2274 # caller: gives us:
2275 # $x->badd(1); => ref x, scalar y
2276 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2277 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2278 # Math::BigInt::badd(1,2); => scalar x, scalar y
2279 # In the last case we check number of arguments to turn it silently into
2280 # $class,1,2. (We can not take '1' as class ;o)
2281 # badd($class,1) is not supported (it should, eventually, try to add undef)
2282 # currently it tries 'Math::BigInt' + 1, which will not work.
2283
2284 # some shortcut for the common cases
2285 # $x->unary_op();
2286 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2287
2288 my $count = abs(shift || 0);
2289
2290 my (@a,$k,$d); # resulting array, temp, and downgrade
2291 if (ref $_[0])
2292 {
2293 # okay, got object as first
2294 $a[0] = ref $_[0];
2295 }
2296 else
2297 {
2298 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2299 $a[0] = $class;
2300 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2301 }
2302
2303 no strict 'refs';
2304 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2305 if (defined ${"$a[0]::downgrade"})
2306 {
2307 $d = ${"$a[0]::downgrade"};
2308 ${"$a[0]::downgrade"} = undef;
2309 }
2310
2311 my $up = ${"$a[0]::upgrade"};
2312 #print "Now in objectify, my class is today $a[0], count = $count\n";
2313 if ($count == 0)
2314 {
2315 while (@_)
2316 {
2317 $k = shift;
2318 if (!ref($k))
2319 {
2320 $k = $a[0]->new($k);
2321 }
2322 elsif (!defined $up && ref($k) ne $a[0])
2323 {
2324 # foreign object, try to convert to integer
2325 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2326 }
2327 push @a,$k;
2328 }
2329 }
2330 else
2331 {
2332 while ($count > 0)
2333 {
2334 $count--;
2335 $k = shift;
2336 if (!ref($k))
2337 {
2338 $k = $a[0]->new($k);
2339 }
2340 elsif (!defined $up && ref($k) ne $a[0])
2341 {
2342 # foreign object, try to convert to integer
2343 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2344 }
2345 push @a,$k;
2346 }
2347 push @a,@_; # return other params, too
2348 }
2349 if (! wantarray)
2350 {
2351 require Carp; Carp::croak ("$class objectify needs list context");
2352 }
2353 ${"$a[0]::downgrade"} = $d;
2354 @a;
2355 }
2356
2357sub _register_callback
2358 {
2359 my ($class,$callback) = @_;
2360
2361 if (ref($callback) ne 'CODE')
2362 {
2363 require Carp;
2364 Carp::croak ("$callback is not a coderef");
2365 }
2366 $CALLBACKS{$class} = $callback;
2367 }
2368
2369sub import
2370 {
2371 my $self = shift;
2372
2373 $IMPORT++; # remember we did import()
2374 my @a; my $l = scalar @_;
2375 for ( my $i = 0; $i < $l ; $i++ )
2376 {
2377 if ($_[$i] eq ':constant')
2378 {
2379 # this causes overlord er load to step in
2380 overload::constant
2381 integer => sub { $self->new(shift) },
2382 binary => sub { $self->new(shift) };
2383 }
2384 elsif ($_[$i] eq 'upgrade')
2385 {
2386 # this causes upgrading
2387 $upgrade = $_[$i+1]; # or undef to disable
2388 $i++;
2389 }
2390 elsif ($_[$i] =~ /^lib$/i)
2391 {
2392 # this causes a different low lib to take care...
2393 $CALC = $_[$i+1] || '';
2394 $i++;
2395 }
2396 else
2397 {
2398 push @a, $_[$i];
2399 }
2400 }
2401 # any non :constant stuff is handled by our parent, Exporter
2402 if (@a > 0)
2403 {
2404 require Exporter;
2405
2406 $self->SUPER::import(@a); # need it for subclasses
2407 $self->export_to_level(1,$self,@a); # need it for MBF
2408 }
2409
2410 # try to load core math lib
2411 my @c = split /\s*,\s*/,$CALC;
2412 foreach (@c)
2413 {
2414 $_ =~ tr/a-zA-Z0-9://cd; # limit to sane characters
2415 }
2416 push @c, 'FastCalc', 'Calc'; # if all fail, try these
2417 $CALC = ''; # signal error
2418 foreach my $lib (@c)
2419 {
2420 next if ($lib || '') eq '';
2421 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2422 $lib =~ s/\.pm$//;
2423 if ($] < 5.006)
2424 {
2425 # Perl < 5.6.0 dies with "out of memory!" when eval("") and ':constant' is
2426 # used in the same script, or eval("") inside import().
2427 my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
2428 my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
2429 require File::Spec;
2430 $file = File::Spec->catfile (@parts, $file);
2431 eval { require "$file"; $lib->import( @c ); }
2432 }
2433 else
2434 {
2435 eval "use $lib qw/@c/;";
2436 }
2437 if ($@ eq '')
2438 {
2439 my $ok = 1;
2440 # loaded it ok, see if the api_version() is high enough
2441 if ($lib->can('api_version') && $lib->api_version() >= 1.0)
2442 {
2443 $ok = 0;
2444 # api_version matches, check if it really provides anything we need
2445 for my $method (qw/
2446 one two ten
2447 str num
2448 add mul div sub dec inc
2449 acmp len digit is_one is_zero is_even is_odd
2450 is_two is_ten
2451 new copy check from_hex from_bin as_hex as_bin zeros
2452 rsft lsft xor and or
2453 mod sqrt root fac pow modinv modpow log_int gcd
2454 /)
2455 {
2456 if (!$lib->can("_$method"))
2457 {
2458 if (($WARN{$lib}||0) < 2)
2459 {
2460 require Carp;
2461 Carp::carp ("$lib is missing method '_$method'");
2462 $WARN{$lib} = 1; # still warn about the lib
2463 }
2464 $ok++; last;
2465 }
2466 }
2467 }
2468 if ($ok == 0)
2469 {
2470 $CALC = $lib;
2471 last; # found a usable one, break
2472 }
2473 else
2474 {
2475 if (($WARN{$lib}||0) < 2)
2476 {
2477 my $ver = eval "\$$lib\::VERSION" || 'unknown';
2478 require Carp;
2479 Carp::carp ("Cannot load outdated $lib v$ver, please upgrade");
2480 $WARN{$lib} = 2; # never warn again
2481 }
2482 }
2483 }
2484 }
2485 if ($CALC eq '')
2486 {
2487 require Carp;
2488 Carp::croak ("Couldn't load any math lib, not even 'Calc.pm'");
2489 }
2490
2491 # notify callbacks
2492 foreach my $class (keys %CALLBACKS)
2493 {
2494 &{$CALLBACKS{$class}}($CALC);
2495 }
2496
2497 # Fill $CAN with the results of $CALC->can(...) for emulating lower math lib
2498 # functions
2499
2500 %CAN = ();
2501 for my $method (qw/ signed_and signed_or signed_xor /)
2502 {
2503 $CAN{$method} = $CALC->can("_$method") ? 1 : 0;
2504 }
2505
2506 # import done
2507 }
2508
2509sub __from_hex
2510 {
2511 # internal
2512 # convert a (ref to) big hex string to BigInt, return undef for error
2513 my $hs = shift;
2514
2515 my $x = Math::BigInt->bzero();
2516
2517 # strip underscores
2518 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2519 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2520
2521 return $x->bnan() if $hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2522
2523 my $sign = '+'; $sign = '-' if $hs =~ /^-/;
2524
2525 $hs =~ s/^[+-]//; # strip sign
2526 $x->{value} = $CALC->_from_hex($hs);
2527 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2528 $x;
2529 }
2530
2531sub __from_bin
2532 {
2533 # internal
2534 # convert a (ref to) big binary string to BigInt, return undef for error
2535 my $bs = shift;
2536
2537 my $x = Math::BigInt->bzero();
2538 # strip underscores
2539 $bs =~ s/([01])_([01])/$1$2/g;
2540 $bs =~ s/([01])_([01])/$1$2/g;
2541 return $x->bnan() if $bs !~ /^[+-]?0b[01]+$/;
2542
2543 my $sign = '+'; $sign = '-' if $bs =~ /^\-/;
2544 $bs =~ s/^[+-]//; # strip sign
2545
2546 $x->{value} = $CALC->_from_bin($bs);
2547 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2548 $x;
2549 }
2550
2551sub _split
2552 {
2553 # input: num_str; output: undef for invalid or
2554 # (\$mantissa_sign,\$mantissa_value,\$mantissa_fraction,\$exp_sign,\$exp_value)
2555 # Internal, take apart a string and return the pieces.
2556 # Strip leading/trailing whitespace, leading zeros, underscore and reject
2557 # invalid input.
2558 my $x = shift;
2559
2560 # strip white space at front, also extranous leading zeros
2561 $x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2562 $x =~ s/^\s+//; # but this will
2563 $x =~ s/\s+$//g; # strip white space at end
2564
2565 # shortcut, if nothing to split, return early
2566 if ($x =~ /^[+-]?\d+\z/)
2567 {
2568 $x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2569 return (\$sign, \$x, \'', \'', \0);
2570 }
2571
2572 # invalid starting char?
2573 return if $x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2574
2575 return __from_hex($x) if $x =~ /^[\-\+]?0x/; # hex string
2576 return __from_bin($x) if $x =~ /^[\-\+]?0b/; # binary string
2577
2578 # strip underscores between digits
2579 $x =~ s/(\d)_(\d)/$1$2/g;
2580 $x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3
2581
2582 # some possible inputs:
2583 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2584 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 # 0e999
2585
2586 my ($m,$e,$last) = split /[Ee]/,$x;
2587 return if defined $last; # last defined => 1e2E3 or others
2588 $e = '0' if !defined $e || $e eq "";
2589
2590 # sign,value for exponent,mantint,mantfrac
2591 my ($es,$ev,$mis,$miv,$mfv);
2592 # valid exponent?
2593 if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2594 {
2595 $es = $1; $ev = $2;
2596 # valid mantissa?
2597 return if $m eq '.' || $m eq '';
2598 my ($mi,$mf,$lastf) = split /\./,$m;
2599 return if defined $lastf; # lastf defined => 1.2.3 or others
2600 $mi = '0' if !defined $mi;
2601 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2602 $mf = '0' if !defined $mf || $mf eq '';
2603 if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2604 {
2605 $mis = $1||'+'; $miv = $2;
2606 return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
2607 $mfv = $1;
2608 # handle the 0e999 case here
2609 $ev = 0 if $miv eq '0' && $mfv eq '';
2610 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2611 }
2612 }
2613 return; # NaN, not a number
2614 }
2615
2616##############################################################################
2617# internal calculation routines (others are in Math::BigInt::Calc etc)
2618
2619sub __lcm
2620 {
2621 # (BINT or num_str, BINT or num_str) return BINT
2622 # does modify first argument
2623 # LCM
2624
2625 my ($x,$ty) = @_;
2626 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2627 my $method = ref($x) . '::bgcd';
2628 no strict 'refs';
2629 $x * $ty / &$method($x,$ty);
2630 }
2631
2632###############################################################################
2633# this method returns 0 if the object can be modified, or 1 if not.
2634# We use a fast constant sub() here, to avoid costly calls. Subclasses
2635# may override it with special code (f.i. Math::BigInt::Constant does so)
2636
2637sub modify () { 0; }
2638
26391;
2640__END__
2641
2642=pod
2643
2644=head1 NAME
2645
2646Math::BigInt - Arbitrary size integer/float math package
2647
2648=head1 SYNOPSIS
2649
2650 use Math::BigInt;
2651
2652 # or make it faster: install (optional) Math::BigInt::GMP
2653 # and always use (it will fall back to pure Perl if the
2654 # GMP library is not installed):
2655
2656 use Math::BigInt lib => 'GMP';
2657
2658 my $str = '1234567890';
2659 my @values = (64,74,18);
2660 my $n = 1; my $sign = '-';
2661
2662 # Number creation
2663 $x = Math::BigInt->new($str); # defaults to 0
2664 $y = $x->copy(); # make a true copy
2665 $nan = Math::BigInt->bnan(); # create a NotANumber
2666 $zero = Math::BigInt->bzero(); # create a +0
2667 $inf = Math::BigInt->binf(); # create a +inf
2668 $inf = Math::BigInt->binf('-'); # create a -inf
2669 $one = Math::BigInt->bone(); # create a +1
2670 $one = Math::BigInt->bone('-'); # create a -1
2671
2672 # Testing (don't modify their arguments)
2673 # (return true if the condition is met, otherwise false)
2674
2675 $x->is_zero(); # if $x is +0
2676 $x->is_nan(); # if $x is NaN
2677 $x->is_one(); # if $x is +1
2678 $x->is_one('-'); # if $x is -1
2679 $x->is_odd(); # if $x is odd
2680 $x->is_even(); # if $x is even
2681 $x->is_pos(); # if $x >= 0
2682 $x->is_neg(); # if $x < 0
2683 $x->is_inf($sign); # if $x is +inf, or -inf (sign is default '+')
2684 $x->is_int(); # if $x is an integer (not a float)
2685
2686 # comparing and digit/sign extration
2687 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2688 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2689 $x->sign(); # return the sign, either +,- or NaN
2690 $x->digit($n); # return the nth digit, counting from right
2691 $x->digit(-$n); # return the nth digit, counting from left
2692
2693 # The following all modify their first argument. If you want to preserve
2694 # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
2695 # neccessary when mixing $a = $b assigments with non-overloaded math.
2696
2697 $x->bzero(); # set $x to 0
2698 $x->bnan(); # set $x to NaN
2699 $x->bone(); # set $x to +1
2700 $x->bone('-'); # set $x to -1
2701 $x->binf(); # set $x to inf
2702 $x->binf('-'); # set $x to -inf
2703
2704 $x->bneg(); # negation
2705 $x->babs(); # absolute value
2706 $x->bnorm(); # normalize (no-op in BigInt)
2707 $x->bnot(); # two's complement (bit wise not)
2708 $x->binc(); # increment $x by 1
2709 $x->bdec(); # decrement $x by 1
2710
2711 $x->badd($y); # addition (add $y to $x)
2712 $x->bsub($y); # subtraction (subtract $y from $x)
2713 $x->bmul($y); # multiplication (multiply $x by $y)
2714 $x->bdiv($y); # divide, set $x to quotient
2715 # return (quo,rem) or quo if scalar
2716
2717 $x->bmod($y); # modulus (x % y)
2718 $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
2719 $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
2720
2721 $x->bpow($y); # power of arguments (x ** y)
2722 $x->blsft($y); # left shift
2723 $x->brsft($y); # right shift
2724 $x->blsft($y,$n); # left shift, by base $n (like 10)
2725 $x->brsft($y,$n); # right shift, by base $n (like 10)
2726
2727 $x->band($y); # bitwise and
2728 $x->bior($y); # bitwise inclusive or
2729 $x->bxor($y); # bitwise exclusive or
2730 $x->bnot(); # bitwise not (two's complement)
2731
2732 $x->bsqrt(); # calculate square-root
2733 $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
2734 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2735
2736 $x->round($A,$P,$mode); # round to accuracy or precision using mode $mode
2737 $x->bround($n); # accuracy: preserve $n digits
2738 $x->bfround($n); # round to $nth digit, no-op for BigInts
2739
2740 # The following do not modify their arguments in BigInt (are no-ops),
2741 # but do so in BigFloat:
2742
2743 $x->bfloor(); # return integer less or equal than $x
2744 $x->bceil(); # return integer greater or equal than $x
2745
2746 # The following do not modify their arguments:
2747
2748 # greatest common divisor (no OO style)
2749 my $gcd = Math::BigInt::bgcd(@values);
2750 # lowest common multiplicator (no OO style)
2751 my $lcm = Math::BigInt::blcm(@values);
2752
2753 $x->length(); # return number of digits in number
2754 ($xl,$f) = $x->length(); # length of number and length of fraction part,
2755 # latter is always 0 digits long for BigInts
2756
2757 $x->exponent(); # return exponent as BigInt
2758 $x->mantissa(); # return (signed) mantissa as BigInt
2759 $x->parts(); # return (mantissa,exponent) as BigInt
2760 $x->copy(); # make a true copy of $x (unlike $y = $x;)
2761 $x->as_int(); # return as BigInt (in BigInt: same as copy())
2762 $x->numify(); # return as scalar (might overflow!)
2763
2764 # conversation to string (do not modify their argument)
2765 $x->bstr(); # normalized string (e.g. '3')
2766 $x->bsstr(); # norm. string in scientific notation (e.g. '3E0')
2767 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
2768 $x->as_bin(); # as signed binary string with prefixed 0b
2769
2770
2771 # precision and accuracy (see section about rounding for more)
2772 $x->precision(); # return P of $x (or global, if P of $x undef)
2773 $x->precision($n); # set P of $x to $n
2774 $x->accuracy(); # return A of $x (or global, if A of $x undef)
2775 $x->accuracy($n); # set A $x to $n
2776
2777 # Global methods
2778 Math::BigInt->precision(); # get/set global P for all BigInt objects
2779 Math::BigInt->accuracy(); # get/set global A for all BigInt objects
2780 Math::BigInt->round_mode(); # get/set global round mode, one of
2781 # 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
2782 Math::BigInt->config(); # return hash containing configuration
2783
2784=head1 DESCRIPTION
2785
2786All operators (inlcuding basic math operations) are overloaded if you
2787declare your big integers as
2788
2789 $i = new Math::BigInt '123_456_789_123_456_789';
2790
2791Operations with overloaded operators preserve the arguments which is
2792exactly what you expect.
2793
2794=over 2
2795
2796=item Input
2797
2798Input values to these routines may be any string, that looks like a number
2799and results in an integer, including hexadecimal and binary numbers.
2800
2801Scalars holding numbers may also be passed, but note that non-integer numbers
2802may already have lost precision due to the conversation to float. Quote
2803your input if you want BigInt to see all the digits:
2804
2805 $x = Math::BigInt->new(12345678890123456789); # bad
2806 $x = Math::BigInt->new('12345678901234567890'); # good
2807
2808You can include one underscore between any two digits.
2809
2810This means integer values like 1.01E2 or even 1000E-2 are also accepted.
2811Non-integer values result in NaN.
2812
2813Currently, Math::BigInt::new() defaults to 0, while Math::BigInt::new('')
2814results in 'NaN'. This might change in the future, so use always the following
2815explicit forms to get a zero or NaN:
2816
2817 $zero = Math::BigInt->bzero();
2818 $nan = Math::BigInt->bnan();
2819
2820C<bnorm()> on a BigInt object is now effectively a no-op, since the numbers
2821are always stored in normalized form. If passed a string, creates a BigInt
2822object from the input.
2823
2824=item Output
2825
2826Output values are BigInt objects (normalized), except for the methods which
2827return a string (see L<SYNOPSIS>).
2828
2829Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
2830C<is_nan()>, etc.) return true or false, while others (C<bcmp()>, C<bacmp()>)
2831return either undef (if NaN is involved), <0, 0 or >0 and are suited for sort.
2832
2833=back
2834
2835=head1 METHODS
2836
2837Each of the methods below (except config(), accuracy() and precision())
2838accepts three additional parameters. These arguments C<$A>, C<$P> and C<$R>
2839are C<accuracy>, C<precision> and C<round_mode>. Please see the section about
2840L<ACCURACY and PRECISION> for more information.
2841
2842=head2 config
2843
2844 use Data::Dumper;
2845
2846 print Dumper ( Math::BigInt->config() );
2847 print Math::BigInt->config()->{lib},"\n";
2848
2849Returns a hash containing the configuration, e.g. the version number, lib
2850loaded etc. The following hash keys are currently filled in with the
2851appropriate information.
2852
2853 key Description
2854 Example
2855 ============================================================
2856 lib Name of the low-level math library
2857 Math::BigInt::Calc
2858 lib_version Version of low-level math library (see 'lib')
2859 0.30
2860 class The class name of config() you just called
2861 Math::BigInt
2862 upgrade To which class math operations might be upgraded
2863 Math::BigFloat
2864 downgrade To which class math operations might be downgraded
2865 undef
2866 precision Global precision
2867 undef
2868 accuracy Global accuracy
2869 undef
2870 round_mode Global round mode
2871 even
2872 version version number of the class you used
2873 1.61
2874 div_scale Fallback acccuracy for div
2875 40
2876 trap_nan If true, traps creation of NaN via croak()
2877 1
2878 trap_inf If true, traps creation of +inf/-inf via croak()
2879 1
2880
2881The following values can be set by passing C<config()> a reference to a hash:
2882
2883 trap_inf trap_nan
2884 upgrade downgrade precision accuracy round_mode div_scale
2885
2886Example:
2887
2888 $new_cfg = Math::BigInt->config( { trap_inf => 1, precision => 5 } );
2889
2890=head2 accuracy
2891
2892 $x->accuracy(5); # local for $x
2893 CLASS->accuracy(5); # global for all members of CLASS
2894 # Note: This also applies to new()!
2895
2896 $A = $x->accuracy(); # read out accuracy that affects $x
2897 $A = CLASS->accuracy(); # read out global accuracy
2898
2899Set or get the global or local accuracy, aka how many significant digits the
2900results have. If you set a global accuracy, then this also applies to new()!
2901
2902Warning! The accuracy I<sticks>, e.g. once you created a number under the
2903influence of C<< CLASS->accuracy($A) >>, all results from math operations with
2904that number will also be rounded.
2905
2906In most cases, you should probably round the results explicitely using one of
2907L<round()>, L<bround()> or L<bfround()> or by passing the desired accuracy
2908to the math operation as additional parameter:
2909
2910 my $x = Math::BigInt->new(30000);
2911 my $y = Math::BigInt->new(7);
2912 print scalar $x->copy()->bdiv($y, 2); # print 4300
2913 print scalar $x->copy()->bdiv($y)->bround(2); # print 4300
2914
2915Please see the section about L<ACCURACY AND PRECISION> for further details.
2916
2917Value must be greater than zero. Pass an undef value to disable it:
2918
2919 $x->accuracy(undef);
2920 Math::BigInt->accuracy(undef);
2921
2922Returns the current accuracy. For C<$x->accuracy()> it will return either the
2923local accuracy, or if not defined, the global. This means the return value
2924represents the accuracy that will be in effect for $x:
2925
2926 $y = Math::BigInt->new(1234567); # unrounded
2927 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
2928 $x = Math::BigInt->new(123456); # $x will be automatically rounded!
2929 print "$x $y\n"; # '123500 1234567'
2930 print $x->accuracy(),"\n"; # will be 4
2931 print $y->accuracy(),"\n"; # also 4, since global is 4
2932 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
2933 print $x->accuracy(),"\n"; # still 4
2934 print $y->accuracy(),"\n"; # 5, since global is 5
2935
2936Note: Works also for subclasses like Math::BigFloat. Each class has it's own
2937globals separated from Math::BigInt, but it is possible to subclass
2938Math::BigInt and make the globals of the subclass aliases to the ones from
2939Math::BigInt.
2940
2941=head2 precision
2942
2943 $x->precision(-2); # local for $x, round at the second digit right of the dot
2944 $x->precision(2); # ditto, round at the second digit left of the dot
2945
2946 CLASS->precision(5); # Global for all members of CLASS
2947 # This also applies to new()!
2948 CLASS->precision(-5); # ditto
2949
2950 $P = CLASS->precision(); # read out global precision
2951 $P = $x->precision(); # read out precision that affects $x
2952
2953Note: You probably want to use L<accuracy()> instead. With L<accuracy> you
2954set the number of digits each result should have, with L<precision> you
2955set the place where to round!
2956
2957C<precision()> sets or gets the global or local precision, aka at which digit
2958before or after the dot to round all results. A set global precision also
2959applies to all newly created numbers!
2960
2961In Math::BigInt, passing a negative number precision has no effect since no
2962numbers have digits after the dot. In L<Math::BigFloat>, it will round all
2963results to P digits after the dot.
2964
2965Please see the section about L<ACCURACY AND PRECISION> for further details.
2966
2967Pass an undef value to disable it:
2968
2969 $x->precision(undef);
2970 Math::BigInt->precision(undef);
2971
2972Returns the current precision. For C<$x->precision()> it will return either the
2973local precision of $x, or if not defined, the global. This means the return
2974value represents the prevision that will be in effect for $x:
2975
2976 $y = Math::BigInt->new(1234567); # unrounded
2977 print Math::BigInt->precision(4),"\n"; # set 4, print 4
2978 $x = Math::BigInt->new(123456); # will be automatically rounded
2979 print $x; # print "120000"!
2980
2981Note: Works also for subclasses like L<Math::BigFloat>. Each class has its
2982own globals separated from Math::BigInt, but it is possible to subclass
2983Math::BigInt and make the globals of the subclass aliases to the ones from
2984Math::BigInt.
2985
2986=head2 brsft
2987
2988 $x->brsft($y,$n);
2989
2990Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
29912, but others work, too.
2992
2993Right shifting usually amounts to dividing $x by $n ** $y and truncating the
2994result:
2995
2996
2997 $x = Math::BigInt->new(10);
2998 $x->brsft(1); # same as $x >> 1: 5
2999 $x = Math::BigInt->new(1234);
3000 $x->brsft(2,10); # result 12
3001
3002There is one exception, and that is base 2 with negative $x:
3003
3004
3005 $x = Math::BigInt->new(-5);
3006 print $x->brsft(1);
3007
3008This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
3009result).
3010
3011=head2 new
3012
3013 $x = Math::BigInt->new($str,$A,$P,$R);
3014
3015Creates a new BigInt object from a scalar or another BigInt object. The
3016input is accepted as decimal, hex (with leading '0x') or binary (with leading
3017'0b').
3018
3019See L<Input> for more info on accepted input formats.
3020
3021=head2 bnan
3022
3023 $x = Math::BigInt->bnan();
3024
3025Creates a new BigInt object representing NaN (Not A Number).
3026If used on an object, it will set it to NaN:
3027
3028 $x->bnan();
3029
3030=head2 bzero
3031
3032 $x = Math::BigInt->bzero();
3033
3034Creates a new BigInt object representing zero.
3035If used on an object, it will set it to zero:
3036
3037 $x->bzero();
3038
3039=head2 binf
3040
3041 $x = Math::BigInt->binf($sign);
3042
3043Creates a new BigInt object representing infinity. The optional argument is
3044either '-' or '+', indicating whether you want infinity or minus infinity.
3045If used on an object, it will set it to infinity:
3046
3047 $x->binf();
3048 $x->binf('-');
3049
3050=head2 bone
3051
3052 $x = Math::BigInt->binf($sign);
3053
3054Creates a new BigInt object representing one. The optional argument is
3055either '-' or '+', indicating whether you want one or minus one.
3056If used on an object, it will set it to one:
3057
3058 $x->bone(); # +1
3059 $x->bone('-'); # -1
3060
3061=head2 is_one()/is_zero()/is_nan()/is_inf()
3062
3063
3064 $x->is_zero(); # true if arg is +0
3065 $x->is_nan(); # true if arg is NaN
3066 $x->is_one(); # true if arg is +1
3067 $x->is_one('-'); # true if arg is -1
3068 $x->is_inf(); # true if +inf
3069 $x->is_inf('-'); # true if -inf (sign is default '+')
3070
3071These methods all test the BigInt for beeing one specific value and return
3072true or false depending on the input. These are faster than doing something
3073like:
3074
3075 if ($x == 0)
3076
3077=head2 is_pos()/is_neg()
3078
3079 $x->is_pos(); # true if > 0
3080 $x->is_neg(); # true if < 0
3081
3082The methods return true if the argument is positive or negative, respectively.
3083C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
3084C<-inf> is negative. A C<zero> is neither positive nor negative.
3085
3086These methods are only testing the sign, and not the value.
3087
3088C<is_positive()> and C<is_negative()> are aliase to C<is_pos()> and
3089C<is_neg()>, respectively. C<is_positive()> and C<is_negative()> were
3090introduced in v1.36, while C<is_pos()> and C<is_neg()> were only introduced
3091in v1.68.
3092
3093=head2 is_odd()/is_even()/is_int()
3094
3095 $x->is_odd(); # true if odd, false for even
3096 $x->is_even(); # true if even, false for odd
3097 $x->is_int(); # true if $x is an integer
3098
3099The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
3100C<-inf> are not integers and are neither odd nor even.
3101
3102In BigInt, all numbers except C<NaN>, C<+inf> and C<-inf> are integers.
3103
3104=head2 bcmp
3105
3106 $x->bcmp($y);
3107
3108Compares $x with $y and takes the sign into account.
3109Returns -1, 0, 1 or undef.
3110
3111=head2 bacmp
3112
3113 $x->bacmp($y);
3114
3115Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
3116
3117=head2 sign
3118
3119 $x->sign();
3120
3121Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
3122
3123If you want $x to have a certain sign, use one of the following methods:
3124
3125 $x->babs(); # '+'
3126 $x->babs()->bneg(); # '-'
3127 $x->bnan(); # 'NaN'
3128 $x->binf(); # '+inf'
3129 $x->binf('-'); # '-inf'
3130
3131=head2 digit
3132
3133 $x->digit($n); # return the nth digit, counting from right
3134
3135If C<$n> is negative, returns the digit counting from left.
3136
3137=head2 bneg
3138
3139 $x->bneg();
3140
3141Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
3142and '-inf', respectively. Does nothing for NaN or zero.
3143
3144=head2 babs
3145
3146 $x->babs();
3147
3148Set the number to it's absolute value, e.g. change the sign from '-' to '+'
3149and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
3150numbers.
3151
3152=head2 bnorm
3153
3154 $x->bnorm(); # normalize (no-op)
3155
3156=head2 bnot
3157
3158 $x->bnot();
3159
3160Two's complement (bit wise not). This is equivalent to
3161
3162 $x->binc()->bneg();
3163
3164but faster.
3165
3166=head2 binc
3167
3168 $x->binc(); # increment x by 1
3169
3170=head2 bdec
3171
3172 $x->bdec(); # decrement x by 1
3173
3174=head2 badd
3175
3176 $x->badd($y); # addition (add $y to $x)
3177
3178=head2 bsub
3179
3180 $x->bsub($y); # subtraction (subtract $y from $x)
3181
3182=head2 bmul
3183
3184 $x->bmul($y); # multiplication (multiply $x by $y)
3185
3186=head2 bdiv
3187
3188 $x->bdiv($y); # divide, set $x to quotient
3189 # return (quo,rem) or quo if scalar
3190
3191=head2 bmod
3192
3193 $x->bmod($y); # modulus (x % y)
3194
3195=head2 bmodinv
3196
3197 num->bmodinv($mod); # modular inverse
3198
3199Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
3200returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
3201C<bgcd($num, $mod)==1>.
3202
3203=head2 bmodpow
3204
3205 $num->bmodpow($exp,$mod); # modular exponentation
3206 # ($num**$exp % $mod)
3207
3208Returns the value of C<$num> taken to the power C<$exp> in the modulus
3209C<$mod> using binary exponentation. C<bmodpow> is far superior to
3210writing
3211
3212 $num ** $exp % $mod
3213
3214because it is much faster - it reduces internal variables into
3215the modulus whenever possible, so it operates on smaller numbers.
3216
3217C<bmodpow> also supports negative exponents.
3218
3219 bmodpow($num, -1, $mod)
3220
3221is exactly equivalent to
3222
3223 bmodinv($num, $mod)
3224
3225=head2 bpow
3226
3227 $x->bpow($y); # power of arguments (x ** y)
3228
3229=head2 blsft
3230
3231 $x->blsft($y); # left shift
3232 $x->blsft($y,$n); # left shift, in base $n (like 10)
3233
3234=head2 brsft
3235
3236 $x->brsft($y); # right shift
3237 $x->brsft($y,$n); # right shift, in base $n (like 10)
3238
3239=head2 band
3240
3241 $x->band($y); # bitwise and
3242
3243=head2 bior
3244
3245 $x->bior($y); # bitwise inclusive or
3246
3247=head2 bxor
3248
3249 $x->bxor($y); # bitwise exclusive or
3250
3251=head2 bnot
3252
3253 $x->bnot(); # bitwise not (two's complement)
3254
3255=head2 bsqrt
3256
3257 $x->bsqrt(); # calculate square-root
3258
3259=head2 bfac
3260
3261 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3262
3263=head2 round
3264
3265 $x->round($A,$P,$round_mode);
3266
3267Round $x to accuracy C<$A> or precision C<$P> using the round mode
3268C<$round_mode>.
3269
3270=head2 bround
3271
3272 $x->bround($N); # accuracy: preserve $N digits
3273
3274=head2 bfround
3275
3276 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3277
3278=head2 bfloor
3279
3280 $x->bfloor();
3281
3282Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3283does change $x in BigFloat.
3284
3285=head2 bceil
3286
3287 $x->bceil();
3288
3289Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3290does change $x in BigFloat.
3291
3292=head2 bgcd
3293
3294 bgcd(@values); # greatest common divisor (no OO style)
3295
3296=head2 blcm
3297
3298 blcm(@values); # lowest common multiplicator (no OO style)
3299
3300head2 length
3301
3302 $x->length();
3303 ($xl,$fl) = $x->length();
3304
3305Returns the number of digits in the decimal representation of the number.
3306In list context, returns the length of the integer and fraction part. For
3307BigInt's, the length of the fraction part will always be 0.
3308
3309=head2 exponent
3310
3311 $x->exponent();
3312
3313Return the exponent of $x as BigInt.
3314
3315=head2 mantissa
3316
3317 $x->mantissa();
3318
3319Return the signed mantissa of $x as BigInt.
3320
3321=head2 parts
3322
3323 $x->parts(); # return (mantissa,exponent) as BigInt
3324
3325=head2 copy
3326
3327 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3328
3329=head2 as_int
3330
3331 $x->as_int();
3332
3333Returns $x as a BigInt (truncated towards zero). In BigInt this is the same as
3334C<copy()>.
3335
3336C<as_number()> is an alias to this method. C<as_number> was introduced in
3337v1.22, while C<as_int()> was only introduced in v1.68.
3338
3339=head2 bstr
3340
3341 $x->bstr();
3342
3343Returns a normalized string represantation of C<$x>.
3344
3345=head2 bsstr
3346
3347 $x->bsstr(); # normalized string in scientific notation
3348
3349=head2 as_hex
3350
3351 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3352
3353=head2 as_bin
3354
3355 $x->as_bin(); # as signed binary string with prefixed 0b
3356
3357=head1 ACCURACY and PRECISION
3358
3359Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3360accuracy and precision based rounding, both automatically after every
3361operation, as well as manually.
3362
3363This section describes the accuracy/precision handling in Math::Big* as it
3364used to be and as it is now, complete with an explanation of all terms and
3365abbreviations.
3366
3367Not yet implemented things (but with correct description) are marked with '!',
3368things that need to be answered are marked with '?'.
3369
3370In the next paragraph follows a short description of terms used here (because
3371these may differ from terms used by others people or documentation).
3372
3373During the rest of this document, the shortcuts A (for accuracy), P (for
3374precision), F (fallback) and R (rounding mode) will be used.
3375
3376=head2 Precision P
3377
3378A fixed number of digits before (positive) or after (negative)
3379the decimal point. For example, 123.45 has a precision of -2. 0 means an
3380integer like 123 (or 120). A precision of 2 means two digits to the left
3381of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3382numbers with zeros before the decimal point may have different precisions,
3383because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3384was). It could also have p < 0, when the digits after the decimal point
3385are zero.
3386
3387The string output (of floating point numbers) will be padded with zeros:
3388
3389 Initial value P A Result String
3390 ------------------------------------------------------------
3391 1234.01 -3 1000 1000
3392 1234 -2 1200 1200
3393 1234.5 -1 1230 1230
3394 1234.001 1 1234 1234.0
3395 1234.01 0 1234 1234
3396 1234.01 2 1234.01 1234.01
3397 1234.01 5 1234.01 1234.01000
3398
3399For BigInts, no padding occurs.
3400
3401=head2 Accuracy A
3402
3403Number of significant digits. Leading zeros are not counted. A
3404number may have an accuracy greater than the non-zero digits
3405when there are zeros in it or trailing zeros. For example, 123.456 has
3406A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3407
3408The string output (of floating point numbers) will be padded with zeros:
3409
3410 Initial value P A Result String
3411 ------------------------------------------------------------
3412 1234.01 3 1230 1230
3413 1234.01 6 1234.01 1234.01
3414 1234.1 8 1234.1 1234.1000
3415
3416For BigInts, no padding occurs.
3417
3418=head2 Fallback F
3419
3420When both A and P are undefined, this is used as a fallback accuracy when
3421dividing numbers.
3422
3423=head2 Rounding mode R
3424
3425When rounding a number, different 'styles' or 'kinds'
3426of rounding are possible. (Note that random rounding, as in
3427Math::Round, is not implemented.)
3428
3429=over 2
3430
3431=item 'trunc'
3432
3433truncation invariably removes all digits following the
3434rounding place, replacing them with zeros. Thus, 987.65 rounded
3435to tens (P=1) becomes 980, and rounded to the fourth sigdig
3436becomes 987.6 (A=4). 123.456 rounded to the second place after the
3437decimal point (P=-2) becomes 123.46.
3438
3439All other implemented styles of rounding attempt to round to the
3440"nearest digit." If the digit D immediately to the right of the
3441rounding place (skipping the decimal point) is greater than 5, the
3442number is incremented at the rounding place (possibly causing a
3443cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3444to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3445truncated at the rounding place: e.g. when rounding to units, 0.4
3446rounds to 0, and -19.4 rounds to -19.
3447
3448However the results of other styles of rounding differ if the
3449digit immediately to the right of the rounding place (skipping the
3450decimal point) is 5 and if there are no digits, or no digits other
3451than 0, after that 5. In such cases:
3452
3453=item 'even'
3454
3455rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3456if it is not already. E.g., when rounding to the first sigdig, 0.45
3457becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3458
3459=item 'odd'
3460
3461rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3462it is not already. E.g., when rounding to the first sigdig, 0.45
3463becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3464
3465=item '+inf'
3466
3467round to plus infinity, i.e. always round up. E.g., when
3468rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3469and 0.4501 also becomes 0.5.
3470
3471=item '-inf'
3472
3473round to minus infinity, i.e. always round down. E.g., when
3474rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3475but 0.4501 becomes 0.5.
3476
3477=item 'zero'
3478
3479round to zero, i.e. positive numbers down, negative ones up.
3480E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3481becomes -0.5, but 0.4501 becomes 0.5.
3482
3483=back
3484
3485The handling of A & P in MBI/MBF (the old core code shipped with Perl
3486versions <= 5.7.2) is like this:
3487
3488=over 2
3489
3490=item Precision
3491
3492 * ffround($p) is able to round to $p number of digits after the decimal
3493 point
3494 * otherwise P is unused
3495
3496=item Accuracy (significant digits)
3497
3498 * fround($a) rounds to $a significant digits
3499 * only fdiv() and fsqrt() take A as (optional) paramater
3500 + other operations simply create the same number (fneg etc), or more (fmul)
3501 of digits
3502 + rounding/truncating is only done when explicitly calling one of fround
3503 or ffround, and never for BigInt (not implemented)
3504 * fsqrt() simply hands its accuracy argument over to fdiv.
3505 * the documentation and the comment in the code indicate two different ways
3506 on how fdiv() determines the maximum number of digits it should calculate,
3507 and the actual code does yet another thing
3508 POD:
3509 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3510 Comment:
3511 result has at most max(scale, length(dividend), length(divisor)) digits
3512 Actual code:
3513 scale = max(scale, length(dividend)-1,length(divisor)-1);
3514 scale += length(divisior) - length(dividend);
3515 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3516 Actually, the 'difference' added to the scale is calculated from the
3517 number of "significant digits" in dividend and divisor, which is derived
3518 by looking at the length of the mantissa. Which is wrong, since it includes
3519 the + sign (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops
3520 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3521 assumption that 124 has 3 significant digits, while 120/7 will get you
3522 '17', not '17.1' since 120 is thought to have 2 significant digits.
3523 The rounding after the division then uses the remainder and $y to determine
3524 wether it must round up or down.
3525 ? I have no idea which is the right way. That's why I used a slightly more
3526 ? simple scheme and tweaked the few failing testcases to match it.
3527
3528=back
3529
3530This is how it works now:
3531
3532=over 2
3533
3534=item Setting/Accessing
3535
3536 * You can set the A global via C<< Math::BigInt->accuracy() >> or
3537 C<< Math::BigFloat->accuracy() >> or whatever class you are using.
3538 * You can also set P globally by using C<< Math::SomeClass->precision() >>
3539 likewise.
3540 * Globals are classwide, and not inherited by subclasses.
3541 * to undefine A, use C<< Math::SomeCLass->accuracy(undef); >>
3542 * to undefine P, use C<< Math::SomeClass->precision(undef); >>
3543 * Setting C<< Math::SomeClass->accuracy() >> clears automatically
3544 C<< Math::SomeClass->precision() >>, and vice versa.
3545 * To be valid, A must be > 0, P can have any value.
3546 * If P is negative, this means round to the P'th place to the right of the
3547 decimal point; positive values mean to the left of the decimal point.
3548 P of 0 means round to integer.
3549 * to find out the current global A, use C<< Math::SomeClass->accuracy() >>
3550 * to find out the current global P, use C<< Math::SomeClass->precision() >>
3551 * use C<< $x->accuracy() >> respective C<< $x->precision() >> for the local
3552 setting of C<< $x >>.
3553 * Please note that C<< $x->accuracy() >> respecive C<< $x->precision() >>
3554 return eventually defined global A or P, when C<< $x >>'s A or P is not
3555 set.
3556
3557=item Creating numbers
3558
3559 * When you create a number, you can give it's desired A or P via:
3560 $x = Math::BigInt->new($number,$A,$P);
3561 * Only one of A or P can be defined, otherwise the result is NaN
3562 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
3563 globals (if set) will be used. Thus changing the global defaults later on
3564 will not change the A or P of previously created numbers (i.e., A and P of
3565 $x will be what was in effect when $x was created)
3566 * If given undef for A and P, B<no> rounding will occur, and the globals will
3567 B<not> be used. This is used by subclasses to create numbers without
3568 suffering rounding in the parent. Thus a subclass is able to have it's own
3569 globals enforced upon creation of a number by using
3570 C<< $x = Math::BigInt->new($number,undef,undef) >>:
3571
3572 use Math::BigInt::SomeSubclass;
3573 use Math::BigInt;
3574
3575 Math::BigInt->accuracy(2);
3576 Math::BigInt::SomeSubClass->accuracy(3);
3577 $x = Math::BigInt::SomeSubClass->new(1234);
3578
3579 $x is now 1230, and not 1200. A subclass might choose to implement
3580 this otherwise, e.g. falling back to the parent's A and P.
3581
3582=item Usage
3583
3584 * If A or P are enabled/defined, they are used to round the result of each
3585 operation according to the rules below
3586 * Negative P is ignored in Math::BigInt, since BigInts never have digits
3587 after the decimal point
3588 * Math::BigFloat uses Math::BigInt internally, but setting A or P inside
3589 Math::BigInt as globals does not tamper with the parts of a BigFloat.
3590 A flag is used to mark all Math::BigFloat numbers as 'never round'.
3591
3592=item Precedence
3593
3594 * It only makes sense that a number has only one of A or P at a time.
3595 If you set either A or P on one object, or globally, the other one will
3596 be automatically cleared.
3597 * If two objects are involved in an operation, and one of them has A in
3598 effect, and the other P, this results in an error (NaN).
3599 * A takes precendence over P (Hint: A comes before P).
3600 If neither of them is defined, nothing is used, i.e. the result will have
3601 as many digits as it can (with an exception for fdiv/fsqrt) and will not
3602 be rounded.
3603 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
3604 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
3605 If either the dividend's or the divisor's mantissa has more digits than
3606 the value of F, the higher value will be used instead of F.
3607 This is to limit the digits (A) of the result (just consider what would
3608 happen with unlimited A and P in the case of 1/3 :-)
3609 * fdiv will calculate (at least) 4 more digits than required (determined by
3610 A, P or F), and, if F is not used, round the result
3611 (this will still fail in the case of a result like 0.12345000000001 with A
3612 or P of 5, but this can not be helped - or can it?)
3613 * Thus you can have the math done by on Math::Big* class in two modi:
3614 + never round (this is the default):
3615 This is done by setting A and P to undef. No math operation
3616 will round the result, with fdiv() and fsqrt() as exceptions to guard
3617 against overflows. You must explicitely call bround(), bfround() or
3618 round() (the latter with parameters).
3619 Note: Once you have rounded a number, the settings will 'stick' on it
3620 and 'infect' all other numbers engaged in math operations with it, since
3621 local settings have the highest precedence. So, to get SaferRound[tm],
3622 use a copy() before rounding like this:
3623
3624 $x = Math::BigFloat->new(12.34);
3625 $y = Math::BigFloat->new(98.76);
3626 $z = $x * $y; # 1218.6984
3627 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
3628 $z = $x * $y; # still 1218.6984, without
3629 # copy would have been 1210!
3630
3631 + round after each op:
3632 After each single operation (except for testing like is_zero()), the
3633 method round() is called and the result is rounded appropriately. By
3634 setting proper values for A and P, you can have all-the-same-A or
3635 all-the-same-P modes. For example, Math::Currency might set A to undef,
3636 and P to -2, globally.
3637
3638 ?Maybe an extra option that forbids local A & P settings would be in order,
3639 ?so that intermediate rounding does not 'poison' further math?
3640
3641=item Overriding globals
3642
3643 * you will be able to give A, P and R as an argument to all the calculation
3644 routines; the second parameter is A, the third one is P, and the fourth is
3645 R (shift right by one for binary operations like badd). P is used only if
3646 the first parameter (A) is undefined. These three parameters override the
3647 globals in the order detailed as follows, i.e. the first defined value
3648 wins:
3649 (local: per object, global: global default, parameter: argument to sub)
3650 + parameter A
3651 + parameter P
3652 + local A (if defined on both of the operands: smaller one is taken)
3653 + local P (if defined on both of the operands: bigger one is taken)
3654 + global A
3655 + global P
3656 + global F
3657 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
3658 arguments (A and P) instead of one
3659
3660=item Local settings
3661
3662 * You can set A or P locally by using C<< $x->accuracy() >> or
3663 C<< $x->precision() >>
3664 and thus force different A and P for different objects/numbers.
3665 * Setting A or P this way immediately rounds $x to the new value.
3666 * C<< $x->accuracy() >> clears C<< $x->precision() >>, and vice versa.
3667
3668=item Rounding
3669
3670 * the rounding routines will use the respective global or local settings.
3671 fround()/bround() is for accuracy rounding, while ffround()/bfround()
3672 is for precision
3673 * the two rounding functions take as the second parameter one of the
3674 following rounding modes (R):
3675 'even', 'odd', '+inf', '-inf', 'zero', 'trunc'
3676 * you can set/get the global R by using C<< Math::SomeClass->round_mode() >>
3677 or by setting C<< $Math::SomeClass::round_mode >>
3678 * after each operation, C<< $result->round() >> is called, and the result may
3679 eventually be rounded (that is, if A or P were set either locally,
3680 globally or as parameter to the operation)
3681 * to manually round a number, call C<< $x->round($A,$P,$round_mode); >>
3682 this will round the number by using the appropriate rounding function
3683 and then normalize it.
3684 * rounding modifies the local settings of the number:
3685
3686 $x = Math::BigFloat->new(123.456);
3687 $x->accuracy(5);
3688 $x->bround(4);
3689
3690 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
3691 will be 4 from now on.
3692
3693=item Default values
3694
3695 * R: 'even'
3696 * F: 40
3697 * A: undef
3698 * P: undef
3699
3700=item Remarks
3701
3702 * The defaults are set up so that the new code gives the same results as
3703 the old code (except in a few cases on fdiv):
3704 + Both A and P are undefined and thus will not be used for rounding
3705 after each operation.
3706 + round() is thus a no-op, unless given extra parameters A and P
3707
3708=back
3709
3710=head1 Infinity and Not a Number
3711
3712While BigInt has extensive handling of inf and NaN, certain quirks remain.
3713
3714=over 2
3715
3716=item oct()/hex()
3717
3718These perl routines currently (as of Perl v.5.8.6) cannot handle passed
3719inf.
3720
3721 te@linux:~> perl -wle 'print 2 ** 3333'
3722 inf
3723 te@linux:~> perl -wle 'print 2 ** 3333 == 2 ** 3333'
3724 1
3725 te@linux:~> perl -wle 'print oct(2 ** 3333)'
3726 0
3727 te@linux:~> perl -wle 'print hex(2 ** 3333)'
3728 Illegal hexadecimal digit 'i' ignored at -e line 1.
3729 0
3730
3731The same problems occur if you pass them Math::BigInt->binf() objects. Since
3732overloading these routines is not possible, this cannot be fixed from BigInt.
3733
3734=item ==, !=, <, >, <=, >= with NaNs
3735
3736BigInt's bcmp() routine currently returns undef to signal that a NaN was
3737involved in a comparisation. However, the overload code turns that into
3738either 1 or '' and thus operations like C<< NaN != NaN >> might return
3739wrong values.
3740
3741=item log(-inf)
3742
3743C<< log(-inf) >> is highly weird. Since log(-x)=pi*i+log(x), then
3744log(-inf)=pi*i+inf. However, since the imaginary part is finite, the real
3745infinity "overshadows" it, so the number might as well just be infinity.
3746However, the result is a complex number, and since BigInt/BigFloat can only
3747have real numbers as results, the result is NaN.
3748
3749=item exp(), cos(), sin(), atan2()
3750
3751These all might have problems handling infinity right.
3752
3753=back
3754
3755=head1 INTERNALS
3756
3757The actual numbers are stored as unsigned big integers (with seperate sign).
3758
3759You should neither care about nor depend on the internal representation; it
3760might change without notice. Use B<ONLY> method calls like C<< $x->sign(); >>
3761instead relying on the internal representation.
3762
3763=head2 MATH LIBRARY
3764
3765Math with the numbers is done (by default) by a module called
3766C<Math::BigInt::Calc>. This is equivalent to saying:
3767
3768 use Math::BigInt lib => 'Calc';
3769
3770You can change this by using:
3771
3772 use Math::BigInt lib => 'BitVect';
3773
3774The following would first try to find Math::BigInt::Foo, then
3775Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
3776
3777 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
3778
3779Since Math::BigInt::GMP is in almost all cases faster than Calc (especially in
3780math involving really big numbers, where it is B<much> faster), and there is
3781no penalty if Math::BigInt::GMP is not installed, it is a good idea to always
3782use the following:
3783
3784 use Math::BigInt lib => 'GMP';
3785
3786Different low-level libraries use different formats to store the
3787numbers. You should B<NOT> depend on the number having a specific format
3788internally.
3789
3790See the respective math library module documentation for further details.
3791
3792=head2 SIGN
3793
3794The sign is either '+', '-', 'NaN', '+inf' or '-inf'.
3795
3796A sign of 'NaN' is used to represent the result when input arguments are not
3797numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
3798minus infinity. You will get '+inf' when dividing a positive number by 0, and
3799'-inf' when dividing any negative number by 0.
3800
3801=head2 mantissa(), exponent() and parts()
3802
3803C<mantissa()> and C<exponent()> return the said parts of the BigInt such
3804that:
3805
3806 $m = $x->mantissa();
3807 $e = $x->exponent();
3808 $y = $m * ( 10 ** $e );
3809 print "ok\n" if $x == $y;
3810
3811C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
3812in one go. Both the returned mantissa and exponent have a sign.
3813
3814Currently, for BigInts C<$e> is always 0, except for NaN, +inf and -inf,
3815where it is C<NaN>; and for C<$x == 0>, where it is C<1> (to be compatible
3816with Math::BigFloat's internal representation of a zero as C<0E1>).
3817
3818C<$m> is currently just a copy of the original number. The relation between
3819C<$e> and C<$m> will stay always the same, though their real values might
3820change.
3821
3822=head1 EXAMPLES
3823
3824 use Math::BigInt;
3825
3826 sub bint { Math::BigInt->new(shift); }
3827
3828 $x = Math::BigInt->bstr("1234") # string "1234"
3829 $x = "$x"; # same as bstr()
3830 $x = Math::BigInt->bneg("1234"); # BigInt "-1234"
3831 $x = Math::BigInt->babs("-12345"); # BigInt "12345"
3832 $x = Math::BigInt->bnorm("-0 00"); # BigInt "0"
3833 $x = bint(1) + bint(2); # BigInt "3"
3834 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
3835 $x = bint(1); # BigInt "1"
3836 $x = $x + 5 / 2; # BigInt "3"
3837 $x = $x ** 3; # BigInt "27"
3838 $x *= 2; # BigInt "54"
3839 $x = Math::BigInt->new(0); # BigInt "0"
3840 $x--; # BigInt "-1"
3841 $x = Math::BigInt->badd(4,5) # BigInt "9"
3842 print $x->bsstr(); # 9e+0
3843
3844Examples for rounding:
3845
3846 use Math::BigFloat;
3847 use Test;
3848
3849 $x = Math::BigFloat->new(123.4567);
3850 $y = Math::BigFloat->new(123.456789);
3851 Math::BigFloat->accuracy(4); # no more A than 4
3852
3853 ok ($x->copy()->fround(),123.4); # even rounding
3854 print $x->copy()->fround(),"\n"; # 123.4
3855 Math::BigFloat->round_mode('odd'); # round to odd
3856 print $x->copy()->fround(),"\n"; # 123.5
3857 Math::BigFloat->accuracy(5); # no more A than 5
3858 Math::BigFloat->round_mode('odd'); # round to odd
3859 print $x->copy()->fround(),"\n"; # 123.46
3860 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
3861 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
3862
3863 Math::BigFloat->accuracy(undef); # A not important now
3864 Math::BigFloat->precision(2); # P important
3865 print $x->copy()->bnorm(),"\n"; # 123.46
3866 print $x->copy()->fround(),"\n"; # 123.46
3867
3868Examples for converting:
3869
3870 my $x = Math::BigInt->new('0b1'.'01' x 123);
3871 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
3872
3873=head1 Autocreating constants
3874
3875After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
3876and binary constants in the given scope are converted to C<Math::BigInt>.
3877This conversion happens at compile time.
3878
3879In particular,
3880
3881 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
3882
3883prints the integer value of C<2**100>. Note that without conversion of
3884constants the expression 2**100 will be calculated as perl scalar.
3885
3886Please note that strings and floating point constants are not affected,
3887so that
3888
3889 use Math::BigInt qw/:constant/;
3890
3891 $x = 1234567890123456789012345678901234567890
3892 + 123456789123456789;
3893 $y = '1234567890123456789012345678901234567890'
3894 + '123456789123456789';
3895
3896do not work. You need an explicit Math::BigInt->new() around one of the
3897operands. You should also quote large constants to protect loss of precision:
3898
3899 use Math::BigInt;
3900
3901 $x = Math::BigInt->new('1234567889123456789123456789123456789');
3902
3903Without the quotes Perl would convert the large number to a floating point
3904constant at compile time and then hand the result to BigInt, which results in
3905an truncated result or a NaN.
3906
3907This also applies to integers that look like floating point constants:
3908
3909 use Math::BigInt ':constant';
3910
3911 print ref(123e2),"\n";
3912 print ref(123.2e2),"\n";
3913
3914will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
3915to get this to work.
3916
3917=head1 PERFORMANCE
3918
3919Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
3920must be made in the second case. For long numbers, the copy can eat up to 20%
3921of the work (in the case of addition/subtraction, less for
3922multiplication/division). If $y is very small compared to $x, the form
3923$x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
3924more time then the actual addition.
3925
3926With a technique called copy-on-write, the cost of copying with overload could
3927be minimized or even completely avoided. A test implementation of COW did show
3928performance gains for overloaded math, but introduced a performance loss due
3929to a constant overhead for all other operatons. So Math::BigInt does currently
3930not COW.
3931
3932The rewritten version of this module (vs. v0.01) is slower on certain
3933operations, like C<new()>, C<bstr()> and C<numify()>. The reason are that it
3934does now more work and handles much more cases. The time spent in these
3935operations is usually gained in the other math operations so that code on
3936the average should get (much) faster. If they don't, please contact the author.
3937
3938Some operations may be slower for small numbers, but are significantly faster
3939for big numbers. Other operations are now constant (O(1), like C<bneg()>,
3940C<babs()> etc), instead of O(N) and thus nearly always take much less time.
3941These optimizations were done on purpose.
3942
3943If you find the Calc module to slow, try to install any of the replacement
3944modules and see if they help you.
3945
3946=head2 Alternative math libraries
3947
3948You can use an alternative library to drive Math::BigInt via:
3949
3950 use Math::BigInt lib => 'Module';
3951
3952See L<MATH LIBRARY> for more information.
3953
3954For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
3955
3956=head2 SUBCLASSING
3957
3958=head1 Subclassing Math::BigInt
3959
3960The basic design of Math::BigInt allows simple subclasses with very little
3961work, as long as a few simple rules are followed:
3962
3963=over 2
3964
3965=item *
3966
3967The public API must remain consistent, i.e. if a sub-class is overloading
3968addition, the sub-class must use the same name, in this case badd(). The
3969reason for this is that Math::BigInt is optimized to call the object methods
3970directly.
3971
3972=item *
3973
3974The private object hash keys like C<$x->{sign}> may not be changed, but
3975additional keys can be added, like C<$x->{_custom}>.
3976
3977=item *
3978
3979Accessor functions are available for all existing object hash keys and should
3980be used instead of directly accessing the internal hash keys. The reason for
3981this is that Math::BigInt itself has a pluggable interface which permits it
3982to support different storage methods.
3983
3984=back
3985
3986More complex sub-classes may have to replicate more of the logic internal of
3987Math::BigInt if they need to change more basic behaviors. A subclass that
3988needs to merely change the output only needs to overload C<bstr()>.
3989
3990All other object methods and overloaded functions can be directly inherited
3991from the parent class.
3992
3993At the very minimum, any subclass will need to provide it's own C<new()> and can
3994store additional hash keys in the object. There are also some package globals
3995that must be defined, e.g.:
3996
3997 # Globals
3998 $accuracy = undef;
3999 $precision = -2; # round to 2 decimal places
4000 $round_mode = 'even';
4001 $div_scale = 40;
4002
4003Additionally, you might want to provide the following two globals to allow
4004auto-upgrading and auto-downgrading to work correctly:
4005
4006 $upgrade = undef;
4007 $downgrade = undef;
4008
4009This allows Math::BigInt to correctly retrieve package globals from the
4010subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
4011t/Math/BigFloat/SubClass.pm completely functional subclass examples.
4012
4013Don't forget to
4014
4015 use overload;
4016
4017in your subclass to automatically inherit the overloading from the parent. If
4018you like, you can change part of the overloading, look at Math::String for an
4019example.
4020
4021=head1 UPGRADING
4022
4023When used like this:
4024
4025 use Math::BigInt upgrade => 'Foo::Bar';
4026
4027certain operations will 'upgrade' their calculation and thus the result to
4028the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
4029
4030 use Math::BigInt upgrade => 'Math::BigFloat';
4031
4032As a shortcut, you can use the module C<bignum>:
4033
4034 use bignum;
4035
4036Also good for oneliners:
4037
4038 perl -Mbignum -le 'print 2 ** 255'
4039
4040This makes it possible to mix arguments of different classes (as in 2.5 + 2)
4041as well es preserve accuracy (as in sqrt(3)).
4042
4043Beware: This feature is not fully implemented yet.
4044
4045=head2 Auto-upgrade
4046
4047The following methods upgrade themselves unconditionally; that is if upgrade
4048is in effect, they will always hand up their work:
4049
4050=over 2
4051
4052=item bsqrt()
4053
4054=item div()
4055
4056=item blog()
4057
4058=back
4059
4060Beware: This list is not complete.
4061
4062All other methods upgrade themselves only when one (or all) of their
4063arguments are of the class mentioned in $upgrade (This might change in later
4064versions to a more sophisticated scheme):
4065
4066=head1 BUGS
4067
4068=over 2
4069
4070=item broot() does not work
4071
4072The broot() function in BigInt may only work for small values. This will be
4073fixed in a later version.
4074
4075=item Out of Memory!
4076
4077Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
4078C<eval()> in your code will crash with "Out of memory". This is probably an
4079overload/exporter bug. You can workaround by not having C<eval()>
4080and ':constant' at the same time or upgrade your Perl to a newer version.
4081
4082=item Fails to load Calc on Perl prior 5.6.0
4083
4084Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
4085will fall back to eval { require ... } when loading the math lib on Perls
4086prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
4087filesystems using a different seperator.
4088
4089=back
4090
4091=head1 CAVEATS
4092
4093Some things might not work as you expect them. Below is documented what is
4094known to be troublesome:
4095
4096=over 1
4097
4098=item bstr(), bsstr() and 'cmp'
4099
4100Both C<bstr()> and C<bsstr()> as well as automated stringify via overload now
4101drop the leading '+'. The old code would return '+3', the new returns '3'.
4102This is to be consistent with Perl and to make C<cmp> (especially with
4103overloading) to work as you expect. It also solves problems with C<Test.pm>,
4104because it's C<ok()> uses 'eq' internally.
4105
4106Mark Biggar said, when asked about to drop the '+' altogether, or make only
4107C<cmp> work:
4108
4109 I agree (with the first alternative), don't add the '+' on positive
4110 numbers. It's not as important anymore with the new internal
4111 form for numbers. It made doing things like abs and neg easier,
4112 but those have to be done differently now anyway.
4113
4114So, the following examples will now work all as expected:
4115
4116 use Test;
4117 BEGIN { plan tests => 1 }
4118 use Math::BigInt;
4119
4120 my $x = new Math::BigInt 3*3;
4121 my $y = new Math::BigInt 3*3;
4122
4123 ok ($x,3*3);
4124 print "$x eq 9" if $x eq $y;
4125 print "$x eq 9" if $x eq '9';
4126 print "$x eq 9" if $x eq 3*3;
4127
4128Additionally, the following still works:
4129
4130 print "$x == 9" if $x == $y;
4131 print "$x == 9" if $x == 9;
4132 print "$x == 9" if $x == 3*3;
4133
4134There is now a C<bsstr()> method to get the string in scientific notation aka
4135C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
4136for comparisation, but Perl will represent some numbers as 100 and others
4137as 1e+308. If in doubt, convert both arguments to Math::BigInt before
4138comparing them as strings:
4139
4140 use Test;
4141 BEGIN { plan tests => 3 }
4142 use Math::BigInt;
4143
4144 $x = Math::BigInt->new('1e56'); $y = 1e56;
4145 ok ($x,$y); # will fail
4146 ok ($x->bsstr(),$y); # okay
4147 $y = Math::BigInt->new($y);
4148 ok ($x,$y); # okay
4149
4150Alternatively, simple use C<< <=> >> for comparisations, this will get it
4151always right. There is not yet a way to get a number automatically represented
4152as a string that matches exactly the way Perl represents it.
4153
4154See also the section about L<Infinity and Not a Number> for problems in
4155comparing NaNs.
4156
4157=item int()
4158
4159C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
4160Perl scalar:
4161
4162 $x = Math::BigInt->new(123);
4163 $y = int($x); # BigInt 123
4164 $x = Math::BigFloat->new(123.45);
4165 $y = int($x); # BigInt 123
4166
4167In all Perl versions you can use C<as_number()> or C<as_int> for the same
4168effect:
4169
4170 $x = Math::BigFloat->new(123.45);
4171 $y = $x->as_number(); # BigInt 123
4172 $y = $x->as_int(); # ditto
4173
4174This also works for other subclasses, like Math::String.
4175
4176It is yet unlcear whether overloaded int() should return a scalar or a BigInt.
4177
4178If you want a real Perl scalar, use C<numify()>:
4179
4180 $y = $x->numify(); # 123 as scalar
4181
4182This is seldom necessary, though, because this is done automatically, like
4183when you access an array:
4184
4185 $z = $array[$x]; # does work automatically
4186
4187=item length
4188
4189The following will probably not do what you expect:
4190
4191 $c = Math::BigInt->new(123);
4192 print $c->length(),"\n"; # prints 30
4193
4194It prints both the number of digits in the number and in the fraction part
4195since print calls C<length()> in list context. Use something like:
4196
4197 print scalar $c->length(),"\n"; # prints 3
4198
4199=item bdiv
4200
4201The following will probably not do what you expect:
4202
4203 print $c->bdiv(10000),"\n";
4204
4205It prints both quotient and remainder since print calls C<bdiv()> in list
4206context. Also, C<bdiv()> will modify $c, so be carefull. You probably want
4207to use
4208
4209 print $c / 10000,"\n";
4210 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
4211
4212instead.
4213
4214The quotient is always the greatest integer less than or equal to the
4215real-valued quotient of the two operands, and the remainder (when it is
4216nonzero) always has the same sign as the second operand; so, for
4217example,
4218
4219 1 / 4 => ( 0, 1)
4220 1 / -4 => (-1,-3)
4221 -3 / 4 => (-1, 1)
4222 -3 / -4 => ( 0,-3)
4223 -11 / 2 => (-5,1)
4224 11 /-2 => (-5,-1)
4225
4226As a consequence, the behavior of the operator % agrees with the
4227behavior of Perl's built-in % operator (as documented in the perlop
4228manpage), and the equation
4229
4230 $x == ($x / $y) * $y + ($x % $y)
4231
4232holds true for any $x and $y, which justifies calling the two return
4233values of bdiv() the quotient and remainder. The only exception to this rule
4234are when $y == 0 and $x is negative, then the remainder will also be
4235negative. See below under "infinity handling" for the reasoning behing this.
4236
4237Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
4238not change BigInt's way to do things. This is because under 'use integer' Perl
4239will do what the underlying C thinks is right and this is different for each
4240system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
4241the author to implement it ;)
4242
4243=item infinity handling
4244
4245Here are some examples that explain the reasons why certain results occur while
4246handling infinity:
4247
4248The following table shows the result of the division and the remainder, so that
4249the equation above holds true. Some "ordinary" cases are strewn in to show more
4250clearly the reasoning:
4251
4252 A / B = C, R so that C * B + R = A
4253 =========================================================
4254 5 / 8 = 0, 5 0 * 8 + 5 = 5
4255 0 / 8 = 0, 0 0 * 8 + 0 = 0
4256 0 / inf = 0, 0 0 * inf + 0 = 0
4257 0 /-inf = 0, 0 0 * -inf + 0 = 0
4258 5 / inf = 0, 5 0 * inf + 5 = 5
4259 5 /-inf = 0, 5 0 * -inf + 5 = 5
4260 -5/ inf = 0, -5 0 * inf + -5 = -5
4261 -5/-inf = 0, -5 0 * -inf + -5 = -5
4262 inf/ 5 = inf, 0 inf * 5 + 0 = inf
4263 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
4264 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
4265 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
4266 5/ 5 = 1, 0 1 * 5 + 0 = 5
4267 -5/ -5 = 1, 0 1 * -5 + 0 = -5
4268 inf/ inf = 1, 0 1 * inf + 0 = inf
4269 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
4270 inf/-inf = -1, 0 -1 * -inf + 0 = inf
4271 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
4272 8/ 0 = inf, 8 inf * 0 + 8 = 8
4273 inf/ 0 = inf, inf inf * 0 + inf = inf
4274 0/ 0 = NaN
4275
4276These cases below violate the "remainder has the sign of the second of the two
4277arguments", since they wouldn't match up otherwise.
4278
4279 A / B = C, R so that C * B + R = A
4280 ========================================================
4281 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
4282 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
4283
4284=item Modifying and =
4285
4286Beware of:
4287
4288 $x = Math::BigFloat->new(5);
4289 $y = $x;
4290
4291It will not do what you think, e.g. making a copy of $x. Instead it just makes
4292a second reference to the B<same> object and stores it in $y. Thus anything
4293that modifies $x (except overloaded operators) will modify $y, and vice versa.
4294Or in other words, C<=> is only safe if you modify your BigInts only via
4295overloaded math. As soon as you use a method call it breaks:
4296
4297 $x->bmul(2);
4298 print "$x, $y\n"; # prints '10, 10'
4299
4300If you want a true copy of $x, use:
4301
4302 $y = $x->copy();
4303
4304You can also chain the calls like this, this will make first a copy and then
4305multiply it by 2:
4306
4307 $y = $x->copy()->bmul(2);
4308
4309See also the documentation for overload.pm regarding C<=>.
4310
4311=item bpow
4312
4313C<bpow()> (and the rounding functions) now modifies the first argument and
4314returns it, unlike the old code which left it alone and only returned the
4315result. This is to be consistent with C<badd()> etc. The first three will
4316modify $x, the last one won't:
4317
4318 print bpow($x,$i),"\n"; # modify $x
4319 print $x->bpow($i),"\n"; # ditto
4320 print $x **= $i,"\n"; # the same
4321 print $x ** $i,"\n"; # leave $x alone
4322
4323The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4324
4325=item Overloading -$x
4326
4327The following:
4328
4329 $x = -$x;
4330
4331is slower than
4332
4333 $x->bneg();
4334
4335since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4336needs to preserve $x since it does not know that it later will get overwritten.
4337This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4338
4339=item Mixing different object types
4340
4341In Perl you will get a floating point value if you do one of the following:
4342
4343 $float = 5.0 + 2;
4344 $float = 2 + 5.0;
4345 $float = 5 / 2;
4346
4347With overloaded math, only the first two variants will result in a BigFloat:
4348
4349 use Math::BigInt;
4350 use Math::BigFloat;
4351
4352 $mbf = Math::BigFloat->new(5);
4353 $mbi2 = Math::BigInteger->new(5);
4354 $mbi = Math::BigInteger->new(2);
4355
4356 # what actually gets called:
4357 $float = $mbf + $mbi; # $mbf->badd()
4358 $float = $mbf / $mbi; # $mbf->bdiv()
4359 $integer = $mbi + $mbf; # $mbi->badd()
4360 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4361 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4362
4363This is because math with overloaded operators follows the first (dominating)
4364operand, and the operation of that is called and returns thus the result. So,
4365Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4366the result should be a Math::BigFloat or the second operant is one.
4367
4368To get a Math::BigFloat you either need to call the operation manually,
4369make sure the operands are already of the proper type or casted to that type
4370via Math::BigFloat->new():
4371
4372 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4373
4374Beware of simple "casting" the entire expression, this would only convert
4375the already computed result:
4376
4377 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4378
4379Beware also of the order of more complicated expressions like:
4380
4381 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4382 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4383
4384If in doubt, break the expression into simpler terms, or cast all operands
4385to the desired resulting type.
4386
4387Scalar values are a bit different, since:
4388
4389 $float = 2 + $mbf;
4390 $float = $mbf + 2;
4391
4392will both result in the proper type due to the way the overloaded math works.
4393
4394This section also applies to other overloaded math packages, like Math::String.
4395
4396One solution to you problem might be autoupgrading|upgrading. See the
4397pragmas L<bignum>, L<bigint> and L<bigrat> for an easy way to do this.
4398
4399=item bsqrt()
4400
4401C<bsqrt()> works only good if the result is a big integer, e.g. the square
4402root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4403mode. The reason is that the result is always truncated to an integer.
4404
4405If you want a better approximation of the square root, then use:
4406
4407 $x = Math::BigFloat->new(12);
4408 Math::BigFloat->precision(0);
4409 Math::BigFloat->round_mode('even');
4410 print $x->copy->bsqrt(),"\n"; # 4
4411
4412 Math::BigFloat->precision(2);
4413 print $x->bsqrt(),"\n"; # 3.46
4414 print $x->bsqrt(3),"\n"; # 3.464
4415
4416=item brsft()
4417
4418For negative numbers in base see also L<brsft|brsft>.
4419
4420=back
4421
4422=head1 LICENSE
4423
4424This program is free software; you may redistribute it and/or modify it under
4425the same terms as Perl itself.
4426
4427=head1 SEE ALSO
4428
4429L<Math::BigFloat>, L<Math::BigRat> and L<Math::Big> as well as
4430L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4431
4432The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest
4433because they solve the autoupgrading/downgrading issue, at least partly.
4434
4435The package at
4436L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4437more documentation including a full version history, testcases, empty
4438subclass files and benchmarks.
4439
4440=head1 AUTHORS
4441
4442Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4443Completely rewritten by Tels http://bloodgate.com in late 2000, 2001 - 2004
4444and still at it in 2005.
4445
4446Many people contributed in one or more ways to the final beast, see the file
4447CREDITS for an (uncomplete) list. If you miss your name, please drop me a
4448mail. Thank you!
4449
4450=cut
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