4 # "Mike had an infinite amount to do and a negative amount of time in which
5 # to do it." - Before and After
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
13 # _f : flags, used by MBF to flag parts of a float as untouchable
15 # Remember not to take shortcuts ala $xs = $x->{value}; $CALC->foo($xs); since
16 # underlying lib might change the reference!
18 my $class = "Math::BigInt";
24 @EXPORT_OK = qw(objectify bgcd blcm);
26 # _trap_inf and _trap_nan are internal and should never be accessed from the
28 use vars qw/$round_mode $accuracy $precision $div_scale $rnd_mode
29 $upgrade $downgrade $_trap_nan $_trap_inf/;
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.
37 # For overloaded ops with only one argument we simple use $_[0]->copy() to
38 # preserve the argument.
40 # Thus inheritance of overload operators becomes possible and transparent for
41 # our subclasses without the need to repeat the entire overload section there.
44 '=' => sub { $_[0]->copy(); },
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]); },
58 '**=' => sub { $_[0]->bpow($_[1]); },
59 '<<=' => sub { $_[0]->blsft($_[1]); },
60 '>>=' => sub { $_[0]->brsft($_[1]); },
62 # not supported by Perl yet
63 '..' => \&_pointpoint,
65 '<=>' => sub { my $rc = $_[2] ?
66 ref($_[0])->bcmp($_[1],$_[0]) :
68 $rc = 1 unless defined $rc;
71 # we need '>=' to get things like "1 >= NaN" right:
72 '>=' => sub { my $rc = $_[2] ?
73 ref($_[0])->bcmp($_[1],$_[0]) :
75 # if there was a NaN involved, return false
76 return '' unless defined $rc;
81 "$_[1]" cmp $_[0]->bstr() :
82 $_[0]->bstr() cmp "$_[1]" },
84 # make cos()/sin()/atan2() "work" with BigInt's or subclasses
85 'cos' => sub { cos($_[0]->numify()) },
86 'sin' => sub { sin($_[0]->numify()) },
87 'atan2' => sub { $_[2] ?
88 atan2($_[1],$_[0]->numify()) :
89 atan2($_[0]->numify(),$_[1]) },
91 # are not yet overloadable
92 #'hex' => sub { print "hex"; $_[0]; },
93 #'oct' => sub { print "oct"; $_[0]; },
95 # log(N) is log(N, e), where e is Euler's number
96 'log' => sub { $_[0]->copy()->blog($_[1], undef); },
97 'exp' => sub { $_[0]->copy()->bexp($_[1]); },
98 'int' => sub { $_[0]->copy(); },
99 'neg' => sub { $_[0]->copy()->bneg(); },
100 'abs' => sub { $_[0]->copy()->babs(); },
101 'sqrt' => sub { $_[0]->copy()->bsqrt(); },
102 '~' => sub { $_[0]->copy()->bnot(); },
104 # for subtract it's a bit tricky to not modify b: b-a => -a+b
105 '-' => sub { my $c = $_[0]->copy; $_[2] ?
106 $c->bneg()->badd( $_[1]) :
108 '+' => sub { $_[0]->copy()->badd($_[1]); },
109 '*' => sub { $_[0]->copy()->bmul($_[1]); },
112 $_[2] ? ref($_[0])->new($_[1])->bdiv($_[0]) : $_[0]->copy->bdiv($_[1]);
115 $_[2] ? ref($_[0])->new($_[1])->bmod($_[0]) : $_[0]->copy->bmod($_[1]);
118 $_[2] ? ref($_[0])->new($_[1])->bpow($_[0]) : $_[0]->copy->bpow($_[1]);
121 $_[2] ? ref($_[0])->new($_[1])->blsft($_[0]) : $_[0]->copy->blsft($_[1]);
124 $_[2] ? ref($_[0])->new($_[1])->brsft($_[0]) : $_[0]->copy->brsft($_[1]);
127 $_[2] ? ref($_[0])->new($_[1])->band($_[0]) : $_[0]->copy->band($_[1]);
130 $_[2] ? ref($_[0])->new($_[1])->bior($_[0]) : $_[0]->copy->bior($_[1]);
133 $_[2] ? ref($_[0])->new($_[1])->bxor($_[0]) : $_[0]->copy->bxor($_[1]);
136 # can modify arg of ++ and --, so avoid a copy() for speed, but don't
137 # use $_[0]->bone(), it would modify $_[0] to be 1!
138 '++' => sub { $_[0]->binc() },
139 '--' => sub { $_[0]->bdec() },
141 # if overloaded, O(1) instead of O(N) and twice as fast for small numbers
143 # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/
144 # v5.6.1 dumps on this: return !$_[0]->is_zero() || undef; :-(
146 $t = 1 if !$_[0]->is_zero();
150 # the original qw() does not work with the TIESCALAR below, why?
151 # Order of arguments unsignificant
152 '""' => sub { $_[0]->bstr(); },
153 '0+' => sub { $_[0]->numify(); }
156 ##############################################################################
157 # global constants, flags and accessory
159 # These vars are public, but their direct usage is not recommended, use the
160 # accessor methods instead
162 $round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'
167 $upgrade = undef; # default is no upgrade
168 $downgrade = undef; # default is no downgrade
170 # These are internally, and not to be used from the outside at all
172 $_trap_nan = 0; # are NaNs ok? set w/ config()
173 $_trap_inf = 0; # are infs ok? set w/ config()
174 my $nan = 'NaN'; # constants for easier life
176 my $CALC = 'Math::BigInt::FastCalc'; # module to do the low level math
177 # default is FastCalc.pm
178 my $IMPORT = 0; # was import() called yet?
179 # used to make require work
180 my %WARN; # warn only once for low-level libs
181 my %CAN; # cache for $CALC->can(...)
182 my %CALLBACKS; # callbacks to notify on lib loads
183 my $EMU_LIB = 'Math/BigInt/CalcEmu.pm'; # emulate low-level math
185 ##############################################################################
186 # the old code had $rnd_mode, so we need to support it, too
189 sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; }
190 sub FETCH { return $round_mode; }
191 sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
195 # tie to enable $rnd_mode to work transparently
196 tie $rnd_mode, 'Math::BigInt';
198 # set up some handy alias names
199 *as_int = \&as_number;
200 *is_pos = \&is_positive;
201 *is_neg = \&is_negative;
204 ##############################################################################
209 # make Class->round_mode() work
211 my $class = ref($self) || $self || __PACKAGE__;
215 if ($m !~ /^(even|odd|\+inf|\-inf|zero|trunc|common)$/)
217 require Carp; Carp::croak ("Unknown round mode '$m'");
219 return ${"${class}::round_mode"} = $m;
221 ${"${class}::round_mode"};
227 # make Class->upgrade() work
229 my $class = ref($self) || $self || __PACKAGE__;
230 # need to set new value?
233 return ${"${class}::upgrade"} = $_[0];
235 ${"${class}::upgrade"};
241 # make Class->downgrade() work
243 my $class = ref($self) || $self || __PACKAGE__;
244 # need to set new value?
247 return ${"${class}::downgrade"} = $_[0];
249 ${"${class}::downgrade"};
255 # make Class->div_scale() work
257 my $class = ref($self) || $self || __PACKAGE__;
262 require Carp; Carp::croak ('div_scale must be greater than zero');
264 ${"${class}::div_scale"} = $_[0];
266 ${"${class}::div_scale"};
271 # $x->accuracy($a); ref($x) $a
272 # $x->accuracy(); ref($x)
273 # Class->accuracy(); class
274 # Class->accuracy($a); class $a
277 my $class = ref($x) || $x || __PACKAGE__;
280 # need to set new value?
284 # convert objects to scalars to avoid deep recursion. If object doesn't
285 # have numify(), then hopefully it will have overloading for int() and
286 # boolean test without wandering into a deep recursion path...
287 $a = $a->numify() if ref($a) && $a->can('numify');
291 # also croak on non-numerical
295 Carp::croak ('Argument to accuracy must be greater than zero');
299 require Carp; Carp::croak ('Argument to accuracy must be an integer');
304 # $object->accuracy() or fallback to global
305 $x->bround($a) if $a; # not for undef, 0
306 $x->{_a} = $a; # set/overwrite, even if not rounded
307 delete $x->{_p}; # clear P
308 $a = ${"${class}::accuracy"} unless defined $a; # proper return value
312 ${"${class}::accuracy"} = $a; # set global A
313 ${"${class}::precision"} = undef; # clear global P
315 return $a; # shortcut
319 # $object->accuracy() or fallback to global
320 $a = $x->{_a} if ref($x);
321 # but don't return global undef, when $x's accuracy is 0!
322 $a = ${"${class}::accuracy"} if !defined $a;
328 # $x->precision($p); ref($x) $p
329 # $x->precision(); ref($x)
330 # Class->precision(); class
331 # Class->precision($p); class $p
334 my $class = ref($x) || $x || __PACKAGE__;
340 # convert objects to scalars to avoid deep recursion. If object doesn't
341 # have numify(), then hopefully it will have overloading for int() and
342 # boolean test without wandering into a deep recursion path...
343 $p = $p->numify() if ref($p) && $p->can('numify');
344 if ((defined $p) && (int($p) != $p))
346 require Carp; Carp::croak ('Argument to precision must be an integer');
350 # $object->precision() or fallback to global
351 $x->bfround($p) if $p; # not for undef, 0
352 $x->{_p} = $p; # set/overwrite, even if not rounded
353 delete $x->{_a}; # clear A
354 $p = ${"${class}::precision"} unless defined $p; # proper return value
358 ${"${class}::precision"} = $p; # set global P
359 ${"${class}::accuracy"} = undef; # clear global A
361 return $p; # shortcut
365 # $object->precision() or fallback to global
366 $p = $x->{_p} if ref($x);
367 # but don't return global undef, when $x's precision is 0!
368 $p = ${"${class}::precision"} if !defined $p;
374 # return (or set) configuration data as hash ref
375 my $class = shift || 'Math::BigInt';
378 if (@_ > 1 || (@_ == 1 && (ref($_[0]) eq 'HASH')))
380 # try to set given options as arguments from hash
383 if (ref($args) ne 'HASH')
387 # these values can be "set"
391 upgrade downgrade precision accuracy round_mode div_scale/
394 $set_args->{$key} = $args->{$key} if exists $args->{$key};
395 delete $args->{$key};
400 Carp::croak ("Illegal key(s) '",
401 join("','",keys %$args),"' passed to $class\->config()");
403 foreach my $key (keys %$set_args)
405 if ($key =~ /^trap_(inf|nan)\z/)
407 ${"${class}::_trap_$1"} = ($set_args->{"trap_$1"} ? 1 : 0);
410 # use a call instead of just setting the $variable to check argument
411 $class->$key($set_args->{$key});
415 # now return actual configuration
419 lib_version => ${"${CALC}::VERSION"},
421 trap_nan => ${"${class}::_trap_nan"},
422 trap_inf => ${"${class}::_trap_inf"},
423 version => ${"${class}::VERSION"},
426 upgrade downgrade precision accuracy round_mode div_scale
429 $cfg->{$key} = ${"${class}::$key"};
431 if (@_ == 1 && (ref($_[0]) ne 'HASH'))
433 # calls of the style config('lib') return just this value
434 return $cfg->{$_[0]};
441 # select accuracy parameter based on precedence,
442 # used by bround() and bfround(), may return undef for scale (means no op)
443 my ($x,$scale,$mode) = @_;
445 $scale = $x->{_a} unless defined $scale;
450 $scale = ${ $class . '::accuracy' } unless defined $scale;
451 $mode = ${ $class . '::round_mode' } unless defined $mode;
458 # select precision parameter based on precedence,
459 # used by bround() and bfround(), may return undef for scale (means no op)
460 my ($x,$scale,$mode) = @_;
462 $scale = $x->{_p} unless defined $scale;
467 $scale = ${ $class . '::precision' } unless defined $scale;
468 $mode = ${ $class . '::round_mode' } unless defined $mode;
473 ##############################################################################
481 # if two arguments, the first one is the class to "swallow" subclasses
489 return unless ref($x); # only for objects
491 my $self = bless {}, $c;
493 $self->{sign} = $x->{sign};
494 $self->{value} = $CALC->_copy($x->{value});
495 $self->{_a} = $x->{_a} if defined $x->{_a};
496 $self->{_p} = $x->{_p} if defined $x->{_p};
502 # create a new BigInt object from a string or another BigInt object.
503 # see hash keys documented at top
505 # the argument could be an object, so avoid ||, && etc on it, this would
506 # cause costly overloaded code to be called. The only allowed ops are
509 my ($class,$wanted,$a,$p,$r) = @_;
511 # avoid numify-calls by not using || on $wanted!
512 return $class->bzero($a,$p) if !defined $wanted; # default to 0
513 return $class->copy($wanted,$a,$p,$r)
514 if ref($wanted) && $wanted->isa($class); # MBI or subclass
516 $class->import() if $IMPORT == 0; # make require work
518 my $self = bless {}, $class;
520 # shortcut for "normal" numbers
521 if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*\z/))
523 $self->{sign} = $1 || '+';
525 if ($wanted =~ /^[+-]/)
527 # remove sign without touching wanted to make it work with constants
528 my $t = $wanted; $t =~ s/^[+-]//;
529 $self->{value} = $CALC->_new($t);
533 $self->{value} = $CALC->_new($wanted);
536 if ( (defined $a) || (defined $p)
537 || (defined ${"${class}::precision"})
538 || (defined ${"${class}::accuracy"})
541 $self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p);
546 # handle '+inf', '-inf' first
547 if ($wanted =~ /^[+-]?inf\z/)
549 $self->{sign} = $wanted; # set a default sign for bstr()
550 return $self->binf($wanted);
552 # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign
553 my ($mis,$miv,$mfv,$es,$ev) = _split($wanted);
558 require Carp; Carp::croak("$wanted is not a number in $class");
560 $self->{value} = $CALC->_zero();
561 $self->{sign} = $nan;
566 # _from_hex or _from_bin
567 $self->{value} = $mis->{value};
568 $self->{sign} = $mis->{sign};
569 return $self; # throw away $mis
571 # make integer from mantissa by adjusting exp, then convert to bigint
572 $self->{sign} = $$mis; # store sign
573 $self->{value} = $CALC->_zero(); # for all the NaN cases
574 my $e = int("$$es$$ev"); # exponent (avoid recursion)
577 my $diff = $e - CORE::length($$mfv);
578 if ($diff < 0) # Not integer
582 require Carp; Carp::croak("$wanted not an integer in $class");
585 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
586 $self->{sign} = $nan;
590 # adjust fraction and add it to value
591 #print "diff > 0 $$miv\n";
592 $$miv = $$miv . ($$mfv . '0' x $diff);
597 if ($$mfv ne '') # e <= 0
599 # fraction and negative/zero E => NOI
602 require Carp; Carp::croak("$wanted not an integer in $class");
604 #print "NOI 2 \$\$mfv '$$mfv'\n";
605 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
606 $self->{sign} = $nan;
610 # xE-y, and empty mfv
613 if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
617 require Carp; Carp::croak("$wanted not an integer in $class");
620 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
621 $self->{sign} = $nan;
625 $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
626 $self->{value} = $CALC->_new($$miv) if $self->{sign} =~ /^[+-]$/;
627 # if any of the globals is set, use them to round and store them inside $self
628 # do not round for new($x,undef,undef) since that is used by MBF to signal
630 $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p;
636 # create a bigint 'NaN', if given a BigInt, set it to 'NaN'
638 $self = $class if !defined $self;
641 my $c = $self; $self = {}; bless $self, $c;
644 if (${"${class}::_trap_nan"})
647 Carp::croak ("Tried to set $self to NaN in $class\::bnan()");
649 $self->import() if $IMPORT == 0; # make require work
650 return if $self->modify('bnan');
651 if ($self->can('_bnan'))
653 # use subclass to initialize
658 # otherwise do our own thing
659 $self->{value} = $CALC->_zero();
661 $self->{sign} = $nan;
662 delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly
668 # create a bigint '+-inf', if given a BigInt, set it to '+-inf'
669 # the sign is either '+', or if given, used from there
671 my $sign = shift; $sign = '+' if !defined $sign || $sign !~ /^-(inf)?$/;
672 $self = $class if !defined $self;
675 my $c = $self; $self = {}; bless $self, $c;
678 if (${"${class}::_trap_inf"})
681 Carp::croak ("Tried to set $self to +-inf in $class\::binf()");
683 $self->import() if $IMPORT == 0; # make require work
684 return if $self->modify('binf');
685 if ($self->can('_binf'))
687 # use subclass to initialize
692 # otherwise do our own thing
693 $self->{value} = $CALC->_zero();
695 $sign = $sign . 'inf' if $sign !~ /inf$/; # - => -inf
696 $self->{sign} = $sign;
697 ($self->{_a},$self->{_p}) = @_; # take over requested rounding
703 # create a bigint '+0', if given a BigInt, set it to 0
705 $self = __PACKAGE__ if !defined $self;
709 my $c = $self; $self = {}; bless $self, $c;
711 $self->import() if $IMPORT == 0; # make require work
712 return if $self->modify('bzero');
714 if ($self->can('_bzero'))
716 # use subclass to initialize
721 # otherwise do our own thing
722 $self->{value} = $CALC->_zero();
729 # call like: $x->bzero($a,$p,$r,$y);
730 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
735 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
737 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
745 # create a bigint '+1' (or -1 if given sign '-'),
746 # if given a BigInt, set it to +1 or -1, respectively
748 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
749 $self = $class if !defined $self;
753 my $c = $self; $self = {}; bless $self, $c;
755 $self->import() if $IMPORT == 0; # make require work
756 return if $self->modify('bone');
758 if ($self->can('_bone'))
760 # use subclass to initialize
765 # otherwise do our own thing
766 $self->{value} = $CALC->_one();
768 $self->{sign} = $sign;
773 # call like: $x->bone($sign,$a,$p,$r,$y);
774 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
778 # call like: $x->bone($sign,$a,$p,$r);
780 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
782 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
788 ##############################################################################
789 # string conversation
793 # (ref to BFLOAT or num_str ) return num_str
794 # Convert number from internal format to scientific string format.
795 # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
796 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
798 if ($x->{sign} !~ /^[+-]$/)
800 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
803 my ($m,$e) = $x->parts();
804 #$m->bstr() . 'e+' . $e->bstr(); # e can only be positive in BigInt
805 # 'e+' because E can only be positive in BigInt
806 $m->bstr() . 'e+' . $CALC->_str($e->{value});
811 # make a string from bigint object
812 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
814 if ($x->{sign} !~ /^[+-]$/)
816 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
819 my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
820 $es.$CALC->_str($x->{value});
825 # Make a "normal" scalar from a BigInt object
826 my $x = shift; $x = $class->new($x) unless ref $x;
828 return $x->bstr() if $x->{sign} !~ /^[+-]$/;
829 my $num = $CALC->_num($x->{value});
830 return -$num if $x->{sign} eq '-';
834 ##############################################################################
835 # public stuff (usually prefixed with "b")
839 # return the sign of the number: +/-/-inf/+inf/NaN
840 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
845 sub _find_round_parameters
847 # After any operation or when calling round(), the result is rounded by
848 # regarding the A & P from arguments, local parameters, or globals.
850 # !!!!!!! If you change this, remember to change round(), too! !!!!!!!!!!
852 # This procedure finds the round parameters, but it is for speed reasons
853 # duplicated in round. Otherwise, it is tested by the testsuite and used
856 # returns ($self) or ($self,$a,$p,$r) - sets $self to NaN of both A and P
857 # were requested/defined (locally or globally or both)
859 my ($self,$a,$p,$r,@args) = @_;
860 # $a accuracy, if given by caller
861 # $p precision, if given by caller
862 # $r round_mode, if given by caller
863 # @args all 'other' arguments (0 for unary, 1 for binary ops)
865 my $c = ref($self); # find out class of argument(s)
868 # convert to normal scalar for speed and correctness in inner parts
869 $a = $a->numify() if defined $a && ref($a);
870 $p = $p->numify() if defined $a && ref($p);
872 # now pick $a or $p, but only if we have got "arguments"
875 foreach ($self,@args)
877 # take the defined one, or if both defined, the one that is smaller
878 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
883 # even if $a is defined, take $p, to signal error for both defined
884 foreach ($self,@args)
886 # take the defined one, or if both defined, the one that is bigger
888 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
891 # if still none defined, use globals (#2)
892 $a = ${"$c\::accuracy"} unless defined $a;
893 $p = ${"$c\::precision"} unless defined $p;
895 # A == 0 is useless, so undef it to signal no rounding
896 $a = undef if defined $a && $a == 0;
899 return ($self) unless defined $a || defined $p; # early out
901 # set A and set P is an fatal error
902 return ($self->bnan()) if defined $a && defined $p; # error
904 $r = ${"$c\::round_mode"} unless defined $r;
905 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc|common)$/)
907 require Carp; Carp::croak ("Unknown round mode '$r'");
915 # Round $self according to given parameters, or given second argument's
916 # parameters or global defaults
918 # for speed reasons, _find_round_parameters is embeded here:
920 my ($self,$a,$p,$r,@args) = @_;
921 # $a accuracy, if given by caller
922 # $p precision, if given by caller
923 # $r round_mode, if given by caller
924 # @args all 'other' arguments (0 for unary, 1 for binary ops)
926 my $c = ref($self); # find out class of argument(s)
929 # now pick $a or $p, but only if we have got "arguments"
932 foreach ($self,@args)
934 # take the defined one, or if both defined, the one that is smaller
935 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
940 # even if $a is defined, take $p, to signal error for both defined
941 foreach ($self,@args)
943 # take the defined one, or if both defined, the one that is bigger
945 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
948 # if still none defined, use globals (#2)
949 $a = ${"$c\::accuracy"} unless defined $a;
950 $p = ${"$c\::precision"} unless defined $p;
952 # A == 0 is useless, so undef it to signal no rounding
953 $a = undef if defined $a && $a == 0;
956 return $self unless defined $a || defined $p; # early out
958 # set A and set P is an fatal error
959 return $self->bnan() if defined $a && defined $p;
961 $r = ${"$c\::round_mode"} unless defined $r;
962 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc|common)$/)
964 require Carp; Carp::croak ("Unknown round mode '$r'");
967 # now round, by calling either fround or ffround:
970 $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a;
972 else # both can't be undefined due to early out
974 $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p;
976 # bround() or bfround() already callled bnorm() if nec.
982 # (numstr or BINT) return BINT
983 # Normalize number -- no-op here
984 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
990 # (BINT or num_str) return BINT
991 # make number absolute, or return absolute BINT from string
992 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
994 return $x if $x->modify('babs');
995 # post-normalized abs for internal use (does nothing for NaN)
996 $x->{sign} =~ s/^-/+/;
1002 # (BINT or num_str) return BINT
1003 # negate number or make a negated number from string
1004 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1006 return $x if $x->modify('bneg');
1008 # for +0 dont negate (to have always normalized +0). Does nothing for 'NaN'
1009 $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $CALC->_is_zero($x->{value}));
1015 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
1016 # (BINT or num_str, BINT or num_str) return cond_code
1019 my ($self,$x,$y) = (ref($_[0]),@_);
1021 # objectify is costly, so avoid it
1022 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1024 ($self,$x,$y) = objectify(2,@_);
1027 return $upgrade->bcmp($x,$y) if defined $upgrade &&
1028 ((!$x->isa($self)) || (!$y->isa($self)));
1030 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1032 # handle +-inf and NaN
1033 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1034 return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
1035 return +1 if $x->{sign} eq '+inf';
1036 return -1 if $x->{sign} eq '-inf';
1037 return -1 if $y->{sign} eq '+inf';
1040 # check sign for speed first
1041 return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
1042 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
1044 # have same sign, so compare absolute values. Don't make tests for zero here
1045 # because it's actually slower than testin in Calc (especially w/ Pari et al)
1047 # post-normalized compare for internal use (honors signs)
1048 if ($x->{sign} eq '+')
1050 # $x and $y both > 0
1051 return $CALC->_acmp($x->{value},$y->{value});
1055 $CALC->_acmp($y->{value},$x->{value}); # swaped acmp (lib returns 0,1,-1)
1060 # Compares 2 values, ignoring their signs.
1061 # Returns one of undef, <0, =0, >0. (suitable for sort)
1062 # (BINT, BINT) return cond_code
1065 my ($self,$x,$y) = (ref($_[0]),@_);
1066 # objectify is costly, so avoid it
1067 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1069 ($self,$x,$y) = objectify(2,@_);
1072 return $upgrade->bacmp($x,$y) if defined $upgrade &&
1073 ((!$x->isa($self)) || (!$y->isa($self)));
1075 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1077 # handle +-inf and NaN
1078 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1079 return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
1080 return 1 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} !~ /^[+-]inf$/;
1083 $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
1088 # add second arg (BINT or string) to first (BINT) (modifies first)
1089 # return result as BINT
1092 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1093 # objectify is costly, so avoid it
1094 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1096 ($self,$x,$y,@r) = objectify(2,@_);
1099 return $x if $x->modify('badd');
1100 return $upgrade->badd($upgrade->new($x),$upgrade->new($y),@r) if defined $upgrade &&
1101 ((!$x->isa($self)) || (!$y->isa($self)));
1103 $r[3] = $y; # no push!
1104 # inf and NaN handling
1105 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1108 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1110 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1112 # +inf++inf or -inf+-inf => same, rest is NaN
1113 return $x if $x->{sign} eq $y->{sign};
1116 # +-inf + something => +inf
1117 # something +-inf => +-inf
1118 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
1122 my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
1126 $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
1130 my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
1133 $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
1138 # speedup, if equal, set result to 0
1139 $x->{value} = $CALC->_zero();
1144 $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
1152 # (BINT or num_str, BINT or num_str) return BINT
1153 # subtract second arg from first, modify first
1156 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1158 # objectify is costly, so avoid it
1159 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1161 ($self,$x,$y,@r) = objectify(2,@_);
1164 return $x if $x->modify('bsub');
1166 return $upgrade->new($x)->bsub($upgrade->new($y),@r) if defined $upgrade &&
1167 ((!$x->isa($self)) || (!$y->isa($self)));
1169 return $x->round(@r) if $y->is_zero();
1171 # To correctly handle the lone special case $x->bsub($x), we note the sign
1172 # of $x, then flip the sign from $y, and if the sign of $x did change, too,
1173 # then we caught the special case:
1174 my $xsign = $x->{sign};
1175 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
1176 if ($xsign ne $x->{sign})
1178 # special case of $x->bsub($x) results in 0
1179 return $x->bzero(@r) if $xsign =~ /^[+-]$/;
1180 return $x->bnan(); # NaN, -inf, +inf
1182 $x->badd($y,@r); # badd does not leave internal zeros
1183 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
1184 $x; # already rounded by badd() or no round nec.
1189 # increment arg by one
1190 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1191 return $x if $x->modify('binc');
1193 if ($x->{sign} eq '+')
1195 $x->{value} = $CALC->_inc($x->{value});
1196 return $x->round($a,$p,$r);
1198 elsif ($x->{sign} eq '-')
1200 $x->{value} = $CALC->_dec($x->{value});
1201 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1202 return $x->round($a,$p,$r);
1204 # inf, nan handling etc
1205 $x->badd($self->bone(),$a,$p,$r); # badd does round
1210 # decrement arg by one
1211 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1212 return $x if $x->modify('bdec');
1214 if ($x->{sign} eq '-')
1217 $x->{value} = $CALC->_inc($x->{value});
1221 return $x->badd($self->bone('-'),@r) unless $x->{sign} eq '+'; # inf or NaN
1223 if ($CALC->_is_zero($x->{value}))
1226 $x->{value} = $CALC->_one(); $x->{sign} = '-'; # 0 => -1
1231 $x->{value} = $CALC->_dec($x->{value});
1239 # calculate $x = $a ** $base + $b and return $a (e.g. the log() to base
1243 my ($self,$x,$base,@r) = (undef,@_);
1244 # objectify is costly, so avoid it
1245 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1247 ($self,$x,$base,@r) = objectify(1,ref($x),@_);
1250 return $x if $x->modify('blog');
1252 $base = $self->new($base) if defined $base && !ref $base;
1254 # inf, -inf, NaN, <0 => NaN
1256 if $x->{sign} ne '+' || (defined $base && $base->{sign} ne '+');
1258 return $upgrade->blog($upgrade->new($x),$base,@r) if
1261 # fix for bug #24969:
1262 # the default base is e (Euler's number) which is not an integer
1265 require Math::BigFloat;
1266 my $u = Math::BigFloat->blog(Math::BigFloat->new($x))->as_int();
1267 # modify $x in place
1268 $x->{value} = $u->{value};
1269 $x->{sign} = $u->{sign};
1273 my ($rc,$exact) = $CALC->_log_int($x->{value},$base->{value});
1274 return $x->bnan() unless defined $rc; # not possible to take log?
1281 # Calculate n over k (binomial coefficient or "choose" function) as integer.
1283 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1285 # objectify is costly, so avoid it
1286 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1288 ($self,$x,$y,@r) = objectify(2,@_);
1291 return $x if $x->modify('bnok');
1292 return $x->bnan() if $x->{sign} eq 'NaN' || $y->{sign} eq 'NaN';
1293 return $x->binf() if $x->{sign} eq '+inf';
1295 # k > n or k < 0 => 0
1296 my $cmp = $x->bacmp($y);
1297 return $x->bzero() if $cmp < 0 || $y->{sign} =~ /^-/;
1299 return $x->bone(@r) if $cmp == 0;
1301 if ($CALC->can('_nok'))
1303 $x->{value} = $CALC->_nok($x->{value},$y->{value});
1307 # ( 7 ) 7! 7*6*5 * 4*3*2*1 7 * 6 * 5
1308 # ( - ) = --------- = --------------- = ---------
1309 # ( 3 ) 3! (7-3)! 3*2*1 * 4*3*2*1 3 * 2 * 1
1311 # compute n - k + 2 (so we start with 5 in the example above)
1316 my $r = $z->copy(); $z->binc();
1317 my $d = $self->new(2);
1318 while ($z->bacmp($x) <= 0) # f < x ?
1320 $r->bmul($z); $r->bdiv($d);
1321 $z->binc(); $d->binc();
1323 $x->{value} = $r->{value}; $x->{sign} = '+';
1325 else { $x->bone(); }
1332 # Calculate e ** $x (Euler's number to the power of X), truncated to
1334 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1335 return $x if $x->modify('bexp');
1337 # inf, -inf, NaN, <0 => NaN
1338 return $x->bnan() if $x->{sign} eq 'NaN';
1339 return $x->bone() if $x->is_zero();
1340 return $x if $x->{sign} eq '+inf';
1341 return $x->bzero() if $x->{sign} eq '-inf';
1345 # run through Math::BigFloat unless told otherwise
1346 require Math::BigFloat unless defined $upgrade;
1347 local $upgrade = 'Math::BigFloat' unless defined $upgrade;
1348 # calculate result, truncate it to integer
1349 $u = $upgrade->bexp($upgrade->new($x),@r);
1352 if (!defined $upgrade)
1355 # modify $x in place
1356 $x->{value} = $u->{value};
1364 # (BINT or num_str, BINT or num_str) return BINT
1365 # does not modify arguments, but returns new object
1366 # Lowest Common Multiplicator
1368 my $y = shift; my ($x);
1375 $x = $class->new($y);
1380 my $y = shift; $y = $self->new($y) if !ref ($y);
1388 # (BINT or num_str, BINT or num_str) return BINT
1389 # does not modify arguments, but returns new object
1390 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1393 $y = $class->new($y) if !ref($y);
1395 my $x = $y->copy()->babs(); # keep arguments
1396 return $x->bnan() if $x->{sign} !~ /^[+-]$/; # x NaN?
1400 $y = shift; $y = $self->new($y) if !ref($y);
1401 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1402 $x->{value} = $CALC->_gcd($x->{value},$y->{value});
1403 last if $CALC->_is_one($x->{value});
1410 # (num_str or BINT) return BINT
1411 # represent ~x as twos-complement number
1412 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1413 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1415 return $x if $x->modify('bnot');
1416 $x->binc()->bneg(); # binc already does round
1419 ##############################################################################
1420 # is_foo test routines
1421 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1425 # return true if arg (BINT or num_str) is zero (array '+', '0')
1426 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1428 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1429 $CALC->_is_zero($x->{value});
1434 # return true if arg (BINT or num_str) is NaN
1435 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1437 $x->{sign} eq $nan ? 1 : 0;
1442 # return true if arg (BINT or num_str) is +-inf
1443 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1447 $sign = '[+-]inf' if $sign eq ''; # +- doesn't matter, only that's inf
1448 $sign = "[$1]inf" if $sign =~ /^([+-])(inf)?$/; # extract '+' or '-'
1449 return $x->{sign} =~ /^$sign$/ ? 1 : 0;
1451 $x->{sign} =~ /^[+-]inf$/ ? 1 : 0; # only +-inf is infinity
1456 # return true if arg (BINT or num_str) is +1, or -1 if sign is given
1457 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1459 $sign = '+' if !defined $sign || $sign ne '-';
1461 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1462 $CALC->_is_one($x->{value});
1467 # return true when arg (BINT or num_str) is odd, false for even
1468 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1470 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1471 $CALC->_is_odd($x->{value});
1476 # return true when arg (BINT or num_str) is even, false for odd
1477 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1479 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1480 $CALC->_is_even($x->{value});
1485 # return true when arg (BINT or num_str) is positive (>= 0)
1486 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1488 return 1 if $x->{sign} eq '+inf'; # +inf is positive
1490 # 0+ is neither positive nor negative
1491 ($x->{sign} eq '+' && !$x->is_zero()) ? 1 : 0;
1496 # return true when arg (BINT or num_str) is negative (< 0)
1497 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1499 $x->{sign} =~ /^-/ ? 1 : 0; # -inf is negative, but NaN is not
1504 # return true when arg (BINT or num_str) is an integer
1505 # always true for BigInt, but different for BigFloats
1506 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1508 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1511 ###############################################################################
1515 # multiply two numbers -- stolen from Knuth Vol 2 pg 233
1516 # (BINT or num_str, BINT or num_str) return BINT
1519 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1520 # objectify is costly, so avoid it
1521 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1523 ($self,$x,$y,@r) = objectify(2,@_);
1526 return $x if $x->modify('bmul');
1528 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1531 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1533 return $x->bnan() if $x->is_zero() || $y->is_zero();
1534 # result will always be +-inf:
1535 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1536 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1537 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1538 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1539 return $x->binf('-');
1542 return $upgrade->bmul($x,$upgrade->new($y),@r)
1543 if defined $upgrade && !$y->isa($self);
1545 $r[3] = $y; # no push here
1547 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1549 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1550 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1557 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1558 my ($self,$x,$y) = @_;
1560 # NaN if x == NaN or y == NaN or x==y==0
1561 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1562 if (($x->is_nan() || $y->is_nan()) ||
1563 ($x->is_zero() && $y->is_zero()));
1565 # +-inf / +-inf == NaN, reminder also NaN
1566 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1568 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1570 # x / +-inf => 0, remainder x (works even if x == 0)
1571 if ($y->{sign} =~ /^[+-]inf$/)
1573 my $t = $x->copy(); # bzero clobbers up $x
1574 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1577 # 5 / 0 => +inf, -6 / 0 => -inf
1578 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1579 # exception: -8 / 0 has remainder -8, not 8
1580 # exception: -inf / 0 has remainder -inf, not inf
1583 # +-inf / 0 => special case for -inf
1584 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1585 if (!$x->is_zero() && !$x->is_inf())
1587 my $t = $x->copy(); # binf clobbers up $x
1589 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1593 # last case: +-inf / ordinary number
1595 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1597 return wantarray ? ($x,$self->bzero()) : $x;
1602 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1603 # (BINT,BINT) (quo,rem) or BINT (only rem)
1606 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1607 # objectify is costly, so avoid it
1608 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1610 ($self,$x,$y,@r) = objectify(2,@_);
1613 return $x if $x->modify('bdiv');
1615 return $self->_div_inf($x,$y)
1616 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1618 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1619 if defined $upgrade;
1621 $r[3] = $y; # no push!
1623 # calc new sign and in case $y == +/- 1, return $x
1624 my $xsign = $x->{sign}; # keep
1625 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1629 my $rem = $self->bzero();
1630 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1631 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1632 $rem->{_a} = $x->{_a};
1633 $rem->{_p} = $x->{_p};
1635 if (! $CALC->_is_zero($rem->{value}))
1637 $rem->{sign} = $y->{sign};
1638 $rem = $y->copy()->bsub($rem) if $xsign ne $y->{sign}; # one of them '-'
1642 $rem->{sign} = '+'; # dont leave -0
1648 $x->{value} = $CALC->_div($x->{value},$y->{value});
1649 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1654 ###############################################################################
1659 # modulus (or remainder)
1660 # (BINT or num_str, BINT or num_str) return BINT
1663 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1664 # objectify is costly, so avoid it
1665 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1667 ($self,$x,$y,@r) = objectify(2,@_);
1670 return $x if $x->modify('bmod');
1671 $r[3] = $y; # no push!
1672 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1674 my ($d,$r) = $self->_div_inf($x,$y);
1675 $x->{sign} = $r->{sign};
1676 $x->{value} = $r->{value};
1677 return $x->round(@r);
1680 # calc new sign and in case $y == +/- 1, return $x
1681 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1682 if (!$CALC->_is_zero($x->{value}))
1684 $x->{value} = $CALC->_sub($y->{value},$x->{value},1) # $y-$x
1685 if ($x->{sign} ne $y->{sign});
1686 $x->{sign} = $y->{sign};
1690 $x->{sign} = '+'; # dont leave -0
1697 # Modular inverse. given a number which is (hopefully) relatively
1698 # prime to the modulus, calculate its inverse using Euclid's
1699 # alogrithm. If the number is not relatively prime to the modulus
1700 # (i.e. their gcd is not one) then NaN is returned.
1703 my ($self,$x,$y,@r) = (undef,@_);
1704 # objectify is costly, so avoid it
1705 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1707 ($self,$x,$y,@r) = objectify(2,@_);
1710 return $x if $x->modify('bmodinv');
1713 if ($y->{sign} ne '+' # -, NaN, +inf, -inf
1714 || $x->is_zero() # or num == 0
1715 || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1718 # put least residue into $x if $x was negative, and thus make it positive
1719 $x->bmod($y) if $x->{sign} eq '-';
1722 ($x->{value},$sign) = $CALC->_modinv($x->{value},$y->{value});
1723 return $x->bnan() if !defined $x->{value}; # in case no GCD found
1724 return $x if !defined $sign; # already real result
1725 $x->{sign} = $sign; # flip/flop see below
1726 $x->bmod($y); # calc real result
1732 # takes a very large number to a very large exponent in a given very
1733 # large modulus, quickly, thanks to binary exponentation. supports
1734 # negative exponents.
1735 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1737 return $num if $num->modify('bmodpow');
1739 # check modulus for valid values
1740 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1741 || $mod->is_zero());
1743 # check exponent for valid values
1744 if ($exp->{sign} =~ /\w/)
1746 # i.e., if it's NaN, +inf, or -inf...
1747 return $num->bnan();
1750 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
1752 # check num for valid values (also NaN if there was no inverse but $exp < 0)
1753 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1755 # $mod is positive, sign on $exp is ignored, result also positive
1756 $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
1760 ###############################################################################
1764 # (BINT or num_str, BINT or num_str) return BINT
1765 # compute factorial number from $x, modify $x in place
1766 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1768 return $x if $x->modify('bfac') || $x->{sign} eq '+inf'; # inf => inf
1769 return $x->bnan() if $x->{sign} ne '+'; # NaN, <0 etc => NaN
1771 $x->{value} = $CALC->_fac($x->{value});
1777 # (BINT or num_str, BINT or num_str) return BINT
1778 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1779 # modifies first argument
1782 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1783 # objectify is costly, so avoid it
1784 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1786 ($self,$x,$y,@r) = objectify(2,@_);
1789 return $x if $x->modify('bpow');
1791 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1794 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1796 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1802 if ($x->{sign} =~ /^[+-]inf/)
1805 return $x->bnan() if $y->is_zero();
1806 # -inf ** -1 => 1/inf => 0
1807 return $x->bzero() if $y->is_one('-') && $x->is_negative();
1810 return $x if $x->{sign} eq '+inf';
1812 # -inf ** Y => -inf if Y is odd
1813 return $x if $y->is_odd();
1819 return $x if $x->is_one();
1822 return $x if $x->is_zero() && $y->{sign} =~ /^[+]/;
1825 return $x->binf() if $x->is_zero();
1828 return $x->bnan() if $x->is_one('-') && $y->{sign} =~ /^[-]/;
1831 return $x->bzero() if $x->{sign} eq '-' && $y->{sign} =~ /^[-]/;
1834 return $x->bnan() if $x->{sign} eq '-';
1837 return $x->binf() if $y->{sign} =~ /^[+]/;
1842 return $upgrade->bpow($upgrade->new($x),$y,@r)
1843 if defined $upgrade && (!$y->isa($self) || $y->{sign} eq '-');
1845 $r[3] = $y; # no push!
1847 # cases 0 ** Y, X ** 0, X ** 1, 1 ** Y are handled by Calc or Emu
1850 $new_sign = $y->is_odd() ? '-' : '+' if ($x->{sign} ne '+');
1852 # 0 ** -7 => ( 1 / (0 ** 7)) => 1 / 0 => +inf
1854 if $y->{sign} eq '-' && $x->{sign} eq '+' && $CALC->_is_zero($x->{value});
1855 # 1 ** -y => 1 / (1 ** |y|)
1856 # so do test for negative $y after above's clause
1857 return $x->bnan() if $y->{sign} eq '-' && !$CALC->_is_one($x->{value});
1859 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1860 $x->{sign} = $new_sign;
1861 $x->{sign} = '+' if $CALC->_is_zero($y->{value});
1867 # (BINT or num_str, BINT or num_str) return BINT
1868 # compute x << y, base n, y >= 0
1871 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1872 # objectify is costly, so avoid it
1873 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1875 ($self,$x,$y,$n,@r) = objectify(2,@_);
1878 return $x if $x->modify('blsft');
1879 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1880 return $x->round(@r) if $y->is_zero();
1882 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1884 $x->{value} = $CALC->_lsft($x->{value},$y->{value},$n);
1890 # (BINT or num_str, BINT or num_str) return BINT
1891 # compute x >> y, base n, y >= 0
1894 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1895 # objectify is costly, so avoid it
1896 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1898 ($self,$x,$y,$n,@r) = objectify(2,@_);
1901 return $x if $x->modify('brsft');
1902 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1903 return $x->round(@r) if $y->is_zero();
1904 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
1906 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1908 # this only works for negative numbers when shifting in base 2
1909 if (($x->{sign} eq '-') && ($n == 2))
1911 return $x->round(@r) if $x->is_one('-'); # -1 => -1
1914 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1915 # but perhaps there is a better emulation for two's complement shift...
1916 # if $y != 1, we must simulate it by doing:
1917 # convert to bin, flip all bits, shift, and be done
1918 $x->binc(); # -3 => -2
1919 my $bin = $x->as_bin();
1920 $bin =~ s/^-0b//; # strip '-0b' prefix
1921 $bin =~ tr/10/01/; # flip bits
1923 if ($y >= CORE::length($bin))
1925 $bin = '0'; # shifting to far right creates -1
1926 # 0, because later increment makes
1927 # that 1, attached '-' makes it '-1'
1928 # because -1 >> x == -1 !
1932 $bin =~ s/.{$y}$//; # cut off at the right side
1933 $bin = '1' . $bin; # extend left side by one dummy '1'
1934 $bin =~ tr/10/01/; # flip bits back
1936 my $res = $self->new('0b'.$bin); # add prefix and convert back
1937 $res->binc(); # remember to increment
1938 $x->{value} = $res->{value}; # take over value
1939 return $x->round(@r); # we are done now, magic, isn't?
1941 # x < 0, n == 2, y == 1
1942 $x->bdec(); # n == 2, but $y == 1: this fixes it
1945 $x->{value} = $CALC->_rsft($x->{value},$y->{value},$n);
1951 #(BINT or num_str, BINT or num_str) return BINT
1955 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1956 # objectify is costly, so avoid it
1957 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1959 ($self,$x,$y,@r) = objectify(2,@_);
1962 return $x if $x->modify('band');
1964 $r[3] = $y; # no push!
1966 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1968 my $sx = $x->{sign} eq '+' ? 1 : -1;
1969 my $sy = $y->{sign} eq '+' ? 1 : -1;
1971 if ($sx == 1 && $sy == 1)
1973 $x->{value} = $CALC->_and($x->{value},$y->{value});
1974 return $x->round(@r);
1977 if ($CAN{signed_and})
1979 $x->{value} = $CALC->_signed_and($x->{value},$y->{value},$sx,$sy);
1980 return $x->round(@r);
1984 __emu_band($self,$x,$y,$sx,$sy,@r);
1989 #(BINT or num_str, BINT or num_str) return BINT
1993 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1994 # objectify is costly, so avoid it
1995 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1997 ($self,$x,$y,@r) = objectify(2,@_);
2000 return $x if $x->modify('bior');
2001 $r[3] = $y; # no push!
2003 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
2005 my $sx = $x->{sign} eq '+' ? 1 : -1;
2006 my $sy = $y->{sign} eq '+' ? 1 : -1;
2008 # the sign of X follows the sign of X, e.g. sign of Y irrelevant for bior()
2010 # don't use lib for negative values
2011 if ($sx == 1 && $sy == 1)
2013 $x->{value} = $CALC->_or($x->{value},$y->{value});
2014 return $x->round(@r);
2017 # if lib can do negative values, let it handle this
2018 if ($CAN{signed_or})
2020 $x->{value} = $CALC->_signed_or($x->{value},$y->{value},$sx,$sy);
2021 return $x->round(@r);
2025 __emu_bior($self,$x,$y,$sx,$sy,@r);
2030 #(BINT or num_str, BINT or num_str) return BINT
2034 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2035 # objectify is costly, so avoid it
2036 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2038 ($self,$x,$y,@r) = objectify(2,@_);
2041 return $x if $x->modify('bxor');
2042 $r[3] = $y; # no push!
2044 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
2046 my $sx = $x->{sign} eq '+' ? 1 : -1;
2047 my $sy = $y->{sign} eq '+' ? 1 : -1;
2049 # don't use lib for negative values
2050 if ($sx == 1 && $sy == 1)
2052 $x->{value} = $CALC->_xor($x->{value},$y->{value});
2053 return $x->round(@r);
2056 # if lib can do negative values, let it handle this
2057 if ($CAN{signed_xor})
2059 $x->{value} = $CALC->_signed_xor($x->{value},$y->{value},$sx,$sy);
2060 return $x->round(@r);
2064 __emu_bxor($self,$x,$y,$sx,$sy,@r);
2069 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2071 my $e = $CALC->_len($x->{value});
2072 wantarray ? ($e,0) : $e;
2077 # return the nth decimal digit, negative values count backward, 0 is right
2078 my ($self,$x,$n) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2080 $n = $n->numify() if ref($n);
2081 $CALC->_digit($x->{value},$n||0);
2086 # return the amount of trailing zeros in $x (as scalar)
2088 $x = $class->new($x) unless ref $x;
2090 return 0 if $x->{sign} !~ /^[+-]$/; # NaN, inf, -inf etc
2092 $CALC->_zeros($x->{value}); # must handle odd values, 0 etc
2097 # calculate square root of $x
2098 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2100 return $x if $x->modify('bsqrt');
2102 return $x->bnan() if $x->{sign} !~ /^\+/; # -x or -inf or NaN => NaN
2103 return $x if $x->{sign} eq '+inf'; # sqrt(+inf) == inf
2105 return $upgrade->bsqrt($x,@r) if defined $upgrade;
2107 $x->{value} = $CALC->_sqrt($x->{value});
2113 # calculate $y'th root of $x
2116 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2118 $y = $self->new(2) unless defined $y;
2120 # objectify is costly, so avoid it
2121 if ((!ref($x)) || (ref($x) ne ref($y)))
2123 ($self,$x,$y,@r) = objectify(2,$self || $class,@_);
2126 return $x if $x->modify('broot');
2128 # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
2129 return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
2130 $y->{sign} !~ /^\+$/;
2132 return $x->round(@r)
2133 if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
2135 return $upgrade->new($x)->broot($upgrade->new($y),@r) if defined $upgrade;
2137 $x->{value} = $CALC->_root($x->{value},$y->{value});
2143 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
2144 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2146 if ($x->{sign} !~ /^[+-]$/)
2148 my $s = $x->{sign}; $s =~ s/^[+-]//; # NaN, -inf,+inf => NaN or inf
2149 return $self->new($s);
2151 return $self->bone() if $x->is_zero();
2153 # 12300 => 2 trailing zeros => exponent is 2
2154 $self->new( $CALC->_zeros($x->{value}) );
2159 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
2160 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2162 if ($x->{sign} !~ /^[+-]$/)
2164 # for NaN, +inf, -inf: keep the sign
2165 return $self->new($x->{sign});
2167 my $m = $x->copy(); delete $m->{_p}; delete $m->{_a};
2169 # that's a bit inefficient:
2170 my $zeros = $CALC->_zeros($m->{value});
2171 $m->brsft($zeros,10) if $zeros != 0;
2177 # return a copy of both the exponent and the mantissa
2178 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2180 ($x->mantissa(),$x->exponent());
2183 ##############################################################################
2184 # rounding functions
2188 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
2189 # $n == 0 || $n == 1 => round to integer
2190 my $x = shift; my $self = ref($x) || $x; $x = $self->new($x) unless ref $x;
2192 my ($scale,$mode) = $x->_scale_p(@_);
2194 return $x if !defined $scale || $x->modify('bfround'); # no-op
2196 # no-op for BigInts if $n <= 0
2197 $x->bround( $x->length()-$scale, $mode) if $scale > 0;
2199 delete $x->{_a}; # delete to save memory
2200 $x->{_p} = $scale; # store new _p
2204 sub _scan_for_nonzero
2206 # internal, used by bround() to scan for non-zeros after a '5'
2207 my ($x,$pad,$xs,$len) = @_;
2209 return 0 if $len == 1; # "5" is trailed by invisible zeros
2210 my $follow = $pad - 1;
2211 return 0 if $follow > $len || $follow < 1;
2213 # use the string form to check whether only '0's follow or not
2214 substr ($xs,-$follow) =~ /[^0]/ ? 1 : 0;
2219 # Exists to make life easier for switch between MBF and MBI (should we
2220 # autoload fxxx() like MBF does for bxxx()?)
2221 my $x = shift; $x = $class->new($x) unless ref $x;
2227 # accuracy: +$n preserve $n digits from left,
2228 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
2230 # and overwrite the rest with 0's, return normalized number
2231 # do not return $x->bnorm(), but $x
2233 my $x = shift; $x = $class->new($x) unless ref $x;
2234 my ($scale,$mode) = $x->_scale_a(@_);
2235 return $x if !defined $scale || $x->modify('bround'); # no-op
2237 if ($x->is_zero() || $scale == 0)
2239 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2242 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2244 # we have fewer digits than we want to scale to
2245 my $len = $x->length();
2246 # convert $scale to a scalar in case it is an object (put's a limit on the
2247 # number length, but this would already limited by memory constraints), makes
2249 $scale = $scale->numify() if ref ($scale);
2251 # scale < 0, but > -len (not >=!)
2252 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2254 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2258 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2259 my ($pad,$digit_round,$digit_after);
2260 $pad = $len - $scale;
2261 $pad = abs($scale-1) if $scale < 0;
2263 # do not use digit(), it is very costly for binary => decimal
2264 # getting the entire string is also costly, but we need to do it only once
2265 my $xs = $CALC->_str($x->{value});
2268 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2269 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2270 $digit_round = '0'; $digit_round = substr($xs,$pl,1) if $pad <= $len;
2271 $pl++; $pl ++ if $pad >= $len;
2272 $digit_after = '0'; $digit_after = substr($xs,$pl,1) if $pad > 0;
2274 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2275 # closer at the remaining digits of the original $x, remember decision
2276 my $round_up = 1; # default round up
2278 ($mode eq 'trunc') || # trunc by round down
2279 ($digit_after =~ /[01234]/) || # round down anyway,
2281 ($digit_after eq '5') && # not 5000...0000
2282 ($x->_scan_for_nonzero($pad,$xs,$len) == 0) &&
2284 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2285 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2286 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2287 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2288 ($mode eq 'zero') # round down if zero, sign adjusted below
2290 my $put_back = 0; # not yet modified
2292 if (($pad > 0) && ($pad <= $len))
2294 substr($xs,-$pad,$pad) = '0' x $pad; # replace with '00...'
2295 $put_back = 1; # need to put back
2299 $x->bzero(); # round to '0'
2302 if ($round_up) # what gave test above?
2304 $put_back = 1; # need to put back
2305 $pad = $len, $xs = '0' x $pad if $scale < 0; # tlr: whack 0.51=>1.0
2307 # we modify directly the string variant instead of creating a number and
2308 # adding it, since that is faster (we already have the string)
2309 my $c = 0; $pad ++; # for $pad == $len case
2310 while ($pad <= $len)
2312 $c = substr($xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2313 substr($xs,-$pad,1) = $c; $pad++;
2314 last if $c != 0; # no overflow => early out
2316 $xs = '1'.$xs if $c == 0;
2319 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back, if needed
2321 $x->{_a} = $scale if $scale >= 0;
2324 $x->{_a} = $len+$scale;
2325 $x->{_a} = 0 if $scale < -$len;
2332 # return integer less or equal then number; no-op since it's already integer
2333 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2340 # return integer greater or equal then number; no-op since it's already int
2341 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2348 # An object might be asked to return itself as bigint on certain overloaded
2349 # operations. This does exactly this, so that sub classes can simple inherit
2350 # it or override with their own integer conversion routine.
2356 # return as hex string, with prefixed 0x
2357 my $x = shift; $x = $class->new($x) if !ref($x);
2359 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2362 $s = $x->{sign} if $x->{sign} eq '-';
2363 $s . $CALC->_as_hex($x->{value});
2368 # return as binary string, with prefixed 0b
2369 my $x = shift; $x = $class->new($x) if !ref($x);
2371 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2373 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2374 return $s . $CALC->_as_bin($x->{value});
2379 # return as octal string, with prefixed 0
2380 my $x = shift; $x = $class->new($x) if !ref($x);
2382 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2384 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2385 return $s . $CALC->_as_oct($x->{value});
2388 ##############################################################################
2389 # private stuff (internal use only)
2393 # check for strings, if yes, return objects instead
2395 # the first argument is number of args objectify() should look at it will
2396 # return $count+1 elements, the first will be a classname. This is because
2397 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2398 # useless objects being created and thrown away. So we cannot simple loop
2399 # over @_. If the given count is 0, all arguments will be used.
2401 # If the second arg is a ref, use it as class.
2402 # If not, try to use it as classname, unless undef, then use $class
2403 # (aka Math::BigInt). The latter shouldn't happen,though.
2406 # $x->badd(1); => ref x, scalar y
2407 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2408 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2409 # Math::BigInt::badd(1,2); => scalar x, scalar y
2410 # In the last case we check number of arguments to turn it silently into
2411 # $class,1,2. (We can not take '1' as class ;o)
2412 # badd($class,1) is not supported (it should, eventually, try to add undef)
2413 # currently it tries 'Math::BigInt' + 1, which will not work.
2415 # some shortcut for the common cases
2417 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2419 my $count = abs(shift || 0);
2421 my (@a,$k,$d); # resulting array, temp, and downgrade
2424 # okay, got object as first
2429 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2431 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2435 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2436 if (defined ${"$a[0]::downgrade"})
2438 $d = ${"$a[0]::downgrade"};
2439 ${"$a[0]::downgrade"} = undef;
2442 my $up = ${"$a[0]::upgrade"};
2443 # print STDERR "# Now in objectify, my class is today $a[0], count = $count\n";
2451 $k = $a[0]->new($k);
2453 elsif (!defined $up && ref($k) ne $a[0])
2455 # foreign object, try to convert to integer
2456 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2469 $k = $a[0]->new($k);
2471 elsif (!defined $up && ref($k) ne $a[0])
2473 # foreign object, try to convert to integer
2474 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2478 push @a,@_; # return other params, too
2482 require Carp; Carp::croak ("$class objectify needs list context");
2484 ${"$a[0]::downgrade"} = $d;
2488 sub _register_callback
2490 my ($class,$callback) = @_;
2492 if (ref($callback) ne 'CODE')
2495 Carp::croak ("$callback is not a coderef");
2497 $CALLBACKS{$class} = $callback;
2504 $IMPORT++; # remember we did import()
2505 my @a; my $l = scalar @_;
2506 my $warn_or_die = 0; # 0 - no warn, 1 - warn, 2 - die
2507 for ( my $i = 0; $i < $l ; $i++ )
2509 if ($_[$i] eq ':constant')
2511 # this causes overlord er load to step in
2513 integer => sub { $self->new(shift) },
2514 binary => sub { $self->new(shift) };
2516 elsif ($_[$i] eq 'upgrade')
2518 # this causes upgrading
2519 $upgrade = $_[$i+1]; # or undef to disable
2522 elsif ($_[$i] =~ /^(lib|try|only)\z/)
2524 # this causes a different low lib to take care...
2525 $CALC = $_[$i+1] || '';
2526 # lib => 1 (warn on fallback), try => 0 (no warn), only => 2 (die on fallback)
2527 $warn_or_die = 1 if $_[$i] eq 'lib';
2528 $warn_or_die = 2 if $_[$i] eq 'only';
2536 # any non :constant stuff is handled by our parent, Exporter
2541 $self->SUPER::import(@a); # need it for subclasses
2542 $self->export_to_level(1,$self,@a); # need it for MBF
2545 # try to load core math lib
2546 my @c = split /\s*,\s*/,$CALC;
2549 $_ =~ tr/a-zA-Z0-9://cd; # limit to sane characters
2551 push @c, \'FastCalc', \'Calc' # if all fail, try these
2552 if $warn_or_die < 2; # but not for "only"
2553 $CALC = ''; # signal error
2556 # fallback libraries are "marked" as \'string', extract string if nec.
2557 my $lib = $l; $lib = $$l if ref($l);
2559 next if ($lib || '') eq '';
2560 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2564 # Perl < 5.6.0 dies with "out of memory!" when eval("") and ':constant' is
2565 # used in the same script, or eval("") inside import().
2566 my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
2567 my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
2569 $file = File::Spec->catfile (@parts, $file);
2570 eval { require "$file"; $lib->import( @c ); }
2574 eval "use $lib qw/@c/;";
2579 # loaded it ok, see if the api_version() is high enough
2580 if ($lib->can('api_version') && $lib->api_version() >= 1.0)
2583 # api_version matches, check if it really provides anything we need
2587 add mul div sub dec inc
2588 acmp len digit is_one is_zero is_even is_odd
2590 zeros new copy check
2591 from_hex from_oct from_bin as_hex as_bin as_oct
2592 rsft lsft xor and or
2593 mod sqrt root fac pow modinv modpow log_int gcd
2596 if (!$lib->can("_$method"))
2598 if (($WARN{$lib}||0) < 2)
2601 Carp::carp ("$lib is missing method '_$method'");
2602 $WARN{$lib} = 1; # still warn about the lib
2611 if ($warn_or_die > 0 && ref($l))
2614 my $msg = "Math::BigInt: couldn't load specified math lib(s), fallback to $lib";
2615 Carp::carp ($msg) if $warn_or_die == 1;
2616 Carp::croak ($msg) if $warn_or_die == 2;
2618 last; # found a usable one, break
2622 if (($WARN{$lib}||0) < 2)
2624 my $ver = eval "\$$lib\::VERSION" || 'unknown';
2626 Carp::carp ("Cannot load outdated $lib v$ver, please upgrade");
2627 $WARN{$lib} = 2; # never warn again
2635 if ($warn_or_die == 2)
2637 Carp::croak ("Couldn't load specified math lib(s) and fallback disallowed");
2641 Carp::croak ("Couldn't load any math lib(s), not even fallback to Calc.pm");
2646 foreach my $class (keys %CALLBACKS)
2648 &{$CALLBACKS{$class}}($CALC);
2651 # Fill $CAN with the results of $CALC->can(...) for emulating lower math lib
2655 for my $method (qw/ signed_and signed_or signed_xor /)
2657 $CAN{$method} = $CALC->can("_$method") ? 1 : 0;
2665 # create a bigint from a hexadecimal string
2666 my ($self, $hs) = @_;
2668 my $rc = $self->__from_hex($hs);
2670 return $self->bnan() unless defined $rc;
2677 # create a bigint from a hexadecimal string
2678 my ($self, $bs) = @_;
2680 my $rc = $self->__from_bin($bs);
2682 return $self->bnan() unless defined $rc;
2689 # create a bigint from a hexadecimal string
2690 my ($self, $os) = @_;
2692 my $x = $self->bzero();
2695 $os =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2696 $os =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2698 return $x->bnan() if $os !~ /^[\-\+]?0[0-9]+$/;
2700 my $sign = '+'; $sign = '-' if $os =~ /^-/;
2702 $os =~ s/^[+-]//; # strip sign
2703 $x->{value} = $CALC->_from_oct($os);
2704 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2711 # convert a (ref to) big hex string to BigInt, return undef for error
2714 my $x = Math::BigInt->bzero();
2717 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2718 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2720 return $x->bnan() if $hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2722 my $sign = '+'; $sign = '-' if $hs =~ /^-/;
2724 $hs =~ s/^[+-]//; # strip sign
2725 $x->{value} = $CALC->_from_hex($hs);
2726 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2733 # convert a (ref to) big binary string to BigInt, return undef for error
2736 my $x = Math::BigInt->bzero();
2739 $bs =~ s/([01])_([01])/$1$2/g;
2740 $bs =~ s/([01])_([01])/$1$2/g;
2741 return $x->bnan() if $bs !~ /^[+-]?0b[01]+$/;
2743 my $sign = '+'; $sign = '-' if $bs =~ /^\-/;
2744 $bs =~ s/^[+-]//; # strip sign
2746 $x->{value} = $CALC->_from_bin($bs);
2747 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2753 # input: num_str; output: undef for invalid or
2754 # (\$mantissa_sign,\$mantissa_value,\$mantissa_fraction,\$exp_sign,\$exp_value)
2755 # Internal, take apart a string and return the pieces.
2756 # Strip leading/trailing whitespace, leading zeros, underscore and reject
2760 # strip white space at front, also extranous leading zeros
2761 $x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2762 $x =~ s/^\s+//; # but this will
2763 $x =~ s/\s+$//g; # strip white space at end
2765 # shortcut, if nothing to split, return early
2766 if ($x =~ /^[+-]?[0-9]+\z/)
2768 $x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2769 return (\$sign, \$x, \'', \'', \0);
2772 # invalid starting char?
2773 return if $x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2775 return __from_hex($x) if $x =~ /^[\-\+]?0x/; # hex string
2776 return __from_bin($x) if $x =~ /^[\-\+]?0b/; # binary string
2778 # strip underscores between digits
2779 $x =~ s/([0-9])_([0-9])/$1$2/g;
2780 $x =~ s/([0-9])_([0-9])/$1$2/g; # do twice for 1_2_3
2782 # some possible inputs:
2783 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2784 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 # 0e999
2786 my ($m,$e,$last) = split /[Ee]/,$x;
2787 return if defined $last; # last defined => 1e2E3 or others
2788 $e = '0' if !defined $e || $e eq "";
2790 # sign,value for exponent,mantint,mantfrac
2791 my ($es,$ev,$mis,$miv,$mfv);
2793 if ($e =~ /^([+-]?)0*([0-9]+)$/) # strip leading zeros
2797 return if $m eq '.' || $m eq '';
2798 my ($mi,$mf,$lastf) = split /\./,$m;
2799 return if defined $lastf; # lastf defined => 1.2.3 or others
2800 $mi = '0' if !defined $mi;
2801 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2802 $mf = '0' if !defined $mf || $mf eq '';
2803 if ($mi =~ /^([+-]?)0*([0-9]+)$/) # strip leading zeros
2805 $mis = $1||'+'; $miv = $2;
2806 return unless ($mf =~ /^([0-9]*?)0*$/); # strip trailing zeros
2808 # handle the 0e999 case here
2809 $ev = 0 if $miv eq '0' && $mfv eq '';
2810 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2813 return; # NaN, not a number
2816 ##############################################################################
2817 # internal calculation routines (others are in Math::BigInt::Calc etc)
2821 # (BINT or num_str, BINT or num_str) return BINT
2822 # does modify first argument
2826 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2827 my $method = ref($x) . '::bgcd';
2829 $x * $ty / &$method($x,$ty);
2832 ###############################################################################
2833 # this method returns 0 if the object can be modified, or 1 if not.
2834 # We use a fast constant sub() here, to avoid costly calls. Subclasses
2835 # may override it with special code (f.i. Math::BigInt::Constant does so)
2837 sub modify () { 0; }
2846 Math::BigInt - Arbitrary size integer/float math package
2852 # or make it faster: install (optional) Math::BigInt::GMP
2853 # and always use (it will fall back to pure Perl if the
2854 # GMP library is not installed):
2856 # will warn if Math::BigInt::GMP cannot be found
2857 use Math::BigInt lib => 'GMP';
2859 # to supress the warning use this:
2860 # use Math::BigInt try => 'GMP';
2862 my $str = '1234567890';
2863 my @values = (64,74,18);
2864 my $n = 1; my $sign = '-';
2867 $x = Math::BigInt->new($str); # defaults to 0
2868 $y = $x->copy(); # make a true copy
2869 $nan = Math::BigInt->bnan(); # create a NotANumber
2870 $zero = Math::BigInt->bzero(); # create a +0
2871 $inf = Math::BigInt->binf(); # create a +inf
2872 $inf = Math::BigInt->binf('-'); # create a -inf
2873 $one = Math::BigInt->bone(); # create a +1
2874 $one = Math::BigInt->bone('-'); # create a -1
2876 $h = Math::BigInt->new('0x123'); # from hexadecimal
2877 $b = Math::BigInt->new('0b101'); # from binary
2878 $o = Math::BigInt->from_oct('0101'); # from octal
2880 # Testing (don't modify their arguments)
2881 # (return true if the condition is met, otherwise false)
2883 $x->is_zero(); # if $x is +0
2884 $x->is_nan(); # if $x is NaN
2885 $x->is_one(); # if $x is +1
2886 $x->is_one('-'); # if $x is -1
2887 $x->is_odd(); # if $x is odd
2888 $x->is_even(); # if $x is even
2889 $x->is_pos(); # if $x >= 0
2890 $x->is_neg(); # if $x < 0
2891 $x->is_inf($sign); # if $x is +inf, or -inf (sign is default '+')
2892 $x->is_int(); # if $x is an integer (not a float)
2894 # comparing and digit/sign extraction
2895 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2896 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2897 $x->sign(); # return the sign, either +,- or NaN
2898 $x->digit($n); # return the nth digit, counting from right
2899 $x->digit(-$n); # return the nth digit, counting from left
2901 # The following all modify their first argument. If you want to preserve
2902 # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
2903 # necessary when mixing $a = $b assignments with non-overloaded math.
2905 $x->bzero(); # set $x to 0
2906 $x->bnan(); # set $x to NaN
2907 $x->bone(); # set $x to +1
2908 $x->bone('-'); # set $x to -1
2909 $x->binf(); # set $x to inf
2910 $x->binf('-'); # set $x to -inf
2912 $x->bneg(); # negation
2913 $x->babs(); # absolute value
2914 $x->bnorm(); # normalize (no-op in BigInt)
2915 $x->bnot(); # two's complement (bit wise not)
2916 $x->binc(); # increment $x by 1
2917 $x->bdec(); # decrement $x by 1
2919 $x->badd($y); # addition (add $y to $x)
2920 $x->bsub($y); # subtraction (subtract $y from $x)
2921 $x->bmul($y); # multiplication (multiply $x by $y)
2922 $x->bdiv($y); # divide, set $x to quotient
2923 # return (quo,rem) or quo if scalar
2925 $x->bmod($y); # modulus (x % y)
2926 $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
2927 $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
2929 $x->bpow($y); # power of arguments (x ** y)
2930 $x->blsft($y); # left shift in base 2
2931 $x->brsft($y); # right shift in base 2
2932 # returns (quo,rem) or quo if in scalar context
2933 $x->blsft($y,$n); # left shift by $y places in base $n
2934 $x->brsft($y,$n); # right shift by $y places in base $n
2935 # returns (quo,rem) or quo if in scalar context
2937 $x->band($y); # bitwise and
2938 $x->bior($y); # bitwise inclusive or
2939 $x->bxor($y); # bitwise exclusive or
2940 $x->bnot(); # bitwise not (two's complement)
2942 $x->bsqrt(); # calculate square-root
2943 $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
2944 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2946 $x->bnok($y); # x over y (binomial coefficient n over k)
2948 $x->blog(); # logarithm of $x to base e (Euler's number)
2949 $x->blog($base); # logarithm of $x to base $base (f.i. 2)
2950 $x->bexp(); # calculate e ** $x where e is Euler's number
2952 $x->round($A,$P,$mode); # round to accuracy or precision using mode $mode
2953 $x->bround($n); # accuracy: preserve $n digits
2954 $x->bfround($n); # round to $nth digit, no-op for BigInts
2956 # The following do not modify their arguments in BigInt (are no-ops),
2957 # but do so in BigFloat:
2959 $x->bfloor(); # return integer less or equal than $x
2960 $x->bceil(); # return integer greater or equal than $x
2962 # The following do not modify their arguments:
2964 # greatest common divisor (no OO style)
2965 my $gcd = Math::BigInt::bgcd(@values);
2966 # lowest common multiplicator (no OO style)
2967 my $lcm = Math::BigInt::blcm(@values);
2969 $x->length(); # return number of digits in number
2970 ($xl,$f) = $x->length(); # length of number and length of fraction part,
2971 # latter is always 0 digits long for BigInts
2973 $x->exponent(); # return exponent as BigInt
2974 $x->mantissa(); # return (signed) mantissa as BigInt
2975 $x->parts(); # return (mantissa,exponent) as BigInt
2976 $x->copy(); # make a true copy of $x (unlike $y = $x;)
2977 $x->as_int(); # return as BigInt (in BigInt: same as copy())
2978 $x->numify(); # return as scalar (might overflow!)
2980 # conversation to string (do not modify their argument)
2981 $x->bstr(); # normalized string (e.g. '3')
2982 $x->bsstr(); # norm. string in scientific notation (e.g. '3E0')
2983 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
2984 $x->as_bin(); # as signed binary string with prefixed 0b
2985 $x->as_oct(); # as signed octal string with prefixed 0
2988 # precision and accuracy (see section about rounding for more)
2989 $x->precision(); # return P of $x (or global, if P of $x undef)
2990 $x->precision($n); # set P of $x to $n
2991 $x->accuracy(); # return A of $x (or global, if A of $x undef)
2992 $x->accuracy($n); # set A $x to $n
2995 Math::BigInt->precision(); # get/set global P for all BigInt objects
2996 Math::BigInt->accuracy(); # get/set global A for all BigInt objects
2997 Math::BigInt->round_mode(); # get/set global round mode, one of
2998 # 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'
2999 Math::BigInt->config(); # return hash containing configuration
3003 All operators (including basic math operations) are overloaded if you
3004 declare your big integers as
3006 $i = new Math::BigInt '123_456_789_123_456_789';
3008 Operations with overloaded operators preserve the arguments which is
3009 exactly what you expect.
3015 Input values to these routines may be any string, that looks like a number
3016 and results in an integer, including hexadecimal and binary numbers.
3018 Scalars holding numbers may also be passed, but note that non-integer numbers
3019 may already have lost precision due to the conversation to float. Quote
3020 your input if you want BigInt to see all the digits:
3022 $x = Math::BigInt->new(12345678890123456789); # bad
3023 $x = Math::BigInt->new('12345678901234567890'); # good
3025 You can include one underscore between any two digits.
3027 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
3028 Non-integer values result in NaN.
3030 Hexadecimal (prefixed with "0x") and binary numbers (prefixed with "0b")
3031 are accepted, too. Please note that octal numbers are not recognized
3032 by new(), so the following will print "123":
3034 perl -MMath::BigInt -le 'print Math::BigInt->new("0123")'
3036 To convert an octal number, use from_oct();
3038 perl -MMath::BigInt -le 'print Math::BigInt->from_oct("0123")'
3040 Currently, Math::BigInt::new() defaults to 0, while Math::BigInt::new('')
3041 results in 'NaN'. This might change in the future, so use always the following
3042 explicit forms to get a zero or NaN:
3044 $zero = Math::BigInt->bzero();
3045 $nan = Math::BigInt->bnan();
3047 C<bnorm()> on a BigInt object is now effectively a no-op, since the numbers
3048 are always stored in normalized form. If passed a string, creates a BigInt
3049 object from the input.
3053 Output values are BigInt objects (normalized), except for the methods which
3054 return a string (see L<SYNOPSIS>).
3056 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
3057 C<is_nan()>, etc.) return true or false, while others (C<bcmp()>, C<bacmp()>)
3058 return either undef (if NaN is involved), <0, 0 or >0 and are suited for sort.
3064 Each of the methods below (except config(), accuracy() and precision())
3065 accepts three additional parameters. These arguments C<$A>, C<$P> and C<$R>
3066 are C<accuracy>, C<precision> and C<round_mode>. Please see the section about
3067 L<ACCURACY and PRECISION> for more information.
3073 print Dumper ( Math::BigInt->config() );
3074 print Math::BigInt->config()->{lib},"\n";
3076 Returns a hash containing the configuration, e.g. the version number, lib
3077 loaded etc. The following hash keys are currently filled in with the
3078 appropriate information.
3082 ============================================================
3083 lib Name of the low-level math library
3085 lib_version Version of low-level math library (see 'lib')
3087 class The class name of config() you just called
3089 upgrade To which class math operations might be upgraded
3091 downgrade To which class math operations might be downgraded
3093 precision Global precision
3095 accuracy Global accuracy
3097 round_mode Global round mode
3099 version version number of the class you used
3101 div_scale Fallback accuracy for div
3103 trap_nan If true, traps creation of NaN via croak()
3105 trap_inf If true, traps creation of +inf/-inf via croak()
3108 The following values can be set by passing C<config()> a reference to a hash:
3111 upgrade downgrade precision accuracy round_mode div_scale
3115 $new_cfg = Math::BigInt->config( { trap_inf => 1, precision => 5 } );
3119 $x->accuracy(5); # local for $x
3120 CLASS->accuracy(5); # global for all members of CLASS
3121 # Note: This also applies to new()!
3123 $A = $x->accuracy(); # read out accuracy that affects $x
3124 $A = CLASS->accuracy(); # read out global accuracy
3126 Set or get the global or local accuracy, aka how many significant digits the
3127 results have. If you set a global accuracy, then this also applies to new()!
3129 Warning! The accuracy I<sticks>, e.g. once you created a number under the
3130 influence of C<< CLASS->accuracy($A) >>, all results from math operations with
3131 that number will also be rounded.
3133 In most cases, you should probably round the results explicitly using one of
3134 L<round()>, L<bround()> or L<bfround()> or by passing the desired accuracy
3135 to the math operation as additional parameter:
3137 my $x = Math::BigInt->new(30000);
3138 my $y = Math::BigInt->new(7);
3139 print scalar $x->copy()->bdiv($y, 2); # print 4300
3140 print scalar $x->copy()->bdiv($y)->bround(2); # print 4300
3142 Please see the section about L<ACCURACY AND PRECISION> for further details.
3144 Value must be greater than zero. Pass an undef value to disable it:
3146 $x->accuracy(undef);
3147 Math::BigInt->accuracy(undef);
3149 Returns the current accuracy. For C<$x->accuracy()> it will return either the
3150 local accuracy, or if not defined, the global. This means the return value
3151 represents the accuracy that will be in effect for $x:
3153 $y = Math::BigInt->new(1234567); # unrounded
3154 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
3155 $x = Math::BigInt->new(123456); # $x will be automatically rounded!
3156 print "$x $y\n"; # '123500 1234567'
3157 print $x->accuracy(),"\n"; # will be 4
3158 print $y->accuracy(),"\n"; # also 4, since global is 4
3159 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
3160 print $x->accuracy(),"\n"; # still 4
3161 print $y->accuracy(),"\n"; # 5, since global is 5
3163 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
3164 globals separated from Math::BigInt, but it is possible to subclass
3165 Math::BigInt and make the globals of the subclass aliases to the ones from
3170 $x->precision(-2); # local for $x, round at the second digit right of the dot
3171 $x->precision(2); # ditto, round at the second digit left of the dot
3173 CLASS->precision(5); # Global for all members of CLASS
3174 # This also applies to new()!
3175 CLASS->precision(-5); # ditto
3177 $P = CLASS->precision(); # read out global precision
3178 $P = $x->precision(); # read out precision that affects $x
3180 Note: You probably want to use L<accuracy()> instead. With L<accuracy> you
3181 set the number of digits each result should have, with L<precision> you
3182 set the place where to round!
3184 C<precision()> sets or gets the global or local precision, aka at which digit
3185 before or after the dot to round all results. A set global precision also
3186 applies to all newly created numbers!
3188 In Math::BigInt, passing a negative number precision has no effect since no
3189 numbers have digits after the dot. In L<Math::BigFloat>, it will round all
3190 results to P digits after the dot.
3192 Please see the section about L<ACCURACY AND PRECISION> for further details.
3194 Pass an undef value to disable it:
3196 $x->precision(undef);
3197 Math::BigInt->precision(undef);
3199 Returns the current precision. For C<$x->precision()> it will return either the
3200 local precision of $x, or if not defined, the global. This means the return
3201 value represents the prevision that will be in effect for $x:
3203 $y = Math::BigInt->new(1234567); # unrounded
3204 print Math::BigInt->precision(4),"\n"; # set 4, print 4
3205 $x = Math::BigInt->new(123456); # will be automatically rounded
3206 print $x; # print "120000"!
3208 Note: Works also for subclasses like L<Math::BigFloat>. Each class has its
3209 own globals separated from Math::BigInt, but it is possible to subclass
3210 Math::BigInt and make the globals of the subclass aliases to the ones from
3217 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
3218 2, but others work, too.
3220 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
3224 $x = Math::BigInt->new(10);
3225 $x->brsft(1); # same as $x >> 1: 5
3226 $x = Math::BigInt->new(1234);
3227 $x->brsft(2,10); # result 12
3229 There is one exception, and that is base 2 with negative $x:
3232 $x = Math::BigInt->new(-5);
3235 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
3240 $x = Math::BigInt->new($str,$A,$P,$R);
3242 Creates a new BigInt object from a scalar or another BigInt object. The
3243 input is accepted as decimal, hex (with leading '0x') or binary (with leading
3246 See L<Input> for more info on accepted input formats.
3250 $x = Math::BigIn->from_oct("0775"); # input is octal
3254 $x = Math::BigIn->from_hex("0xcafe"); # input is hexadecimal
3258 $x = Math::BigIn->from_oct("0x10011"); # input is binary
3262 $x = Math::BigInt->bnan();
3264 Creates a new BigInt object representing NaN (Not A Number).
3265 If used on an object, it will set it to NaN:
3271 $x = Math::BigInt->bzero();
3273 Creates a new BigInt object representing zero.
3274 If used on an object, it will set it to zero:
3280 $x = Math::BigInt->binf($sign);
3282 Creates a new BigInt object representing infinity. The optional argument is
3283 either '-' or '+', indicating whether you want infinity or minus infinity.
3284 If used on an object, it will set it to infinity:
3291 $x = Math::BigInt->binf($sign);
3293 Creates a new BigInt object representing one. The optional argument is
3294 either '-' or '+', indicating whether you want one or minus one.
3295 If used on an object, it will set it to one:
3300 =head2 is_one()/is_zero()/is_nan()/is_inf()
3303 $x->is_zero(); # true if arg is +0
3304 $x->is_nan(); # true if arg is NaN
3305 $x->is_one(); # true if arg is +1
3306 $x->is_one('-'); # true if arg is -1
3307 $x->is_inf(); # true if +inf
3308 $x->is_inf('-'); # true if -inf (sign is default '+')
3310 These methods all test the BigInt for being one specific value and return
3311 true or false depending on the input. These are faster than doing something
3316 =head2 is_pos()/is_neg()/is_positive()/is_negative()
3318 $x->is_pos(); # true if > 0
3319 $x->is_neg(); # true if < 0
3321 The methods return true if the argument is positive or negative, respectively.
3322 C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
3323 C<-inf> is negative. A C<zero> is neither positive nor negative.
3325 These methods are only testing the sign, and not the value.
3327 C<is_positive()> and C<is_negative()> are aliases to C<is_pos()> and
3328 C<is_neg()>, respectively. C<is_positive()> and C<is_negative()> were
3329 introduced in v1.36, while C<is_pos()> and C<is_neg()> were only introduced
3332 =head2 is_odd()/is_even()/is_int()
3334 $x->is_odd(); # true if odd, false for even
3335 $x->is_even(); # true if even, false for odd
3336 $x->is_int(); # true if $x is an integer
3338 The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
3339 C<-inf> are not integers and are neither odd nor even.
3341 In BigInt, all numbers except C<NaN>, C<+inf> and C<-inf> are integers.
3347 Compares $x with $y and takes the sign into account.
3348 Returns -1, 0, 1 or undef.
3354 Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
3360 Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
3362 If you want $x to have a certain sign, use one of the following methods:
3365 $x->babs()->bneg(); # '-'
3367 $x->binf(); # '+inf'
3368 $x->binf('-'); # '-inf'
3372 $x->digit($n); # return the nth digit, counting from right
3374 If C<$n> is negative, returns the digit counting from left.
3380 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
3381 and '-inf', respectively. Does nothing for NaN or zero.
3387 Set the number to its absolute value, e.g. change the sign from '-' to '+'
3388 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
3393 $x->bnorm(); # normalize (no-op)
3399 Two's complement (bitwise not). This is equivalent to
3407 $x->binc(); # increment x by 1
3411 $x->bdec(); # decrement x by 1
3415 $x->badd($y); # addition (add $y to $x)
3419 $x->bsub($y); # subtraction (subtract $y from $x)
3423 $x->bmul($y); # multiplication (multiply $x by $y)
3427 $x->bdiv($y); # divide, set $x to quotient
3428 # return (quo,rem) or quo if scalar
3432 $x->bmod($y); # modulus (x % y)
3436 num->bmodinv($mod); # modular inverse
3438 Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
3439 returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
3440 C<bgcd($num, $mod)==1>.
3444 $num->bmodpow($exp,$mod); # modular exponentation
3445 # ($num**$exp % $mod)
3447 Returns the value of C<$num> taken to the power C<$exp> in the modulus
3448 C<$mod> using binary exponentation. C<bmodpow> is far superior to
3453 because it is much faster - it reduces internal variables into
3454 the modulus whenever possible, so it operates on smaller numbers.
3456 C<bmodpow> also supports negative exponents.
3458 bmodpow($num, -1, $mod)
3460 is exactly equivalent to
3466 $x->bpow($y); # power of arguments (x ** y)
3470 $x->blog($base, $accuracy); # logarithm of x to the base $base
3472 If C<$base> is not defined, Euler's number (e) is used:
3474 print $x->blog(undef, 100); # log(x) to 100 digits
3478 $x->bexp($accuracy); # calculate e ** X
3480 Calculates the expression C<e ** $x> where C<e> is Euler's number.
3482 This method was added in v1.82 of Math::BigInt (April 2007).
3488 $x->bnok($y); # x over y (binomial coefficient n over k)
3490 Calculates the binomial coefficient n over k, also called the "choose"
3491 function. The result is equivalent to:
3497 This method was added in v1.84 of Math::BigInt (April 2007).
3501 $x->blsft($y); # left shift in base 2
3502 $x->blsft($y,$n); # left shift, in base $n (like 10)
3506 $x->brsft($y); # right shift in base 2
3507 $x->brsft($y,$n); # right shift, in base $n (like 10)
3511 $x->band($y); # bitwise and
3515 $x->bior($y); # bitwise inclusive or
3519 $x->bxor($y); # bitwise exclusive or
3523 $x->bnot(); # bitwise not (two's complement)
3527 $x->bsqrt(); # calculate square-root
3531 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3535 $x->round($A,$P,$round_mode);
3537 Round $x to accuracy C<$A> or precision C<$P> using the round mode
3542 $x->bround($N); # accuracy: preserve $N digits
3546 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3552 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3553 does change $x in BigFloat.
3559 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3560 does change $x in BigFloat.
3564 bgcd(@values); # greatest common divisor (no OO style)
3568 blcm(@values); # lowest common multiplicator (no OO style)
3573 ($xl,$fl) = $x->length();
3575 Returns the number of digits in the decimal representation of the number.
3576 In list context, returns the length of the integer and fraction part. For
3577 BigInt's, the length of the fraction part will always be 0.
3583 Return the exponent of $x as BigInt.
3589 Return the signed mantissa of $x as BigInt.
3593 $x->parts(); # return (mantissa,exponent) as BigInt
3597 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3599 =head2 as_int()/as_number()
3603 Returns $x as a BigInt (truncated towards zero). In BigInt this is the same as
3606 C<as_number()> is an alias to this method. C<as_number> was introduced in
3607 v1.22, while C<as_int()> was only introduced in v1.68.
3613 Returns a normalized string representation of C<$x>.
3617 $x->bsstr(); # normalized string in scientific notation
3621 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3625 $x->as_bin(); # as signed binary string with prefixed 0b
3629 $x->as_oct(); # as signed octal string with prefixed 0
3635 This returns a normal Perl scalar from $x. It is used automatically
3636 whenever a scalar is needed, for instance in array index operations.
3638 This loses precision, to avoid this use L<as_int()> instead.
3642 $x->modify('bpowd');
3644 This method returns 0 if the object can be modified with the given
3645 peration, or 1 if not.
3647 This is used for instance by L<Math::BigInt::Constant>.
3649 =head2 upgrade()/downgrade()
3651 Set/get the class for downgrade/upgrade operations. Thuis is used
3652 for instance by L<bignum>. The defaults are '', thus the following
3653 operation will create a BigInt, not a BigFloat:
3655 my $i = Math::BigInt->new(123);
3656 my $f = Math::BigFloat->new('123.1');
3658 print $i + $f,"\n"; # print 246
3662 Set/get the number of digits for the default precision in divide
3667 Set/get the current round mode.
3669 =head1 ACCURACY and PRECISION
3671 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3672 accuracy and precision based rounding, both automatically after every
3673 operation, as well as manually.
3675 This section describes the accuracy/precision handling in Math::Big* as it
3676 used to be and as it is now, complete with an explanation of all terms and
3679 Not yet implemented things (but with correct description) are marked with '!',
3680 things that need to be answered are marked with '?'.
3682 In the next paragraph follows a short description of terms used here (because
3683 these may differ from terms used by others people or documentation).
3685 During the rest of this document, the shortcuts A (for accuracy), P (for
3686 precision), F (fallback) and R (rounding mode) will be used.
3690 A fixed number of digits before (positive) or after (negative)
3691 the decimal point. For example, 123.45 has a precision of -2. 0 means an
3692 integer like 123 (or 120). A precision of 2 means two digits to the left
3693 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3694 numbers with zeros before the decimal point may have different precisions,
3695 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3696 was). It could also have p < 0, when the digits after the decimal point
3699 The string output (of floating point numbers) will be padded with zeros:
3701 Initial value P A Result String
3702 ------------------------------------------------------------
3703 1234.01 -3 1000 1000
3706 1234.001 1 1234 1234.0
3708 1234.01 2 1234.01 1234.01
3709 1234.01 5 1234.01 1234.01000
3711 For BigInts, no padding occurs.
3715 Number of significant digits. Leading zeros are not counted. A
3716 number may have an accuracy greater than the non-zero digits
3717 when there are zeros in it or trailing zeros. For example, 123.456 has
3718 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3720 The string output (of floating point numbers) will be padded with zeros:
3722 Initial value P A Result String
3723 ------------------------------------------------------------
3725 1234.01 6 1234.01 1234.01
3726 1234.1 8 1234.1 1234.1000
3728 For BigInts, no padding occurs.
3732 When both A and P are undefined, this is used as a fallback accuracy when
3735 =head2 Rounding mode R
3737 When rounding a number, different 'styles' or 'kinds'
3738 of rounding are possible. (Note that random rounding, as in
3739 Math::Round, is not implemented.)
3745 truncation invariably removes all digits following the
3746 rounding place, replacing them with zeros. Thus, 987.65 rounded
3747 to tens (P=1) becomes 980, and rounded to the fourth sigdig
3748 becomes 987.6 (A=4). 123.456 rounded to the second place after the
3749 decimal point (P=-2) becomes 123.46.
3751 All other implemented styles of rounding attempt to round to the
3752 "nearest digit." If the digit D immediately to the right of the
3753 rounding place (skipping the decimal point) is greater than 5, the
3754 number is incremented at the rounding place (possibly causing a
3755 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3756 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3757 truncated at the rounding place: e.g. when rounding to units, 0.4
3758 rounds to 0, and -19.4 rounds to -19.
3760 However the results of other styles of rounding differ if the
3761 digit immediately to the right of the rounding place (skipping the
3762 decimal point) is 5 and if there are no digits, or no digits other
3763 than 0, after that 5. In such cases:
3767 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3768 if it is not already. E.g., when rounding to the first sigdig, 0.45
3769 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3773 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3774 it is not already. E.g., when rounding to the first sigdig, 0.45
3775 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3779 round to plus infinity, i.e. always round up. E.g., when
3780 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3781 and 0.4501 also becomes 0.5.
3785 round to minus infinity, i.e. always round down. E.g., when
3786 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3787 but 0.4501 becomes 0.5.
3791 round to zero, i.e. positive numbers down, negative ones up.
3792 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3793 becomes -0.5, but 0.4501 becomes 0.5.
3797 round up if the digit immediately to the right of the rounding place
3798 is 5 or greater, otherwise round down. E.g., 0.15 becomes 0.2 and
3803 The handling of A & P in MBI/MBF (the old core code shipped with Perl
3804 versions <= 5.7.2) is like this:
3810 * ffround($p) is able to round to $p number of digits after the decimal
3812 * otherwise P is unused
3814 =item Accuracy (significant digits)
3816 * fround($a) rounds to $a significant digits
3817 * only fdiv() and fsqrt() take A as (optional) paramater
3818 + other operations simply create the same number (fneg etc), or more (fmul)
3820 + rounding/truncating is only done when explicitly calling one of fround
3821 or ffround, and never for BigInt (not implemented)
3822 * fsqrt() simply hands its accuracy argument over to fdiv.
3823 * the documentation and the comment in the code indicate two different ways
3824 on how fdiv() determines the maximum number of digits it should calculate,
3825 and the actual code does yet another thing
3827 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3829 result has at most max(scale, length(dividend), length(divisor)) digits
3831 scale = max(scale, length(dividend)-1,length(divisor)-1);
3832 scale += length(divisor) - length(dividend);
3833 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3834 Actually, the 'difference' added to the scale is calculated from the
3835 number of "significant digits" in dividend and divisor, which is derived
3836 by looking at the length of the mantissa. Which is wrong, since it includes
3837 the + sign (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops
3838 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3839 assumption that 124 has 3 significant digits, while 120/7 will get you
3840 '17', not '17.1' since 120 is thought to have 2 significant digits.
3841 The rounding after the division then uses the remainder and $y to determine
3842 wether it must round up or down.
3843 ? I have no idea which is the right way. That's why I used a slightly more
3844 ? simple scheme and tweaked the few failing testcases to match it.
3848 This is how it works now:
3852 =item Setting/Accessing
3854 * You can set the A global via C<< Math::BigInt->accuracy() >> or
3855 C<< Math::BigFloat->accuracy() >> or whatever class you are using.
3856 * You can also set P globally by using C<< Math::SomeClass->precision() >>
3858 * Globals are classwide, and not inherited by subclasses.
3859 * to undefine A, use C<< Math::SomeCLass->accuracy(undef); >>
3860 * to undefine P, use C<< Math::SomeClass->precision(undef); >>
3861 * Setting C<< Math::SomeClass->accuracy() >> clears automatically
3862 C<< Math::SomeClass->precision() >>, and vice versa.
3863 * To be valid, A must be > 0, P can have any value.
3864 * If P is negative, this means round to the P'th place to the right of the
3865 decimal point; positive values mean to the left of the decimal point.
3866 P of 0 means round to integer.
3867 * to find out the current global A, use C<< Math::SomeClass->accuracy() >>
3868 * to find out the current global P, use C<< Math::SomeClass->precision() >>
3869 * use C<< $x->accuracy() >> respective C<< $x->precision() >> for the local
3870 setting of C<< $x >>.
3871 * Please note that C<< $x->accuracy() >> respective C<< $x->precision() >>
3872 return eventually defined global A or P, when C<< $x >>'s A or P is not
3875 =item Creating numbers
3877 * When you create a number, you can give the desired A or P via:
3878 $x = Math::BigInt->new($number,$A,$P);
3879 * Only one of A or P can be defined, otherwise the result is NaN
3880 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
3881 globals (if set) will be used. Thus changing the global defaults later on
3882 will not change the A or P of previously created numbers (i.e., A and P of
3883 $x will be what was in effect when $x was created)
3884 * If given undef for A and P, B<no> rounding will occur, and the globals will
3885 B<not> be used. This is used by subclasses to create numbers without
3886 suffering rounding in the parent. Thus a subclass is able to have its own
3887 globals enforced upon creation of a number by using
3888 C<< $x = Math::BigInt->new($number,undef,undef) >>:
3890 use Math::BigInt::SomeSubclass;
3893 Math::BigInt->accuracy(2);
3894 Math::BigInt::SomeSubClass->accuracy(3);
3895 $x = Math::BigInt::SomeSubClass->new(1234);
3897 $x is now 1230, and not 1200. A subclass might choose to implement
3898 this otherwise, e.g. falling back to the parent's A and P.
3902 * If A or P are enabled/defined, they are used to round the result of each
3903 operation according to the rules below
3904 * Negative P is ignored in Math::BigInt, since BigInts never have digits
3905 after the decimal point
3906 * Math::BigFloat uses Math::BigInt internally, but setting A or P inside
3907 Math::BigInt as globals does not tamper with the parts of a BigFloat.
3908 A flag is used to mark all Math::BigFloat numbers as 'never round'.
3912 * It only makes sense that a number has only one of A or P at a time.
3913 If you set either A or P on one object, or globally, the other one will
3914 be automatically cleared.
3915 * If two objects are involved in an operation, and one of them has A in
3916 effect, and the other P, this results in an error (NaN).
3917 * A takes precedence over P (Hint: A comes before P).
3918 If neither of them is defined, nothing is used, i.e. the result will have
3919 as many digits as it can (with an exception for fdiv/fsqrt) and will not
3921 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
3922 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
3923 If either the dividend's or the divisor's mantissa has more digits than
3924 the value of F, the higher value will be used instead of F.
3925 This is to limit the digits (A) of the result (just consider what would
3926 happen with unlimited A and P in the case of 1/3 :-)
3927 * fdiv will calculate (at least) 4 more digits than required (determined by
3928 A, P or F), and, if F is not used, round the result
3929 (this will still fail in the case of a result like 0.12345000000001 with A
3930 or P of 5, but this can not be helped - or can it?)
3931 * Thus you can have the math done by on Math::Big* class in two modi:
3932 + never round (this is the default):
3933 This is done by setting A and P to undef. No math operation
3934 will round the result, with fdiv() and fsqrt() as exceptions to guard
3935 against overflows. You must explicitly call bround(), bfround() or
3936 round() (the latter with parameters).
3937 Note: Once you have rounded a number, the settings will 'stick' on it
3938 and 'infect' all other numbers engaged in math operations with it, since
3939 local settings have the highest precedence. So, to get SaferRound[tm],
3940 use a copy() before rounding like this:
3942 $x = Math::BigFloat->new(12.34);
3943 $y = Math::BigFloat->new(98.76);
3944 $z = $x * $y; # 1218.6984
3945 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
3946 $z = $x * $y; # still 1218.6984, without
3947 # copy would have been 1210!
3949 + round after each op:
3950 After each single operation (except for testing like is_zero()), the
3951 method round() is called and the result is rounded appropriately. By
3952 setting proper values for A and P, you can have all-the-same-A or
3953 all-the-same-P modes. For example, Math::Currency might set A to undef,
3954 and P to -2, globally.
3956 ?Maybe an extra option that forbids local A & P settings would be in order,
3957 ?so that intermediate rounding does not 'poison' further math?
3959 =item Overriding globals
3961 * you will be able to give A, P and R as an argument to all the calculation
3962 routines; the second parameter is A, the third one is P, and the fourth is
3963 R (shift right by one for binary operations like badd). P is used only if
3964 the first parameter (A) is undefined. These three parameters override the
3965 globals in the order detailed as follows, i.e. the first defined value
3967 (local: per object, global: global default, parameter: argument to sub)
3970 + local A (if defined on both of the operands: smaller one is taken)
3971 + local P (if defined on both of the operands: bigger one is taken)
3975 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
3976 arguments (A and P) instead of one
3978 =item Local settings
3980 * You can set A or P locally by using C<< $x->accuracy() >> or
3981 C<< $x->precision() >>
3982 and thus force different A and P for different objects/numbers.
3983 * Setting A or P this way immediately rounds $x to the new value.
3984 * C<< $x->accuracy() >> clears C<< $x->precision() >>, and vice versa.
3988 * the rounding routines will use the respective global or local settings.
3989 fround()/bround() is for accuracy rounding, while ffround()/bfround()
3991 * the two rounding functions take as the second parameter one of the
3992 following rounding modes (R):
3993 'even', 'odd', '+inf', '-inf', 'zero', 'trunc', 'common'
3994 * you can set/get the global R by using C<< Math::SomeClass->round_mode() >>
3995 or by setting C<< $Math::SomeClass::round_mode >>
3996 * after each operation, C<< $result->round() >> is called, and the result may
3997 eventually be rounded (that is, if A or P were set either locally,
3998 globally or as parameter to the operation)
3999 * to manually round a number, call C<< $x->round($A,$P,$round_mode); >>
4000 this will round the number by using the appropriate rounding function
4001 and then normalize it.
4002 * rounding modifies the local settings of the number:
4004 $x = Math::BigFloat->new(123.456);
4008 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
4009 will be 4 from now on.
4011 =item Default values
4020 * The defaults are set up so that the new code gives the same results as
4021 the old code (except in a few cases on fdiv):
4022 + Both A and P are undefined and thus will not be used for rounding
4023 after each operation.
4024 + round() is thus a no-op, unless given extra parameters A and P
4028 =head1 Infinity and Not a Number
4030 While BigInt has extensive handling of inf and NaN, certain quirks remain.
4036 These perl routines currently (as of Perl v.5.8.6) cannot handle passed
4039 te@linux:~> perl -wle 'print 2 ** 3333'
4041 te@linux:~> perl -wle 'print 2 ** 3333 == 2 ** 3333'
4043 te@linux:~> perl -wle 'print oct(2 ** 3333)'
4045 te@linux:~> perl -wle 'print hex(2 ** 3333)'
4046 Illegal hexadecimal digit 'i' ignored at -e line 1.
4049 The same problems occur if you pass them Math::BigInt->binf() objects. Since
4050 overloading these routines is not possible, this cannot be fixed from BigInt.
4052 =item ==, !=, <, >, <=, >= with NaNs
4054 BigInt's bcmp() routine currently returns undef to signal that a NaN was
4055 involved in a comparison. However, the overload code turns that into
4056 either 1 or '' and thus operations like C<< NaN != NaN >> might return
4061 C<< log(-inf) >> is highly weird. Since log(-x)=pi*i+log(x), then
4062 log(-inf)=pi*i+inf. However, since the imaginary part is finite, the real
4063 infinity "overshadows" it, so the number might as well just be infinity.
4064 However, the result is a complex number, and since BigInt/BigFloat can only
4065 have real numbers as results, the result is NaN.
4067 =item exp(), cos(), sin(), atan2()
4069 These all might have problems handling infinity right.
4075 The actual numbers are stored as unsigned big integers (with seperate sign).
4077 You should neither care about nor depend on the internal representation; it
4078 might change without notice. Use B<ONLY> method calls like C<< $x->sign(); >>
4079 instead relying on the internal representation.
4083 Math with the numbers is done (by default) by a module called
4084 C<Math::BigInt::Calc>. This is equivalent to saying:
4086 use Math::BigInt lib => 'Calc';
4088 You can change this by using:
4090 use Math::BigInt lib => 'BitVect';
4092 The following would first try to find Math::BigInt::Foo, then
4093 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
4095 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
4097 Since Math::BigInt::GMP is in almost all cases faster than Calc (especially in
4098 math involving really big numbers, where it is B<much> faster), and there is
4099 no penalty if Math::BigInt::GMP is not installed, it is a good idea to always
4102 use Math::BigInt lib => 'GMP';
4104 Different low-level libraries use different formats to store the
4105 numbers. You should B<NOT> depend on the number having a specific format
4108 See the respective math library module documentation for further details.
4112 The sign is either '+', '-', 'NaN', '+inf' or '-inf'.
4114 A sign of 'NaN' is used to represent the result when input arguments are not
4115 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
4116 minus infinity. You will get '+inf' when dividing a positive number by 0, and
4117 '-inf' when dividing any negative number by 0.
4119 =head2 mantissa(), exponent() and parts()
4121 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
4124 $m = $x->mantissa();
4125 $e = $x->exponent();
4126 $y = $m * ( 10 ** $e );
4127 print "ok\n" if $x == $y;
4129 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
4130 in one go. Both the returned mantissa and exponent have a sign.
4132 Currently, for BigInts C<$e> is always 0, except +inf and -inf, where it is
4133 C<+inf>; and for NaN, where it is C<NaN>; and for C<$x == 0>, where it is C<1>
4134 (to be compatible with Math::BigFloat's internal representation of a zero as
4137 C<$m> is currently just a copy of the original number. The relation between
4138 C<$e> and C<$m> will stay always the same, though their real values might
4145 sub bint { Math::BigInt->new(shift); }
4147 $x = Math::BigInt->bstr("1234") # string "1234"
4148 $x = "$x"; # same as bstr()
4149 $x = Math::BigInt->bneg("1234"); # BigInt "-1234"
4150 $x = Math::BigInt->babs("-12345"); # BigInt "12345"
4151 $x = Math::BigInt->bnorm("-0.00"); # BigInt "0"
4152 $x = bint(1) + bint(2); # BigInt "3"
4153 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
4154 $x = bint(1); # BigInt "1"
4155 $x = $x + 5 / 2; # BigInt "3"
4156 $x = $x ** 3; # BigInt "27"
4157 $x *= 2; # BigInt "54"
4158 $x = Math::BigInt->new(0); # BigInt "0"
4160 $x = Math::BigInt->badd(4,5) # BigInt "9"
4161 print $x->bsstr(); # 9e+0
4163 Examples for rounding:
4168 $x = Math::BigFloat->new(123.4567);
4169 $y = Math::BigFloat->new(123.456789);
4170 Math::BigFloat->accuracy(4); # no more A than 4
4172 ok ($x->copy()->fround(),123.4); # even rounding
4173 print $x->copy()->fround(),"\n"; # 123.4
4174 Math::BigFloat->round_mode('odd'); # round to odd
4175 print $x->copy()->fround(),"\n"; # 123.5
4176 Math::BigFloat->accuracy(5); # no more A than 5
4177 Math::BigFloat->round_mode('odd'); # round to odd
4178 print $x->copy()->fround(),"\n"; # 123.46
4179 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
4180 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
4182 Math::BigFloat->accuracy(undef); # A not important now
4183 Math::BigFloat->precision(2); # P important
4184 print $x->copy()->bnorm(),"\n"; # 123.46
4185 print $x->copy()->fround(),"\n"; # 123.46
4187 Examples for converting:
4189 my $x = Math::BigInt->new('0b1'.'01' x 123);
4190 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
4192 =head1 Autocreating constants
4194 After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
4195 and binary constants in the given scope are converted to C<Math::BigInt>.
4196 This conversion happens at compile time.
4200 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
4202 prints the integer value of C<2**100>. Note that without conversion of
4203 constants the expression 2**100 will be calculated as perl scalar.
4205 Please note that strings and floating point constants are not affected,
4208 use Math::BigInt qw/:constant/;
4210 $x = 1234567890123456789012345678901234567890
4211 + 123456789123456789;
4212 $y = '1234567890123456789012345678901234567890'
4213 + '123456789123456789';
4215 do not work. You need an explicit Math::BigInt->new() around one of the
4216 operands. You should also quote large constants to protect loss of precision:
4220 $x = Math::BigInt->new('1234567889123456789123456789123456789');
4222 Without the quotes Perl would convert the large number to a floating point
4223 constant at compile time and then hand the result to BigInt, which results in
4224 an truncated result or a NaN.
4226 This also applies to integers that look like floating point constants:
4228 use Math::BigInt ':constant';
4230 print ref(123e2),"\n";
4231 print ref(123.2e2),"\n";
4233 will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
4234 to get this to work.
4238 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
4239 must be made in the second case. For long numbers, the copy can eat up to 20%
4240 of the work (in the case of addition/subtraction, less for
4241 multiplication/division). If $y is very small compared to $x, the form
4242 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
4243 more time then the actual addition.
4245 With a technique called copy-on-write, the cost of copying with overload could
4246 be minimized or even completely avoided. A test implementation of COW did show
4247 performance gains for overloaded math, but introduced a performance loss due
4248 to a constant overhead for all other operations. So Math::BigInt does currently
4251 The rewritten version of this module (vs. v0.01) is slower on certain
4252 operations, like C<new()>, C<bstr()> and C<numify()>. The reason are that it
4253 does now more work and handles much more cases. The time spent in these
4254 operations is usually gained in the other math operations so that code on
4255 the average should get (much) faster. If they don't, please contact the author.
4257 Some operations may be slower for small numbers, but are significantly faster
4258 for big numbers. Other operations are now constant (O(1), like C<bneg()>,
4259 C<babs()> etc), instead of O(N) and thus nearly always take much less time.
4260 These optimizations were done on purpose.
4262 If you find the Calc module to slow, try to install any of the replacement
4263 modules and see if they help you.
4265 =head2 Alternative math libraries
4267 You can use an alternative library to drive Math::BigInt via:
4269 use Math::BigInt lib => 'Module';
4271 See L<MATH LIBRARY> for more information.
4273 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
4277 =head1 Subclassing Math::BigInt
4279 The basic design of Math::BigInt allows simple subclasses with very little
4280 work, as long as a few simple rules are followed:
4286 The public API must remain consistent, i.e. if a sub-class is overloading
4287 addition, the sub-class must use the same name, in this case badd(). The
4288 reason for this is that Math::BigInt is optimized to call the object methods
4293 The private object hash keys like C<$x->{sign}> may not be changed, but
4294 additional keys can be added, like C<$x->{_custom}>.
4298 Accessor functions are available for all existing object hash keys and should
4299 be used instead of directly accessing the internal hash keys. The reason for
4300 this is that Math::BigInt itself has a pluggable interface which permits it
4301 to support different storage methods.
4305 More complex sub-classes may have to replicate more of the logic internal of
4306 Math::BigInt if they need to change more basic behaviors. A subclass that
4307 needs to merely change the output only needs to overload C<bstr()>.
4309 All other object methods and overloaded functions can be directly inherited
4310 from the parent class.
4312 At the very minimum, any subclass will need to provide its own C<new()> and can
4313 store additional hash keys in the object. There are also some package globals
4314 that must be defined, e.g.:
4318 $precision = -2; # round to 2 decimal places
4319 $round_mode = 'even';
4322 Additionally, you might want to provide the following two globals to allow
4323 auto-upgrading and auto-downgrading to work correctly:
4328 This allows Math::BigInt to correctly retrieve package globals from the
4329 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
4330 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
4336 in your subclass to automatically inherit the overloading from the parent. If
4337 you like, you can change part of the overloading, look at Math::String for an
4342 When used like this:
4344 use Math::BigInt upgrade => 'Foo::Bar';
4346 certain operations will 'upgrade' their calculation and thus the result to
4347 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
4349 use Math::BigInt upgrade => 'Math::BigFloat';
4351 As a shortcut, you can use the module C<bignum>:
4355 Also good for oneliners:
4357 perl -Mbignum -le 'print 2 ** 255'
4359 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
4360 as well es preserve accuracy (as in sqrt(3)).
4362 Beware: This feature is not fully implemented yet.
4366 The following methods upgrade themselves unconditionally; that is if upgrade
4367 is in effect, they will always hand up their work:
4381 Beware: This list is not complete.
4383 All other methods upgrade themselves only when one (or all) of their
4384 arguments are of the class mentioned in $upgrade (This might change in later
4385 versions to a more sophisticated scheme):
4391 =item broot() does not work
4393 The broot() function in BigInt may only work for small values. This will be
4394 fixed in a later version.
4396 =item Out of Memory!
4398 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
4399 C<eval()> in your code will crash with "Out of memory". This is probably an
4400 overload/exporter bug. You can workaround by not having C<eval()>
4401 and ':constant' at the same time or upgrade your Perl to a newer version.
4403 =item Fails to load Calc on Perl prior 5.6.0
4405 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
4406 will fall back to eval { require ... } when loading the math lib on Perls
4407 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
4408 filesystems using a different seperator.
4414 Some things might not work as you expect them. Below is documented what is
4415 known to be troublesome:
4419 =item bstr(), bsstr() and 'cmp'
4421 Both C<bstr()> and C<bsstr()> as well as automated stringify via overload now
4422 drop the leading '+'. The old code would return '+3', the new returns '3'.
4423 This is to be consistent with Perl and to make C<cmp> (especially with
4424 overloading) to work as you expect. It also solves problems with C<Test.pm>,
4425 because its C<ok()> uses 'eq' internally.
4427 Mark Biggar said, when asked about to drop the '+' altogether, or make only
4430 I agree (with the first alternative), don't add the '+' on positive
4431 numbers. It's not as important anymore with the new internal
4432 form for numbers. It made doing things like abs and neg easier,
4433 but those have to be done differently now anyway.
4435 So, the following examples will now work all as expected:
4438 BEGIN { plan tests => 1 }
4441 my $x = new Math::BigInt 3*3;
4442 my $y = new Math::BigInt 3*3;
4445 print "$x eq 9" if $x eq $y;
4446 print "$x eq 9" if $x eq '9';
4447 print "$x eq 9" if $x eq 3*3;
4449 Additionally, the following still works:
4451 print "$x == 9" if $x == $y;
4452 print "$x == 9" if $x == 9;
4453 print "$x == 9" if $x == 3*3;
4455 There is now a C<bsstr()> method to get the string in scientific notation aka
4456 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
4457 for comparison, but Perl will represent some numbers as 100 and others
4458 as 1e+308. If in doubt, convert both arguments to Math::BigInt before
4459 comparing them as strings:
4462 BEGIN { plan tests => 3 }
4465 $x = Math::BigInt->new('1e56'); $y = 1e56;
4466 ok ($x,$y); # will fail
4467 ok ($x->bsstr(),$y); # okay
4468 $y = Math::BigInt->new($y);
4471 Alternatively, simple use C<< <=> >> for comparisons, this will get it
4472 always right. There is not yet a way to get a number automatically represented
4473 as a string that matches exactly the way Perl represents it.
4475 See also the section about L<Infinity and Not a Number> for problems in
4480 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
4483 $x = Math::BigInt->new(123);
4484 $y = int($x); # BigInt 123
4485 $x = Math::BigFloat->new(123.45);
4486 $y = int($x); # BigInt 123
4488 In all Perl versions you can use C<as_number()> or C<as_int> for the same
4491 $x = Math::BigFloat->new(123.45);
4492 $y = $x->as_number(); # BigInt 123
4493 $y = $x->as_int(); # ditto
4495 This also works for other subclasses, like Math::String.
4497 If you want a real Perl scalar, use C<numify()>:
4499 $y = $x->numify(); # 123 as scalar
4501 This is seldom necessary, though, because this is done automatically, like
4502 when you access an array:
4504 $z = $array[$x]; # does work automatically
4508 The following will probably not do what you expect:
4510 $c = Math::BigInt->new(123);
4511 print $c->length(),"\n"; # prints 30
4513 It prints both the number of digits in the number and in the fraction part
4514 since print calls C<length()> in list context. Use something like:
4516 print scalar $c->length(),"\n"; # prints 3
4520 The following will probably not do what you expect:
4522 print $c->bdiv(10000),"\n";
4524 It prints both quotient and remainder since print calls C<bdiv()> in list
4525 context. Also, C<bdiv()> will modify $c, so be careful. You probably want
4528 print $c / 10000,"\n";
4529 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
4533 The quotient is always the greatest integer less than or equal to the
4534 real-valued quotient of the two operands, and the remainder (when it is
4535 nonzero) always has the same sign as the second operand; so, for
4545 As a consequence, the behavior of the operator % agrees with the
4546 behavior of Perl's built-in % operator (as documented in the perlop
4547 manpage), and the equation
4549 $x == ($x / $y) * $y + ($x % $y)
4551 holds true for any $x and $y, which justifies calling the two return
4552 values of bdiv() the quotient and remainder. The only exception to this rule
4553 are when $y == 0 and $x is negative, then the remainder will also be
4554 negative. See below under "infinity handling" for the reasoning behind this.
4556 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
4557 not change BigInt's way to do things. This is because under 'use integer' Perl
4558 will do what the underlying C thinks is right and this is different for each
4559 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
4560 the author to implement it ;)
4562 =item infinity handling
4564 Here are some examples that explain the reasons why certain results occur while
4567 The following table shows the result of the division and the remainder, so that
4568 the equation above holds true. Some "ordinary" cases are strewn in to show more
4569 clearly the reasoning:
4571 A / B = C, R so that C * B + R = A
4572 =========================================================
4573 5 / 8 = 0, 5 0 * 8 + 5 = 5
4574 0 / 8 = 0, 0 0 * 8 + 0 = 0
4575 0 / inf = 0, 0 0 * inf + 0 = 0
4576 0 /-inf = 0, 0 0 * -inf + 0 = 0
4577 5 / inf = 0, 5 0 * inf + 5 = 5
4578 5 /-inf = 0, 5 0 * -inf + 5 = 5
4579 -5/ inf = 0, -5 0 * inf + -5 = -5
4580 -5/-inf = 0, -5 0 * -inf + -5 = -5
4581 inf/ 5 = inf, 0 inf * 5 + 0 = inf
4582 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
4583 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
4584 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
4585 5/ 5 = 1, 0 1 * 5 + 0 = 5
4586 -5/ -5 = 1, 0 1 * -5 + 0 = -5
4587 inf/ inf = 1, 0 1 * inf + 0 = inf
4588 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
4589 inf/-inf = -1, 0 -1 * -inf + 0 = inf
4590 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
4591 8/ 0 = inf, 8 inf * 0 + 8 = 8
4592 inf/ 0 = inf, inf inf * 0 + inf = inf
4595 These cases below violate the "remainder has the sign of the second of the two
4596 arguments", since they wouldn't match up otherwise.
4598 A / B = C, R so that C * B + R = A
4599 ========================================================
4600 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
4601 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
4603 =item Modifying and =
4607 $x = Math::BigFloat->new(5);
4610 It will not do what you think, e.g. making a copy of $x. Instead it just makes
4611 a second reference to the B<same> object and stores it in $y. Thus anything
4612 that modifies $x (except overloaded operators) will modify $y, and vice versa.
4613 Or in other words, C<=> is only safe if you modify your BigInts only via
4614 overloaded math. As soon as you use a method call it breaks:
4617 print "$x, $y\n"; # prints '10, 10'
4619 If you want a true copy of $x, use:
4623 You can also chain the calls like this, this will make first a copy and then
4626 $y = $x->copy()->bmul(2);
4628 See also the documentation for overload.pm regarding C<=>.
4632 C<bpow()> (and the rounding functions) now modifies the first argument and
4633 returns it, unlike the old code which left it alone and only returned the
4634 result. This is to be consistent with C<badd()> etc. The first three will
4635 modify $x, the last one won't:
4637 print bpow($x,$i),"\n"; # modify $x
4638 print $x->bpow($i),"\n"; # ditto
4639 print $x **= $i,"\n"; # the same
4640 print $x ** $i,"\n"; # leave $x alone
4642 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4644 =item Overloading -$x
4654 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4655 needs to preserve $x since it does not know that it later will get overwritten.
4656 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4658 =item Mixing different object types
4660 In Perl you will get a floating point value if you do one of the following:
4666 With overloaded math, only the first two variants will result in a BigFloat:
4671 $mbf = Math::BigFloat->new(5);
4672 $mbi2 = Math::BigInteger->new(5);
4673 $mbi = Math::BigInteger->new(2);
4675 # what actually gets called:
4676 $float = $mbf + $mbi; # $mbf->badd()
4677 $float = $mbf / $mbi; # $mbf->bdiv()
4678 $integer = $mbi + $mbf; # $mbi->badd()
4679 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4680 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4682 This is because math with overloaded operators follows the first (dominating)
4683 operand, and the operation of that is called and returns thus the result. So,
4684 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4685 the result should be a Math::BigFloat or the second operant is one.
4687 To get a Math::BigFloat you either need to call the operation manually,
4688 make sure the operands are already of the proper type or casted to that type
4689 via Math::BigFloat->new():
4691 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4693 Beware of simple "casting" the entire expression, this would only convert
4694 the already computed result:
4696 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4698 Beware also of the order of more complicated expressions like:
4700 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4701 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4703 If in doubt, break the expression into simpler terms, or cast all operands
4704 to the desired resulting type.
4706 Scalar values are a bit different, since:
4711 will both result in the proper type due to the way the overloaded math works.
4713 This section also applies to other overloaded math packages, like Math::String.
4715 One solution to you problem might be autoupgrading|upgrading. See the
4716 pragmas L<bignum>, L<bigint> and L<bigrat> for an easy way to do this.
4720 C<bsqrt()> works only good if the result is a big integer, e.g. the square
4721 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4722 mode. The reason is that the result is always truncated to an integer.
4724 If you want a better approximation of the square root, then use:
4726 $x = Math::BigFloat->new(12);
4727 Math::BigFloat->precision(0);
4728 Math::BigFloat->round_mode('even');
4729 print $x->copy->bsqrt(),"\n"; # 4
4731 Math::BigFloat->precision(2);
4732 print $x->bsqrt(),"\n"; # 3.46
4733 print $x->bsqrt(3),"\n"; # 3.464
4737 For negative numbers in base see also L<brsft|brsft>.
4743 This program is free software; you may redistribute it and/or modify it under
4744 the same terms as Perl itself.
4748 L<Math::BigFloat>, L<Math::BigRat> and L<Math::Big> as well as
4749 L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4751 The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest
4752 because they solve the autoupgrading/downgrading issue, at least partly.
4755 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4756 more documentation including a full version history, testcases, empty
4757 subclass files and benchmarks.
4761 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4762 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001 - 2006
4763 and still at it in 2007.
4765 Many people contributed in one or more ways to the final beast, see the file
4766 CREDITS for an (incomplete) list. If you miss your name, please drop me a