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 ##############################################################################
478 # if two arguments, the first one is the class to "swallow" subclasses
482 sign => $_[1]->{sign},
483 value => $CALC->_copy($_[1]->{value}),
486 $self->{_a} = $_[1]->{_a} if defined $_[1]->{_a};
487 $self->{_p} = $_[1]->{_p} if defined $_[1]->{_p};
492 sign => $_[0]->{sign},
493 value => $CALC->_copy($_[0]->{value}),
496 $self->{_a} = $_[0]->{_a} if defined $_[0]->{_a};
497 $self->{_p} = $_[0]->{_p} if defined $_[0]->{_p};
503 # create a new BigInt object from a string or another BigInt object.
504 # see hash keys documented at top
506 # the argument could be an object, so avoid ||, && etc on it, this would
507 # cause costly overloaded code to be called. The only allowed ops are
510 my ($class,$wanted,$a,$p,$r) = @_;
512 # avoid numify-calls by not using || on $wanted!
513 return $class->bzero($a,$p) if !defined $wanted; # default to 0
514 return $class->copy($wanted,$a,$p,$r)
515 if ref($wanted) && $wanted->isa($class); # MBI or subclass
517 $class->import() if $IMPORT == 0; # make require work
519 my $self = bless {}, $class;
521 # shortcut for "normal" numbers
522 if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*\z/))
524 $self->{sign} = $1 || '+';
526 if ($wanted =~ /^[+-]/)
528 # remove sign without touching wanted to make it work with constants
529 my $t = $wanted; $t =~ s/^[+-]//;
530 $self->{value} = $CALC->_new($t);
534 $self->{value} = $CALC->_new($wanted);
537 if ( (defined $a) || (defined $p)
538 || (defined ${"${class}::precision"})
539 || (defined ${"${class}::accuracy"})
542 $self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p);
547 # handle '+inf', '-inf' first
548 if ($wanted =~ /^[+-]?inf\z/)
550 $self->{sign} = $wanted; # set a default sign for bstr()
551 return $self->binf($wanted);
553 # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign
554 my ($mis,$miv,$mfv,$es,$ev) = _split($wanted);
559 require Carp; Carp::croak("$wanted is not a number in $class");
561 $self->{value} = $CALC->_zero();
562 $self->{sign} = $nan;
567 # _from_hex or _from_bin
568 $self->{value} = $mis->{value};
569 $self->{sign} = $mis->{sign};
570 return $self; # throw away $mis
572 # make integer from mantissa by adjusting exp, then convert to bigint
573 $self->{sign} = $$mis; # store sign
574 $self->{value} = $CALC->_zero(); # for all the NaN cases
575 my $e = int("$$es$$ev"); # exponent (avoid recursion)
578 my $diff = $e - CORE::length($$mfv);
579 if ($diff < 0) # Not integer
583 require Carp; Carp::croak("$wanted not an integer in $class");
586 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
587 $self->{sign} = $nan;
591 # adjust fraction and add it to value
592 #print "diff > 0 $$miv\n";
593 $$miv = $$miv . ($$mfv . '0' x $diff);
598 if ($$mfv ne '') # e <= 0
600 # fraction and negative/zero E => NOI
603 require Carp; Carp::croak("$wanted not an integer in $class");
605 #print "NOI 2 \$\$mfv '$$mfv'\n";
606 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
607 $self->{sign} = $nan;
611 # xE-y, and empty mfv
614 if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
618 require Carp; Carp::croak("$wanted not an integer in $class");
621 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
622 $self->{sign} = $nan;
626 $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
627 $self->{value} = $CALC->_new($$miv) if $self->{sign} =~ /^[+-]$/;
628 # if any of the globals is set, use them to round and store them inside $self
629 # do not round for new($x,undef,undef) since that is used by MBF to signal
631 $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p;
637 # create a bigint 'NaN', if given a BigInt, set it to 'NaN'
639 $self = $class if !defined $self;
642 my $c = $self; $self = {}; bless $self, $c;
645 if (${"${class}::_trap_nan"})
648 Carp::croak ("Tried to set $self to NaN in $class\::bnan()");
650 $self->import() if $IMPORT == 0; # make require work
651 return if $self->modify('bnan');
652 if ($self->can('_bnan'))
654 # use subclass to initialize
659 # otherwise do our own thing
660 $self->{value} = $CALC->_zero();
662 $self->{sign} = $nan;
663 delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly
669 # create a bigint '+-inf', if given a BigInt, set it to '+-inf'
670 # the sign is either '+', or if given, used from there
672 my $sign = shift; $sign = '+' if !defined $sign || $sign !~ /^-(inf)?$/;
673 $self = $class if !defined $self;
676 my $c = $self; $self = {}; bless $self, $c;
679 if (${"${class}::_trap_inf"})
682 Carp::croak ("Tried to set $self to +-inf in $class\::binf()");
684 $self->import() if $IMPORT == 0; # make require work
685 return if $self->modify('binf');
686 if ($self->can('_binf'))
688 # use subclass to initialize
693 # otherwise do our own thing
694 $self->{value} = $CALC->_zero();
696 $sign = $sign . 'inf' if $sign !~ /inf$/; # - => -inf
697 $self->{sign} = $sign;
698 ($self->{_a},$self->{_p}) = @_; # take over requested rounding
704 # create a bigint '+0', if given a BigInt, set it to 0
706 $self = __PACKAGE__ if !defined $self;
710 my $c = $self; $self = {}; bless $self, $c;
712 $self->import() if $IMPORT == 0; # make require work
713 return if $self->modify('bzero');
715 if ($self->can('_bzero'))
717 # use subclass to initialize
722 # otherwise do our own thing
723 $self->{value} = $CALC->_zero();
730 # call like: $x->bzero($a,$p,$r,$y);
731 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
736 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
738 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
746 # create a bigint '+1' (or -1 if given sign '-'),
747 # if given a BigInt, set it to +1 or -1, respectively
749 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
750 $self = $class if !defined $self;
754 my $c = $self; $self = {}; bless $self, $c;
756 $self->import() if $IMPORT == 0; # make require work
757 return if $self->modify('bone');
759 if ($self->can('_bone'))
761 # use subclass to initialize
766 # otherwise do our own thing
767 $self->{value} = $CALC->_one();
769 $self->{sign} = $sign;
774 # call like: $x->bone($sign,$a,$p,$r,$y);
775 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
779 # call like: $x->bone($sign,$a,$p,$r);
781 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
783 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
789 ##############################################################################
790 # string conversation
794 # (ref to BFLOAT or num_str ) return num_str
795 # Convert number from internal format to scientific string format.
796 # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
797 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
799 if ($x->{sign} !~ /^[+-]$/)
801 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
804 my ($m,$e) = $x->parts();
805 #$m->bstr() . 'e+' . $e->bstr(); # e can only be positive in BigInt
806 # 'e+' because E can only be positive in BigInt
807 $m->bstr() . 'e+' . $CALC->_str($e->{value});
812 # make a string from bigint object
813 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
815 if ($x->{sign} !~ /^[+-]$/)
817 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
820 my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
821 $es.$CALC->_str($x->{value});
826 # Make a "normal" scalar from a BigInt object
827 my $x = shift; $x = $class->new($x) unless ref $x;
829 return $x->bstr() if $x->{sign} !~ /^[+-]$/;
830 my $num = $CALC->_num($x->{value});
831 return -$num if $x->{sign} eq '-';
835 ##############################################################################
836 # public stuff (usually prefixed with "b")
840 # return the sign of the number: +/-/-inf/+inf/NaN
841 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
846 sub _find_round_parameters
848 # After any operation or when calling round(), the result is rounded by
849 # regarding the A & P from arguments, local parameters, or globals.
851 # !!!!!!! If you change this, remember to change round(), too! !!!!!!!!!!
853 # This procedure finds the round parameters, but it is for speed reasons
854 # duplicated in round. Otherwise, it is tested by the testsuite and used
857 # returns ($self) or ($self,$a,$p,$r) - sets $self to NaN of both A and P
858 # were requested/defined (locally or globally or both)
860 my ($self,$a,$p,$r,@args) = @_;
861 # $a accuracy, if given by caller
862 # $p precision, if given by caller
863 # $r round_mode, if given by caller
864 # @args all 'other' arguments (0 for unary, 1 for binary ops)
866 my $c = ref($self); # find out class of argument(s)
869 # convert to normal scalar for speed and correctness in inner parts
870 $a = $a->can('numify') ? $a->numify() : "$a" if defined $a && ref($a);
871 $p = $p->can('numify') ? $p->numify() : "$p" if defined $p && ref($p);
873 # now pick $a or $p, but only if we have got "arguments"
876 foreach ($self,@args)
878 # take the defined one, or if both defined, the one that is smaller
879 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
884 # even if $a is defined, take $p, to signal error for both defined
885 foreach ($self,@args)
887 # take the defined one, or if both defined, the one that is bigger
889 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
892 # if still none defined, use globals (#2)
893 $a = ${"$c\::accuracy"} unless defined $a;
894 $p = ${"$c\::precision"} unless defined $p;
896 # A == 0 is useless, so undef it to signal no rounding
897 $a = undef if defined $a && $a == 0;
900 return ($self) unless defined $a || defined $p; # early out
902 # set A and set P is an fatal error
903 return ($self->bnan()) if defined $a && defined $p; # error
905 $r = ${"$c\::round_mode"} unless defined $r;
906 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc|common)$/)
908 require Carp; Carp::croak ("Unknown round mode '$r'");
916 # Round $self according to given parameters, or given second argument's
917 # parameters or global defaults
919 # for speed reasons, _find_round_parameters is embeded here:
921 my ($self,$a,$p,$r,@args) = @_;
922 # $a accuracy, if given by caller
923 # $p precision, if given by caller
924 # $r round_mode, if given by caller
925 # @args all 'other' arguments (0 for unary, 1 for binary ops)
927 my $c = ref($self); # find out class of argument(s)
930 # now pick $a or $p, but only if we have got "arguments"
933 foreach ($self,@args)
935 # take the defined one, or if both defined, the one that is smaller
936 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
941 # even if $a is defined, take $p, to signal error for both defined
942 foreach ($self,@args)
944 # take the defined one, or if both defined, the one that is bigger
946 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
949 # if still none defined, use globals (#2)
950 $a = ${"$c\::accuracy"} unless defined $a;
951 $p = ${"$c\::precision"} unless defined $p;
953 # A == 0 is useless, so undef it to signal no rounding
954 $a = undef if defined $a && $a == 0;
957 return $self unless defined $a || defined $p; # early out
959 # set A and set P is an fatal error
960 return $self->bnan() if defined $a && defined $p;
962 $r = ${"$c\::round_mode"} unless defined $r;
963 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc|common)$/)
965 require Carp; Carp::croak ("Unknown round mode '$r'");
968 # now round, by calling either fround or ffround:
971 $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a;
973 else # both can't be undefined due to early out
975 $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p;
977 # bround() or bfround() already callled bnorm() if nec.
983 # (numstr or BINT) return BINT
984 # Normalize number -- no-op here
985 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
991 # (BINT or num_str) return BINT
992 # make number absolute, or return absolute BINT from string
993 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
995 return $x if $x->modify('babs');
996 # post-normalized abs for internal use (does nothing for NaN)
997 $x->{sign} =~ s/^-/+/;
1003 # (BINT or num_str) return BINT
1004 # negate number or make a negated number from string
1005 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1007 return $x if $x->modify('bneg');
1009 # for +0 dont negate (to have always normalized +0). Does nothing for 'NaN'
1010 $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $CALC->_is_zero($x->{value}));
1016 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
1017 # (BINT or num_str, BINT or num_str) return cond_code
1020 my ($self,$x,$y) = (ref($_[0]),@_);
1022 # objectify is costly, so avoid it
1023 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1025 ($self,$x,$y) = objectify(2,@_);
1028 return $upgrade->bcmp($x,$y) if defined $upgrade &&
1029 ((!$x->isa($self)) || (!$y->isa($self)));
1031 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1033 # handle +-inf and NaN
1034 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1035 return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
1036 return +1 if $x->{sign} eq '+inf';
1037 return -1 if $x->{sign} eq '-inf';
1038 return -1 if $y->{sign} eq '+inf';
1041 # check sign for speed first
1042 return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
1043 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
1045 # have same sign, so compare absolute values. Don't make tests for zero here
1046 # because it's actually slower than testin in Calc (especially w/ Pari et al)
1048 # post-normalized compare for internal use (honors signs)
1049 if ($x->{sign} eq '+')
1051 # $x and $y both > 0
1052 return $CALC->_acmp($x->{value},$y->{value});
1056 $CALC->_acmp($y->{value},$x->{value}); # swaped acmp (lib returns 0,1,-1)
1061 # Compares 2 values, ignoring their signs.
1062 # Returns one of undef, <0, =0, >0. (suitable for sort)
1063 # (BINT, BINT) return cond_code
1066 my ($self,$x,$y) = (ref($_[0]),@_);
1067 # objectify is costly, so avoid it
1068 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1070 ($self,$x,$y) = objectify(2,@_);
1073 return $upgrade->bacmp($x,$y) if defined $upgrade &&
1074 ((!$x->isa($self)) || (!$y->isa($self)));
1076 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1078 # handle +-inf and NaN
1079 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1080 return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
1081 return 1 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} !~ /^[+-]inf$/;
1084 $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
1089 # add second arg (BINT or string) to first (BINT) (modifies first)
1090 # return result as BINT
1093 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1094 # objectify is costly, so avoid it
1095 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1097 ($self,$x,$y,@r) = objectify(2,@_);
1100 return $x if $x->modify('badd');
1101 return $upgrade->badd($upgrade->new($x),$upgrade->new($y),@r) if defined $upgrade &&
1102 ((!$x->isa($self)) || (!$y->isa($self)));
1104 $r[3] = $y; # no push!
1105 # inf and NaN handling
1106 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1109 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1111 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1113 # +inf++inf or -inf+-inf => same, rest is NaN
1114 return $x if $x->{sign} eq $y->{sign};
1117 # +-inf + something => +inf
1118 # something +-inf => +-inf
1119 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
1123 my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
1127 $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
1131 my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
1134 $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
1139 # speedup, if equal, set result to 0
1140 $x->{value} = $CALC->_zero();
1145 $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
1153 # (BINT or num_str, BINT or num_str) return BINT
1154 # subtract second arg from first, modify first
1157 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1159 # objectify is costly, so avoid it
1160 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1162 ($self,$x,$y,@r) = objectify(2,@_);
1165 return $x if $x->modify('bsub');
1167 return $upgrade->new($x)->bsub($upgrade->new($y),@r) if defined $upgrade &&
1168 ((!$x->isa($self)) || (!$y->isa($self)));
1170 return $x->round(@r) if $y->is_zero();
1172 # To correctly handle the lone special case $x->bsub($x), we note the sign
1173 # of $x, then flip the sign from $y, and if the sign of $x did change, too,
1174 # then we caught the special case:
1175 my $xsign = $x->{sign};
1176 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
1177 if ($xsign ne $x->{sign})
1179 # special case of $x->bsub($x) results in 0
1180 return $x->bzero(@r) if $xsign =~ /^[+-]$/;
1181 return $x->bnan(); # NaN, -inf, +inf
1183 $x->badd($y,@r); # badd does not leave internal zeros
1184 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
1185 $x; # already rounded by badd() or no round nec.
1190 # increment arg by one
1191 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1192 return $x if $x->modify('binc');
1194 if ($x->{sign} eq '+')
1196 $x->{value} = $CALC->_inc($x->{value});
1197 return $x->round($a,$p,$r);
1199 elsif ($x->{sign} eq '-')
1201 $x->{value} = $CALC->_dec($x->{value});
1202 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1203 return $x->round($a,$p,$r);
1205 # inf, nan handling etc
1206 $x->badd($self->bone(),$a,$p,$r); # badd does round
1211 # decrement arg by one
1212 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1213 return $x if $x->modify('bdec');
1215 if ($x->{sign} eq '-')
1218 $x->{value} = $CALC->_inc($x->{value});
1222 return $x->badd($self->bone('-'),@r) unless $x->{sign} eq '+'; # inf or NaN
1224 if ($CALC->_is_zero($x->{value}))
1227 $x->{value} = $CALC->_one(); $x->{sign} = '-'; # 0 => -1
1232 $x->{value} = $CALC->_dec($x->{value});
1240 # calculate $x = $a ** $base + $b and return $a (e.g. the log() to base
1244 my ($self,$x,$base,@r) = (undef,@_);
1245 # objectify is costly, so avoid it
1246 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1248 ($self,$x,$base,@r) = objectify(1,ref($x),@_);
1251 return $x if $x->modify('blog');
1253 $base = $self->new($base) if defined $base && !ref $base;
1255 # inf, -inf, NaN, <0 => NaN
1257 if $x->{sign} ne '+' || (defined $base && $base->{sign} ne '+');
1259 return $upgrade->blog($upgrade->new($x),$base,@r) if
1262 # fix for bug #24969:
1263 # the default base is e (Euler's number) which is not an integer
1266 require Math::BigFloat;
1267 my $u = Math::BigFloat->blog(Math::BigFloat->new($x))->as_int();
1268 # modify $x in place
1269 $x->{value} = $u->{value};
1270 $x->{sign} = $u->{sign};
1274 my ($rc,$exact) = $CALC->_log_int($x->{value},$base->{value});
1275 return $x->bnan() unless defined $rc; # not possible to take log?
1282 # Calculate n over k (binomial coefficient or "choose" function) as integer.
1284 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1286 # objectify is costly, so avoid it
1287 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1289 ($self,$x,$y,@r) = objectify(2,@_);
1292 return $x if $x->modify('bnok');
1293 return $x->bnan() if $x->{sign} eq 'NaN' || $y->{sign} eq 'NaN';
1294 return $x->binf() if $x->{sign} eq '+inf';
1296 # k > n or k < 0 => 0
1297 my $cmp = $x->bacmp($y);
1298 return $x->bzero() if $cmp < 0 || $y->{sign} =~ /^-/;
1300 return $x->bone(@r) if $cmp == 0;
1302 if ($CALC->can('_nok'))
1304 $x->{value} = $CALC->_nok($x->{value},$y->{value});
1308 # ( 7 ) 7! 7*6*5 * 4*3*2*1 7 * 6 * 5
1309 # ( - ) = --------- = --------------- = ---------
1310 # ( 3 ) 3! (7-3)! 3*2*1 * 4*3*2*1 3 * 2 * 1
1312 # compute n - k + 2 (so we start with 5 in the example above)
1317 my $r = $z->copy(); $z->binc();
1318 my $d = $self->new(2);
1319 while ($z->bacmp($x) <= 0) # f < x ?
1321 $r->bmul($z); $r->bdiv($d);
1322 $z->binc(); $d->binc();
1324 $x->{value} = $r->{value}; $x->{sign} = '+';
1326 else { $x->bone(); }
1333 # Calculate e ** $x (Euler's number to the power of X), truncated to
1335 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1336 return $x if $x->modify('bexp');
1338 # inf, -inf, NaN, <0 => NaN
1339 return $x->bnan() if $x->{sign} eq 'NaN';
1340 return $x->bone() if $x->is_zero();
1341 return $x if $x->{sign} eq '+inf';
1342 return $x->bzero() if $x->{sign} eq '-inf';
1346 # run through Math::BigFloat unless told otherwise
1347 require Math::BigFloat unless defined $upgrade;
1348 local $upgrade = 'Math::BigFloat' unless defined $upgrade;
1349 # calculate result, truncate it to integer
1350 $u = $upgrade->bexp($upgrade->new($x),@r);
1353 if (!defined $upgrade)
1356 # modify $x in place
1357 $x->{value} = $u->{value};
1365 # (BINT or num_str, BINT or num_str) return BINT
1366 # does not modify arguments, but returns new object
1367 # Lowest Common Multiplicator
1369 my $y = shift; my ($x);
1376 $x = $class->new($y);
1381 my $y = shift; $y = $self->new($y) if !ref ($y);
1389 # (BINT or num_str, BINT or num_str) return BINT
1390 # does not modify arguments, but returns new object
1391 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1394 $y = $class->new($y) if !ref($y);
1396 my $x = $y->copy()->babs(); # keep arguments
1397 return $x->bnan() if $x->{sign} !~ /^[+-]$/; # x NaN?
1401 $y = shift; $y = $self->new($y) if !ref($y);
1402 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1403 $x->{value} = $CALC->_gcd($x->{value},$y->{value});
1404 last if $CALC->_is_one($x->{value});
1411 # (num_str or BINT) return BINT
1412 # represent ~x as twos-complement number
1413 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1414 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1416 return $x if $x->modify('bnot');
1417 $x->binc()->bneg(); # binc already does round
1420 ##############################################################################
1421 # is_foo test routines
1422 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1426 # return true if arg (BINT or num_str) is zero (array '+', '0')
1427 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1429 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1430 $CALC->_is_zero($x->{value});
1435 # return true if arg (BINT or num_str) is NaN
1436 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1438 $x->{sign} eq $nan ? 1 : 0;
1443 # return true if arg (BINT or num_str) is +-inf
1444 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1448 $sign = '[+-]inf' if $sign eq ''; # +- doesn't matter, only that's inf
1449 $sign = "[$1]inf" if $sign =~ /^([+-])(inf)?$/; # extract '+' or '-'
1450 return $x->{sign} =~ /^$sign$/ ? 1 : 0;
1452 $x->{sign} =~ /^[+-]inf$/ ? 1 : 0; # only +-inf is infinity
1457 # return true if arg (BINT or num_str) is +1, or -1 if sign is given
1458 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1460 $sign = '+' if !defined $sign || $sign ne '-';
1462 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1463 $CALC->_is_one($x->{value});
1468 # return true when arg (BINT or num_str) is odd, false for even
1469 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1471 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1472 $CALC->_is_odd($x->{value});
1477 # return true when arg (BINT or num_str) is even, false for odd
1478 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1480 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1481 $CALC->_is_even($x->{value});
1486 # return true when arg (BINT or num_str) is positive (>= 0)
1487 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1489 return 1 if $x->{sign} eq '+inf'; # +inf is positive
1491 # 0+ is neither positive nor negative
1492 ($x->{sign} eq '+' && !$x->is_zero()) ? 1 : 0;
1497 # return true when arg (BINT or num_str) is negative (< 0)
1498 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1500 $x->{sign} =~ /^-/ ? 1 : 0; # -inf is negative, but NaN is not
1505 # return true when arg (BINT or num_str) is an integer
1506 # always true for BigInt, but different for BigFloats
1507 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1509 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1512 ###############################################################################
1516 # multiply two numbers -- stolen from Knuth Vol 2 pg 233
1517 # (BINT or num_str, BINT or num_str) return BINT
1520 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1521 # objectify is costly, so avoid it
1522 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1524 ($self,$x,$y,@r) = objectify(2,@_);
1527 return $x if $x->modify('bmul');
1529 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1532 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1534 return $x->bnan() if $x->is_zero() || $y->is_zero();
1535 # result will always be +-inf:
1536 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1537 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1538 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1539 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1540 return $x->binf('-');
1543 return $upgrade->bmul($x,$upgrade->new($y),@r)
1544 if defined $upgrade && !$y->isa($self);
1546 $r[3] = $y; # no push here
1548 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1550 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1551 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1558 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1559 my ($self,$x,$y) = @_;
1561 # NaN if x == NaN or y == NaN or x==y==0
1562 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1563 if (($x->is_nan() || $y->is_nan()) ||
1564 ($x->is_zero() && $y->is_zero()));
1566 # +-inf / +-inf == NaN, reminder also NaN
1567 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1569 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1571 # x / +-inf => 0, remainder x (works even if x == 0)
1572 if ($y->{sign} =~ /^[+-]inf$/)
1574 my $t = $x->copy(); # bzero clobbers up $x
1575 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1578 # 5 / 0 => +inf, -6 / 0 => -inf
1579 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1580 # exception: -8 / 0 has remainder -8, not 8
1581 # exception: -inf / 0 has remainder -inf, not inf
1584 # +-inf / 0 => special case for -inf
1585 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1586 if (!$x->is_zero() && !$x->is_inf())
1588 my $t = $x->copy(); # binf clobbers up $x
1590 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1594 # last case: +-inf / ordinary number
1596 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1598 return wantarray ? ($x,$self->bzero()) : $x;
1603 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1604 # (BINT,BINT) (quo,rem) or BINT (only rem)
1607 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1608 # objectify is costly, so avoid it
1609 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1611 ($self,$x,$y,@r) = objectify(2,@_);
1614 return $x if $x->modify('bdiv');
1616 return $self->_div_inf($x,$y)
1617 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1619 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1620 if defined $upgrade;
1622 $r[3] = $y; # no push!
1624 # calc new sign and in case $y == +/- 1, return $x
1625 my $xsign = $x->{sign}; # keep
1626 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1630 my $rem = $self->bzero();
1631 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1632 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1633 $rem->{_a} = $x->{_a};
1634 $rem->{_p} = $x->{_p};
1636 if (! $CALC->_is_zero($rem->{value}))
1638 $rem->{sign} = $y->{sign};
1639 $rem = $y->copy()->bsub($rem) if $xsign ne $y->{sign}; # one of them '-'
1643 $rem->{sign} = '+'; # dont leave -0
1649 $x->{value} = $CALC->_div($x->{value},$y->{value});
1650 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1655 ###############################################################################
1660 # modulus (or remainder)
1661 # (BINT or num_str, BINT or num_str) return BINT
1664 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1665 # objectify is costly, so avoid it
1666 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1668 ($self,$x,$y,@r) = objectify(2,@_);
1671 return $x if $x->modify('bmod');
1672 $r[3] = $y; # no push!
1673 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1675 my ($d,$r) = $self->_div_inf($x,$y);
1676 $x->{sign} = $r->{sign};
1677 $x->{value} = $r->{value};
1678 return $x->round(@r);
1681 # calc new sign and in case $y == +/- 1, return $x
1682 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1683 if (!$CALC->_is_zero($x->{value}))
1685 $x->{value} = $CALC->_sub($y->{value},$x->{value},1) # $y-$x
1686 if ($x->{sign} ne $y->{sign});
1687 $x->{sign} = $y->{sign};
1691 $x->{sign} = '+'; # dont leave -0
1698 # Modular inverse. given a number which is (hopefully) relatively
1699 # prime to the modulus, calculate its inverse using Euclid's
1700 # alogrithm. If the number is not relatively prime to the modulus
1701 # (i.e. their gcd is not one) then NaN is returned.
1704 my ($self,$x,$y,@r) = (undef,@_);
1705 # objectify is costly, so avoid it
1706 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1708 ($self,$x,$y,@r) = objectify(2,@_);
1711 return $x if $x->modify('bmodinv');
1714 if ($y->{sign} ne '+' # -, NaN, +inf, -inf
1715 || $x->is_zero() # or num == 0
1716 || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1719 # put least residue into $x if $x was negative, and thus make it positive
1720 $x->bmod($y) if $x->{sign} eq '-';
1723 ($x->{value},$sign) = $CALC->_modinv($x->{value},$y->{value});
1724 return $x->bnan() if !defined $x->{value}; # in case no GCD found
1725 return $x if !defined $sign; # already real result
1726 $x->{sign} = $sign; # flip/flop see below
1727 $x->bmod($y); # calc real result
1733 # takes a very large number to a very large exponent in a given very
1734 # large modulus, quickly, thanks to binary exponentation. supports
1735 # negative exponents.
1736 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1738 return $num if $num->modify('bmodpow');
1740 # check modulus for valid values
1741 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1742 || $mod->is_zero());
1744 # check exponent for valid values
1745 if ($exp->{sign} =~ /\w/)
1747 # i.e., if it's NaN, +inf, or -inf...
1748 return $num->bnan();
1751 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
1753 # check num for valid values (also NaN if there was no inverse but $exp < 0)
1754 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1756 # $mod is positive, sign on $exp is ignored, result also positive
1757 $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
1761 ###############################################################################
1765 # (BINT or num_str, BINT or num_str) return BINT
1766 # compute factorial number from $x, modify $x in place
1767 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1769 return $x if $x->modify('bfac') || $x->{sign} eq '+inf'; # inf => inf
1770 return $x->bnan() if $x->{sign} ne '+'; # NaN, <0 etc => NaN
1772 $x->{value} = $CALC->_fac($x->{value});
1778 # (BINT or num_str, BINT or num_str) return BINT
1779 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1780 # modifies first argument
1783 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1784 # objectify is costly, so avoid it
1785 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1787 ($self,$x,$y,@r) = objectify(2,@_);
1790 return $x if $x->modify('bpow');
1792 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1795 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1797 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1803 if ($x->{sign} =~ /^[+-]inf/)
1806 return $x->bnan() if $y->is_zero();
1807 # -inf ** -1 => 1/inf => 0
1808 return $x->bzero() if $y->is_one('-') && $x->is_negative();
1811 return $x if $x->{sign} eq '+inf';
1813 # -inf ** Y => -inf if Y is odd
1814 return $x if $y->is_odd();
1820 return $x if $x->is_one();
1823 return $x if $x->is_zero() && $y->{sign} =~ /^[+]/;
1826 return $x->binf() if $x->is_zero();
1829 return $x->bnan() if $x->is_one('-') && $y->{sign} =~ /^[-]/;
1832 return $x->bzero() if $x->{sign} eq '-' && $y->{sign} =~ /^[-]/;
1835 return $x->bnan() if $x->{sign} eq '-';
1838 return $x->binf() if $y->{sign} =~ /^[+]/;
1843 return $upgrade->bpow($upgrade->new($x),$y,@r)
1844 if defined $upgrade && (!$y->isa($self) || $y->{sign} eq '-');
1846 $r[3] = $y; # no push!
1848 # cases 0 ** Y, X ** 0, X ** 1, 1 ** Y are handled by Calc or Emu
1851 $new_sign = $y->is_odd() ? '-' : '+' if ($x->{sign} ne '+');
1853 # 0 ** -7 => ( 1 / (0 ** 7)) => 1 / 0 => +inf
1855 if $y->{sign} eq '-' && $x->{sign} eq '+' && $CALC->_is_zero($x->{value});
1856 # 1 ** -y => 1 / (1 ** |y|)
1857 # so do test for negative $y after above's clause
1858 return $x->bnan() if $y->{sign} eq '-' && !$CALC->_is_one($x->{value});
1860 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1861 $x->{sign} = $new_sign;
1862 $x->{sign} = '+' if $CALC->_is_zero($y->{value});
1868 # (BINT or num_str, BINT or num_str) return BINT
1869 # compute x << y, base n, y >= 0
1872 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1873 # objectify is costly, so avoid it
1874 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1876 ($self,$x,$y,$n,@r) = objectify(2,@_);
1879 return $x if $x->modify('blsft');
1880 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1881 return $x->round(@r) if $y->is_zero();
1883 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1885 $x->{value} = $CALC->_lsft($x->{value},$y->{value},$n);
1891 # (BINT or num_str, BINT or num_str) return BINT
1892 # compute x >> y, base n, y >= 0
1895 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1896 # objectify is costly, so avoid it
1897 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1899 ($self,$x,$y,$n,@r) = objectify(2,@_);
1902 return $x if $x->modify('brsft');
1903 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1904 return $x->round(@r) if $y->is_zero();
1905 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
1907 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1909 # this only works for negative numbers when shifting in base 2
1910 if (($x->{sign} eq '-') && ($n == 2))
1912 return $x->round(@r) if $x->is_one('-'); # -1 => -1
1915 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1916 # but perhaps there is a better emulation for two's complement shift...
1917 # if $y != 1, we must simulate it by doing:
1918 # convert to bin, flip all bits, shift, and be done
1919 $x->binc(); # -3 => -2
1920 my $bin = $x->as_bin();
1921 $bin =~ s/^-0b//; # strip '-0b' prefix
1922 $bin =~ tr/10/01/; # flip bits
1924 if ($y >= CORE::length($bin))
1926 $bin = '0'; # shifting to far right creates -1
1927 # 0, because later increment makes
1928 # that 1, attached '-' makes it '-1'
1929 # because -1 >> x == -1 !
1933 $bin =~ s/.{$y}$//; # cut off at the right side
1934 $bin = '1' . $bin; # extend left side by one dummy '1'
1935 $bin =~ tr/10/01/; # flip bits back
1937 my $res = $self->new('0b'.$bin); # add prefix and convert back
1938 $res->binc(); # remember to increment
1939 $x->{value} = $res->{value}; # take over value
1940 return $x->round(@r); # we are done now, magic, isn't?
1942 # x < 0, n == 2, y == 1
1943 $x->bdec(); # n == 2, but $y == 1: this fixes it
1946 $x->{value} = $CALC->_rsft($x->{value},$y->{value},$n);
1952 #(BINT or num_str, BINT or num_str) return BINT
1956 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1957 # objectify is costly, so avoid it
1958 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1960 ($self,$x,$y,@r) = objectify(2,@_);
1963 return $x if $x->modify('band');
1965 $r[3] = $y; # no push!
1967 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1969 my $sx = $x->{sign} eq '+' ? 1 : -1;
1970 my $sy = $y->{sign} eq '+' ? 1 : -1;
1972 if ($sx == 1 && $sy == 1)
1974 $x->{value} = $CALC->_and($x->{value},$y->{value});
1975 return $x->round(@r);
1978 if ($CAN{signed_and})
1980 $x->{value} = $CALC->_signed_and($x->{value},$y->{value},$sx,$sy);
1981 return $x->round(@r);
1985 __emu_band($self,$x,$y,$sx,$sy,@r);
1990 #(BINT or num_str, BINT or num_str) return BINT
1994 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1995 # objectify is costly, so avoid it
1996 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1998 ($self,$x,$y,@r) = objectify(2,@_);
2001 return $x if $x->modify('bior');
2002 $r[3] = $y; # no push!
2004 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
2006 my $sx = $x->{sign} eq '+' ? 1 : -1;
2007 my $sy = $y->{sign} eq '+' ? 1 : -1;
2009 # the sign of X follows the sign of X, e.g. sign of Y irrelevant for bior()
2011 # don't use lib for negative values
2012 if ($sx == 1 && $sy == 1)
2014 $x->{value} = $CALC->_or($x->{value},$y->{value});
2015 return $x->round(@r);
2018 # if lib can do negative values, let it handle this
2019 if ($CAN{signed_or})
2021 $x->{value} = $CALC->_signed_or($x->{value},$y->{value},$sx,$sy);
2022 return $x->round(@r);
2026 __emu_bior($self,$x,$y,$sx,$sy,@r);
2031 #(BINT or num_str, BINT or num_str) return BINT
2035 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2036 # objectify is costly, so avoid it
2037 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2039 ($self,$x,$y,@r) = objectify(2,@_);
2042 return $x if $x->modify('bxor');
2043 $r[3] = $y; # no push!
2045 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
2047 my $sx = $x->{sign} eq '+' ? 1 : -1;
2048 my $sy = $y->{sign} eq '+' ? 1 : -1;
2050 # don't use lib for negative values
2051 if ($sx == 1 && $sy == 1)
2053 $x->{value} = $CALC->_xor($x->{value},$y->{value});
2054 return $x->round(@r);
2057 # if lib can do negative values, let it handle this
2058 if ($CAN{signed_xor})
2060 $x->{value} = $CALC->_signed_xor($x->{value},$y->{value},$sx,$sy);
2061 return $x->round(@r);
2065 __emu_bxor($self,$x,$y,$sx,$sy,@r);
2070 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2072 my $e = $CALC->_len($x->{value});
2073 wantarray ? ($e,0) : $e;
2078 # return the nth decimal digit, negative values count backward, 0 is right
2079 my ($self,$x,$n) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2081 $n = $n->numify() if ref($n);
2082 $CALC->_digit($x->{value},$n||0);
2087 # return the amount of trailing zeros in $x (as scalar)
2089 $x = $class->new($x) unless ref $x;
2091 return 0 if $x->{sign} !~ /^[+-]$/; # NaN, inf, -inf etc
2093 $CALC->_zeros($x->{value}); # must handle odd values, 0 etc
2098 # calculate square root of $x
2099 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2101 return $x if $x->modify('bsqrt');
2103 return $x->bnan() if $x->{sign} !~ /^\+/; # -x or -inf or NaN => NaN
2104 return $x if $x->{sign} eq '+inf'; # sqrt(+inf) == inf
2106 return $upgrade->bsqrt($x,@r) if defined $upgrade;
2108 $x->{value} = $CALC->_sqrt($x->{value});
2114 # calculate $y'th root of $x
2117 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2119 $y = $self->new(2) unless defined $y;
2121 # objectify is costly, so avoid it
2122 if ((!ref($x)) || (ref($x) ne ref($y)))
2124 ($self,$x,$y,@r) = objectify(2,$self || $class,@_);
2127 return $x if $x->modify('broot');
2129 # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
2130 return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
2131 $y->{sign} !~ /^\+$/;
2133 return $x->round(@r)
2134 if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
2136 return $upgrade->new($x)->broot($upgrade->new($y),@r) if defined $upgrade;
2138 $x->{value} = $CALC->_root($x->{value},$y->{value});
2144 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
2145 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2147 if ($x->{sign} !~ /^[+-]$/)
2149 my $s = $x->{sign}; $s =~ s/^[+-]//; # NaN, -inf,+inf => NaN or inf
2150 return $self->new($s);
2152 return $self->bone() if $x->is_zero();
2154 # 12300 => 2 trailing zeros => exponent is 2
2155 $self->new( $CALC->_zeros($x->{value}) );
2160 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
2161 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2163 if ($x->{sign} !~ /^[+-]$/)
2165 # for NaN, +inf, -inf: keep the sign
2166 return $self->new($x->{sign});
2168 my $m = $x->copy(); delete $m->{_p}; delete $m->{_a};
2170 # that's a bit inefficient:
2171 my $zeros = $CALC->_zeros($m->{value});
2172 $m->brsft($zeros,10) if $zeros != 0;
2178 # return a copy of both the exponent and the mantissa
2179 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2181 ($x->mantissa(),$x->exponent());
2184 ##############################################################################
2185 # rounding functions
2189 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
2190 # $n == 0 || $n == 1 => round to integer
2191 my $x = shift; my $self = ref($x) || $x; $x = $self->new($x) unless ref $x;
2193 my ($scale,$mode) = $x->_scale_p(@_);
2195 return $x if !defined $scale || $x->modify('bfround'); # no-op
2197 # no-op for BigInts if $n <= 0
2198 $x->bround( $x->length()-$scale, $mode) if $scale > 0;
2200 delete $x->{_a}; # delete to save memory
2201 $x->{_p} = $scale; # store new _p
2205 sub _scan_for_nonzero
2207 # internal, used by bround() to scan for non-zeros after a '5'
2208 my ($x,$pad,$xs,$len) = @_;
2210 return 0 if $len == 1; # "5" is trailed by invisible zeros
2211 my $follow = $pad - 1;
2212 return 0 if $follow > $len || $follow < 1;
2214 # use the string form to check whether only '0's follow or not
2215 substr ($xs,-$follow) =~ /[^0]/ ? 1 : 0;
2220 # Exists to make life easier for switch between MBF and MBI (should we
2221 # autoload fxxx() like MBF does for bxxx()?)
2222 my $x = shift; $x = $class->new($x) unless ref $x;
2228 # accuracy: +$n preserve $n digits from left,
2229 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
2231 # and overwrite the rest with 0's, return normalized number
2232 # do not return $x->bnorm(), but $x
2234 my $x = shift; $x = $class->new($x) unless ref $x;
2235 my ($scale,$mode) = $x->_scale_a(@_);
2236 return $x if !defined $scale || $x->modify('bround'); # no-op
2238 if ($x->is_zero() || $scale == 0)
2240 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2243 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2245 # we have fewer digits than we want to scale to
2246 my $len = $x->length();
2247 # convert $scale to a scalar in case it is an object (put's a limit on the
2248 # number length, but this would already limited by memory constraints), makes
2250 $scale = $scale->numify() if ref ($scale);
2252 # scale < 0, but > -len (not >=!)
2253 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2255 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2259 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2260 my ($pad,$digit_round,$digit_after);
2261 $pad = $len - $scale;
2262 $pad = abs($scale-1) if $scale < 0;
2264 # do not use digit(), it is very costly for binary => decimal
2265 # getting the entire string is also costly, but we need to do it only once
2266 my $xs = $CALC->_str($x->{value});
2269 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2270 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2271 $digit_round = '0'; $digit_round = substr($xs,$pl,1) if $pad <= $len;
2272 $pl++; $pl ++ if $pad >= $len;
2273 $digit_after = '0'; $digit_after = substr($xs,$pl,1) if $pad > 0;
2275 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2276 # closer at the remaining digits of the original $x, remember decision
2277 my $round_up = 1; # default round up
2279 ($mode eq 'trunc') || # trunc by round down
2280 ($digit_after =~ /[01234]/) || # round down anyway,
2282 ($digit_after eq '5') && # not 5000...0000
2283 ($x->_scan_for_nonzero($pad,$xs,$len) == 0) &&
2285 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2286 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2287 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2288 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2289 ($mode eq 'zero') # round down if zero, sign adjusted below
2291 my $put_back = 0; # not yet modified
2293 if (($pad > 0) && ($pad <= $len))
2295 substr($xs,-$pad,$pad) = '0' x $pad; # replace with '00...'
2296 $put_back = 1; # need to put back
2300 $x->bzero(); # round to '0'
2303 if ($round_up) # what gave test above?
2305 $put_back = 1; # need to put back
2306 $pad = $len, $xs = '0' x $pad if $scale < 0; # tlr: whack 0.51=>1.0
2308 # we modify directly the string variant instead of creating a number and
2309 # adding it, since that is faster (we already have the string)
2310 my $c = 0; $pad ++; # for $pad == $len case
2311 while ($pad <= $len)
2313 $c = substr($xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2314 substr($xs,-$pad,1) = $c; $pad++;
2315 last if $c != 0; # no overflow => early out
2317 $xs = '1'.$xs if $c == 0;
2320 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back, if needed
2322 $x->{_a} = $scale if $scale >= 0;
2325 $x->{_a} = $len+$scale;
2326 $x->{_a} = 0 if $scale < -$len;
2333 # return integer less or equal then number; no-op since it's already integer
2334 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2341 # return integer greater or equal then number; no-op since it's already int
2342 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2349 # An object might be asked to return itself as bigint on certain overloaded
2350 # operations. This does exactly this, so that sub classes can simple inherit
2351 # it or override with their own integer conversion routine.
2357 # return as hex string, with prefixed 0x
2358 my $x = shift; $x = $class->new($x) if !ref($x);
2360 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2363 $s = $x->{sign} if $x->{sign} eq '-';
2364 $s . $CALC->_as_hex($x->{value});
2369 # return as binary string, with prefixed 0b
2370 my $x = shift; $x = $class->new($x) if !ref($x);
2372 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2374 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2375 return $s . $CALC->_as_bin($x->{value});
2380 # return as octal string, with prefixed 0
2381 my $x = shift; $x = $class->new($x) if !ref($x);
2383 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2385 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2386 return $s . $CALC->_as_oct($x->{value});
2389 ##############################################################################
2390 # private stuff (internal use only)
2394 # check for strings, if yes, return objects instead
2396 # the first argument is number of args objectify() should look at it will
2397 # return $count+1 elements, the first will be a classname. This is because
2398 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2399 # useless objects being created and thrown away. So we cannot simple loop
2400 # over @_. If the given count is 0, all arguments will be used.
2402 # If the second arg is a ref, use it as class.
2403 # If not, try to use it as classname, unless undef, then use $class
2404 # (aka Math::BigInt). The latter shouldn't happen,though.
2407 # $x->badd(1); => ref x, scalar y
2408 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2409 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2410 # Math::BigInt::badd(1,2); => scalar x, scalar y
2411 # In the last case we check number of arguments to turn it silently into
2412 # $class,1,2. (We can not take '1' as class ;o)
2413 # badd($class,1) is not supported (it should, eventually, try to add undef)
2414 # currently it tries 'Math::BigInt' + 1, which will not work.
2416 # some shortcut for the common cases
2418 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2420 my $count = abs(shift || 0);
2422 my (@a,$k,$d); # resulting array, temp, and downgrade
2425 # okay, got object as first
2430 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2432 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2436 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2437 if (defined ${"$a[0]::downgrade"})
2439 $d = ${"$a[0]::downgrade"};
2440 ${"$a[0]::downgrade"} = undef;
2443 my $up = ${"$a[0]::upgrade"};
2444 # print STDERR "# Now in objectify, my class is today $a[0], count = $count\n";
2452 $k = $a[0]->new($k);
2454 elsif (!defined $up && ref($k) ne $a[0])
2456 # foreign object, try to convert to integer
2457 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2470 $k = $a[0]->new($k);
2472 elsif (!defined $up && ref($k) ne $a[0])
2474 # foreign object, try to convert to integer
2475 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2479 push @a,@_; # return other params, too
2483 require Carp; Carp::croak ("$class objectify needs list context");
2485 ${"$a[0]::downgrade"} = $d;
2489 sub _register_callback
2491 my ($class,$callback) = @_;
2493 if (ref($callback) ne 'CODE')
2496 Carp::croak ("$callback is not a coderef");
2498 $CALLBACKS{$class} = $callback;
2505 $IMPORT++; # remember we did import()
2506 my @a; my $l = scalar @_;
2507 my $warn_or_die = 0; # 0 - no warn, 1 - warn, 2 - die
2508 for ( my $i = 0; $i < $l ; $i++ )
2510 if ($_[$i] eq ':constant')
2512 # this causes overlord er load to step in
2514 integer => sub { $self->new(shift) },
2515 binary => sub { $self->new(shift) };
2517 elsif ($_[$i] eq 'upgrade')
2519 # this causes upgrading
2520 $upgrade = $_[$i+1]; # or undef to disable
2523 elsif ($_[$i] =~ /^(lib|try|only)\z/)
2525 # this causes a different low lib to take care...
2526 $CALC = $_[$i+1] || '';
2527 # lib => 1 (warn on fallback), try => 0 (no warn), only => 2 (die on fallback)
2528 $warn_or_die = 1 if $_[$i] eq 'lib';
2529 $warn_or_die = 2 if $_[$i] eq 'only';
2537 # any non :constant stuff is handled by our parent, Exporter
2542 $self->SUPER::import(@a); # need it for subclasses
2543 $self->export_to_level(1,$self,@a); # need it for MBF
2546 # try to load core math lib
2547 my @c = split /\s*,\s*/,$CALC;
2550 $_ =~ tr/a-zA-Z0-9://cd; # limit to sane characters
2552 push @c, \'FastCalc', \'Calc' # if all fail, try these
2553 if $warn_or_die < 2; # but not for "only"
2554 $CALC = ''; # signal error
2557 # fallback libraries are "marked" as \'string', extract string if nec.
2558 my $lib = $l; $lib = $$l if ref($l);
2560 next if ($lib || '') eq '';
2561 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2565 # Perl < 5.6.0 dies with "out of memory!" when eval("") and ':constant' is
2566 # used in the same script, or eval("") inside import().
2567 my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
2568 my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
2570 $file = File::Spec->catfile (@parts, $file);
2571 eval { require "$file"; $lib->import( @c ); }
2575 eval "use $lib qw/@c/;";
2580 # loaded it ok, see if the api_version() is high enough
2581 if ($lib->can('api_version') && $lib->api_version() >= 1.0)
2584 # api_version matches, check if it really provides anything we need
2588 add mul div sub dec inc
2589 acmp len digit is_one is_zero is_even is_odd
2591 zeros new copy check
2592 from_hex from_oct from_bin as_hex as_bin as_oct
2593 rsft lsft xor and or
2594 mod sqrt root fac pow modinv modpow log_int gcd
2597 if (!$lib->can("_$method"))
2599 if (($WARN{$lib}||0) < 2)
2602 Carp::carp ("$lib is missing method '_$method'");
2603 $WARN{$lib} = 1; # still warn about the lib
2612 if ($warn_or_die > 0 && ref($l))
2615 my $msg = "Math::BigInt: couldn't load specified math lib(s), fallback to $lib";
2616 Carp::carp ($msg) if $warn_or_die == 1;
2617 Carp::croak ($msg) if $warn_or_die == 2;
2619 last; # found a usable one, break
2623 if (($WARN{$lib}||0) < 2)
2625 my $ver = eval "\$$lib\::VERSION" || 'unknown';
2627 Carp::carp ("Cannot load outdated $lib v$ver, please upgrade");
2628 $WARN{$lib} = 2; # never warn again
2636 if ($warn_or_die == 2)
2638 Carp::croak ("Couldn't load specified math lib(s) and fallback disallowed");
2642 Carp::croak ("Couldn't load any math lib(s), not even fallback to Calc.pm");
2647 foreach my $class (keys %CALLBACKS)
2649 &{$CALLBACKS{$class}}($CALC);
2652 # Fill $CAN with the results of $CALC->can(...) for emulating lower math lib
2656 for my $method (qw/ signed_and signed_or signed_xor /)
2658 $CAN{$method} = $CALC->can("_$method") ? 1 : 0;
2666 # create a bigint from a hexadecimal string
2667 my ($self, $hs) = @_;
2669 my $rc = $self->__from_hex($hs);
2671 return $self->bnan() unless defined $rc;
2678 # create a bigint from a hexadecimal string
2679 my ($self, $bs) = @_;
2681 my $rc = $self->__from_bin($bs);
2683 return $self->bnan() unless defined $rc;
2690 # create a bigint from a hexadecimal string
2691 my ($self, $os) = @_;
2693 my $x = $self->bzero();
2696 $os =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2697 $os =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2699 return $x->bnan() if $os !~ /^[\-\+]?0[0-9]+$/;
2701 my $sign = '+'; $sign = '-' if $os =~ /^-/;
2703 $os =~ s/^[+-]//; # strip sign
2704 $x->{value} = $CALC->_from_oct($os);
2705 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2712 # convert a (ref to) big hex string to BigInt, return undef for error
2715 my $x = Math::BigInt->bzero();
2718 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2719 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2721 return $x->bnan() if $hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2723 my $sign = '+'; $sign = '-' if $hs =~ /^-/;
2725 $hs =~ s/^[+-]//; # strip sign
2726 $x->{value} = $CALC->_from_hex($hs);
2727 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2734 # convert a (ref to) big binary string to BigInt, return undef for error
2737 my $x = Math::BigInt->bzero();
2740 $bs =~ s/([01])_([01])/$1$2/g;
2741 $bs =~ s/([01])_([01])/$1$2/g;
2742 return $x->bnan() if $bs !~ /^[+-]?0b[01]+$/;
2744 my $sign = '+'; $sign = '-' if $bs =~ /^\-/;
2745 $bs =~ s/^[+-]//; # strip sign
2747 $x->{value} = $CALC->_from_bin($bs);
2748 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2754 # input: num_str; output: undef for invalid or
2755 # (\$mantissa_sign,\$mantissa_value,\$mantissa_fraction,\$exp_sign,\$exp_value)
2756 # Internal, take apart a string and return the pieces.
2757 # Strip leading/trailing whitespace, leading zeros, underscore and reject
2761 # strip white space at front, also extranous leading zeros
2762 $x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2763 $x =~ s/^\s+//; # but this will
2764 $x =~ s/\s+$//g; # strip white space at end
2766 # shortcut, if nothing to split, return early
2767 if ($x =~ /^[+-]?[0-9]+\z/)
2769 $x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2770 return (\$sign, \$x, \'', \'', \0);
2773 # invalid starting char?
2774 return if $x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2776 return __from_hex($x) if $x =~ /^[\-\+]?0x/; # hex string
2777 return __from_bin($x) if $x =~ /^[\-\+]?0b/; # binary string
2779 # strip underscores between digits
2780 $x =~ s/([0-9])_([0-9])/$1$2/g;
2781 $x =~ s/([0-9])_([0-9])/$1$2/g; # do twice for 1_2_3
2783 # some possible inputs:
2784 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2785 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 # 0e999
2787 my ($m,$e,$last) = split /[Ee]/,$x;
2788 return if defined $last; # last defined => 1e2E3 or others
2789 $e = '0' if !defined $e || $e eq "";
2791 # sign,value for exponent,mantint,mantfrac
2792 my ($es,$ev,$mis,$miv,$mfv);
2794 if ($e =~ /^([+-]?)0*([0-9]+)$/) # strip leading zeros
2798 return if $m eq '.' || $m eq '';
2799 my ($mi,$mf,$lastf) = split /\./,$m;
2800 return if defined $lastf; # lastf defined => 1.2.3 or others
2801 $mi = '0' if !defined $mi;
2802 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2803 $mf = '0' if !defined $mf || $mf eq '';
2804 if ($mi =~ /^([+-]?)0*([0-9]+)$/) # strip leading zeros
2806 $mis = $1||'+'; $miv = $2;
2807 return unless ($mf =~ /^([0-9]*?)0*$/); # strip trailing zeros
2809 # handle the 0e999 case here
2810 $ev = 0 if $miv eq '0' && $mfv eq '';
2811 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2814 return; # NaN, not a number
2817 ##############################################################################
2818 # internal calculation routines (others are in Math::BigInt::Calc etc)
2822 # (BINT or num_str, BINT or num_str) return BINT
2823 # does modify first argument
2827 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2828 my $method = ref($x) . '::bgcd';
2830 $x * $ty / &$method($x,$ty);
2833 ###############################################################################
2834 # trigonometric functions
2838 # Calculate PI to N digits. Unless upgrading is in effect, returns the
2839 # result truncated to an integer, that is, always returns '3'.
2843 # called like Math::BigInt::bpi(10);
2844 $n = $self; $self = $class;
2846 $self = ref($self) if ref($self);
2848 return $upgrade->new($n) if defined $upgrade;
2854 ###############################################################################
2855 # this method returns 0 if the object can be modified, or 1 if not.
2856 # We use a fast constant sub() here, to avoid costly calls. Subclasses
2857 # may override it with special code (f.i. Math::BigInt::Constant does so)
2859 sub modify () { 0; }
2868 Math::BigInt - Arbitrary size integer/float math package
2874 # or make it faster: install (optional) Math::BigInt::GMP
2875 # and always use (it will fall back to pure Perl if the
2876 # GMP library is not installed):
2878 # will warn if Math::BigInt::GMP cannot be found
2879 use Math::BigInt lib => 'GMP';
2881 # to supress the warning use this:
2882 # use Math::BigInt try => 'GMP';
2884 my $str = '1234567890';
2885 my @values = (64,74,18);
2886 my $n = 1; my $sign = '-';
2889 my $x = Math::BigInt->new($str); # defaults to 0
2890 my $y = $x->copy(); # make a true copy
2891 my $nan = Math::BigInt->bnan(); # create a NotANumber
2892 my $zero = Math::BigInt->bzero(); # create a +0
2893 my $inf = Math::BigInt->binf(); # create a +inf
2894 my $inf = Math::BigInt->binf('-'); # create a -inf
2895 my $one = Math::BigInt->bone(); # create a +1
2896 my $mone = Math::BigInt->bone('-'); # create a -1
2898 my $pi = Math::BigInt->bpi(); # returns '3'
2899 # see Math::BigFloat::bpi()
2901 $h = Math::BigInt->new('0x123'); # from hexadecimal
2902 $b = Math::BigInt->new('0b101'); # from binary
2903 $o = Math::BigInt->from_oct('0101'); # from octal
2905 # Testing (don't modify their arguments)
2906 # (return true if the condition is met, otherwise false)
2908 $x->is_zero(); # if $x is +0
2909 $x->is_nan(); # if $x is NaN
2910 $x->is_one(); # if $x is +1
2911 $x->is_one('-'); # if $x is -1
2912 $x->is_odd(); # if $x is odd
2913 $x->is_even(); # if $x is even
2914 $x->is_pos(); # if $x >= 0
2915 $x->is_neg(); # if $x < 0
2916 $x->is_inf($sign); # if $x is +inf, or -inf (sign is default '+')
2917 $x->is_int(); # if $x is an integer (not a float)
2919 # comparing and digit/sign extraction
2920 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2921 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2922 $x->sign(); # return the sign, either +,- or NaN
2923 $x->digit($n); # return the nth digit, counting from right
2924 $x->digit(-$n); # return the nth digit, counting from left
2926 # The following all modify their first argument. If you want to preserve
2927 # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
2928 # necessary when mixing $a = $b assignments with non-overloaded math.
2930 $x->bzero(); # set $x to 0
2931 $x->bnan(); # set $x to NaN
2932 $x->bone(); # set $x to +1
2933 $x->bone('-'); # set $x to -1
2934 $x->binf(); # set $x to inf
2935 $x->binf('-'); # set $x to -inf
2937 $x->bneg(); # negation
2938 $x->babs(); # absolute value
2939 $x->bnorm(); # normalize (no-op in BigInt)
2940 $x->bnot(); # two's complement (bit wise not)
2941 $x->binc(); # increment $x by 1
2942 $x->bdec(); # decrement $x by 1
2944 $x->badd($y); # addition (add $y to $x)
2945 $x->bsub($y); # subtraction (subtract $y from $x)
2946 $x->bmul($y); # multiplication (multiply $x by $y)
2947 $x->bdiv($y); # divide, set $x to quotient
2948 # return (quo,rem) or quo if scalar
2950 $x->bmod($y); # modulus (x % y)
2951 $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
2952 $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
2954 $x->bpow($y); # power of arguments (x ** y)
2955 $x->blsft($y); # left shift in base 2
2956 $x->brsft($y); # right shift in base 2
2957 # returns (quo,rem) or quo if in scalar context
2958 $x->blsft($y,$n); # left shift by $y places in base $n
2959 $x->brsft($y,$n); # right shift by $y places in base $n
2960 # returns (quo,rem) or quo if in scalar context
2962 $x->band($y); # bitwise and
2963 $x->bior($y); # bitwise inclusive or
2964 $x->bxor($y); # bitwise exclusive or
2965 $x->bnot(); # bitwise not (two's complement)
2967 $x->bsqrt(); # calculate square-root
2968 $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
2969 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2971 $x->bnok($y); # x over y (binomial coefficient n over k)
2973 $x->blog(); # logarithm of $x to base e (Euler's number)
2974 $x->blog($base); # logarithm of $x to base $base (f.i. 2)
2975 $x->bexp(); # calculate e ** $x where e is Euler's number
2977 $x->round($A,$P,$mode); # round to accuracy or precision using mode $mode
2978 $x->bround($n); # accuracy: preserve $n digits
2979 $x->bfround($n); # round to $nth digit, no-op for BigInts
2981 # The following do not modify their arguments in BigInt (are no-ops),
2982 # but do so in BigFloat:
2984 $x->bfloor(); # return integer less or equal than $x
2985 $x->bceil(); # return integer greater or equal than $x
2987 # The following do not modify their arguments:
2989 # greatest common divisor (no OO style)
2990 my $gcd = Math::BigInt::bgcd(@values);
2991 # lowest common multiplicator (no OO style)
2992 my $lcm = Math::BigInt::blcm(@values);
2994 $x->length(); # return number of digits in number
2995 ($xl,$f) = $x->length(); # length of number and length of fraction part,
2996 # latter is always 0 digits long for BigInts
2998 $x->exponent(); # return exponent as BigInt
2999 $x->mantissa(); # return (signed) mantissa as BigInt
3000 $x->parts(); # return (mantissa,exponent) as BigInt
3001 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3002 $x->as_int(); # return as BigInt (in BigInt: same as copy())
3003 $x->numify(); # return as scalar (might overflow!)
3005 # conversation to string (do not modify their argument)
3006 $x->bstr(); # normalized string (e.g. '3')
3007 $x->bsstr(); # norm. string in scientific notation (e.g. '3E0')
3008 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3009 $x->as_bin(); # as signed binary string with prefixed 0b
3010 $x->as_oct(); # as signed octal string with prefixed 0
3013 # precision and accuracy (see section about rounding for more)
3014 $x->precision(); # return P of $x (or global, if P of $x undef)
3015 $x->precision($n); # set P of $x to $n
3016 $x->accuracy(); # return A of $x (or global, if A of $x undef)
3017 $x->accuracy($n); # set A $x to $n
3020 Math::BigInt->precision(); # get/set global P for all BigInt objects
3021 Math::BigInt->accuracy(); # get/set global A for all BigInt objects
3022 Math::BigInt->round_mode(); # get/set global round mode, one of
3023 # 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'
3024 Math::BigInt->config(); # return hash containing configuration
3028 All operators (including basic math operations) are overloaded if you
3029 declare your big integers as
3031 $i = new Math::BigInt '123_456_789_123_456_789';
3033 Operations with overloaded operators preserve the arguments which is
3034 exactly what you expect.
3040 Input values to these routines may be any string, that looks like a number
3041 and results in an integer, including hexadecimal and binary numbers.
3043 Scalars holding numbers may also be passed, but note that non-integer numbers
3044 may already have lost precision due to the conversation to float. Quote
3045 your input if you want BigInt to see all the digits:
3047 $x = Math::BigInt->new(12345678890123456789); # bad
3048 $x = Math::BigInt->new('12345678901234567890'); # good
3050 You can include one underscore between any two digits.
3052 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
3053 Non-integer values result in NaN.
3055 Hexadecimal (prefixed with "0x") and binary numbers (prefixed with "0b")
3056 are accepted, too. Please note that octal numbers are not recognized
3057 by new(), so the following will print "123":
3059 perl -MMath::BigInt -le 'print Math::BigInt->new("0123")'
3061 To convert an octal number, use from_oct();
3063 perl -MMath::BigInt -le 'print Math::BigInt->from_oct("0123")'
3065 Currently, Math::BigInt::new() defaults to 0, while Math::BigInt::new('')
3066 results in 'NaN'. This might change in the future, so use always the following
3067 explicit forms to get a zero or NaN:
3069 $zero = Math::BigInt->bzero();
3070 $nan = Math::BigInt->bnan();
3072 C<bnorm()> on a BigInt object is now effectively a no-op, since the numbers
3073 are always stored in normalized form. If passed a string, creates a BigInt
3074 object from the input.
3078 Output values are BigInt objects (normalized), except for the methods which
3079 return a string (see L<SYNOPSIS>).
3081 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
3082 C<is_nan()>, etc.) return true or false, while others (C<bcmp()>, C<bacmp()>)
3083 return either undef (if NaN is involved), <0, 0 or >0 and are suited for sort.
3089 Each of the methods below (except config(), accuracy() and precision())
3090 accepts three additional parameters. These arguments C<$A>, C<$P> and C<$R>
3091 are C<accuracy>, C<precision> and C<round_mode>. Please see the section about
3092 L<ACCURACY and PRECISION> for more information.
3098 print Dumper ( Math::BigInt->config() );
3099 print Math::BigInt->config()->{lib},"\n";
3101 Returns a hash containing the configuration, e.g. the version number, lib
3102 loaded etc. The following hash keys are currently filled in with the
3103 appropriate information.
3107 ============================================================
3108 lib Name of the low-level math library
3110 lib_version Version of low-level math library (see 'lib')
3112 class The class name of config() you just called
3114 upgrade To which class math operations might be upgraded
3116 downgrade To which class math operations might be downgraded
3118 precision Global precision
3120 accuracy Global accuracy
3122 round_mode Global round mode
3124 version version number of the class you used
3126 div_scale Fallback accuracy for div
3128 trap_nan If true, traps creation of NaN via croak()
3130 trap_inf If true, traps creation of +inf/-inf via croak()
3133 The following values can be set by passing C<config()> a reference to a hash:
3136 upgrade downgrade precision accuracy round_mode div_scale
3140 $new_cfg = Math::BigInt->config( { trap_inf => 1, precision => 5 } );
3144 $x->accuracy(5); # local for $x
3145 CLASS->accuracy(5); # global for all members of CLASS
3146 # Note: This also applies to new()!
3148 $A = $x->accuracy(); # read out accuracy that affects $x
3149 $A = CLASS->accuracy(); # read out global accuracy
3151 Set or get the global or local accuracy, aka how many significant digits the
3152 results have. If you set a global accuracy, then this also applies to new()!
3154 Warning! The accuracy I<sticks>, e.g. once you created a number under the
3155 influence of C<< CLASS->accuracy($A) >>, all results from math operations with
3156 that number will also be rounded.
3158 In most cases, you should probably round the results explicitly using one of
3159 L<round()>, L<bround()> or L<bfround()> or by passing the desired accuracy
3160 to the math operation as additional parameter:
3162 my $x = Math::BigInt->new(30000);
3163 my $y = Math::BigInt->new(7);
3164 print scalar $x->copy()->bdiv($y, 2); # print 4300
3165 print scalar $x->copy()->bdiv($y)->bround(2); # print 4300
3167 Please see the section about L<ACCURACY AND PRECISION> for further details.
3169 Value must be greater than zero. Pass an undef value to disable it:
3171 $x->accuracy(undef);
3172 Math::BigInt->accuracy(undef);
3174 Returns the current accuracy. For C<$x->accuracy()> it will return either the
3175 local accuracy, or if not defined, the global. This means the return value
3176 represents the accuracy that will be in effect for $x:
3178 $y = Math::BigInt->new(1234567); # unrounded
3179 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
3180 $x = Math::BigInt->new(123456); # $x will be automatically rounded!
3181 print "$x $y\n"; # '123500 1234567'
3182 print $x->accuracy(),"\n"; # will be 4
3183 print $y->accuracy(),"\n"; # also 4, since global is 4
3184 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
3185 print $x->accuracy(),"\n"; # still 4
3186 print $y->accuracy(),"\n"; # 5, since global is 5
3188 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
3189 globals separated from Math::BigInt, but it is possible to subclass
3190 Math::BigInt and make the globals of the subclass aliases to the ones from
3195 $x->precision(-2); # local for $x, round at the second digit right of the dot
3196 $x->precision(2); # ditto, round at the second digit left of the dot
3198 CLASS->precision(5); # Global for all members of CLASS
3199 # This also applies to new()!
3200 CLASS->precision(-5); # ditto
3202 $P = CLASS->precision(); # read out global precision
3203 $P = $x->precision(); # read out precision that affects $x
3205 Note: You probably want to use L<accuracy()> instead. With L<accuracy> you
3206 set the number of digits each result should have, with L<precision> you
3207 set the place where to round!
3209 C<precision()> sets or gets the global or local precision, aka at which digit
3210 before or after the dot to round all results. A set global precision also
3211 applies to all newly created numbers!
3213 In Math::BigInt, passing a negative number precision has no effect since no
3214 numbers have digits after the dot. In L<Math::BigFloat>, it will round all
3215 results to P digits after the dot.
3217 Please see the section about L<ACCURACY AND PRECISION> for further details.
3219 Pass an undef value to disable it:
3221 $x->precision(undef);
3222 Math::BigInt->precision(undef);
3224 Returns the current precision. For C<$x->precision()> it will return either the
3225 local precision of $x, or if not defined, the global. This means the return
3226 value represents the prevision that will be in effect for $x:
3228 $y = Math::BigInt->new(1234567); # unrounded
3229 print Math::BigInt->precision(4),"\n"; # set 4, print 4
3230 $x = Math::BigInt->new(123456); # will be automatically rounded
3231 print $x; # print "120000"!
3233 Note: Works also for subclasses like L<Math::BigFloat>. Each class has its
3234 own globals separated from Math::BigInt, but it is possible to subclass
3235 Math::BigInt and make the globals of the subclass aliases to the ones from
3242 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
3243 2, but others work, too.
3245 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
3249 $x = Math::BigInt->new(10);
3250 $x->brsft(1); # same as $x >> 1: 5
3251 $x = Math::BigInt->new(1234);
3252 $x->brsft(2,10); # result 12
3254 There is one exception, and that is base 2 with negative $x:
3257 $x = Math::BigInt->new(-5);
3260 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
3265 $x = Math::BigInt->new($str,$A,$P,$R);
3267 Creates a new BigInt object from a scalar or another BigInt object. The
3268 input is accepted as decimal, hex (with leading '0x') or binary (with leading
3271 See L<Input> for more info on accepted input formats.
3275 $x = Math::BigIn->from_oct("0775"); # input is octal
3279 $x = Math::BigIn->from_hex("0xcafe"); # input is hexadecimal
3283 $x = Math::BigIn->from_oct("0x10011"); # input is binary
3287 $x = Math::BigInt->bnan();
3289 Creates a new BigInt object representing NaN (Not A Number).
3290 If used on an object, it will set it to NaN:
3296 $x = Math::BigInt->bzero();
3298 Creates a new BigInt object representing zero.
3299 If used on an object, it will set it to zero:
3305 $x = Math::BigInt->binf($sign);
3307 Creates a new BigInt object representing infinity. The optional argument is
3308 either '-' or '+', indicating whether you want infinity or minus infinity.
3309 If used on an object, it will set it to infinity:
3316 $x = Math::BigInt->binf($sign);
3318 Creates a new BigInt object representing one. The optional argument is
3319 either '-' or '+', indicating whether you want one or minus one.
3320 If used on an object, it will set it to one:
3325 =head2 is_one()/is_zero()/is_nan()/is_inf()
3328 $x->is_zero(); # true if arg is +0
3329 $x->is_nan(); # true if arg is NaN
3330 $x->is_one(); # true if arg is +1
3331 $x->is_one('-'); # true if arg is -1
3332 $x->is_inf(); # true if +inf
3333 $x->is_inf('-'); # true if -inf (sign is default '+')
3335 These methods all test the BigInt for being one specific value and return
3336 true or false depending on the input. These are faster than doing something
3341 =head2 is_pos()/is_neg()/is_positive()/is_negative()
3343 $x->is_pos(); # true if > 0
3344 $x->is_neg(); # true if < 0
3346 The methods return true if the argument is positive or negative, respectively.
3347 C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
3348 C<-inf> is negative. A C<zero> is neither positive nor negative.
3350 These methods are only testing the sign, and not the value.
3352 C<is_positive()> and C<is_negative()> are aliases to C<is_pos()> and
3353 C<is_neg()>, respectively. C<is_positive()> and C<is_negative()> were
3354 introduced in v1.36, while C<is_pos()> and C<is_neg()> were only introduced
3357 =head2 is_odd()/is_even()/is_int()
3359 $x->is_odd(); # true if odd, false for even
3360 $x->is_even(); # true if even, false for odd
3361 $x->is_int(); # true if $x is an integer
3363 The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
3364 C<-inf> are not integers and are neither odd nor even.
3366 In BigInt, all numbers except C<NaN>, C<+inf> and C<-inf> are integers.
3372 Compares $x with $y and takes the sign into account.
3373 Returns -1, 0, 1 or undef.
3379 Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
3385 Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
3387 If you want $x to have a certain sign, use one of the following methods:
3390 $x->babs()->bneg(); # '-'
3392 $x->binf(); # '+inf'
3393 $x->binf('-'); # '-inf'
3397 $x->digit($n); # return the nth digit, counting from right
3399 If C<$n> is negative, returns the digit counting from left.
3405 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
3406 and '-inf', respectively. Does nothing for NaN or zero.
3412 Set the number to its absolute value, e.g. change the sign from '-' to '+'
3413 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
3418 $x->bnorm(); # normalize (no-op)
3424 Two's complement (bitwise not). This is equivalent to
3432 $x->binc(); # increment x by 1
3436 $x->bdec(); # decrement x by 1
3440 $x->badd($y); # addition (add $y to $x)
3444 $x->bsub($y); # subtraction (subtract $y from $x)
3448 $x->bmul($y); # multiplication (multiply $x by $y)
3452 $x->bdiv($y); # divide, set $x to quotient
3453 # return (quo,rem) or quo if scalar
3457 $x->bmod($y); # modulus (x % y)
3461 num->bmodinv($mod); # modular inverse
3463 Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
3464 returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
3465 C<bgcd($num, $mod)==1>.
3469 $num->bmodpow($exp,$mod); # modular exponentation
3470 # ($num**$exp % $mod)
3472 Returns the value of C<$num> taken to the power C<$exp> in the modulus
3473 C<$mod> using binary exponentation. C<bmodpow> is far superior to
3478 because it is much faster - it reduces internal variables into
3479 the modulus whenever possible, so it operates on smaller numbers.
3481 C<bmodpow> also supports negative exponents.
3483 bmodpow($num, -1, $mod)
3485 is exactly equivalent to
3491 $x->bpow($y); # power of arguments (x ** y)
3495 $x->blog($base, $accuracy); # logarithm of x to the base $base
3497 If C<$base> is not defined, Euler's number (e) is used:
3499 print $x->blog(undef, 100); # log(x) to 100 digits
3503 $x->bexp($accuracy); # calculate e ** X
3505 Calculates the expression C<e ** $x> where C<e> is Euler's number.
3507 This method was added in v1.82 of Math::BigInt (April 2007).
3513 $x->bnok($y); # x over y (binomial coefficient n over k)
3515 Calculates the binomial coefficient n over k, also called the "choose"
3516 function. The result is equivalent to:
3522 This method was added in v1.84 of Math::BigInt (April 2007).
3526 print Math::BigInt->bpi(100), "\n"; # 3
3528 Returns PI truncated to an integer, with the argument being ignored. that
3529 is it always returns C<3>.
3531 If upgrading is in effect, returns PI to N digits (including the "3"
3535 use Math::BigInt upgrade => Math::BigFloat;
3536 print Math::BigInt->bpi(3), "\n"; # 3.14
3537 print Math::BigInt->bpi(100), "\n"; # 3.1415....
3539 This method was added in v1.87 of Math::BigInt (June 2007).
3543 $x->blsft($y); # left shift in base 2
3544 $x->blsft($y,$n); # left shift, in base $n (like 10)
3548 $x->brsft($y); # right shift in base 2
3549 $x->brsft($y,$n); # right shift, in base $n (like 10)
3553 $x->band($y); # bitwise and
3557 $x->bior($y); # bitwise inclusive or
3561 $x->bxor($y); # bitwise exclusive or
3565 $x->bnot(); # bitwise not (two's complement)
3569 $x->bsqrt(); # calculate square-root
3573 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3577 $x->round($A,$P,$round_mode);
3579 Round $x to accuracy C<$A> or precision C<$P> using the round mode
3584 $x->bround($N); # accuracy: preserve $N digits
3588 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3594 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3595 does change $x in BigFloat.
3601 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3602 does change $x in BigFloat.
3606 bgcd(@values); # greatest common divisor (no OO style)
3610 blcm(@values); # lowest common multiplicator (no OO style)
3615 ($xl,$fl) = $x->length();
3617 Returns the number of digits in the decimal representation of the number.
3618 In list context, returns the length of the integer and fraction part. For
3619 BigInt's, the length of the fraction part will always be 0.
3625 Return the exponent of $x as BigInt.
3631 Return the signed mantissa of $x as BigInt.
3635 $x->parts(); # return (mantissa,exponent) as BigInt
3639 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3641 =head2 as_int()/as_number()
3645 Returns $x as a BigInt (truncated towards zero). In BigInt this is the same as
3648 C<as_number()> is an alias to this method. C<as_number> was introduced in
3649 v1.22, while C<as_int()> was only introduced in v1.68.
3655 Returns a normalized string representation of C<$x>.
3659 $x->bsstr(); # normalized string in scientific notation
3663 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3667 $x->as_bin(); # as signed binary string with prefixed 0b
3671 $x->as_oct(); # as signed octal string with prefixed 0
3677 This returns a normal Perl scalar from $x. It is used automatically
3678 whenever a scalar is needed, for instance in array index operations.
3680 This loses precision, to avoid this use L<as_int()> instead.
3684 $x->modify('bpowd');
3686 This method returns 0 if the object can be modified with the given
3687 peration, or 1 if not.
3689 This is used for instance by L<Math::BigInt::Constant>.
3691 =head2 upgrade()/downgrade()
3693 Set/get the class for downgrade/upgrade operations. Thuis is used
3694 for instance by L<bignum>. The defaults are '', thus the following
3695 operation will create a BigInt, not a BigFloat:
3697 my $i = Math::BigInt->new(123);
3698 my $f = Math::BigFloat->new('123.1');
3700 print $i + $f,"\n"; # print 246
3704 Set/get the number of digits for the default precision in divide
3709 Set/get the current round mode.
3711 =head1 ACCURACY and PRECISION
3713 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3714 accuracy and precision based rounding, both automatically after every
3715 operation, as well as manually.
3717 This section describes the accuracy/precision handling in Math::Big* as it
3718 used to be and as it is now, complete with an explanation of all terms and
3721 Not yet implemented things (but with correct description) are marked with '!',
3722 things that need to be answered are marked with '?'.
3724 In the next paragraph follows a short description of terms used here (because
3725 these may differ from terms used by others people or documentation).
3727 During the rest of this document, the shortcuts A (for accuracy), P (for
3728 precision), F (fallback) and R (rounding mode) will be used.
3732 A fixed number of digits before (positive) or after (negative)
3733 the decimal point. For example, 123.45 has a precision of -2. 0 means an
3734 integer like 123 (or 120). A precision of 2 means two digits to the left
3735 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3736 numbers with zeros before the decimal point may have different precisions,
3737 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3738 was). It could also have p < 0, when the digits after the decimal point
3741 The string output (of floating point numbers) will be padded with zeros:
3743 Initial value P A Result String
3744 ------------------------------------------------------------
3745 1234.01 -3 1000 1000
3748 1234.001 1 1234 1234.0
3750 1234.01 2 1234.01 1234.01
3751 1234.01 5 1234.01 1234.01000
3753 For BigInts, no padding occurs.
3757 Number of significant digits. Leading zeros are not counted. A
3758 number may have an accuracy greater than the non-zero digits
3759 when there are zeros in it or trailing zeros. For example, 123.456 has
3760 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3762 The string output (of floating point numbers) will be padded with zeros:
3764 Initial value P A Result String
3765 ------------------------------------------------------------
3767 1234.01 6 1234.01 1234.01
3768 1234.1 8 1234.1 1234.1000
3770 For BigInts, no padding occurs.
3774 When both A and P are undefined, this is used as a fallback accuracy when
3777 =head2 Rounding mode R
3779 When rounding a number, different 'styles' or 'kinds'
3780 of rounding are possible. (Note that random rounding, as in
3781 Math::Round, is not implemented.)
3787 truncation invariably removes all digits following the
3788 rounding place, replacing them with zeros. Thus, 987.65 rounded
3789 to tens (P=1) becomes 980, and rounded to the fourth sigdig
3790 becomes 987.6 (A=4). 123.456 rounded to the second place after the
3791 decimal point (P=-2) becomes 123.46.
3793 All other implemented styles of rounding attempt to round to the
3794 "nearest digit." If the digit D immediately to the right of the
3795 rounding place (skipping the decimal point) is greater than 5, the
3796 number is incremented at the rounding place (possibly causing a
3797 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3798 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3799 truncated at the rounding place: e.g. when rounding to units, 0.4
3800 rounds to 0, and -19.4 rounds to -19.
3802 However the results of other styles of rounding differ if the
3803 digit immediately to the right of the rounding place (skipping the
3804 decimal point) is 5 and if there are no digits, or no digits other
3805 than 0, after that 5. In such cases:
3809 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3810 if it is not already. E.g., when rounding to the first sigdig, 0.45
3811 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3815 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3816 it is not already. E.g., when rounding to the first sigdig, 0.45
3817 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3821 round to plus infinity, i.e. always round up. E.g., when
3822 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3823 and 0.4501 also becomes 0.5.
3827 round to minus infinity, i.e. always round down. E.g., when
3828 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3829 but 0.4501 becomes 0.5.
3833 round to zero, i.e. positive numbers down, negative ones up.
3834 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3835 becomes -0.5, but 0.4501 becomes 0.5.
3839 round up if the digit immediately to the right of the rounding place
3840 is 5 or greater, otherwise round down. E.g., 0.15 becomes 0.2 and
3845 The handling of A & P in MBI/MBF (the old core code shipped with Perl
3846 versions <= 5.7.2) is like this:
3852 * ffround($p) is able to round to $p number of digits after the decimal
3854 * otherwise P is unused
3856 =item Accuracy (significant digits)
3858 * fround($a) rounds to $a significant digits
3859 * only fdiv() and fsqrt() take A as (optional) paramater
3860 + other operations simply create the same number (fneg etc), or more (fmul)
3862 + rounding/truncating is only done when explicitly calling one of fround
3863 or ffround, and never for BigInt (not implemented)
3864 * fsqrt() simply hands its accuracy argument over to fdiv.
3865 * the documentation and the comment in the code indicate two different ways
3866 on how fdiv() determines the maximum number of digits it should calculate,
3867 and the actual code does yet another thing
3869 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3871 result has at most max(scale, length(dividend), length(divisor)) digits
3873 scale = max(scale, length(dividend)-1,length(divisor)-1);
3874 scale += length(divisor) - length(dividend);
3875 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3876 Actually, the 'difference' added to the scale is calculated from the
3877 number of "significant digits" in dividend and divisor, which is derived
3878 by looking at the length of the mantissa. Which is wrong, since it includes
3879 the + sign (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops
3880 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3881 assumption that 124 has 3 significant digits, while 120/7 will get you
3882 '17', not '17.1' since 120 is thought to have 2 significant digits.
3883 The rounding after the division then uses the remainder and $y to determine
3884 wether it must round up or down.
3885 ? I have no idea which is the right way. That's why I used a slightly more
3886 ? simple scheme and tweaked the few failing testcases to match it.
3890 This is how it works now:
3894 =item Setting/Accessing
3896 * You can set the A global via C<< Math::BigInt->accuracy() >> or
3897 C<< Math::BigFloat->accuracy() >> or whatever class you are using.
3898 * You can also set P globally by using C<< Math::SomeClass->precision() >>
3900 * Globals are classwide, and not inherited by subclasses.
3901 * to undefine A, use C<< Math::SomeCLass->accuracy(undef); >>
3902 * to undefine P, use C<< Math::SomeClass->precision(undef); >>
3903 * Setting C<< Math::SomeClass->accuracy() >> clears automatically
3904 C<< Math::SomeClass->precision() >>, and vice versa.
3905 * To be valid, A must be > 0, P can have any value.
3906 * If P is negative, this means round to the P'th place to the right of the
3907 decimal point; positive values mean to the left of the decimal point.
3908 P of 0 means round to integer.
3909 * to find out the current global A, use C<< Math::SomeClass->accuracy() >>
3910 * to find out the current global P, use C<< Math::SomeClass->precision() >>
3911 * use C<< $x->accuracy() >> respective C<< $x->precision() >> for the local
3912 setting of C<< $x >>.
3913 * Please note that C<< $x->accuracy() >> respective C<< $x->precision() >>
3914 return eventually defined global A or P, when C<< $x >>'s A or P is not
3917 =item Creating numbers
3919 * When you create a number, you can give the desired A or P via:
3920 $x = Math::BigInt->new($number,$A,$P);
3921 * Only one of A or P can be defined, otherwise the result is NaN
3922 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
3923 globals (if set) will be used. Thus changing the global defaults later on
3924 will not change the A or P of previously created numbers (i.e., A and P of
3925 $x will be what was in effect when $x was created)
3926 * If given undef for A and P, B<no> rounding will occur, and the globals will
3927 B<not> be used. This is used by subclasses to create numbers without
3928 suffering rounding in the parent. Thus a subclass is able to have its own
3929 globals enforced upon creation of a number by using
3930 C<< $x = Math::BigInt->new($number,undef,undef) >>:
3932 use Math::BigInt::SomeSubclass;
3935 Math::BigInt->accuracy(2);
3936 Math::BigInt::SomeSubClass->accuracy(3);
3937 $x = Math::BigInt::SomeSubClass->new(1234);
3939 $x is now 1230, and not 1200. A subclass might choose to implement
3940 this otherwise, e.g. falling back to the parent's A and P.
3944 * If A or P are enabled/defined, they are used to round the result of each
3945 operation according to the rules below
3946 * Negative P is ignored in Math::BigInt, since BigInts never have digits
3947 after the decimal point
3948 * Math::BigFloat uses Math::BigInt internally, but setting A or P inside
3949 Math::BigInt as globals does not tamper with the parts of a BigFloat.
3950 A flag is used to mark all Math::BigFloat numbers as 'never round'.
3954 * It only makes sense that a number has only one of A or P at a time.
3955 If you set either A or P on one object, or globally, the other one will
3956 be automatically cleared.
3957 * If two objects are involved in an operation, and one of them has A in
3958 effect, and the other P, this results in an error (NaN).
3959 * A takes precedence over P (Hint: A comes before P).
3960 If neither of them is defined, nothing is used, i.e. the result will have
3961 as many digits as it can (with an exception for fdiv/fsqrt) and will not
3963 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
3964 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
3965 If either the dividend's or the divisor's mantissa has more digits than
3966 the value of F, the higher value will be used instead of F.
3967 This is to limit the digits (A) of the result (just consider what would
3968 happen with unlimited A and P in the case of 1/3 :-)
3969 * fdiv will calculate (at least) 4 more digits than required (determined by
3970 A, P or F), and, if F is not used, round the result
3971 (this will still fail in the case of a result like 0.12345000000001 with A
3972 or P of 5, but this can not be helped - or can it?)
3973 * Thus you can have the math done by on Math::Big* class in two modi:
3974 + never round (this is the default):
3975 This is done by setting A and P to undef. No math operation
3976 will round the result, with fdiv() and fsqrt() as exceptions to guard
3977 against overflows. You must explicitly call bround(), bfround() or
3978 round() (the latter with parameters).
3979 Note: Once you have rounded a number, the settings will 'stick' on it
3980 and 'infect' all other numbers engaged in math operations with it, since
3981 local settings have the highest precedence. So, to get SaferRound[tm],
3982 use a copy() before rounding like this:
3984 $x = Math::BigFloat->new(12.34);
3985 $y = Math::BigFloat->new(98.76);
3986 $z = $x * $y; # 1218.6984
3987 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
3988 $z = $x * $y; # still 1218.6984, without
3989 # copy would have been 1210!
3991 + round after each op:
3992 After each single operation (except for testing like is_zero()), the
3993 method round() is called and the result is rounded appropriately. By
3994 setting proper values for A and P, you can have all-the-same-A or
3995 all-the-same-P modes. For example, Math::Currency might set A to undef,
3996 and P to -2, globally.
3998 ?Maybe an extra option that forbids local A & P settings would be in order,
3999 ?so that intermediate rounding does not 'poison' further math?
4001 =item Overriding globals
4003 * you will be able to give A, P and R as an argument to all the calculation
4004 routines; the second parameter is A, the third one is P, and the fourth is
4005 R (shift right by one for binary operations like badd). P is used only if
4006 the first parameter (A) is undefined. These three parameters override the
4007 globals in the order detailed as follows, i.e. the first defined value
4009 (local: per object, global: global default, parameter: argument to sub)
4012 + local A (if defined on both of the operands: smaller one is taken)
4013 + local P (if defined on both of the operands: bigger one is taken)
4017 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
4018 arguments (A and P) instead of one
4020 =item Local settings
4022 * You can set A or P locally by using C<< $x->accuracy() >> or
4023 C<< $x->precision() >>
4024 and thus force different A and P for different objects/numbers.
4025 * Setting A or P this way immediately rounds $x to the new value.
4026 * C<< $x->accuracy() >> clears C<< $x->precision() >>, and vice versa.
4030 * the rounding routines will use the respective global or local settings.
4031 fround()/bround() is for accuracy rounding, while ffround()/bfround()
4033 * the two rounding functions take as the second parameter one of the
4034 following rounding modes (R):
4035 'even', 'odd', '+inf', '-inf', 'zero', 'trunc', 'common'
4036 * you can set/get the global R by using C<< Math::SomeClass->round_mode() >>
4037 or by setting C<< $Math::SomeClass::round_mode >>
4038 * after each operation, C<< $result->round() >> is called, and the result may
4039 eventually be rounded (that is, if A or P were set either locally,
4040 globally or as parameter to the operation)
4041 * to manually round a number, call C<< $x->round($A,$P,$round_mode); >>
4042 this will round the number by using the appropriate rounding function
4043 and then normalize it.
4044 * rounding modifies the local settings of the number:
4046 $x = Math::BigFloat->new(123.456);
4050 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
4051 will be 4 from now on.
4053 =item Default values
4062 * The defaults are set up so that the new code gives the same results as
4063 the old code (except in a few cases on fdiv):
4064 + Both A and P are undefined and thus will not be used for rounding
4065 after each operation.
4066 + round() is thus a no-op, unless given extra parameters A and P
4070 =head1 Infinity and Not a Number
4072 While BigInt has extensive handling of inf and NaN, certain quirks remain.
4078 These perl routines currently (as of Perl v.5.8.6) cannot handle passed
4081 te@linux:~> perl -wle 'print 2 ** 3333'
4083 te@linux:~> perl -wle 'print 2 ** 3333 == 2 ** 3333'
4085 te@linux:~> perl -wle 'print oct(2 ** 3333)'
4087 te@linux:~> perl -wle 'print hex(2 ** 3333)'
4088 Illegal hexadecimal digit 'i' ignored at -e line 1.
4091 The same problems occur if you pass them Math::BigInt->binf() objects. Since
4092 overloading these routines is not possible, this cannot be fixed from BigInt.
4094 =item ==, !=, <, >, <=, >= with NaNs
4096 BigInt's bcmp() routine currently returns undef to signal that a NaN was
4097 involved in a comparison. However, the overload code turns that into
4098 either 1 or '' and thus operations like C<< NaN != NaN >> might return
4103 C<< log(-inf) >> is highly weird. Since log(-x)=pi*i+log(x), then
4104 log(-inf)=pi*i+inf. However, since the imaginary part is finite, the real
4105 infinity "overshadows" it, so the number might as well just be infinity.
4106 However, the result is a complex number, and since BigInt/BigFloat can only
4107 have real numbers as results, the result is NaN.
4109 =item exp(), cos(), sin(), atan2()
4111 These all might have problems handling infinity right.
4117 The actual numbers are stored as unsigned big integers (with seperate sign).
4119 You should neither care about nor depend on the internal representation; it
4120 might change without notice. Use B<ONLY> method calls like C<< $x->sign(); >>
4121 instead relying on the internal representation.
4125 Math with the numbers is done (by default) by a module called
4126 C<Math::BigInt::Calc>. This is equivalent to saying:
4128 use Math::BigInt lib => 'Calc';
4130 You can change this by using:
4132 use Math::BigInt lib => 'BitVect';
4134 The following would first try to find Math::BigInt::Foo, then
4135 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
4137 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
4139 Since Math::BigInt::GMP is in almost all cases faster than Calc (especially in
4140 math involving really big numbers, where it is B<much> faster), and there is
4141 no penalty if Math::BigInt::GMP is not installed, it is a good idea to always
4144 use Math::BigInt lib => 'GMP';
4146 Different low-level libraries use different formats to store the
4147 numbers. You should B<NOT> depend on the number having a specific format
4150 See the respective math library module documentation for further details.
4154 The sign is either '+', '-', 'NaN', '+inf' or '-inf'.
4156 A sign of 'NaN' is used to represent the result when input arguments are not
4157 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
4158 minus infinity. You will get '+inf' when dividing a positive number by 0, and
4159 '-inf' when dividing any negative number by 0.
4161 =head2 mantissa(), exponent() and parts()
4163 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
4166 $m = $x->mantissa();
4167 $e = $x->exponent();
4168 $y = $m * ( 10 ** $e );
4169 print "ok\n" if $x == $y;
4171 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
4172 in one go. Both the returned mantissa and exponent have a sign.
4174 Currently, for BigInts C<$e> is always 0, except +inf and -inf, where it is
4175 C<+inf>; and for NaN, where it is C<NaN>; and for C<$x == 0>, where it is C<1>
4176 (to be compatible with Math::BigFloat's internal representation of a zero as
4179 C<$m> is currently just a copy of the original number. The relation between
4180 C<$e> and C<$m> will stay always the same, though their real values might
4187 sub bint { Math::BigInt->new(shift); }
4189 $x = Math::BigInt->bstr("1234") # string "1234"
4190 $x = "$x"; # same as bstr()
4191 $x = Math::BigInt->bneg("1234"); # BigInt "-1234"
4192 $x = Math::BigInt->babs("-12345"); # BigInt "12345"
4193 $x = Math::BigInt->bnorm("-0.00"); # BigInt "0"
4194 $x = bint(1) + bint(2); # BigInt "3"
4195 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
4196 $x = bint(1); # BigInt "1"
4197 $x = $x + 5 / 2; # BigInt "3"
4198 $x = $x ** 3; # BigInt "27"
4199 $x *= 2; # BigInt "54"
4200 $x = Math::BigInt->new(0); # BigInt "0"
4202 $x = Math::BigInt->badd(4,5) # BigInt "9"
4203 print $x->bsstr(); # 9e+0
4205 Examples for rounding:
4210 $x = Math::BigFloat->new(123.4567);
4211 $y = Math::BigFloat->new(123.456789);
4212 Math::BigFloat->accuracy(4); # no more A than 4
4214 ok ($x->copy()->fround(),123.4); # even rounding
4215 print $x->copy()->fround(),"\n"; # 123.4
4216 Math::BigFloat->round_mode('odd'); # round to odd
4217 print $x->copy()->fround(),"\n"; # 123.5
4218 Math::BigFloat->accuracy(5); # no more A than 5
4219 Math::BigFloat->round_mode('odd'); # round to odd
4220 print $x->copy()->fround(),"\n"; # 123.46
4221 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
4222 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
4224 Math::BigFloat->accuracy(undef); # A not important now
4225 Math::BigFloat->precision(2); # P important
4226 print $x->copy()->bnorm(),"\n"; # 123.46
4227 print $x->copy()->fround(),"\n"; # 123.46
4229 Examples for converting:
4231 my $x = Math::BigInt->new('0b1'.'01' x 123);
4232 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
4234 =head1 Autocreating constants
4236 After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
4237 and binary constants in the given scope are converted to C<Math::BigInt>.
4238 This conversion happens at compile time.
4242 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
4244 prints the integer value of C<2**100>. Note that without conversion of
4245 constants the expression 2**100 will be calculated as perl scalar.
4247 Please note that strings and floating point constants are not affected,
4250 use Math::BigInt qw/:constant/;
4252 $x = 1234567890123456789012345678901234567890
4253 + 123456789123456789;
4254 $y = '1234567890123456789012345678901234567890'
4255 + '123456789123456789';
4257 do not work. You need an explicit Math::BigInt->new() around one of the
4258 operands. You should also quote large constants to protect loss of precision:
4262 $x = Math::BigInt->new('1234567889123456789123456789123456789');
4264 Without the quotes Perl would convert the large number to a floating point
4265 constant at compile time and then hand the result to BigInt, which results in
4266 an truncated result or a NaN.
4268 This also applies to integers that look like floating point constants:
4270 use Math::BigInt ':constant';
4272 print ref(123e2),"\n";
4273 print ref(123.2e2),"\n";
4275 will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
4276 to get this to work.
4280 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
4281 must be made in the second case. For long numbers, the copy can eat up to 20%
4282 of the work (in the case of addition/subtraction, less for
4283 multiplication/division). If $y is very small compared to $x, the form
4284 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
4285 more time then the actual addition.
4287 With a technique called copy-on-write, the cost of copying with overload could
4288 be minimized or even completely avoided. A test implementation of COW did show
4289 performance gains for overloaded math, but introduced a performance loss due
4290 to a constant overhead for all other operations. So Math::BigInt does currently
4293 The rewritten version of this module (vs. v0.01) is slower on certain
4294 operations, like C<new()>, C<bstr()> and C<numify()>. The reason are that it
4295 does now more work and handles much more cases. The time spent in these
4296 operations is usually gained in the other math operations so that code on
4297 the average should get (much) faster. If they don't, please contact the author.
4299 Some operations may be slower for small numbers, but are significantly faster
4300 for big numbers. Other operations are now constant (O(1), like C<bneg()>,
4301 C<babs()> etc), instead of O(N) and thus nearly always take much less time.
4302 These optimizations were done on purpose.
4304 If you find the Calc module to slow, try to install any of the replacement
4305 modules and see if they help you.
4307 =head2 Alternative math libraries
4309 You can use an alternative library to drive Math::BigInt via:
4311 use Math::BigInt lib => 'Module';
4313 See L<MATH LIBRARY> for more information.
4315 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
4319 =head1 Subclassing Math::BigInt
4321 The basic design of Math::BigInt allows simple subclasses with very little
4322 work, as long as a few simple rules are followed:
4328 The public API must remain consistent, i.e. if a sub-class is overloading
4329 addition, the sub-class must use the same name, in this case badd(). The
4330 reason for this is that Math::BigInt is optimized to call the object methods
4335 The private object hash keys like C<$x->{sign}> may not be changed, but
4336 additional keys can be added, like C<$x->{_custom}>.
4340 Accessor functions are available for all existing object hash keys and should
4341 be used instead of directly accessing the internal hash keys. The reason for
4342 this is that Math::BigInt itself has a pluggable interface which permits it
4343 to support different storage methods.
4347 More complex sub-classes may have to replicate more of the logic internal of
4348 Math::BigInt if they need to change more basic behaviors. A subclass that
4349 needs to merely change the output only needs to overload C<bstr()>.
4351 All other object methods and overloaded functions can be directly inherited
4352 from the parent class.
4354 At the very minimum, any subclass will need to provide its own C<new()> and can
4355 store additional hash keys in the object. There are also some package globals
4356 that must be defined, e.g.:
4360 $precision = -2; # round to 2 decimal places
4361 $round_mode = 'even';
4364 Additionally, you might want to provide the following two globals to allow
4365 auto-upgrading and auto-downgrading to work correctly:
4370 This allows Math::BigInt to correctly retrieve package globals from the
4371 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
4372 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
4378 in your subclass to automatically inherit the overloading from the parent. If
4379 you like, you can change part of the overloading, look at Math::String for an
4384 When used like this:
4386 use Math::BigInt upgrade => 'Foo::Bar';
4388 certain operations will 'upgrade' their calculation and thus the result to
4389 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
4391 use Math::BigInt upgrade => 'Math::BigFloat';
4393 As a shortcut, you can use the module C<bignum>:
4397 Also good for oneliners:
4399 perl -Mbignum -le 'print 2 ** 255'
4401 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
4402 as well es preserve accuracy (as in sqrt(3)).
4404 Beware: This feature is not fully implemented yet.
4408 The following methods upgrade themselves unconditionally; that is if upgrade
4409 is in effect, they will always hand up their work:
4423 Beware: This list is not complete.
4425 All other methods upgrade themselves only when one (or all) of their
4426 arguments are of the class mentioned in $upgrade (This might change in later
4427 versions to a more sophisticated scheme):
4431 C<Math::BigInt> exports nothing by default, but can export the following methods:
4440 =item broot() does not work
4442 The broot() function in BigInt may only work for small values. This will be
4443 fixed in a later version.
4445 =item Out of Memory!
4447 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
4448 C<eval()> in your code will crash with "Out of memory". This is probably an
4449 overload/exporter bug. You can workaround by not having C<eval()>
4450 and ':constant' at the same time or upgrade your Perl to a newer version.
4452 =item Fails to load Calc on Perl prior 5.6.0
4454 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
4455 will fall back to eval { require ... } when loading the math lib on Perls
4456 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
4457 filesystems using a different seperator.
4463 Some things might not work as you expect them. Below is documented what is
4464 known to be troublesome:
4468 =item bstr(), bsstr() and 'cmp'
4470 Both C<bstr()> and C<bsstr()> as well as automated stringify via overload now
4471 drop the leading '+'. The old code would return '+3', the new returns '3'.
4472 This is to be consistent with Perl and to make C<cmp> (especially with
4473 overloading) to work as you expect. It also solves problems with C<Test.pm>,
4474 because its C<ok()> uses 'eq' internally.
4476 Mark Biggar said, when asked about to drop the '+' altogether, or make only
4479 I agree (with the first alternative), don't add the '+' on positive
4480 numbers. It's not as important anymore with the new internal
4481 form for numbers. It made doing things like abs and neg easier,
4482 but those have to be done differently now anyway.
4484 So, the following examples will now work all as expected:
4487 BEGIN { plan tests => 1 }
4490 my $x = new Math::BigInt 3*3;
4491 my $y = new Math::BigInt 3*3;
4494 print "$x eq 9" if $x eq $y;
4495 print "$x eq 9" if $x eq '9';
4496 print "$x eq 9" if $x eq 3*3;
4498 Additionally, the following still works:
4500 print "$x == 9" if $x == $y;
4501 print "$x == 9" if $x == 9;
4502 print "$x == 9" if $x == 3*3;
4504 There is now a C<bsstr()> method to get the string in scientific notation aka
4505 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
4506 for comparison, but Perl will represent some numbers as 100 and others
4507 as 1e+308. If in doubt, convert both arguments to Math::BigInt before
4508 comparing them as strings:
4511 BEGIN { plan tests => 3 }
4514 $x = Math::BigInt->new('1e56'); $y = 1e56;
4515 ok ($x,$y); # will fail
4516 ok ($x->bsstr(),$y); # okay
4517 $y = Math::BigInt->new($y);
4520 Alternatively, simple use C<< <=> >> for comparisons, this will get it
4521 always right. There is not yet a way to get a number automatically represented
4522 as a string that matches exactly the way Perl represents it.
4524 See also the section about L<Infinity and Not a Number> for problems in
4529 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
4532 $x = Math::BigInt->new(123);
4533 $y = int($x); # BigInt 123
4534 $x = Math::BigFloat->new(123.45);
4535 $y = int($x); # BigInt 123
4537 In all Perl versions you can use C<as_number()> or C<as_int> for the same
4540 $x = Math::BigFloat->new(123.45);
4541 $y = $x->as_number(); # BigInt 123
4542 $y = $x->as_int(); # ditto
4544 This also works for other subclasses, like Math::String.
4546 If you want a real Perl scalar, use C<numify()>:
4548 $y = $x->numify(); # 123 as scalar
4550 This is seldom necessary, though, because this is done automatically, like
4551 when you access an array:
4553 $z = $array[$x]; # does work automatically
4557 The following will probably not do what you expect:
4559 $c = Math::BigInt->new(123);
4560 print $c->length(),"\n"; # prints 30
4562 It prints both the number of digits in the number and in the fraction part
4563 since print calls C<length()> in list context. Use something like:
4565 print scalar $c->length(),"\n"; # prints 3
4569 The following will probably not do what you expect:
4571 print $c->bdiv(10000),"\n";
4573 It prints both quotient and remainder since print calls C<bdiv()> in list
4574 context. Also, C<bdiv()> will modify $c, so be careful. You probably want
4577 print $c / 10000,"\n";
4578 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
4582 The quotient is always the greatest integer less than or equal to the
4583 real-valued quotient of the two operands, and the remainder (when it is
4584 nonzero) always has the same sign as the second operand; so, for
4594 As a consequence, the behavior of the operator % agrees with the
4595 behavior of Perl's built-in % operator (as documented in the perlop
4596 manpage), and the equation
4598 $x == ($x / $y) * $y + ($x % $y)
4600 holds true for any $x and $y, which justifies calling the two return
4601 values of bdiv() the quotient and remainder. The only exception to this rule
4602 are when $y == 0 and $x is negative, then the remainder will also be
4603 negative. See below under "infinity handling" for the reasoning behind this.
4605 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
4606 not change BigInt's way to do things. This is because under 'use integer' Perl
4607 will do what the underlying C thinks is right and this is different for each
4608 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
4609 the author to implement it ;)
4611 =item infinity handling
4613 Here are some examples that explain the reasons why certain results occur while
4616 The following table shows the result of the division and the remainder, so that
4617 the equation above holds true. Some "ordinary" cases are strewn in to show more
4618 clearly the reasoning:
4620 A / B = C, R so that C * B + R = A
4621 =========================================================
4622 5 / 8 = 0, 5 0 * 8 + 5 = 5
4623 0 / 8 = 0, 0 0 * 8 + 0 = 0
4624 0 / inf = 0, 0 0 * inf + 0 = 0
4625 0 /-inf = 0, 0 0 * -inf + 0 = 0
4626 5 / inf = 0, 5 0 * inf + 5 = 5
4627 5 /-inf = 0, 5 0 * -inf + 5 = 5
4628 -5/ inf = 0, -5 0 * inf + -5 = -5
4629 -5/-inf = 0, -5 0 * -inf + -5 = -5
4630 inf/ 5 = inf, 0 inf * 5 + 0 = inf
4631 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
4632 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
4633 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
4634 5/ 5 = 1, 0 1 * 5 + 0 = 5
4635 -5/ -5 = 1, 0 1 * -5 + 0 = -5
4636 inf/ inf = 1, 0 1 * inf + 0 = inf
4637 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
4638 inf/-inf = -1, 0 -1 * -inf + 0 = inf
4639 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
4640 8/ 0 = inf, 8 inf * 0 + 8 = 8
4641 inf/ 0 = inf, inf inf * 0 + inf = inf
4644 These cases below violate the "remainder has the sign of the second of the two
4645 arguments", since they wouldn't match up otherwise.
4647 A / B = C, R so that C * B + R = A
4648 ========================================================
4649 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
4650 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
4652 =item Modifying and =
4656 $x = Math::BigFloat->new(5);
4659 It will not do what you think, e.g. making a copy of $x. Instead it just makes
4660 a second reference to the B<same> object and stores it in $y. Thus anything
4661 that modifies $x (except overloaded operators) will modify $y, and vice versa.
4662 Or in other words, C<=> is only safe if you modify your BigInts only via
4663 overloaded math. As soon as you use a method call it breaks:
4666 print "$x, $y\n"; # prints '10, 10'
4668 If you want a true copy of $x, use:
4672 You can also chain the calls like this, this will make first a copy and then
4675 $y = $x->copy()->bmul(2);
4677 See also the documentation for overload.pm regarding C<=>.
4681 C<bpow()> (and the rounding functions) now modifies the first argument and
4682 returns it, unlike the old code which left it alone and only returned the
4683 result. This is to be consistent with C<badd()> etc. The first three will
4684 modify $x, the last one won't:
4686 print bpow($x,$i),"\n"; # modify $x
4687 print $x->bpow($i),"\n"; # ditto
4688 print $x **= $i,"\n"; # the same
4689 print $x ** $i,"\n"; # leave $x alone
4691 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4693 =item Overloading -$x
4703 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4704 needs to preserve $x since it does not know that it later will get overwritten.
4705 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4707 =item Mixing different object types
4709 In Perl you will get a floating point value if you do one of the following:
4715 With overloaded math, only the first two variants will result in a BigFloat:
4720 $mbf = Math::BigFloat->new(5);
4721 $mbi2 = Math::BigInteger->new(5);
4722 $mbi = Math::BigInteger->new(2);
4724 # what actually gets called:
4725 $float = $mbf + $mbi; # $mbf->badd()
4726 $float = $mbf / $mbi; # $mbf->bdiv()
4727 $integer = $mbi + $mbf; # $mbi->badd()
4728 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4729 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4731 This is because math with overloaded operators follows the first (dominating)
4732 operand, and the operation of that is called and returns thus the result. So,
4733 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4734 the result should be a Math::BigFloat or the second operant is one.
4736 To get a Math::BigFloat you either need to call the operation manually,
4737 make sure the operands are already of the proper type or casted to that type
4738 via Math::BigFloat->new():
4740 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4742 Beware of simple "casting" the entire expression, this would only convert
4743 the already computed result:
4745 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4747 Beware also of the order of more complicated expressions like:
4749 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4750 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4752 If in doubt, break the expression into simpler terms, or cast all operands
4753 to the desired resulting type.
4755 Scalar values are a bit different, since:
4760 will both result in the proper type due to the way the overloaded math works.
4762 This section also applies to other overloaded math packages, like Math::String.
4764 One solution to you problem might be autoupgrading|upgrading. See the
4765 pragmas L<bignum>, L<bigint> and L<bigrat> for an easy way to do this.
4769 C<bsqrt()> works only good if the result is a big integer, e.g. the square
4770 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4771 mode. The reason is that the result is always truncated to an integer.
4773 If you want a better approximation of the square root, then use:
4775 $x = Math::BigFloat->new(12);
4776 Math::BigFloat->precision(0);
4777 Math::BigFloat->round_mode('even');
4778 print $x->copy->bsqrt(),"\n"; # 4
4780 Math::BigFloat->precision(2);
4781 print $x->bsqrt(),"\n"; # 3.46
4782 print $x->bsqrt(3),"\n"; # 3.464
4786 For negative numbers in base see also L<brsft|brsft>.
4792 This program is free software; you may redistribute it and/or modify it under
4793 the same terms as Perl itself.
4797 L<Math::BigFloat>, L<Math::BigRat> and L<Math::Big> as well as
4798 L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4800 The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest
4801 because they solve the autoupgrading/downgrading issue, at least partly.
4804 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4805 more documentation including a full version history, testcases, empty
4806 subclass files and benchmarks.
4810 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4811 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001 - 2006
4812 and still at it in 2007.
4814 Many people contributed in one or more ways to the final beast, see the file
4815 CREDITS for an (incomplete) list. If you miss your name, please drop me a