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 # we might need '==' and '!=' to get things like "NaN == NaN" right
66 '<=>' => sub { $_[2] ?
67 ref($_[0])->bcmp($_[1],$_[0]) :
68 $_[0]->bcmp($_[1]); },
71 "$_[1]" cmp $_[0]->bstr() :
72 $_[0]->bstr() cmp "$_[1]" },
74 # make cos()/sin()/exp() "work" with BigInt's or subclasses
75 'cos' => sub { cos($_[0]->numify()) },
76 'sin' => sub { sin($_[0]->numify()) },
77 'exp' => sub { exp($_[0]->numify()) },
78 'atan2' => sub { $_[2] ?
79 atan2($_[1],$_[0]->numify()) :
80 atan2($_[0]->numify(),$_[1]) },
82 # are not yet overloadable
83 #'hex' => sub { print "hex"; $_[0]; },
84 #'oct' => sub { print "oct"; $_[0]; },
86 'log' => sub { $_[0]->copy()->blog($_[1]); },
87 'int' => sub { $_[0]->copy(); },
88 'neg' => sub { $_[0]->copy()->bneg(); },
89 'abs' => sub { $_[0]->copy()->babs(); },
90 'sqrt' => sub { $_[0]->copy()->bsqrt(); },
91 '~' => sub { $_[0]->copy()->bnot(); },
93 # for subtract it's a bit tricky to not modify b: b-a => -a+b
94 '-' => sub { my $c = $_[0]->copy; $_[2] ?
95 $c->bneg()->badd( $_[1]) :
97 '+' => sub { $_[0]->copy()->badd($_[1]); },
98 '*' => sub { $_[0]->copy()->bmul($_[1]); },
101 $_[2] ? ref($_[0])->new($_[1])->bdiv($_[0]) : $_[0]->copy->bdiv($_[1]);
104 $_[2] ? ref($_[0])->new($_[1])->bmod($_[0]) : $_[0]->copy->bmod($_[1]);
107 $_[2] ? ref($_[0])->new($_[1])->bpow($_[0]) : $_[0]->copy->bpow($_[1]);
110 $_[2] ? ref($_[0])->new($_[1])->blsft($_[0]) : $_[0]->copy->blsft($_[1]);
113 $_[2] ? ref($_[0])->new($_[1])->brsft($_[0]) : $_[0]->copy->brsft($_[1]);
116 $_[2] ? ref($_[0])->new($_[1])->band($_[0]) : $_[0]->copy->band($_[1]);
119 $_[2] ? ref($_[0])->new($_[1])->bior($_[0]) : $_[0]->copy->bior($_[1]);
122 $_[2] ? ref($_[0])->new($_[1])->bxor($_[0]) : $_[0]->copy->bxor($_[1]);
125 # can modify arg of ++ and --, so avoid a copy() for speed, but don't
126 # use $_[0]->bone(), it would modify $_[0] to be 1!
127 '++' => sub { $_[0]->binc() },
128 '--' => sub { $_[0]->bdec() },
130 # if overloaded, O(1) instead of O(N) and twice as fast for small numbers
132 # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/
133 # v5.6.1 dumps on this: return !$_[0]->is_zero() || undef; :-(
135 $t = 1 if !$_[0]->is_zero();
139 # the original qw() does not work with the TIESCALAR below, why?
140 # Order of arguments unsignificant
141 '""' => sub { $_[0]->bstr(); },
142 '0+' => sub { $_[0]->numify(); }
145 ##############################################################################
146 # global constants, flags and accessory
148 # These vars are public, but their direct usage is not recommended, use the
149 # accessor methods instead
151 $round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
156 $upgrade = undef; # default is no upgrade
157 $downgrade = undef; # default is no downgrade
159 # These are internally, and not to be used from the outside at all
161 $_trap_nan = 0; # are NaNs ok? set w/ config()
162 $_trap_inf = 0; # are infs ok? set w/ config()
163 my $nan = 'NaN'; # constants for easier life
165 my $CALC = 'Math::BigInt::FastCalc'; # module to do the low level math
166 # default is FastCalc.pm
167 my $IMPORT = 0; # was import() called yet?
168 # used to make require work
169 my %WARN; # warn only once for low-level libs
170 my %CAN; # cache for $CALC->can(...)
171 my %CALLBACKS; # callbacks to notify on lib loads
172 my $EMU_LIB = 'Math/BigInt/CalcEmu.pm'; # emulate low-level math
174 ##############################################################################
175 # the old code had $rnd_mode, so we need to support it, too
178 sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; }
179 sub FETCH { return $round_mode; }
180 sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
184 # tie to enable $rnd_mode to work transparently
185 tie $rnd_mode, 'Math::BigInt';
187 # set up some handy alias names
188 *as_int = \&as_number;
189 *is_pos = \&is_positive;
190 *is_neg = \&is_negative;
193 ##############################################################################
198 # make Class->round_mode() work
200 my $class = ref($self) || $self || __PACKAGE__;
204 if ($m !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
206 require Carp; Carp::croak ("Unknown round mode '$m'");
208 return ${"${class}::round_mode"} = $m;
210 ${"${class}::round_mode"};
216 # make Class->upgrade() work
218 my $class = ref($self) || $self || __PACKAGE__;
219 # need to set new value?
222 return ${"${class}::upgrade"} = $_[0];
224 ${"${class}::upgrade"};
230 # make Class->downgrade() work
232 my $class = ref($self) || $self || __PACKAGE__;
233 # need to set new value?
236 return ${"${class}::downgrade"} = $_[0];
238 ${"${class}::downgrade"};
244 # make Class->div_scale() work
246 my $class = ref($self) || $self || __PACKAGE__;
251 require Carp; Carp::croak ('div_scale must be greater than zero');
253 ${"${class}::div_scale"} = $_[0];
255 ${"${class}::div_scale"};
260 # $x->accuracy($a); ref($x) $a
261 # $x->accuracy(); ref($x)
262 # Class->accuracy(); class
263 # Class->accuracy($a); class $a
266 my $class = ref($x) || $x || __PACKAGE__;
269 # need to set new value?
273 # convert objects to scalars to avoid deep recursion. If object doesn't
274 # have numify(), then hopefully it will have overloading for int() and
275 # boolean test without wandering into a deep recursion path...
276 $a = $a->numify() if ref($a) && $a->can('numify');
280 # also croak on non-numerical
284 Carp::croak ('Argument to accuracy must be greater than zero');
288 require Carp; Carp::croak ('Argument to accuracy must be an integer');
293 # $object->accuracy() or fallback to global
294 $x->bround($a) if $a; # not for undef, 0
295 $x->{_a} = $a; # set/overwrite, even if not rounded
296 delete $x->{_p}; # clear P
297 $a = ${"${class}::accuracy"} unless defined $a; # proper return value
301 ${"${class}::accuracy"} = $a; # set global A
302 ${"${class}::precision"} = undef; # clear global P
304 return $a; # shortcut
308 # $object->accuracy() or fallback to global
309 $a = $x->{_a} if ref($x);
310 # but don't return global undef, when $x's accuracy is 0!
311 $a = ${"${class}::accuracy"} if !defined $a;
317 # $x->precision($p); ref($x) $p
318 # $x->precision(); ref($x)
319 # Class->precision(); class
320 # Class->precision($p); class $p
323 my $class = ref($x) || $x || __PACKAGE__;
329 # convert objects to scalars to avoid deep recursion. If object doesn't
330 # have numify(), then hopefully it will have overloading for int() and
331 # boolean test without wandering into a deep recursion path...
332 $p = $p->numify() if ref($p) && $p->can('numify');
333 if ((defined $p) && (int($p) != $p))
335 require Carp; Carp::croak ('Argument to precision must be an integer');
339 # $object->precision() or fallback to global
340 $x->bfround($p) if $p; # not for undef, 0
341 $x->{_p} = $p; # set/overwrite, even if not rounded
342 delete $x->{_a}; # clear A
343 $p = ${"${class}::precision"} unless defined $p; # proper return value
347 ${"${class}::precision"} = $p; # set global P
348 ${"${class}::accuracy"} = undef; # clear global A
350 return $p; # shortcut
354 # $object->precision() or fallback to global
355 $p = $x->{_p} if ref($x);
356 # but don't return global undef, when $x's precision is 0!
357 $p = ${"${class}::precision"} if !defined $p;
363 # return (or set) configuration data as hash ref
364 my $class = shift || 'Math::BigInt';
369 # try to set given options as arguments from hash
372 if (ref($args) ne 'HASH')
376 # these values can be "set"
380 upgrade downgrade precision accuracy round_mode div_scale/
383 $set_args->{$key} = $args->{$key} if exists $args->{$key};
384 delete $args->{$key};
389 Carp::croak ("Illegal key(s) '",
390 join("','",keys %$args),"' passed to $class\->config()");
392 foreach my $key (keys %$set_args)
394 if ($key =~ /^trap_(inf|nan)\z/)
396 ${"${class}::_trap_$1"} = ($set_args->{"trap_$1"} ? 1 : 0);
399 # use a call instead of just setting the $variable to check argument
400 $class->$key($set_args->{$key});
404 # now return actual configuration
408 lib_version => ${"${CALC}::VERSION"},
410 trap_nan => ${"${class}::_trap_nan"},
411 trap_inf => ${"${class}::_trap_inf"},
412 version => ${"${class}::VERSION"},
415 upgrade downgrade precision accuracy round_mode div_scale
418 $cfg->{$key} = ${"${class}::$key"};
425 # select accuracy parameter based on precedence,
426 # used by bround() and bfround(), may return undef for scale (means no op)
427 my ($x,$scale,$mode) = @_;
429 $scale = $x->{_a} unless defined $scale;
434 $scale = ${ $class . '::accuracy' } unless defined $scale;
435 $mode = ${ $class . '::round_mode' } unless defined $mode;
442 # select precision parameter based on precedence,
443 # used by bround() and bfround(), may return undef for scale (means no op)
444 my ($x,$scale,$mode) = @_;
446 $scale = $x->{_p} unless defined $scale;
451 $scale = ${ $class . '::precision' } unless defined $scale;
452 $mode = ${ $class . '::round_mode' } unless defined $mode;
457 ##############################################################################
465 # if two arguments, the first one is the class to "swallow" subclasses
473 return unless ref($x); # only for objects
475 my $self = bless {}, $c;
477 $self->{sign} = $x->{sign};
478 $self->{value} = $CALC->_copy($x->{value});
479 $self->{_a} = $x->{_a} if defined $x->{_a};
480 $self->{_p} = $x->{_p} if defined $x->{_p};
486 # create a new BigInt object from a string or another BigInt object.
487 # see hash keys documented at top
489 # the argument could be an object, so avoid ||, && etc on it, this would
490 # cause costly overloaded code to be called. The only allowed ops are
493 my ($class,$wanted,$a,$p,$r) = @_;
495 # avoid numify-calls by not using || on $wanted!
496 return $class->bzero($a,$p) if !defined $wanted; # default to 0
497 return $class->copy($wanted,$a,$p,$r)
498 if ref($wanted) && $wanted->isa($class); # MBI or subclass
500 $class->import() if $IMPORT == 0; # make require work
502 my $self = bless {}, $class;
504 # shortcut for "normal" numbers
505 if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*\z/))
507 $self->{sign} = $1 || '+';
509 if ($wanted =~ /^[+-]/)
511 # remove sign without touching wanted to make it work with constants
512 my $t = $wanted; $t =~ s/^[+-]//;
513 $self->{value} = $CALC->_new($t);
517 $self->{value} = $CALC->_new($wanted);
520 if ( (defined $a) || (defined $p)
521 || (defined ${"${class}::precision"})
522 || (defined ${"${class}::accuracy"})
525 $self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p);
530 # handle '+inf', '-inf' first
531 if ($wanted =~ /^[+-]?inf\z/)
533 $self->{sign} = $wanted; # set a default sign for bstr()
534 return $self->binf($wanted);
536 # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign
537 my ($mis,$miv,$mfv,$es,$ev) = _split($wanted);
542 require Carp; Carp::croak("$wanted is not a number in $class");
544 $self->{value} = $CALC->_zero();
545 $self->{sign} = $nan;
550 # _from_hex or _from_bin
551 $self->{value} = $mis->{value};
552 $self->{sign} = $mis->{sign};
553 return $self; # throw away $mis
555 # make integer from mantissa by adjusting exp, then convert to bigint
556 $self->{sign} = $$mis; # store sign
557 $self->{value} = $CALC->_zero(); # for all the NaN cases
558 my $e = int("$$es$$ev"); # exponent (avoid recursion)
561 my $diff = $e - CORE::length($$mfv);
562 if ($diff < 0) # Not integer
566 require Carp; Carp::croak("$wanted not an integer in $class");
569 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
570 $self->{sign} = $nan;
574 # adjust fraction and add it to value
575 #print "diff > 0 $$miv\n";
576 $$miv = $$miv . ($$mfv . '0' x $diff);
581 if ($$mfv ne '') # e <= 0
583 # fraction and negative/zero E => NOI
586 require Carp; Carp::croak("$wanted not an integer in $class");
588 #print "NOI 2 \$\$mfv '$$mfv'\n";
589 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
590 $self->{sign} = $nan;
594 # xE-y, and empty mfv
597 if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
601 require Carp; Carp::croak("$wanted not an integer in $class");
604 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
605 $self->{sign} = $nan;
609 $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
610 $self->{value} = $CALC->_new($$miv) if $self->{sign} =~ /^[+-]$/;
611 # if any of the globals is set, use them to round and store them inside $self
612 # do not round for new($x,undef,undef) since that is used by MBF to signal
614 $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p;
620 # create a bigint 'NaN', if given a BigInt, set it to 'NaN'
622 $self = $class if !defined $self;
625 my $c = $self; $self = {}; bless $self, $c;
628 if (${"${class}::_trap_nan"})
631 Carp::croak ("Tried to set $self to NaN in $class\::bnan()");
633 $self->import() if $IMPORT == 0; # make require work
634 return if $self->modify('bnan');
635 if ($self->can('_bnan'))
637 # use subclass to initialize
642 # otherwise do our own thing
643 $self->{value} = $CALC->_zero();
645 $self->{sign} = $nan;
646 delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly
652 # create a bigint '+-inf', if given a BigInt, set it to '+-inf'
653 # the sign is either '+', or if given, used from there
655 my $sign = shift; $sign = '+' if !defined $sign || $sign !~ /^-(inf)?$/;
656 $self = $class if !defined $self;
659 my $c = $self; $self = {}; bless $self, $c;
662 if (${"${class}::_trap_inf"})
665 Carp::croak ("Tried to set $self to +-inf in $class\::binf()");
667 $self->import() if $IMPORT == 0; # make require work
668 return if $self->modify('binf');
669 if ($self->can('_binf'))
671 # use subclass to initialize
676 # otherwise do our own thing
677 $self->{value} = $CALC->_zero();
679 $sign = $sign . 'inf' if $sign !~ /inf$/; # - => -inf
680 $self->{sign} = $sign;
681 ($self->{_a},$self->{_p}) = @_; # take over requested rounding
687 # create a bigint '+0', if given a BigInt, set it to 0
689 $self = __PACKAGE__ if !defined $self;
693 my $c = $self; $self = {}; bless $self, $c;
695 $self->import() if $IMPORT == 0; # make require work
696 return if $self->modify('bzero');
698 if ($self->can('_bzero'))
700 # use subclass to initialize
705 # otherwise do our own thing
706 $self->{value} = $CALC->_zero();
713 # call like: $x->bzero($a,$p,$r,$y);
714 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
719 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
721 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
729 # create a bigint '+1' (or -1 if given sign '-'),
730 # if given a BigInt, set it to +1 or -1, respectively
732 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
733 $self = $class if !defined $self;
737 my $c = $self; $self = {}; bless $self, $c;
739 $self->import() if $IMPORT == 0; # make require work
740 return if $self->modify('bone');
742 if ($self->can('_bone'))
744 # use subclass to initialize
749 # otherwise do our own thing
750 $self->{value} = $CALC->_one();
752 $self->{sign} = $sign;
757 # call like: $x->bone($sign,$a,$p,$r,$y);
758 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
762 # call like: $x->bone($sign,$a,$p,$r);
764 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
766 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
772 ##############################################################################
773 # string conversation
777 # (ref to BFLOAT or num_str ) return num_str
778 # Convert number from internal format to scientific string format.
779 # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
780 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
782 if ($x->{sign} !~ /^[+-]$/)
784 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
787 my ($m,$e) = $x->parts();
788 #$m->bstr() . 'e+' . $e->bstr(); # e can only be positive in BigInt
789 # 'e+' because E can only be positive in BigInt
790 $m->bstr() . 'e+' . $CALC->_str($e->{value});
795 # make a string from bigint object
796 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
798 if ($x->{sign} !~ /^[+-]$/)
800 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
803 my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
804 $es.$CALC->_str($x->{value});
809 # Make a "normal" scalar from a BigInt object
810 my $x = shift; $x = $class->new($x) unless ref $x;
812 return $x->bstr() if $x->{sign} !~ /^[+-]$/;
813 my $num = $CALC->_num($x->{value});
814 return -$num if $x->{sign} eq '-';
818 ##############################################################################
819 # public stuff (usually prefixed with "b")
823 # return the sign of the number: +/-/-inf/+inf/NaN
824 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
829 sub _find_round_parameters
831 # After any operation or when calling round(), the result is rounded by
832 # regarding the A & P from arguments, local parameters, or globals.
834 # !!!!!!! If you change this, remember to change round(), too! !!!!!!!!!!
836 # This procedure finds the round parameters, but it is for speed reasons
837 # duplicated in round. Otherwise, it is tested by the testsuite and used
840 # returns ($self) or ($self,$a,$p,$r) - sets $self to NaN of both A and P
841 # were requested/defined (locally or globally or both)
843 my ($self,$a,$p,$r,@args) = @_;
844 # $a accuracy, if given by caller
845 # $p precision, if given by caller
846 # $r round_mode, if given by caller
847 # @args all 'other' arguments (0 for unary, 1 for binary ops)
849 my $c = ref($self); # find out class of argument(s)
852 # now pick $a or $p, but only if we have got "arguments"
855 foreach ($self,@args)
857 # take the defined one, or if both defined, the one that is smaller
858 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
863 # even if $a is defined, take $p, to signal error for both defined
864 foreach ($self,@args)
866 # take the defined one, or if both defined, the one that is bigger
868 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
871 # if still none defined, use globals (#2)
872 $a = ${"$c\::accuracy"} unless defined $a;
873 $p = ${"$c\::precision"} unless defined $p;
875 # A == 0 is useless, so undef it to signal no rounding
876 $a = undef if defined $a && $a == 0;
879 return ($self) unless defined $a || defined $p; # early out
881 # set A and set P is an fatal error
882 return ($self->bnan()) if defined $a && defined $p; # error
884 $r = ${"$c\::round_mode"} unless defined $r;
885 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
887 require Carp; Carp::croak ("Unknown round mode '$r'");
895 # Round $self according to given parameters, or given second argument's
896 # parameters or global defaults
898 # for speed reasons, _find_round_parameters is embeded here:
900 my ($self,$a,$p,$r,@args) = @_;
901 # $a accuracy, if given by caller
902 # $p precision, if given by caller
903 # $r round_mode, if given by caller
904 # @args all 'other' arguments (0 for unary, 1 for binary ops)
906 my $c = ref($self); # find out class of argument(s)
909 # now pick $a or $p, but only if we have got "arguments"
912 foreach ($self,@args)
914 # take the defined one, or if both defined, the one that is smaller
915 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
920 # even if $a is defined, take $p, to signal error for both defined
921 foreach ($self,@args)
923 # take the defined one, or if both defined, the one that is bigger
925 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
928 # if still none defined, use globals (#2)
929 $a = ${"$c\::accuracy"} unless defined $a;
930 $p = ${"$c\::precision"} unless defined $p;
932 # A == 0 is useless, so undef it to signal no rounding
933 $a = undef if defined $a && $a == 0;
936 return $self unless defined $a || defined $p; # early out
938 # set A and set P is an fatal error
939 return $self->bnan() if defined $a && defined $p;
941 $r = ${"$c\::round_mode"} unless defined $r;
942 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
944 require Carp; Carp::croak ("Unknown round mode '$r'");
947 # now round, by calling either fround or ffround:
950 $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a;
952 else # both can't be undefined due to early out
954 $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p;
956 # bround() or bfround() already callled bnorm() if necc.
962 # (numstr or BINT) return BINT
963 # Normalize number -- no-op here
964 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
970 # (BINT or num_str) return BINT
971 # make number absolute, or return absolute BINT from string
972 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
974 return $x if $x->modify('babs');
975 # post-normalized abs for internal use (does nothing for NaN)
976 $x->{sign} =~ s/^-/+/;
982 # (BINT or num_str) return BINT
983 # negate number or make a negated number from string
984 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
986 return $x if $x->modify('bneg');
988 # for +0 dont negate (to have always normalized +0). Does nothing for 'NaN'
989 $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $CALC->_is_zero($x->{value}));
995 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
996 # (BINT or num_str, BINT or num_str) return cond_code
999 my ($self,$x,$y) = (ref($_[0]),@_);
1001 # objectify is costly, so avoid it
1002 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1004 ($self,$x,$y) = objectify(2,@_);
1007 return $upgrade->bcmp($x,$y) if defined $upgrade &&
1008 ((!$x->isa($self)) || (!$y->isa($self)));
1010 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1012 # handle +-inf and NaN
1013 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1014 return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
1015 return +1 if $x->{sign} eq '+inf';
1016 return -1 if $x->{sign} eq '-inf';
1017 return -1 if $y->{sign} eq '+inf';
1020 # check sign for speed first
1021 return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
1022 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
1024 # have same sign, so compare absolute values. Don't make tests for zero here
1025 # because it's actually slower than testin in Calc (especially w/ Pari et al)
1027 # post-normalized compare for internal use (honors signs)
1028 if ($x->{sign} eq '+')
1030 # $x and $y both > 0
1031 return $CALC->_acmp($x->{value},$y->{value});
1035 $CALC->_acmp($y->{value},$x->{value}); # swaped acmp (lib returns 0,1,-1)
1040 # Compares 2 values, ignoring their signs.
1041 # Returns one of undef, <0, =0, >0. (suitable for sort)
1042 # (BINT, BINT) return cond_code
1045 my ($self,$x,$y) = (ref($_[0]),@_);
1046 # objectify is costly, so avoid it
1047 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1049 ($self,$x,$y) = objectify(2,@_);
1052 return $upgrade->bacmp($x,$y) if defined $upgrade &&
1053 ((!$x->isa($self)) || (!$y->isa($self)));
1055 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1057 # handle +-inf and NaN
1058 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1059 return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
1060 return 1 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} !~ /^[+-]inf$/;
1063 $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
1068 # add second arg (BINT or string) to first (BINT) (modifies first)
1069 # return result as BINT
1072 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1073 # objectify is costly, so avoid it
1074 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1076 ($self,$x,$y,@r) = objectify(2,@_);
1079 return $x if $x->modify('badd');
1080 return $upgrade->badd($upgrade->new($x),$upgrade->new($y),@r) if defined $upgrade &&
1081 ((!$x->isa($self)) || (!$y->isa($self)));
1083 $r[3] = $y; # no push!
1084 # inf and NaN handling
1085 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1088 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1090 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1092 # +inf++inf or -inf+-inf => same, rest is NaN
1093 return $x if $x->{sign} eq $y->{sign};
1096 # +-inf + something => +inf
1097 # something +-inf => +-inf
1098 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
1102 my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
1106 $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
1110 my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
1113 $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
1118 # speedup, if equal, set result to 0
1119 $x->{value} = $CALC->_zero();
1124 $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
1132 # (BINT or num_str, BINT or num_str) return BINT
1133 # subtract second arg from first, modify first
1136 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1137 # objectify is costly, so avoid it
1138 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1140 ($self,$x,$y,@r) = objectify(2,@_);
1143 return $x if $x->modify('bsub');
1145 return $upgrade->new($x)->bsub($upgrade->new($y),@r) if defined $upgrade &&
1146 ((!$x->isa($self)) || (!$y->isa($self)));
1148 return $x->round(@r) if $y->is_zero();
1150 # To correctly handle the lone special case $x->bsub($x), we note the sign
1151 # of $x, then flip the sign from $y, and if the sign of $x did change, too,
1152 # then we caught the special case:
1153 my $xsign = $x->{sign};
1154 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
1155 if ($xsign ne $x->{sign})
1157 # special case of $x->bsub($x) results in 0
1158 return $x->bzero(@r) if $xsign =~ /^[+-]$/;
1159 return $x->bnan(); # NaN, -inf, +inf
1161 $x->badd($y,@r); # badd does not leave internal zeros
1162 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
1163 $x; # already rounded by badd() or no round necc.
1168 # increment arg by one
1169 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1170 return $x if $x->modify('binc');
1172 if ($x->{sign} eq '+')
1174 $x->{value} = $CALC->_inc($x->{value});
1175 return $x->round($a,$p,$r);
1177 elsif ($x->{sign} eq '-')
1179 $x->{value} = $CALC->_dec($x->{value});
1180 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1181 return $x->round($a,$p,$r);
1183 # inf, nan handling etc
1184 $x->badd($self->bone(),$a,$p,$r); # badd does round
1189 # decrement arg by one
1190 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1191 return $x if $x->modify('bdec');
1193 if ($x->{sign} eq '-')
1196 $x->{value} = $CALC->_inc($x->{value});
1200 return $x->badd($self->bone('-'),@r) unless $x->{sign} eq '+'; # inf or NaN
1202 if ($CALC->_is_zero($x->{value}))
1205 $x->{value} = $CALC->_one(); $x->{sign} = '-'; # 0 => -1
1210 $x->{value} = $CALC->_dec($x->{value});
1218 # calculate $x = $a ** $base + $b and return $a (e.g. the log() to base
1222 my ($self,$x,$base,@r) = (undef,@_);
1223 # objectify is costly, so avoid it
1224 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1226 ($self,$x,$base,@r) = objectify(1,ref($x),@_);
1229 return $x if $x->modify('blog');
1231 # inf, -inf, NaN, <0 => NaN
1233 if $x->{sign} ne '+' || (defined $base && $base->{sign} ne '+');
1235 return $upgrade->blog($upgrade->new($x),$base,@r) if
1238 my ($rc,$exact) = $CALC->_log_int($x->{value},$base->{value});
1239 return $x->bnan() unless defined $rc; # not possible to take log?
1246 # (BINT or num_str, BINT or num_str) return BINT
1247 # does not modify arguments, but returns new object
1248 # Lowest Common Multiplicator
1250 my $y = shift; my ($x);
1257 $x = $class->new($y);
1262 my $y = shift; $y = $self->new($y) if !ref ($y);
1270 # (BINT or num_str, BINT or num_str) return BINT
1271 # does not modify arguments, but returns new object
1272 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1275 $y = $class->new($y) if !ref($y);
1277 my $x = $y->copy()->babs(); # keep arguments
1278 return $x->bnan() if $x->{sign} !~ /^[+-]$/; # x NaN?
1282 $y = shift; $y = $self->new($y) if !ref($y);
1283 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1284 $x->{value} = $CALC->_gcd($x->{value},$y->{value});
1285 last if $CALC->_is_one($x->{value});
1292 # (num_str or BINT) return BINT
1293 # represent ~x as twos-complement number
1294 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1295 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1297 return $x if $x->modify('bnot');
1298 $x->binc()->bneg(); # binc already does round
1301 ##############################################################################
1302 # is_foo test routines
1303 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1307 # return true if arg (BINT or num_str) is zero (array '+', '0')
1308 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1310 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1311 $CALC->_is_zero($x->{value});
1316 # return true if arg (BINT or num_str) is NaN
1317 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1319 $x->{sign} eq $nan ? 1 : 0;
1324 # return true if arg (BINT or num_str) is +-inf
1325 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1329 $sign = '[+-]inf' if $sign eq ''; # +- doesn't matter, only that's inf
1330 $sign = "[$1]inf" if $sign =~ /^([+-])(inf)?$/; # extract '+' or '-'
1331 return $x->{sign} =~ /^$sign$/ ? 1 : 0;
1333 $x->{sign} =~ /^[+-]inf$/ ? 1 : 0; # only +-inf is infinity
1338 # return true if arg (BINT or num_str) is +1, or -1 if sign is given
1339 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1341 $sign = '+' if !defined $sign || $sign ne '-';
1343 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1344 $CALC->_is_one($x->{value});
1349 # return true when arg (BINT or num_str) is odd, false for even
1350 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1352 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1353 $CALC->_is_odd($x->{value});
1358 # return true when arg (BINT or num_str) is even, false for odd
1359 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1361 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1362 $CALC->_is_even($x->{value});
1367 # return true when arg (BINT or num_str) is positive (>= 0)
1368 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1370 return 1 if $x->{sign} eq '+inf'; # +inf is positive
1372 # 0+ is neither positive nor negative
1373 ($x->{sign} eq '+' && !$x->is_zero()) ? 1 : 0;
1378 # return true when arg (BINT or num_str) is negative (< 0)
1379 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1381 $x->{sign} =~ /^-/ ? 1 : 0; # -inf is negative, but NaN is not
1386 # return true when arg (BINT or num_str) is an integer
1387 # always true for BigInt, but different for BigFloats
1388 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1390 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1393 ###############################################################################
1397 # multiply two numbers -- stolen from Knuth Vol 2 pg 233
1398 # (BINT or num_str, BINT or num_str) return BINT
1401 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1402 # objectify is costly, so avoid it
1403 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1405 ($self,$x,$y,@r) = objectify(2,@_);
1408 return $x if $x->modify('bmul');
1410 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1413 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1415 return $x->bnan() if $x->is_zero() || $y->is_zero();
1416 # result will always be +-inf:
1417 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1418 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1419 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1420 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1421 return $x->binf('-');
1424 return $upgrade->bmul($x,$upgrade->new($y),@r)
1425 if defined $upgrade && !$y->isa($self);
1427 $r[3] = $y; # no push here
1429 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1431 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1432 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1439 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1440 my ($self,$x,$y) = @_;
1442 # NaN if x == NaN or y == NaN or x==y==0
1443 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1444 if (($x->is_nan() || $y->is_nan()) ||
1445 ($x->is_zero() && $y->is_zero()));
1447 # +-inf / +-inf == NaN, reminder also NaN
1448 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1450 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1452 # x / +-inf => 0, remainder x (works even if x == 0)
1453 if ($y->{sign} =~ /^[+-]inf$/)
1455 my $t = $x->copy(); # bzero clobbers up $x
1456 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1459 # 5 / 0 => +inf, -6 / 0 => -inf
1460 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1461 # exception: -8 / 0 has remainder -8, not 8
1462 # exception: -inf / 0 has remainder -inf, not inf
1465 # +-inf / 0 => special case for -inf
1466 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1467 if (!$x->is_zero() && !$x->is_inf())
1469 my $t = $x->copy(); # binf clobbers up $x
1471 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1475 # last case: +-inf / ordinary number
1477 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1479 return wantarray ? ($x,$self->bzero()) : $x;
1484 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1485 # (BINT,BINT) (quo,rem) or BINT (only rem)
1488 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1489 # objectify is costly, so avoid it
1490 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1492 ($self,$x,$y,@r) = objectify(2,@_);
1495 return $x if $x->modify('bdiv');
1497 return $self->_div_inf($x,$y)
1498 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1500 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1501 if defined $upgrade;
1503 $r[3] = $y; # no push!
1505 # calc new sign and in case $y == +/- 1, return $x
1506 my $xsign = $x->{sign}; # keep
1507 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1511 my $rem = $self->bzero();
1512 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1513 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1514 $rem->{_a} = $x->{_a};
1515 $rem->{_p} = $x->{_p};
1517 if (! $CALC->_is_zero($rem->{value}))
1519 $rem->{sign} = $y->{sign};
1520 $rem = $y->copy()->bsub($rem) if $xsign ne $y->{sign}; # one of them '-'
1524 $rem->{sign} = '+'; # dont leave -0
1530 $x->{value} = $CALC->_div($x->{value},$y->{value});
1531 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1536 ###############################################################################
1541 # modulus (or remainder)
1542 # (BINT or num_str, BINT or num_str) return BINT
1545 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1546 # objectify is costly, so avoid it
1547 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1549 ($self,$x,$y,@r) = objectify(2,@_);
1552 return $x if $x->modify('bmod');
1553 $r[3] = $y; # no push!
1554 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1556 my ($d,$r) = $self->_div_inf($x,$y);
1557 $x->{sign} = $r->{sign};
1558 $x->{value} = $r->{value};
1559 return $x->round(@r);
1562 # calc new sign and in case $y == +/- 1, return $x
1563 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1564 if (!$CALC->_is_zero($x->{value}))
1566 $x->{value} = $CALC->_sub($y->{value},$x->{value},1) # $y-$x
1567 if ($x->{sign} ne $y->{sign});
1568 $x->{sign} = $y->{sign};
1572 $x->{sign} = '+'; # dont leave -0
1579 # Modular inverse. given a number which is (hopefully) relatively
1580 # prime to the modulus, calculate its inverse using Euclid's
1581 # alogrithm. If the number is not relatively prime to the modulus
1582 # (i.e. their gcd is not one) then NaN is returned.
1585 my ($self,$x,$y,@r) = (undef,@_);
1586 # objectify is costly, so avoid it
1587 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1589 ($self,$x,$y,@r) = objectify(2,@_);
1592 return $x if $x->modify('bmodinv');
1595 if ($y->{sign} ne '+' # -, NaN, +inf, -inf
1596 || $x->is_zero() # or num == 0
1597 || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1600 # put least residue into $x if $x was negative, and thus make it positive
1601 $x->bmod($y) if $x->{sign} eq '-';
1604 ($x->{value},$sign) = $CALC->_modinv($x->{value},$y->{value});
1605 return $x->bnan() if !defined $x->{value}; # in case no GCD found
1606 return $x if !defined $sign; # already real result
1607 $x->{sign} = $sign; # flip/flop see below
1608 $x->bmod($y); # calc real result
1614 # takes a very large number to a very large exponent in a given very
1615 # large modulus, quickly, thanks to binary exponentation. supports
1616 # negative exponents.
1617 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1619 return $num if $num->modify('bmodpow');
1621 # check modulus for valid values
1622 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1623 || $mod->is_zero());
1625 # check exponent for valid values
1626 if ($exp->{sign} =~ /\w/)
1628 # i.e., if it's NaN, +inf, or -inf...
1629 return $num->bnan();
1632 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
1634 # check num for valid values (also NaN if there was no inverse but $exp < 0)
1635 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1637 # $mod is positive, sign on $exp is ignored, result also positive
1638 $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
1642 ###############################################################################
1646 # (BINT or num_str, BINT or num_str) return BINT
1647 # compute factorial number from $x, modify $x in place
1648 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1650 return $x if $x->modify('bfac') || $x->{sign} eq '+inf'; # inf => inf
1651 return $x->bnan() if $x->{sign} ne '+'; # NaN, <0 etc => NaN
1653 $x->{value} = $CALC->_fac($x->{value});
1659 # (BINT or num_str, BINT or num_str) return BINT
1660 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1661 # modifies first argument
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('bpow');
1673 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1676 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1678 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1684 if ($x->{sign} =~ /^[+-]inf/)
1687 return $x->bnan() if $y->is_zero();
1688 # -inf ** -1 => 1/inf => 0
1689 return $x->bzero() if $y->is_one('-') && $x->is_negative();
1692 return $x if $x->{sign} eq '+inf';
1694 # -inf ** Y => -inf if Y is odd
1695 return $x if $y->is_odd();
1701 return $x if $x->is_one();
1704 return $x if $x->is_zero() && $y->{sign} =~ /^[+]/;
1707 return $x->binf() if $x->is_zero();
1710 return $x->bnan() if $x->is_one('-') && $y->{sign} =~ /^[-]/;
1713 return $x->bzero() if $x->{sign} eq '-' && $y->{sign} =~ /^[-]/;
1716 return $x->bnan() if $x->{sign} eq '-';
1719 return $x->binf() if $y->{sign} =~ /^[+]/;
1724 return $upgrade->bpow($upgrade->new($x),$y,@r)
1725 if defined $upgrade && !$y->isa($self);
1727 $r[3] = $y; # no push!
1729 # cases 0 ** Y, X ** 0, X ** 1, 1 ** Y are handled by Calc or Emu
1732 $new_sign = $y->is_odd() ? '-' : '+' if ($x->{sign} ne '+');
1734 # 0 ** -7 => ( 1 / (0 ** 7)) => 1 / 0 => +inf
1736 if $y->{sign} eq '-' && $x->{sign} eq '+' && $CALC->_is_zero($x->{value});
1737 # 1 ** -y => 1 / (1 ** |y|)
1738 # so do test for negative $y after above's clause
1739 return $x->bnan() if $y->{sign} eq '-' && !$CALC->_is_one($x->{value});
1741 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1742 $x->{sign} = $new_sign;
1743 $x->{sign} = '+' if $CALC->_is_zero($y->{value});
1749 # (BINT or num_str, BINT or num_str) return BINT
1750 # compute x << y, base n, y >= 0
1753 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1754 # objectify is costly, so avoid it
1755 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1757 ($self,$x,$y,$n,@r) = objectify(2,@_);
1760 return $x if $x->modify('blsft');
1761 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1762 return $x->round(@r) if $y->is_zero();
1764 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1766 $x->{value} = $CALC->_lsft($x->{value},$y->{value},$n);
1772 # (BINT or num_str, BINT or num_str) return BINT
1773 # compute x >> y, base n, y >= 0
1776 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1777 # objectify is costly, so avoid it
1778 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1780 ($self,$x,$y,$n,@r) = objectify(2,@_);
1783 return $x if $x->modify('brsft');
1784 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1785 return $x->round(@r) if $y->is_zero();
1786 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
1788 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1790 # this only works for negative numbers when shifting in base 2
1791 if (($x->{sign} eq '-') && ($n == 2))
1793 return $x->round(@r) if $x->is_one('-'); # -1 => -1
1796 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1797 # but perhaps there is a better emulation for two's complement shift...
1798 # if $y != 1, we must simulate it by doing:
1799 # convert to bin, flip all bits, shift, and be done
1800 $x->binc(); # -3 => -2
1801 my $bin = $x->as_bin();
1802 $bin =~ s/^-0b//; # strip '-0b' prefix
1803 $bin =~ tr/10/01/; # flip bits
1805 if (CORE::length($bin) <= $y)
1807 $bin = '0'; # shifting to far right creates -1
1808 # 0, because later increment makes
1809 # that 1, attached '-' makes it '-1'
1810 # because -1 >> x == -1 !
1814 $bin =~ s/.{$y}$//; # cut off at the right side
1815 $bin = '1' . $bin; # extend left side by one dummy '1'
1816 $bin =~ tr/10/01/; # flip bits back
1818 my $res = $self->new('0b'.$bin); # add prefix and convert back
1819 $res->binc(); # remember to increment
1820 $x->{value} = $res->{value}; # take over value
1821 return $x->round(@r); # we are done now, magic, isn't?
1823 # x < 0, n == 2, y == 1
1824 $x->bdec(); # n == 2, but $y == 1: this fixes it
1827 $x->{value} = $CALC->_rsft($x->{value},$y->{value},$n);
1833 #(BINT or num_str, BINT or num_str) return BINT
1837 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1838 # objectify is costly, so avoid it
1839 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1841 ($self,$x,$y,@r) = objectify(2,@_);
1844 return $x if $x->modify('band');
1846 $r[3] = $y; # no push!
1848 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1850 my $sx = $x->{sign} eq '+' ? 1 : -1;
1851 my $sy = $y->{sign} eq '+' ? 1 : -1;
1853 if ($sx == 1 && $sy == 1)
1855 $x->{value} = $CALC->_and($x->{value},$y->{value});
1856 return $x->round(@r);
1859 if ($CAN{signed_and})
1861 $x->{value} = $CALC->_signed_and($x->{value},$y->{value},$sx,$sy);
1862 return $x->round(@r);
1866 __emu_band($self,$x,$y,$sx,$sy,@r);
1871 #(BINT or num_str, BINT or num_str) return BINT
1875 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1876 # objectify is costly, so avoid it
1877 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1879 ($self,$x,$y,@r) = objectify(2,@_);
1882 return $x if $x->modify('bior');
1883 $r[3] = $y; # no push!
1885 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1887 my $sx = $x->{sign} eq '+' ? 1 : -1;
1888 my $sy = $y->{sign} eq '+' ? 1 : -1;
1890 # the sign of X follows the sign of X, e.g. sign of Y irrelevant for bior()
1892 # don't use lib for negative values
1893 if ($sx == 1 && $sy == 1)
1895 $x->{value} = $CALC->_or($x->{value},$y->{value});
1896 return $x->round(@r);
1899 # if lib can do negative values, let it handle this
1900 if ($CAN{signed_or})
1902 $x->{value} = $CALC->_signed_or($x->{value},$y->{value},$sx,$sy);
1903 return $x->round(@r);
1907 __emu_bior($self,$x,$y,$sx,$sy,@r);
1912 #(BINT or num_str, BINT or num_str) return BINT
1916 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1917 # objectify is costly, so avoid it
1918 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1920 ($self,$x,$y,@r) = objectify(2,@_);
1923 return $x if $x->modify('bxor');
1924 $r[3] = $y; # no push!
1926 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1928 my $sx = $x->{sign} eq '+' ? 1 : -1;
1929 my $sy = $y->{sign} eq '+' ? 1 : -1;
1931 # don't use lib for negative values
1932 if ($sx == 1 && $sy == 1)
1934 $x->{value} = $CALC->_xor($x->{value},$y->{value});
1935 return $x->round(@r);
1938 # if lib can do negative values, let it handle this
1939 if ($CAN{signed_xor})
1941 $x->{value} = $CALC->_signed_xor($x->{value},$y->{value},$sx,$sy);
1942 return $x->round(@r);
1946 __emu_bxor($self,$x,$y,$sx,$sy,@r);
1951 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1953 my $e = $CALC->_len($x->{value});
1954 wantarray ? ($e,0) : $e;
1959 # return the nth decimal digit, negative values count backward, 0 is right
1960 my ($self,$x,$n) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1962 $n = $n->numify() if ref($n);
1963 $CALC->_digit($x->{value},$n||0);
1968 # return the amount of trailing zeros in $x (as scalar)
1970 $x = $class->new($x) unless ref $x;
1972 return 0 if $x->{sign} !~ /^[+-]$/; # NaN, inf, -inf etc
1974 $CALC->_zeros($x->{value}); # must handle odd values, 0 etc
1979 # calculate square root of $x
1980 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1982 return $x if $x->modify('bsqrt');
1984 return $x->bnan() if $x->{sign} !~ /^\+/; # -x or -inf or NaN => NaN
1985 return $x if $x->{sign} eq '+inf'; # sqrt(+inf) == inf
1987 return $upgrade->bsqrt($x,@r) if defined $upgrade;
1989 $x->{value} = $CALC->_sqrt($x->{value});
1995 # calculate $y'th root of $x
1998 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2000 $y = $self->new(2) unless defined $y;
2002 # objectify is costly, so avoid it
2003 if ((!ref($x)) || (ref($x) ne ref($y)))
2005 ($self,$x,$y,@r) = objectify(2,$self || $class,@_);
2008 return $x if $x->modify('broot');
2010 # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
2011 return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
2012 $y->{sign} !~ /^\+$/;
2014 return $x->round(@r)
2015 if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
2017 return $upgrade->new($x)->broot($upgrade->new($y),@r) if defined $upgrade;
2019 $x->{value} = $CALC->_root($x->{value},$y->{value});
2025 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
2026 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2028 if ($x->{sign} !~ /^[+-]$/)
2030 my $s = $x->{sign}; $s =~ s/^[+-]//; # NaN, -inf,+inf => NaN or inf
2031 return $self->new($s);
2033 return $self->bone() if $x->is_zero();
2035 $self->new($x->_trailing_zeros());
2040 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
2041 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2043 if ($x->{sign} !~ /^[+-]$/)
2045 # for NaN, +inf, -inf: keep the sign
2046 return $self->new($x->{sign});
2048 my $m = $x->copy(); delete $m->{_p}; delete $m->{_a};
2049 # that's a bit inefficient:
2050 my $zeros = $m->_trailing_zeros();
2051 $m->brsft($zeros,10) if $zeros != 0;
2057 # return a copy of both the exponent and the mantissa
2058 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2060 ($x->mantissa(),$x->exponent());
2063 ##############################################################################
2064 # rounding functions
2068 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
2069 # $n == 0 || $n == 1 => round to integer
2070 my $x = shift; my $self = ref($x) || $x; $x = $self->new($x) unless ref $x;
2072 my ($scale,$mode) = $x->_scale_p(@_);
2074 return $x if !defined $scale || $x->modify('bfround'); # no-op
2076 # no-op for BigInts if $n <= 0
2077 $x->bround( $x->length()-$scale, $mode) if $scale > 0;
2079 delete $x->{_a}; # delete to save memory
2080 $x->{_p} = $scale; # store new _p
2084 sub _scan_for_nonzero
2086 # internal, used by bround() to scan for non-zeros after a '5'
2087 my ($x,$pad,$xs,$len) = @_;
2089 return 0 if $len == 1; # "5" is trailed by invisible zeros
2090 my $follow = $pad - 1;
2091 return 0 if $follow > $len || $follow < 1;
2093 # use the string form to check whether only '0's follow or not
2094 substr ($xs,-$follow) =~ /[^0]/ ? 1 : 0;
2099 # Exists to make life easier for switch between MBF and MBI (should we
2100 # autoload fxxx() like MBF does for bxxx()?)
2101 my $x = shift; $x = $class->new($x) unless ref $x;
2107 # accuracy: +$n preserve $n digits from left,
2108 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
2110 # and overwrite the rest with 0's, return normalized number
2111 # do not return $x->bnorm(), but $x
2113 my $x = shift; $x = $class->new($x) unless ref $x;
2114 my ($scale,$mode) = $x->_scale_a(@_);
2115 return $x if !defined $scale || $x->modify('bround'); # no-op
2117 if ($x->is_zero() || $scale == 0)
2119 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2122 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2124 # we have fewer digits than we want to scale to
2125 my $len = $x->length();
2126 # convert $scale to a scalar in case it is an object (put's a limit on the
2127 # number length, but this would already limited by memory constraints), makes
2129 $scale = $scale->numify() if ref ($scale);
2131 # scale < 0, but > -len (not >=!)
2132 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2134 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2138 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2139 my ($pad,$digit_round,$digit_after);
2140 $pad = $len - $scale;
2141 $pad = abs($scale-1) if $scale < 0;
2143 # do not use digit(), it is very costly for binary => decimal
2144 # getting the entire string is also costly, but we need to do it only once
2145 my $xs = $CALC->_str($x->{value});
2148 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2149 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2150 $digit_round = '0'; $digit_round = substr($xs,$pl,1) if $pad <= $len;
2151 $pl++; $pl ++ if $pad >= $len;
2152 $digit_after = '0'; $digit_after = substr($xs,$pl,1) if $pad > 0;
2154 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2155 # closer at the remaining digits of the original $x, remember decision
2156 my $round_up = 1; # default round up
2158 ($mode eq 'trunc') || # trunc by round down
2159 ($digit_after =~ /[01234]/) || # round down anyway,
2161 ($digit_after eq '5') && # not 5000...0000
2162 ($x->_scan_for_nonzero($pad,$xs,$len) == 0) &&
2164 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2165 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2166 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2167 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2168 ($mode eq 'zero') # round down if zero, sign adjusted below
2170 my $put_back = 0; # not yet modified
2172 if (($pad > 0) && ($pad <= $len))
2174 substr($xs,-$pad,$pad) = '0' x $pad; # replace with '00...'
2175 $put_back = 1; # need to put back
2179 $x->bzero(); # round to '0'
2182 if ($round_up) # what gave test above?
2184 $put_back = 1; # need to put back
2185 $pad = $len, $xs = '0' x $pad if $scale < 0; # tlr: whack 0.51=>1.0
2187 # we modify directly the string variant instead of creating a number and
2188 # adding it, since that is faster (we already have the string)
2189 my $c = 0; $pad ++; # for $pad == $len case
2190 while ($pad <= $len)
2192 $c = substr($xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2193 substr($xs,-$pad,1) = $c; $pad++;
2194 last if $c != 0; # no overflow => early out
2196 $xs = '1'.$xs if $c == 0;
2199 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back, if needed
2201 $x->{_a} = $scale if $scale >= 0;
2204 $x->{_a} = $len+$scale;
2205 $x->{_a} = 0 if $scale < -$len;
2212 # return integer less or equal then number; no-op since it's already integer
2213 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2220 # return integer greater or equal then number; no-op since it's already int
2221 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2228 # An object might be asked to return itself as bigint on certain overloaded
2229 # operations, this does exactly this, so that sub classes can simple inherit
2230 # it or override with their own integer conversion routine.
2236 # return as hex string, with prefixed 0x
2237 my $x = shift; $x = $class->new($x) if !ref($x);
2239 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2242 $s = $x->{sign} if $x->{sign} eq '-';
2243 $s . $CALC->_as_hex($x->{value});
2248 # return as binary string, with prefixed 0b
2249 my $x = shift; $x = $class->new($x) if !ref($x);
2251 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2253 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2254 return $s . $CALC->_as_bin($x->{value});
2257 ##############################################################################
2258 # private stuff (internal use only)
2262 # check for strings, if yes, return objects instead
2264 # the first argument is number of args objectify() should look at it will
2265 # return $count+1 elements, the first will be a classname. This is because
2266 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2267 # useless objects being created and thrown away. So we cannot simple loop
2268 # over @_. If the given count is 0, all arguments will be used.
2270 # If the second arg is a ref, use it as class.
2271 # If not, try to use it as classname, unless undef, then use $class
2272 # (aka Math::BigInt). The latter shouldn't happen,though.
2275 # $x->badd(1); => ref x, scalar y
2276 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2277 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2278 # Math::BigInt::badd(1,2); => scalar x, scalar y
2279 # In the last case we check number of arguments to turn it silently into
2280 # $class,1,2. (We can not take '1' as class ;o)
2281 # badd($class,1) is not supported (it should, eventually, try to add undef)
2282 # currently it tries 'Math::BigInt' + 1, which will not work.
2284 # some shortcut for the common cases
2286 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2288 my $count = abs(shift || 0);
2290 my (@a,$k,$d); # resulting array, temp, and downgrade
2293 # okay, got object as first
2298 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2300 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2304 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2305 if (defined ${"$a[0]::downgrade"})
2307 $d = ${"$a[0]::downgrade"};
2308 ${"$a[0]::downgrade"} = undef;
2311 my $up = ${"$a[0]::upgrade"};
2312 #print "Now in objectify, my class is today $a[0], count = $count\n";
2320 $k = $a[0]->new($k);
2322 elsif (!defined $up && ref($k) ne $a[0])
2324 # foreign object, try to convert to integer
2325 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2338 $k = $a[0]->new($k);
2340 elsif (!defined $up && ref($k) ne $a[0])
2342 # foreign object, try to convert to integer
2343 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2347 push @a,@_; # return other params, too
2351 require Carp; Carp::croak ("$class objectify needs list context");
2353 ${"$a[0]::downgrade"} = $d;
2357 sub _register_callback
2359 my ($class,$callback) = @_;
2361 if (ref($callback) ne 'CODE')
2364 Carp::croak ("$callback is not a coderef");
2366 $CALLBACKS{$class} = $callback;
2373 $IMPORT++; # remember we did import()
2374 my @a; my $l = scalar @_;
2375 for ( my $i = 0; $i < $l ; $i++ )
2377 if ($_[$i] eq ':constant')
2379 # this causes overlord er load to step in
2381 integer => sub { $self->new(shift) },
2382 binary => sub { $self->new(shift) };
2384 elsif ($_[$i] eq 'upgrade')
2386 # this causes upgrading
2387 $upgrade = $_[$i+1]; # or undef to disable
2390 elsif ($_[$i] =~ /^lib$/i)
2392 # this causes a different low lib to take care...
2393 $CALC = $_[$i+1] || '';
2401 # any non :constant stuff is handled by our parent, Exporter
2406 $self->SUPER::import(@a); # need it for subclasses
2407 $self->export_to_level(1,$self,@a); # need it for MBF
2410 # try to load core math lib
2411 my @c = split /\s*,\s*/,$CALC;
2414 $_ =~ tr/a-zA-Z0-9://cd; # limit to sane characters
2416 push @c, 'FastCalc', 'Calc'; # if all fail, try these
2417 $CALC = ''; # signal error
2418 foreach my $lib (@c)
2420 next if ($lib || '') eq '';
2421 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2425 # Perl < 5.6.0 dies with "out of memory!" when eval("") and ':constant' is
2426 # used in the same script, or eval("") inside import().
2427 my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
2428 my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
2430 $file = File::Spec->catfile (@parts, $file);
2431 eval { require "$file"; $lib->import( @c ); }
2435 eval "use $lib qw/@c/;";
2440 # loaded it ok, see if the api_version() is high enough
2441 if ($lib->can('api_version') && $lib->api_version() >= 1.0)
2444 # api_version matches, check if it really provides anything we need
2448 add mul div sub dec inc
2449 acmp len digit is_one is_zero is_even is_odd
2451 new copy check from_hex from_bin as_hex as_bin zeros
2452 rsft lsft xor and or
2453 mod sqrt root fac pow modinv modpow log_int gcd
2456 if (!$lib->can("_$method"))
2458 if (($WARN{$lib}||0) < 2)
2461 Carp::carp ("$lib is missing method '_$method'");
2462 $WARN{$lib} = 1; # still warn about the lib
2471 last; # found a usable one, break
2475 if (($WARN{$lib}||0) < 2)
2477 my $ver = eval "\$$lib\::VERSION" || 'unknown';
2479 Carp::carp ("Cannot load outdated $lib v$ver, please upgrade");
2480 $WARN{$lib} = 2; # never warn again
2488 Carp::croak ("Couldn't load any math lib, not even 'Calc.pm'");
2492 foreach my $class (keys %CALLBACKS)
2494 &{$CALLBACKS{$class}}($CALC);
2497 # Fill $CAN with the results of $CALC->can(...) for emulating lower math lib
2501 for my $method (qw/ signed_and signed_or signed_xor /)
2503 $CAN{$method} = $CALC->can("_$method") ? 1 : 0;
2512 # convert a (ref to) big hex string to BigInt, return undef for error
2515 my $x = Math::BigInt->bzero();
2518 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2519 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2521 return $x->bnan() if $hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2523 my $sign = '+'; $sign = '-' if $hs =~ /^-/;
2525 $hs =~ s/^[+-]//; # strip sign
2526 $x->{value} = $CALC->_from_hex($hs);
2527 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2534 # convert a (ref to) big binary string to BigInt, return undef for error
2537 my $x = Math::BigInt->bzero();
2539 $bs =~ s/([01])_([01])/$1$2/g;
2540 $bs =~ s/([01])_([01])/$1$2/g;
2541 return $x->bnan() if $bs !~ /^[+-]?0b[01]+$/;
2543 my $sign = '+'; $sign = '-' if $bs =~ /^\-/;
2544 $bs =~ s/^[+-]//; # strip sign
2546 $x->{value} = $CALC->_from_bin($bs);
2547 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2553 # input: num_str; output: undef for invalid or
2554 # (\$mantissa_sign,\$mantissa_value,\$mantissa_fraction,\$exp_sign,\$exp_value)
2555 # Internal, take apart a string and return the pieces.
2556 # Strip leading/trailing whitespace, leading zeros, underscore and reject
2560 # strip white space at front, also extranous leading zeros
2561 $x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2562 $x =~ s/^\s+//; # but this will
2563 $x =~ s/\s+$//g; # strip white space at end
2565 # shortcut, if nothing to split, return early
2566 if ($x =~ /^[+-]?\d+\z/)
2568 $x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2569 return (\$sign, \$x, \'', \'', \0);
2572 # invalid starting char?
2573 return if $x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2575 return __from_hex($x) if $x =~ /^[\-\+]?0x/; # hex string
2576 return __from_bin($x) if $x =~ /^[\-\+]?0b/; # binary string
2578 # strip underscores between digits
2579 $x =~ s/(\d)_(\d)/$1$2/g;
2580 $x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3
2582 # some possible inputs:
2583 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2584 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 # 0e999
2586 my ($m,$e,$last) = split /[Ee]/,$x;
2587 return if defined $last; # last defined => 1e2E3 or others
2588 $e = '0' if !defined $e || $e eq "";
2590 # sign,value for exponent,mantint,mantfrac
2591 my ($es,$ev,$mis,$miv,$mfv);
2593 if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2597 return if $m eq '.' || $m eq '';
2598 my ($mi,$mf,$lastf) = split /\./,$m;
2599 return if defined $lastf; # lastf defined => 1.2.3 or others
2600 $mi = '0' if !defined $mi;
2601 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2602 $mf = '0' if !defined $mf || $mf eq '';
2603 if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2605 $mis = $1||'+'; $miv = $2;
2606 return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
2608 # handle the 0e999 case here
2609 $ev = 0 if $miv eq '0' && $mfv eq '';
2610 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2613 return; # NaN, not a number
2616 ##############################################################################
2617 # internal calculation routines (others are in Math::BigInt::Calc etc)
2621 # (BINT or num_str, BINT or num_str) return BINT
2622 # does modify first argument
2626 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2627 my $method = ref($x) . '::bgcd';
2629 $x * $ty / &$method($x,$ty);
2632 ###############################################################################
2633 # this method returns 0 if the object can be modified, or 1 if not.
2634 # We use a fast constant sub() here, to avoid costly calls. Subclasses
2635 # may override it with special code (f.i. Math::BigInt::Constant does so)
2637 sub modify () { 0; }
2646 Math::BigInt - Arbitrary size integer/float math package
2652 # or make it faster: install (optional) Math::BigInt::GMP
2653 # and always use (it will fall back to pure Perl if the
2654 # GMP library is not installed):
2656 use Math::BigInt lib => 'GMP';
2658 my $str = '1234567890';
2659 my @values = (64,74,18);
2660 my $n = 1; my $sign = '-';
2663 $x = Math::BigInt->new($str); # defaults to 0
2664 $y = $x->copy(); # make a true copy
2665 $nan = Math::BigInt->bnan(); # create a NotANumber
2666 $zero = Math::BigInt->bzero(); # create a +0
2667 $inf = Math::BigInt->binf(); # create a +inf
2668 $inf = Math::BigInt->binf('-'); # create a -inf
2669 $one = Math::BigInt->bone(); # create a +1
2670 $one = Math::BigInt->bone('-'); # create a -1
2672 # Testing (don't modify their arguments)
2673 # (return true if the condition is met, otherwise false)
2675 $x->is_zero(); # if $x is +0
2676 $x->is_nan(); # if $x is NaN
2677 $x->is_one(); # if $x is +1
2678 $x->is_one('-'); # if $x is -1
2679 $x->is_odd(); # if $x is odd
2680 $x->is_even(); # if $x is even
2681 $x->is_pos(); # if $x >= 0
2682 $x->is_neg(); # if $x < 0
2683 $x->is_inf($sign); # if $x is +inf, or -inf (sign is default '+')
2684 $x->is_int(); # if $x is an integer (not a float)
2686 # comparing and digit/sign extraction
2687 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2688 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2689 $x->sign(); # return the sign, either +,- or NaN
2690 $x->digit($n); # return the nth digit, counting from right
2691 $x->digit(-$n); # return the nth digit, counting from left
2693 # The following all modify their first argument. If you want to preserve
2694 # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
2695 # necessary when mixing $a = $b assignments with non-overloaded math.
2697 $x->bzero(); # set $x to 0
2698 $x->bnan(); # set $x to NaN
2699 $x->bone(); # set $x to +1
2700 $x->bone('-'); # set $x to -1
2701 $x->binf(); # set $x to inf
2702 $x->binf('-'); # set $x to -inf
2704 $x->bneg(); # negation
2705 $x->babs(); # absolute value
2706 $x->bnorm(); # normalize (no-op in BigInt)
2707 $x->bnot(); # two's complement (bit wise not)
2708 $x->binc(); # increment $x by 1
2709 $x->bdec(); # decrement $x by 1
2711 $x->badd($y); # addition (add $y to $x)
2712 $x->bsub($y); # subtraction (subtract $y from $x)
2713 $x->bmul($y); # multiplication (multiply $x by $y)
2714 $x->bdiv($y); # divide, set $x to quotient
2715 # return (quo,rem) or quo if scalar
2717 $x->bmod($y); # modulus (x % y)
2718 $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
2719 $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
2721 $x->bpow($y); # power of arguments (x ** y)
2722 $x->blsft($y); # left shift
2723 $x->brsft($y); # right shift
2724 $x->blsft($y,$n); # left shift, by base $n (like 10)
2725 $x->brsft($y,$n); # right shift, by base $n (like 10)
2727 $x->band($y); # bitwise and
2728 $x->bior($y); # bitwise inclusive or
2729 $x->bxor($y); # bitwise exclusive or
2730 $x->bnot(); # bitwise not (two's complement)
2732 $x->bsqrt(); # calculate square-root
2733 $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
2734 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2736 $x->round($A,$P,$mode); # round to accuracy or precision using mode $mode
2737 $x->bround($n); # accuracy: preserve $n digits
2738 $x->bfround($n); # round to $nth digit, no-op for BigInts
2740 # The following do not modify their arguments in BigInt (are no-ops),
2741 # but do so in BigFloat:
2743 $x->bfloor(); # return integer less or equal than $x
2744 $x->bceil(); # return integer greater or equal than $x
2746 # The following do not modify their arguments:
2748 # greatest common divisor (no OO style)
2749 my $gcd = Math::BigInt::bgcd(@values);
2750 # lowest common multiplicator (no OO style)
2751 my $lcm = Math::BigInt::blcm(@values);
2753 $x->length(); # return number of digits in number
2754 ($xl,$f) = $x->length(); # length of number and length of fraction part,
2755 # latter is always 0 digits long for BigInts
2757 $x->exponent(); # return exponent as BigInt
2758 $x->mantissa(); # return (signed) mantissa as BigInt
2759 $x->parts(); # return (mantissa,exponent) as BigInt
2760 $x->copy(); # make a true copy of $x (unlike $y = $x;)
2761 $x->as_int(); # return as BigInt (in BigInt: same as copy())
2762 $x->numify(); # return as scalar (might overflow!)
2764 # conversation to string (do not modify their argument)
2765 $x->bstr(); # normalized string (e.g. '3')
2766 $x->bsstr(); # norm. string in scientific notation (e.g. '3E0')
2767 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
2768 $x->as_bin(); # as signed binary string with prefixed 0b
2771 # precision and accuracy (see section about rounding for more)
2772 $x->precision(); # return P of $x (or global, if P of $x undef)
2773 $x->precision($n); # set P of $x to $n
2774 $x->accuracy(); # return A of $x (or global, if A of $x undef)
2775 $x->accuracy($n); # set A $x to $n
2778 Math::BigInt->precision(); # get/set global P for all BigInt objects
2779 Math::BigInt->accuracy(); # get/set global A for all BigInt objects
2780 Math::BigInt->round_mode(); # get/set global round mode, one of
2781 # 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
2782 Math::BigInt->config(); # return hash containing configuration
2786 All operators (including basic math operations) are overloaded if you
2787 declare your big integers as
2789 $i = new Math::BigInt '123_456_789_123_456_789';
2791 Operations with overloaded operators preserve the arguments which is
2792 exactly what you expect.
2798 Input values to these routines may be any string, that looks like a number
2799 and results in an integer, including hexadecimal and binary numbers.
2801 Scalars holding numbers may also be passed, but note that non-integer numbers
2802 may already have lost precision due to the conversation to float. Quote
2803 your input if you want BigInt to see all the digits:
2805 $x = Math::BigInt->new(12345678890123456789); # bad
2806 $x = Math::BigInt->new('12345678901234567890'); # good
2808 You can include one underscore between any two digits.
2810 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
2811 Non-integer values result in NaN.
2813 Currently, Math::BigInt::new() defaults to 0, while Math::BigInt::new('')
2814 results in 'NaN'. This might change in the future, so use always the following
2815 explicit forms to get a zero or NaN:
2817 $zero = Math::BigInt->bzero();
2818 $nan = Math::BigInt->bnan();
2820 C<bnorm()> on a BigInt object is now effectively a no-op, since the numbers
2821 are always stored in normalized form. If passed a string, creates a BigInt
2822 object from the input.
2826 Output values are BigInt objects (normalized), except for the methods which
2827 return a string (see L<SYNOPSIS>).
2829 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
2830 C<is_nan()>, etc.) return true or false, while others (C<bcmp()>, C<bacmp()>)
2831 return either undef (if NaN is involved), <0, 0 or >0 and are suited for sort.
2837 Each of the methods below (except config(), accuracy() and precision())
2838 accepts three additional parameters. These arguments C<$A>, C<$P> and C<$R>
2839 are C<accuracy>, C<precision> and C<round_mode>. Please see the section about
2840 L<ACCURACY and PRECISION> for more information.
2846 print Dumper ( Math::BigInt->config() );
2847 print Math::BigInt->config()->{lib},"\n";
2849 Returns a hash containing the configuration, e.g. the version number, lib
2850 loaded etc. The following hash keys are currently filled in with the
2851 appropriate information.
2855 ============================================================
2856 lib Name of the low-level math library
2858 lib_version Version of low-level math library (see 'lib')
2860 class The class name of config() you just called
2862 upgrade To which class math operations might be upgraded
2864 downgrade To which class math operations might be downgraded
2866 precision Global precision
2868 accuracy Global accuracy
2870 round_mode Global round mode
2872 version version number of the class you used
2874 div_scale Fallback accuracy for div
2876 trap_nan If true, traps creation of NaN via croak()
2878 trap_inf If true, traps creation of +inf/-inf via croak()
2881 The following values can be set by passing C<config()> a reference to a hash:
2884 upgrade downgrade precision accuracy round_mode div_scale
2888 $new_cfg = Math::BigInt->config( { trap_inf => 1, precision => 5 } );
2892 $x->accuracy(5); # local for $x
2893 CLASS->accuracy(5); # global for all members of CLASS
2894 # Note: This also applies to new()!
2896 $A = $x->accuracy(); # read out accuracy that affects $x
2897 $A = CLASS->accuracy(); # read out global accuracy
2899 Set or get the global or local accuracy, aka how many significant digits the
2900 results have. If you set a global accuracy, then this also applies to new()!
2902 Warning! The accuracy I<sticks>, e.g. once you created a number under the
2903 influence of C<< CLASS->accuracy($A) >>, all results from math operations with
2904 that number will also be rounded.
2906 In most cases, you should probably round the results explicitly using one of
2907 L<round()>, L<bround()> or L<bfround()> or by passing the desired accuracy
2908 to the math operation as additional parameter:
2910 my $x = Math::BigInt->new(30000);
2911 my $y = Math::BigInt->new(7);
2912 print scalar $x->copy()->bdiv($y, 2); # print 4300
2913 print scalar $x->copy()->bdiv($y)->bround(2); # print 4300
2915 Please see the section about L<ACCURACY AND PRECISION> for further details.
2917 Value must be greater than zero. Pass an undef value to disable it:
2919 $x->accuracy(undef);
2920 Math::BigInt->accuracy(undef);
2922 Returns the current accuracy. For C<$x->accuracy()> it will return either the
2923 local accuracy, or if not defined, the global. This means the return value
2924 represents the accuracy that will be in effect for $x:
2926 $y = Math::BigInt->new(1234567); # unrounded
2927 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
2928 $x = Math::BigInt->new(123456); # $x will be automatically rounded!
2929 print "$x $y\n"; # '123500 1234567'
2930 print $x->accuracy(),"\n"; # will be 4
2931 print $y->accuracy(),"\n"; # also 4, since global is 4
2932 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
2933 print $x->accuracy(),"\n"; # still 4
2934 print $y->accuracy(),"\n"; # 5, since global is 5
2936 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
2937 globals separated from Math::BigInt, but it is possible to subclass
2938 Math::BigInt and make the globals of the subclass aliases to the ones from
2943 $x->precision(-2); # local for $x, round at the second digit right of the dot
2944 $x->precision(2); # ditto, round at the second digit left of the dot
2946 CLASS->precision(5); # Global for all members of CLASS
2947 # This also applies to new()!
2948 CLASS->precision(-5); # ditto
2950 $P = CLASS->precision(); # read out global precision
2951 $P = $x->precision(); # read out precision that affects $x
2953 Note: You probably want to use L<accuracy()> instead. With L<accuracy> you
2954 set the number of digits each result should have, with L<precision> you
2955 set the place where to round!
2957 C<precision()> sets or gets the global or local precision, aka at which digit
2958 before or after the dot to round all results. A set global precision also
2959 applies to all newly created numbers!
2961 In Math::BigInt, passing a negative number precision has no effect since no
2962 numbers have digits after the dot. In L<Math::BigFloat>, it will round all
2963 results to P digits after the dot.
2965 Please see the section about L<ACCURACY AND PRECISION> for further details.
2967 Pass an undef value to disable it:
2969 $x->precision(undef);
2970 Math::BigInt->precision(undef);
2972 Returns the current precision. For C<$x->precision()> it will return either the
2973 local precision of $x, or if not defined, the global. This means the return
2974 value represents the prevision that will be in effect for $x:
2976 $y = Math::BigInt->new(1234567); # unrounded
2977 print Math::BigInt->precision(4),"\n"; # set 4, print 4
2978 $x = Math::BigInt->new(123456); # will be automatically rounded
2979 print $x; # print "120000"!
2981 Note: Works also for subclasses like L<Math::BigFloat>. Each class has its
2982 own globals separated from Math::BigInt, but it is possible to subclass
2983 Math::BigInt and make the globals of the subclass aliases to the ones from
2990 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
2991 2, but others work, too.
2993 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
2997 $x = Math::BigInt->new(10);
2998 $x->brsft(1); # same as $x >> 1: 5
2999 $x = Math::BigInt->new(1234);
3000 $x->brsft(2,10); # result 12
3002 There is one exception, and that is base 2 with negative $x:
3005 $x = Math::BigInt->new(-5);
3008 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
3013 $x = Math::BigInt->new($str,$A,$P,$R);
3015 Creates a new BigInt object from a scalar or another BigInt object. The
3016 input is accepted as decimal, hex (with leading '0x') or binary (with leading
3019 See L<Input> for more info on accepted input formats.
3023 $x = Math::BigInt->bnan();
3025 Creates a new BigInt object representing NaN (Not A Number).
3026 If used on an object, it will set it to NaN:
3032 $x = Math::BigInt->bzero();
3034 Creates a new BigInt object representing zero.
3035 If used on an object, it will set it to zero:
3041 $x = Math::BigInt->binf($sign);
3043 Creates a new BigInt object representing infinity. The optional argument is
3044 either '-' or '+', indicating whether you want infinity or minus infinity.
3045 If used on an object, it will set it to infinity:
3052 $x = Math::BigInt->binf($sign);
3054 Creates a new BigInt object representing one. The optional argument is
3055 either '-' or '+', indicating whether you want one or minus one.
3056 If used on an object, it will set it to one:
3061 =head2 is_one()/is_zero()/is_nan()/is_inf()
3064 $x->is_zero(); # true if arg is +0
3065 $x->is_nan(); # true if arg is NaN
3066 $x->is_one(); # true if arg is +1
3067 $x->is_one('-'); # true if arg is -1
3068 $x->is_inf(); # true if +inf
3069 $x->is_inf('-'); # true if -inf (sign is default '+')
3071 These methods all test the BigInt for being one specific value and return
3072 true or false depending on the input. These are faster than doing something
3077 =head2 is_pos()/is_neg()
3079 $x->is_pos(); # true if > 0
3080 $x->is_neg(); # true if < 0
3082 The methods return true if the argument is positive or negative, respectively.
3083 C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
3084 C<-inf> is negative. A C<zero> is neither positive nor negative.
3086 These methods are only testing the sign, and not the value.
3088 C<is_positive()> and C<is_negative()> are aliases to C<is_pos()> and
3089 C<is_neg()>, respectively. C<is_positive()> and C<is_negative()> were
3090 introduced in v1.36, while C<is_pos()> and C<is_neg()> were only introduced
3093 =head2 is_odd()/is_even()/is_int()
3095 $x->is_odd(); # true if odd, false for even
3096 $x->is_even(); # true if even, false for odd
3097 $x->is_int(); # true if $x is an integer
3099 The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
3100 C<-inf> are not integers and are neither odd nor even.
3102 In BigInt, all numbers except C<NaN>, C<+inf> and C<-inf> are integers.
3108 Compares $x with $y and takes the sign into account.
3109 Returns -1, 0, 1 or undef.
3115 Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
3121 Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
3123 If you want $x to have a certain sign, use one of the following methods:
3126 $x->babs()->bneg(); # '-'
3128 $x->binf(); # '+inf'
3129 $x->binf('-'); # '-inf'
3133 $x->digit($n); # return the nth digit, counting from right
3135 If C<$n> is negative, returns the digit counting from left.
3141 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
3142 and '-inf', respectively. Does nothing for NaN or zero.
3148 Set the number to it's absolute value, e.g. change the sign from '-' to '+'
3149 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
3154 $x->bnorm(); # normalize (no-op)
3160 Two's complement (bit wise not). This is equivalent to
3168 $x->binc(); # increment x by 1
3172 $x->bdec(); # decrement x by 1
3176 $x->badd($y); # addition (add $y to $x)
3180 $x->bsub($y); # subtraction (subtract $y from $x)
3184 $x->bmul($y); # multiplication (multiply $x by $y)
3188 $x->bdiv($y); # divide, set $x to quotient
3189 # return (quo,rem) or quo if scalar
3193 $x->bmod($y); # modulus (x % y)
3197 num->bmodinv($mod); # modular inverse
3199 Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
3200 returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
3201 C<bgcd($num, $mod)==1>.
3205 $num->bmodpow($exp,$mod); # modular exponentation
3206 # ($num**$exp % $mod)
3208 Returns the value of C<$num> taken to the power C<$exp> in the modulus
3209 C<$mod> using binary exponentation. C<bmodpow> is far superior to
3214 because it is much faster - it reduces internal variables into
3215 the modulus whenever possible, so it operates on smaller numbers.
3217 C<bmodpow> also supports negative exponents.
3219 bmodpow($num, -1, $mod)
3221 is exactly equivalent to
3227 $x->bpow($y); # power of arguments (x ** y)
3231 $x->blsft($y); # left shift
3232 $x->blsft($y,$n); # left shift, in base $n (like 10)
3236 $x->brsft($y); # right shift
3237 $x->brsft($y,$n); # right shift, in base $n (like 10)
3241 $x->band($y); # bitwise and
3245 $x->bior($y); # bitwise inclusive or
3249 $x->bxor($y); # bitwise exclusive or
3253 $x->bnot(); # bitwise not (two's complement)
3257 $x->bsqrt(); # calculate square-root
3261 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3265 $x->round($A,$P,$round_mode);
3267 Round $x to accuracy C<$A> or precision C<$P> using the round mode
3272 $x->bround($N); # accuracy: preserve $N digits
3276 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3282 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3283 does change $x in BigFloat.
3289 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3290 does change $x in BigFloat.
3294 bgcd(@values); # greatest common divisor (no OO style)
3298 blcm(@values); # lowest common multiplicator (no OO style)
3303 ($xl,$fl) = $x->length();
3305 Returns the number of digits in the decimal representation of the number.
3306 In list context, returns the length of the integer and fraction part. For
3307 BigInt's, the length of the fraction part will always be 0.
3313 Return the exponent of $x as BigInt.
3319 Return the signed mantissa of $x as BigInt.
3323 $x->parts(); # return (mantissa,exponent) as BigInt
3327 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3333 Returns $x as a BigInt (truncated towards zero). In BigInt this is the same as
3336 C<as_number()> is an alias to this method. C<as_number> was introduced in
3337 v1.22, while C<as_int()> was only introduced in v1.68.
3343 Returns a normalized string representation of C<$x>.
3347 $x->bsstr(); # normalized string in scientific notation
3351 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3355 $x->as_bin(); # as signed binary string with prefixed 0b
3357 =head1 ACCURACY and PRECISION
3359 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3360 accuracy and precision based rounding, both automatically after every
3361 operation, as well as manually.
3363 This section describes the accuracy/precision handling in Math::Big* as it
3364 used to be and as it is now, complete with an explanation of all terms and
3367 Not yet implemented things (but with correct description) are marked with '!',
3368 things that need to be answered are marked with '?'.
3370 In the next paragraph follows a short description of terms used here (because
3371 these may differ from terms used by others people or documentation).
3373 During the rest of this document, the shortcuts A (for accuracy), P (for
3374 precision), F (fallback) and R (rounding mode) will be used.
3378 A fixed number of digits before (positive) or after (negative)
3379 the decimal point. For example, 123.45 has a precision of -2. 0 means an
3380 integer like 123 (or 120). A precision of 2 means two digits to the left
3381 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3382 numbers with zeros before the decimal point may have different precisions,
3383 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3384 was). It could also have p < 0, when the digits after the decimal point
3387 The string output (of floating point numbers) will be padded with zeros:
3389 Initial value P A Result String
3390 ------------------------------------------------------------
3391 1234.01 -3 1000 1000
3394 1234.001 1 1234 1234.0
3396 1234.01 2 1234.01 1234.01
3397 1234.01 5 1234.01 1234.01000
3399 For BigInts, no padding occurs.
3403 Number of significant digits. Leading zeros are not counted. A
3404 number may have an accuracy greater than the non-zero digits
3405 when there are zeros in it or trailing zeros. For example, 123.456 has
3406 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3408 The string output (of floating point numbers) will be padded with zeros:
3410 Initial value P A Result String
3411 ------------------------------------------------------------
3413 1234.01 6 1234.01 1234.01
3414 1234.1 8 1234.1 1234.1000
3416 For BigInts, no padding occurs.
3420 When both A and P are undefined, this is used as a fallback accuracy when
3423 =head2 Rounding mode R
3425 When rounding a number, different 'styles' or 'kinds'
3426 of rounding are possible. (Note that random rounding, as in
3427 Math::Round, is not implemented.)
3433 truncation invariably removes all digits following the
3434 rounding place, replacing them with zeros. Thus, 987.65 rounded
3435 to tens (P=1) becomes 980, and rounded to the fourth sigdig
3436 becomes 987.6 (A=4). 123.456 rounded to the second place after the
3437 decimal point (P=-2) becomes 123.46.
3439 All other implemented styles of rounding attempt to round to the
3440 "nearest digit." If the digit D immediately to the right of the
3441 rounding place (skipping the decimal point) is greater than 5, the
3442 number is incremented at the rounding place (possibly causing a
3443 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3444 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3445 truncated at the rounding place: e.g. when rounding to units, 0.4
3446 rounds to 0, and -19.4 rounds to -19.
3448 However the results of other styles of rounding differ if the
3449 digit immediately to the right of the rounding place (skipping the
3450 decimal point) is 5 and if there are no digits, or no digits other
3451 than 0, after that 5. In such cases:
3455 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3456 if it is not already. E.g., when rounding to the first sigdig, 0.45
3457 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3461 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3462 it is not already. E.g., when rounding to the first sigdig, 0.45
3463 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3467 round to plus infinity, i.e. always round up. E.g., when
3468 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3469 and 0.4501 also becomes 0.5.
3473 round to minus infinity, i.e. always round down. E.g., when
3474 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3475 but 0.4501 becomes 0.5.
3479 round to zero, i.e. positive numbers down, negative ones up.
3480 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3481 becomes -0.5, but 0.4501 becomes 0.5.
3485 The handling of A & P in MBI/MBF (the old core code shipped with Perl
3486 versions <= 5.7.2) is like this:
3492 * ffround($p) is able to round to $p number of digits after the decimal
3494 * otherwise P is unused
3496 =item Accuracy (significant digits)
3498 * fround($a) rounds to $a significant digits
3499 * only fdiv() and fsqrt() take A as (optional) paramater
3500 + other operations simply create the same number (fneg etc), or more (fmul)
3502 + rounding/truncating is only done when explicitly calling one of fround
3503 or ffround, and never for BigInt (not implemented)
3504 * fsqrt() simply hands its accuracy argument over to fdiv.
3505 * the documentation and the comment in the code indicate two different ways
3506 on how fdiv() determines the maximum number of digits it should calculate,
3507 and the actual code does yet another thing
3509 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3511 result has at most max(scale, length(dividend), length(divisor)) digits
3513 scale = max(scale, length(dividend)-1,length(divisor)-1);
3514 scale += length(divisor) - length(dividend);
3515 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3516 Actually, the 'difference' added to the scale is calculated from the
3517 number of "significant digits" in dividend and divisor, which is derived
3518 by looking at the length of the mantissa. Which is wrong, since it includes
3519 the + sign (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops
3520 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3521 assumption that 124 has 3 significant digits, while 120/7 will get you
3522 '17', not '17.1' since 120 is thought to have 2 significant digits.
3523 The rounding after the division then uses the remainder and $y to determine
3524 wether it must round up or down.
3525 ? I have no idea which is the right way. That's why I used a slightly more
3526 ? simple scheme and tweaked the few failing testcases to match it.
3530 This is how it works now:
3534 =item Setting/Accessing
3536 * You can set the A global via C<< Math::BigInt->accuracy() >> or
3537 C<< Math::BigFloat->accuracy() >> or whatever class you are using.
3538 * You can also set P globally by using C<< Math::SomeClass->precision() >>
3540 * Globals are classwide, and not inherited by subclasses.
3541 * to undefine A, use C<< Math::SomeCLass->accuracy(undef); >>
3542 * to undefine P, use C<< Math::SomeClass->precision(undef); >>
3543 * Setting C<< Math::SomeClass->accuracy() >> clears automatically
3544 C<< Math::SomeClass->precision() >>, and vice versa.
3545 * To be valid, A must be > 0, P can have any value.
3546 * If P is negative, this means round to the P'th place to the right of the
3547 decimal point; positive values mean to the left of the decimal point.
3548 P of 0 means round to integer.
3549 * to find out the current global A, use C<< Math::SomeClass->accuracy() >>
3550 * to find out the current global P, use C<< Math::SomeClass->precision() >>
3551 * use C<< $x->accuracy() >> respective C<< $x->precision() >> for the local
3552 setting of C<< $x >>.
3553 * Please note that C<< $x->accuracy() >> respective C<< $x->precision() >>
3554 return eventually defined global A or P, when C<< $x >>'s A or P is not
3557 =item Creating numbers
3559 * When you create a number, you can give it's desired A or P via:
3560 $x = Math::BigInt->new($number,$A,$P);
3561 * Only one of A or P can be defined, otherwise the result is NaN
3562 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
3563 globals (if set) will be used. Thus changing the global defaults later on
3564 will not change the A or P of previously created numbers (i.e., A and P of
3565 $x will be what was in effect when $x was created)
3566 * If given undef for A and P, B<no> rounding will occur, and the globals will
3567 B<not> be used. This is used by subclasses to create numbers without
3568 suffering rounding in the parent. Thus a subclass is able to have it's own
3569 globals enforced upon creation of a number by using
3570 C<< $x = Math::BigInt->new($number,undef,undef) >>:
3572 use Math::BigInt::SomeSubclass;
3575 Math::BigInt->accuracy(2);
3576 Math::BigInt::SomeSubClass->accuracy(3);
3577 $x = Math::BigInt::SomeSubClass->new(1234);
3579 $x is now 1230, and not 1200. A subclass might choose to implement
3580 this otherwise, e.g. falling back to the parent's A and P.
3584 * If A or P are enabled/defined, they are used to round the result of each
3585 operation according to the rules below
3586 * Negative P is ignored in Math::BigInt, since BigInts never have digits
3587 after the decimal point
3588 * Math::BigFloat uses Math::BigInt internally, but setting A or P inside
3589 Math::BigInt as globals does not tamper with the parts of a BigFloat.
3590 A flag is used to mark all Math::BigFloat numbers as 'never round'.
3594 * It only makes sense that a number has only one of A or P at a time.
3595 If you set either A or P on one object, or globally, the other one will
3596 be automatically cleared.
3597 * If two objects are involved in an operation, and one of them has A in
3598 effect, and the other P, this results in an error (NaN).
3599 * A takes precedence over P (Hint: A comes before P).
3600 If neither of them is defined, nothing is used, i.e. the result will have
3601 as many digits as it can (with an exception for fdiv/fsqrt) and will not
3603 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
3604 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
3605 If either the dividend's or the divisor's mantissa has more digits than
3606 the value of F, the higher value will be used instead of F.
3607 This is to limit the digits (A) of the result (just consider what would
3608 happen with unlimited A and P in the case of 1/3 :-)
3609 * fdiv will calculate (at least) 4 more digits than required (determined by
3610 A, P or F), and, if F is not used, round the result
3611 (this will still fail in the case of a result like 0.12345000000001 with A
3612 or P of 5, but this can not be helped - or can it?)
3613 * Thus you can have the math done by on Math::Big* class in two modi:
3614 + never round (this is the default):
3615 This is done by setting A and P to undef. No math operation
3616 will round the result, with fdiv() and fsqrt() as exceptions to guard
3617 against overflows. You must explicitly call bround(), bfround() or
3618 round() (the latter with parameters).
3619 Note: Once you have rounded a number, the settings will 'stick' on it
3620 and 'infect' all other numbers engaged in math operations with it, since
3621 local settings have the highest precedence. So, to get SaferRound[tm],
3622 use a copy() before rounding like this:
3624 $x = Math::BigFloat->new(12.34);
3625 $y = Math::BigFloat->new(98.76);
3626 $z = $x * $y; # 1218.6984
3627 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
3628 $z = $x * $y; # still 1218.6984, without
3629 # copy would have been 1210!
3631 + round after each op:
3632 After each single operation (except for testing like is_zero()), the
3633 method round() is called and the result is rounded appropriately. By
3634 setting proper values for A and P, you can have all-the-same-A or
3635 all-the-same-P modes. For example, Math::Currency might set A to undef,
3636 and P to -2, globally.
3638 ?Maybe an extra option that forbids local A & P settings would be in order,
3639 ?so that intermediate rounding does not 'poison' further math?
3641 =item Overriding globals
3643 * you will be able to give A, P and R as an argument to all the calculation
3644 routines; the second parameter is A, the third one is P, and the fourth is
3645 R (shift right by one for binary operations like badd). P is used only if
3646 the first parameter (A) is undefined. These three parameters override the
3647 globals in the order detailed as follows, i.e. the first defined value
3649 (local: per object, global: global default, parameter: argument to sub)
3652 + local A (if defined on both of the operands: smaller one is taken)
3653 + local P (if defined on both of the operands: bigger one is taken)
3657 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
3658 arguments (A and P) instead of one
3660 =item Local settings
3662 * You can set A or P locally by using C<< $x->accuracy() >> or
3663 C<< $x->precision() >>
3664 and thus force different A and P for different objects/numbers.
3665 * Setting A or P this way immediately rounds $x to the new value.
3666 * C<< $x->accuracy() >> clears C<< $x->precision() >>, and vice versa.
3670 * the rounding routines will use the respective global or local settings.
3671 fround()/bround() is for accuracy rounding, while ffround()/bfround()
3673 * the two rounding functions take as the second parameter one of the
3674 following rounding modes (R):
3675 'even', 'odd', '+inf', '-inf', 'zero', 'trunc'
3676 * you can set/get the global R by using C<< Math::SomeClass->round_mode() >>
3677 or by setting C<< $Math::SomeClass::round_mode >>
3678 * after each operation, C<< $result->round() >> is called, and the result may
3679 eventually be rounded (that is, if A or P were set either locally,
3680 globally or as parameter to the operation)
3681 * to manually round a number, call C<< $x->round($A,$P,$round_mode); >>
3682 this will round the number by using the appropriate rounding function
3683 and then normalize it.
3684 * rounding modifies the local settings of the number:
3686 $x = Math::BigFloat->new(123.456);
3690 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
3691 will be 4 from now on.
3693 =item Default values
3702 * The defaults are set up so that the new code gives the same results as
3703 the old code (except in a few cases on fdiv):
3704 + Both A and P are undefined and thus will not be used for rounding
3705 after each operation.
3706 + round() is thus a no-op, unless given extra parameters A and P
3710 =head1 Infinity and Not a Number
3712 While BigInt has extensive handling of inf and NaN, certain quirks remain.
3718 These perl routines currently (as of Perl v.5.8.6) cannot handle passed
3721 te@linux:~> perl -wle 'print 2 ** 3333'
3723 te@linux:~> perl -wle 'print 2 ** 3333 == 2 ** 3333'
3725 te@linux:~> perl -wle 'print oct(2 ** 3333)'
3727 te@linux:~> perl -wle 'print hex(2 ** 3333)'
3728 Illegal hexadecimal digit 'i' ignored at -e line 1.
3731 The same problems occur if you pass them Math::BigInt->binf() objects. Since
3732 overloading these routines is not possible, this cannot be fixed from BigInt.
3734 =item ==, !=, <, >, <=, >= with NaNs
3736 BigInt's bcmp() routine currently returns undef to signal that a NaN was
3737 involved in a comparison. However, the overload code turns that into
3738 either 1 or '' and thus operations like C<< NaN != NaN >> might return
3743 C<< log(-inf) >> is highly weird. Since log(-x)=pi*i+log(x), then
3744 log(-inf)=pi*i+inf. However, since the imaginary part is finite, the real
3745 infinity "overshadows" it, so the number might as well just be infinity.
3746 However, the result is a complex number, and since BigInt/BigFloat can only
3747 have real numbers as results, the result is NaN.
3749 =item exp(), cos(), sin(), atan2()
3751 These all might have problems handling infinity right.
3757 The actual numbers are stored as unsigned big integers (with seperate sign).
3759 You should neither care about nor depend on the internal representation; it
3760 might change without notice. Use B<ONLY> method calls like C<< $x->sign(); >>
3761 instead relying on the internal representation.
3765 Math with the numbers is done (by default) by a module called
3766 C<Math::BigInt::Calc>. This is equivalent to saying:
3768 use Math::BigInt lib => 'Calc';
3770 You can change this by using:
3772 use Math::BigInt lib => 'BitVect';
3774 The following would first try to find Math::BigInt::Foo, then
3775 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
3777 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
3779 Since Math::BigInt::GMP is in almost all cases faster than Calc (especially in
3780 math involving really big numbers, where it is B<much> faster), and there is
3781 no penalty if Math::BigInt::GMP is not installed, it is a good idea to always
3784 use Math::BigInt lib => 'GMP';
3786 Different low-level libraries use different formats to store the
3787 numbers. You should B<NOT> depend on the number having a specific format
3790 See the respective math library module documentation for further details.
3794 The sign is either '+', '-', 'NaN', '+inf' or '-inf'.
3796 A sign of 'NaN' is used to represent the result when input arguments are not
3797 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
3798 minus infinity. You will get '+inf' when dividing a positive number by 0, and
3799 '-inf' when dividing any negative number by 0.
3801 =head2 mantissa(), exponent() and parts()
3803 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
3806 $m = $x->mantissa();
3807 $e = $x->exponent();
3808 $y = $m * ( 10 ** $e );
3809 print "ok\n" if $x == $y;
3811 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
3812 in one go. Both the returned mantissa and exponent have a sign.
3814 Currently, for BigInts C<$e> is always 0, except for NaN, +inf and -inf,
3815 where it is C<NaN>; and for C<$x == 0>, where it is C<1> (to be compatible
3816 with Math::BigFloat's internal representation of a zero as C<0E1>).
3818 C<$m> is currently just a copy of the original number. The relation between
3819 C<$e> and C<$m> will stay always the same, though their real values might
3826 sub bint { Math::BigInt->new(shift); }
3828 $x = Math::BigInt->bstr("1234") # string "1234"
3829 $x = "$x"; # same as bstr()
3830 $x = Math::BigInt->bneg("1234"); # BigInt "-1234"
3831 $x = Math::BigInt->babs("-12345"); # BigInt "12345"
3832 $x = Math::BigInt->bnorm("-0 00"); # BigInt "0"
3833 $x = bint(1) + bint(2); # BigInt "3"
3834 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
3835 $x = bint(1); # BigInt "1"
3836 $x = $x + 5 / 2; # BigInt "3"
3837 $x = $x ** 3; # BigInt "27"
3838 $x *= 2; # BigInt "54"
3839 $x = Math::BigInt->new(0); # BigInt "0"
3841 $x = Math::BigInt->badd(4,5) # BigInt "9"
3842 print $x->bsstr(); # 9e+0
3844 Examples for rounding:
3849 $x = Math::BigFloat->new(123.4567);
3850 $y = Math::BigFloat->new(123.456789);
3851 Math::BigFloat->accuracy(4); # no more A than 4
3853 ok ($x->copy()->fround(),123.4); # even rounding
3854 print $x->copy()->fround(),"\n"; # 123.4
3855 Math::BigFloat->round_mode('odd'); # round to odd
3856 print $x->copy()->fround(),"\n"; # 123.5
3857 Math::BigFloat->accuracy(5); # no more A than 5
3858 Math::BigFloat->round_mode('odd'); # round to odd
3859 print $x->copy()->fround(),"\n"; # 123.46
3860 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
3861 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
3863 Math::BigFloat->accuracy(undef); # A not important now
3864 Math::BigFloat->precision(2); # P important
3865 print $x->copy()->bnorm(),"\n"; # 123.46
3866 print $x->copy()->fround(),"\n"; # 123.46
3868 Examples for converting:
3870 my $x = Math::BigInt->new('0b1'.'01' x 123);
3871 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
3873 =head1 Autocreating constants
3875 After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
3876 and binary constants in the given scope are converted to C<Math::BigInt>.
3877 This conversion happens at compile time.
3881 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
3883 prints the integer value of C<2**100>. Note that without conversion of
3884 constants the expression 2**100 will be calculated as perl scalar.
3886 Please note that strings and floating point constants are not affected,
3889 use Math::BigInt qw/:constant/;
3891 $x = 1234567890123456789012345678901234567890
3892 + 123456789123456789;
3893 $y = '1234567890123456789012345678901234567890'
3894 + '123456789123456789';
3896 do not work. You need an explicit Math::BigInt->new() around one of the
3897 operands. You should also quote large constants to protect loss of precision:
3901 $x = Math::BigInt->new('1234567889123456789123456789123456789');
3903 Without the quotes Perl would convert the large number to a floating point
3904 constant at compile time and then hand the result to BigInt, which results in
3905 an truncated result or a NaN.
3907 This also applies to integers that look like floating point constants:
3909 use Math::BigInt ':constant';
3911 print ref(123e2),"\n";
3912 print ref(123.2e2),"\n";
3914 will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
3915 to get this to work.
3919 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
3920 must be made in the second case. For long numbers, the copy can eat up to 20%
3921 of the work (in the case of addition/subtraction, less for
3922 multiplication/division). If $y is very small compared to $x, the form
3923 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
3924 more time then the actual addition.
3926 With a technique called copy-on-write, the cost of copying with overload could
3927 be minimized or even completely avoided. A test implementation of COW did show
3928 performance gains for overloaded math, but introduced a performance loss due
3929 to a constant overhead for all other operations. So Math::BigInt does currently
3932 The rewritten version of this module (vs. v0.01) is slower on certain
3933 operations, like C<new()>, C<bstr()> and C<numify()>. The reason are that it
3934 does now more work and handles much more cases. The time spent in these
3935 operations is usually gained in the other math operations so that code on
3936 the average should get (much) faster. If they don't, please contact the author.
3938 Some operations may be slower for small numbers, but are significantly faster
3939 for big numbers. Other operations are now constant (O(1), like C<bneg()>,
3940 C<babs()> etc), instead of O(N) and thus nearly always take much less time.
3941 These optimizations were done on purpose.
3943 If you find the Calc module to slow, try to install any of the replacement
3944 modules and see if they help you.
3946 =head2 Alternative math libraries
3948 You can use an alternative library to drive Math::BigInt via:
3950 use Math::BigInt lib => 'Module';
3952 See L<MATH LIBRARY> for more information.
3954 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
3958 =head1 Subclassing Math::BigInt
3960 The basic design of Math::BigInt allows simple subclasses with very little
3961 work, as long as a few simple rules are followed:
3967 The public API must remain consistent, i.e. if a sub-class is overloading
3968 addition, the sub-class must use the same name, in this case badd(). The
3969 reason for this is that Math::BigInt is optimized to call the object methods
3974 The private object hash keys like C<$x->{sign}> may not be changed, but
3975 additional keys can be added, like C<$x->{_custom}>.
3979 Accessor functions are available for all existing object hash keys and should
3980 be used instead of directly accessing the internal hash keys. The reason for
3981 this is that Math::BigInt itself has a pluggable interface which permits it
3982 to support different storage methods.
3986 More complex sub-classes may have to replicate more of the logic internal of
3987 Math::BigInt if they need to change more basic behaviors. A subclass that
3988 needs to merely change the output only needs to overload C<bstr()>.
3990 All other object methods and overloaded functions can be directly inherited
3991 from the parent class.
3993 At the very minimum, any subclass will need to provide it's own C<new()> and can
3994 store additional hash keys in the object. There are also some package globals
3995 that must be defined, e.g.:
3999 $precision = -2; # round to 2 decimal places
4000 $round_mode = 'even';
4003 Additionally, you might want to provide the following two globals to allow
4004 auto-upgrading and auto-downgrading to work correctly:
4009 This allows Math::BigInt to correctly retrieve package globals from the
4010 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
4011 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
4017 in your subclass to automatically inherit the overloading from the parent. If
4018 you like, you can change part of the overloading, look at Math::String for an
4023 When used like this:
4025 use Math::BigInt upgrade => 'Foo::Bar';
4027 certain operations will 'upgrade' their calculation and thus the result to
4028 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
4030 use Math::BigInt upgrade => 'Math::BigFloat';
4032 As a shortcut, you can use the module C<bignum>:
4036 Also good for oneliners:
4038 perl -Mbignum -le 'print 2 ** 255'
4040 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
4041 as well es preserve accuracy (as in sqrt(3)).
4043 Beware: This feature is not fully implemented yet.
4047 The following methods upgrade themselves unconditionally; that is if upgrade
4048 is in effect, they will always hand up their work:
4060 Beware: This list is not complete.
4062 All other methods upgrade themselves only when one (or all) of their
4063 arguments are of the class mentioned in $upgrade (This might change in later
4064 versions to a more sophisticated scheme):
4070 =item broot() does not work
4072 The broot() function in BigInt may only work for small values. This will be
4073 fixed in a later version.
4075 =item Out of Memory!
4077 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
4078 C<eval()> in your code will crash with "Out of memory". This is probably an
4079 overload/exporter bug. You can workaround by not having C<eval()>
4080 and ':constant' at the same time or upgrade your Perl to a newer version.
4082 =item Fails to load Calc on Perl prior 5.6.0
4084 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
4085 will fall back to eval { require ... } when loading the math lib on Perls
4086 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
4087 filesystems using a different seperator.
4093 Some things might not work as you expect them. Below is documented what is
4094 known to be troublesome:
4098 =item bstr(), bsstr() and 'cmp'
4100 Both C<bstr()> and C<bsstr()> as well as automated stringify via overload now
4101 drop the leading '+'. The old code would return '+3', the new returns '3'.
4102 This is to be consistent with Perl and to make C<cmp> (especially with
4103 overloading) to work as you expect. It also solves problems with C<Test.pm>,
4104 because it's C<ok()> uses 'eq' internally.
4106 Mark Biggar said, when asked about to drop the '+' altogether, or make only
4109 I agree (with the first alternative), don't add the '+' on positive
4110 numbers. It's not as important anymore with the new internal
4111 form for numbers. It made doing things like abs and neg easier,
4112 but those have to be done differently now anyway.
4114 So, the following examples will now work all as expected:
4117 BEGIN { plan tests => 1 }
4120 my $x = new Math::BigInt 3*3;
4121 my $y = new Math::BigInt 3*3;
4124 print "$x eq 9" if $x eq $y;
4125 print "$x eq 9" if $x eq '9';
4126 print "$x eq 9" if $x eq 3*3;
4128 Additionally, the following still works:
4130 print "$x == 9" if $x == $y;
4131 print "$x == 9" if $x == 9;
4132 print "$x == 9" if $x == 3*3;
4134 There is now a C<bsstr()> method to get the string in scientific notation aka
4135 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
4136 for comparison, but Perl will represent some numbers as 100 and others
4137 as 1e+308. If in doubt, convert both arguments to Math::BigInt before
4138 comparing them as strings:
4141 BEGIN { plan tests => 3 }
4144 $x = Math::BigInt->new('1e56'); $y = 1e56;
4145 ok ($x,$y); # will fail
4146 ok ($x->bsstr(),$y); # okay
4147 $y = Math::BigInt->new($y);
4150 Alternatively, simple use C<< <=> >> for comparisons, this will get it
4151 always right. There is not yet a way to get a number automatically represented
4152 as a string that matches exactly the way Perl represents it.
4154 See also the section about L<Infinity and Not a Number> for problems in
4159 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
4162 $x = Math::BigInt->new(123);
4163 $y = int($x); # BigInt 123
4164 $x = Math::BigFloat->new(123.45);
4165 $y = int($x); # BigInt 123
4167 In all Perl versions you can use C<as_number()> or C<as_int> for the same
4170 $x = Math::BigFloat->new(123.45);
4171 $y = $x->as_number(); # BigInt 123
4172 $y = $x->as_int(); # ditto
4174 This also works for other subclasses, like Math::String.
4176 It is yet unclear whether overloaded int() should return a scalar or a BigInt.
4178 If you want a real Perl scalar, use C<numify()>:
4180 $y = $x->numify(); # 123 as scalar
4182 This is seldom necessary, though, because this is done automatically, like
4183 when you access an array:
4185 $z = $array[$x]; # does work automatically
4189 The following will probably not do what you expect:
4191 $c = Math::BigInt->new(123);
4192 print $c->length(),"\n"; # prints 30
4194 It prints both the number of digits in the number and in the fraction part
4195 since print calls C<length()> in list context. Use something like:
4197 print scalar $c->length(),"\n"; # prints 3
4201 The following will probably not do what you expect:
4203 print $c->bdiv(10000),"\n";
4205 It prints both quotient and remainder since print calls C<bdiv()> in list
4206 context. Also, C<bdiv()> will modify $c, so be careful. You probably want
4209 print $c / 10000,"\n";
4210 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
4214 The quotient is always the greatest integer less than or equal to the
4215 real-valued quotient of the two operands, and the remainder (when it is
4216 nonzero) always has the same sign as the second operand; so, for
4226 As a consequence, the behavior of the operator % agrees with the
4227 behavior of Perl's built-in % operator (as documented in the perlop
4228 manpage), and the equation
4230 $x == ($x / $y) * $y + ($x % $y)
4232 holds true for any $x and $y, which justifies calling the two return
4233 values of bdiv() the quotient and remainder. The only exception to this rule
4234 are when $y == 0 and $x is negative, then the remainder will also be
4235 negative. See below under "infinity handling" for the reasoning behind this.
4237 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
4238 not change BigInt's way to do things. This is because under 'use integer' Perl
4239 will do what the underlying C thinks is right and this is different for each
4240 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
4241 the author to implement it ;)
4243 =item infinity handling
4245 Here are some examples that explain the reasons why certain results occur while
4248 The following table shows the result of the division and the remainder, so that
4249 the equation above holds true. Some "ordinary" cases are strewn in to show more
4250 clearly the reasoning:
4252 A / B = C, R so that C * B + R = A
4253 =========================================================
4254 5 / 8 = 0, 5 0 * 8 + 5 = 5
4255 0 / 8 = 0, 0 0 * 8 + 0 = 0
4256 0 / inf = 0, 0 0 * inf + 0 = 0
4257 0 /-inf = 0, 0 0 * -inf + 0 = 0
4258 5 / inf = 0, 5 0 * inf + 5 = 5
4259 5 /-inf = 0, 5 0 * -inf + 5 = 5
4260 -5/ inf = 0, -5 0 * inf + -5 = -5
4261 -5/-inf = 0, -5 0 * -inf + -5 = -5
4262 inf/ 5 = inf, 0 inf * 5 + 0 = inf
4263 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
4264 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
4265 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
4266 5/ 5 = 1, 0 1 * 5 + 0 = 5
4267 -5/ -5 = 1, 0 1 * -5 + 0 = -5
4268 inf/ inf = 1, 0 1 * inf + 0 = inf
4269 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
4270 inf/-inf = -1, 0 -1 * -inf + 0 = inf
4271 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
4272 8/ 0 = inf, 8 inf * 0 + 8 = 8
4273 inf/ 0 = inf, inf inf * 0 + inf = inf
4276 These cases below violate the "remainder has the sign of the second of the two
4277 arguments", since they wouldn't match up otherwise.
4279 A / B = C, R so that C * B + R = A
4280 ========================================================
4281 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
4282 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
4284 =item Modifying and =
4288 $x = Math::BigFloat->new(5);
4291 It will not do what you think, e.g. making a copy of $x. Instead it just makes
4292 a second reference to the B<same> object and stores it in $y. Thus anything
4293 that modifies $x (except overloaded operators) will modify $y, and vice versa.
4294 Or in other words, C<=> is only safe if you modify your BigInts only via
4295 overloaded math. As soon as you use a method call it breaks:
4298 print "$x, $y\n"; # prints '10, 10'
4300 If you want a true copy of $x, use:
4304 You can also chain the calls like this, this will make first a copy and then
4307 $y = $x->copy()->bmul(2);
4309 See also the documentation for overload.pm regarding C<=>.
4313 C<bpow()> (and the rounding functions) now modifies the first argument and
4314 returns it, unlike the old code which left it alone and only returned the
4315 result. This is to be consistent with C<badd()> etc. The first three will
4316 modify $x, the last one won't:
4318 print bpow($x,$i),"\n"; # modify $x
4319 print $x->bpow($i),"\n"; # ditto
4320 print $x **= $i,"\n"; # the same
4321 print $x ** $i,"\n"; # leave $x alone
4323 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4325 =item Overloading -$x
4335 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4336 needs to preserve $x since it does not know that it later will get overwritten.
4337 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4339 =item Mixing different object types
4341 In Perl you will get a floating point value if you do one of the following:
4347 With overloaded math, only the first two variants will result in a BigFloat:
4352 $mbf = Math::BigFloat->new(5);
4353 $mbi2 = Math::BigInteger->new(5);
4354 $mbi = Math::BigInteger->new(2);
4356 # what actually gets called:
4357 $float = $mbf + $mbi; # $mbf->badd()
4358 $float = $mbf / $mbi; # $mbf->bdiv()
4359 $integer = $mbi + $mbf; # $mbi->badd()
4360 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4361 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4363 This is because math with overloaded operators follows the first (dominating)
4364 operand, and the operation of that is called and returns thus the result. So,
4365 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4366 the result should be a Math::BigFloat or the second operant is one.
4368 To get a Math::BigFloat you either need to call the operation manually,
4369 make sure the operands are already of the proper type or casted to that type
4370 via Math::BigFloat->new():
4372 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4374 Beware of simple "casting" the entire expression, this would only convert
4375 the already computed result:
4377 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4379 Beware also of the order of more complicated expressions like:
4381 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4382 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4384 If in doubt, break the expression into simpler terms, or cast all operands
4385 to the desired resulting type.
4387 Scalar values are a bit different, since:
4392 will both result in the proper type due to the way the overloaded math works.
4394 This section also applies to other overloaded math packages, like Math::String.
4396 One solution to you problem might be autoupgrading|upgrading. See the
4397 pragmas L<bignum>, L<bigint> and L<bigrat> for an easy way to do this.
4401 C<bsqrt()> works only good if the result is a big integer, e.g. the square
4402 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4403 mode. The reason is that the result is always truncated to an integer.
4405 If you want a better approximation of the square root, then use:
4407 $x = Math::BigFloat->new(12);
4408 Math::BigFloat->precision(0);
4409 Math::BigFloat->round_mode('even');
4410 print $x->copy->bsqrt(),"\n"; # 4
4412 Math::BigFloat->precision(2);
4413 print $x->bsqrt(),"\n"; # 3.46
4414 print $x->bsqrt(3),"\n"; # 3.464
4418 For negative numbers in base see also L<brsft|brsft>.
4424 This program is free software; you may redistribute it and/or modify it under
4425 the same terms as Perl itself.
4429 L<Math::BigFloat>, L<Math::BigRat> and L<Math::Big> as well as
4430 L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4432 The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest
4433 because they solve the autoupgrading/downgrading issue, at least partly.
4436 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4437 more documentation including a full version history, testcases, empty
4438 subclass files and benchmarks.
4442 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4443 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001 - 2004
4444 and still at it in 2005.
4446 Many people contributed in one or more ways to the final beast, see the file
4447 CREDITS for an (incomplete) list. If you miss your name, please drop me a