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 $class = "Math::BigInt";
23 @ISA = qw( Exporter );
24 @EXPORT_OK = qw( objectify bgcd blcm);
26 # _trap_inf and _trap_nan are internal and should never be accessed from the
28 use vars qw/$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($_[0]->numify()) },
78 'atan2' => sub { atan2($_[0]->numify(),$_[1]) },
80 # are not yet overloadable
81 #'hex' => sub { print "hex"; $_[0]; },
82 #'oct' => sub { print "oct"; $_[0]; },
84 'log' => sub { $_[0]->copy()->blog($_[1]); },
85 'int' => sub { $_[0]->copy(); },
86 'neg' => sub { $_[0]->copy()->bneg(); },
87 'abs' => sub { $_[0]->copy()->babs(); },
88 'sqrt' => sub { $_[0]->copy()->bsqrt(); },
89 '~' => sub { $_[0]->copy()->bnot(); },
91 # for subtract it's a bit tricky to not modify b: b-a => -a+b
92 '-' => sub { my $c = $_[0]->copy; $_[2] ?
93 $c->bneg()->badd( $_[1]) :
95 '+' => sub { $_[0]->copy()->badd($_[1]); },
96 '*' => sub { $_[0]->copy()->bmul($_[1]); },
99 $_[2] ? ref($_[0])->new($_[1])->bdiv($_[0]) : $_[0]->copy->bdiv($_[1]);
102 $_[2] ? ref($_[0])->new($_[1])->bmod($_[0]) : $_[0]->copy->bmod($_[1]);
105 $_[2] ? ref($_[0])->new($_[1])->bpow($_[0]) : $_[0]->copy->bpow($_[1]);
108 $_[2] ? ref($_[0])->new($_[1])->blsft($_[0]) : $_[0]->copy->blsft($_[1]);
111 $_[2] ? ref($_[0])->new($_[1])->brsft($_[0]) : $_[0]->copy->brsft($_[1]);
114 $_[2] ? ref($_[0])->new($_[1])->band($_[0]) : $_[0]->copy->band($_[1]);
117 $_[2] ? ref($_[0])->new($_[1])->bior($_[0]) : $_[0]->copy->bior($_[1]);
120 $_[2] ? ref($_[0])->new($_[1])->bxor($_[0]) : $_[0]->copy->bxor($_[1]);
123 # can modify arg of ++ and --, so avoid a copy() for speed, but don't
124 # use $_[0]->bone(), it would modify $_[0] to be 1!
125 '++' => sub { $_[0]->binc() },
126 '--' => sub { $_[0]->bdec() },
128 # if overloaded, O(1) instead of O(N) and twice as fast for small numbers
130 # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/
131 # v5.6.1 dumps on this: return !$_[0]->is_zero() || undef; :-(
133 $t = 1 if !$_[0]->is_zero();
137 # the original qw() does not work with the TIESCALAR below, why?
138 # Order of arguments unsignificant
139 '""' => sub { $_[0]->bstr(); },
140 '0+' => sub { $_[0]->numify(); }
143 ##############################################################################
144 # global constants, flags and accessory
146 # These vars are public, but their direct usage is not recommended, use the
147 # accessor methods instead
149 $round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
154 $upgrade = undef; # default is no upgrade
155 $downgrade = undef; # default is no downgrade
157 # These are internally, and not to be used from the outside at all
159 $_trap_nan = 0; # are NaNs ok? set w/ config()
160 $_trap_inf = 0; # are infs ok? set w/ config()
161 my $nan = 'NaN'; # constants for easier life
163 my $CALC = 'Math::BigInt::Calc'; # module to do the low level math
165 my $IMPORT = 0; # was import() called yet?
166 # used to make require work
167 my %WARN; # warn only once for low-level libs
168 my %CAN; # cache for $CALC->can(...)
169 my %CALLBACKS; # callbacks to notify on lib loads
170 my $EMU_LIB = 'Math/BigInt/CalcEmu.pm'; # emulate low-level math
172 ##############################################################################
173 # the old code had $rnd_mode, so we need to support it, too
176 sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; }
177 sub FETCH { return $round_mode; }
178 sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
182 # tie to enable $rnd_mode to work transparently
183 tie $rnd_mode, 'Math::BigInt';
185 # set up some handy alias names
186 *as_int = \&as_number;
187 *is_pos = \&is_positive;
188 *is_neg = \&is_negative;
191 ##############################################################################
196 # make Class->round_mode() work
198 my $class = ref($self) || $self || __PACKAGE__;
202 if ($m !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
204 require Carp; Carp::croak ("Unknown round mode '$m'");
206 return ${"${class}::round_mode"} = $m;
208 ${"${class}::round_mode"};
214 # make Class->upgrade() work
216 my $class = ref($self) || $self || __PACKAGE__;
217 # need to set new value?
220 return ${"${class}::upgrade"} = $_[0];
222 ${"${class}::upgrade"};
228 # make Class->downgrade() work
230 my $class = ref($self) || $self || __PACKAGE__;
231 # need to set new value?
234 return ${"${class}::downgrade"} = $_[0];
236 ${"${class}::downgrade"};
242 # make Class->div_scale() work
244 my $class = ref($self) || $self || __PACKAGE__;
249 require Carp; Carp::croak ('div_scale must be greater than zero');
251 ${"${class}::div_scale"} = $_[0];
253 ${"${class}::div_scale"};
258 # $x->accuracy($a); ref($x) $a
259 # $x->accuracy(); ref($x)
260 # Class->accuracy(); class
261 # Class->accuracy($a); class $a
264 my $class = ref($x) || $x || __PACKAGE__;
267 # need to set new value?
271 # convert objects to scalars to avoid deep recursion. If object doesn't
272 # have numify(), then hopefully it will have overloading for int() and
273 # boolean test without wandering into a deep recursion path...
274 $a = $a->numify() if ref($a) && $a->can('numify');
278 # also croak on non-numerical
282 Carp::croak ('Argument to accuracy must be greater than zero');
286 require Carp; Carp::croak ('Argument to accuracy must be an integer');
291 # $object->accuracy() or fallback to global
292 $x->bround($a) if $a; # not for undef, 0
293 $x->{_a} = $a; # set/overwrite, even if not rounded
294 delete $x->{_p}; # clear P
295 $a = ${"${class}::accuracy"} unless defined $a; # proper return value
299 ${"${class}::accuracy"} = $a; # set global A
300 ${"${class}::precision"} = undef; # clear global P
302 return $a; # shortcut
306 # $object->accuracy() or fallback to global
307 $a = $x->{_a} if ref($x);
308 # but don't return global undef, when $x's accuracy is 0!
309 $a = ${"${class}::accuracy"} if !defined $a;
315 # $x->precision($p); ref($x) $p
316 # $x->precision(); ref($x)
317 # Class->precision(); class
318 # Class->precision($p); class $p
321 my $class = ref($x) || $x || __PACKAGE__;
327 # convert objects to scalars to avoid deep recursion. If object doesn't
328 # have numify(), then hopefully it will have overloading for int() and
329 # boolean test without wandering into a deep recursion path...
330 $p = $p->numify() if ref($p) && $p->can('numify');
331 if ((defined $p) && (int($p) != $p))
333 require Carp; Carp::croak ('Argument to precision must be an integer');
337 # $object->precision() or fallback to global
338 $x->bfround($p) if $p; # not for undef, 0
339 $x->{_p} = $p; # set/overwrite, even if not rounded
340 delete $x->{_a}; # clear A
341 $p = ${"${class}::precision"} unless defined $p; # proper return value
345 ${"${class}::precision"} = $p; # set global P
346 ${"${class}::accuracy"} = undef; # clear global A
348 return $p; # shortcut
352 # $object->precision() or fallback to global
353 $p = $x->{_p} if ref($x);
354 # but don't return global undef, when $x's precision is 0!
355 $p = ${"${class}::precision"} if !defined $p;
361 # return (or set) configuration data as hash ref
362 my $class = shift || 'Math::BigInt';
367 # try to set given options as arguments from hash
370 if (ref($args) ne 'HASH')
374 # these values can be "set"
378 upgrade downgrade precision accuracy round_mode div_scale/
381 $set_args->{$key} = $args->{$key} if exists $args->{$key};
382 delete $args->{$key};
387 Carp::croak ("Illegal key(s) '",
388 join("','",keys %$args),"' passed to $class\->config()");
390 foreach my $key (keys %$set_args)
392 if ($key =~ /^trap_(inf|nan)\z/)
394 ${"${class}::_trap_$1"} = ($set_args->{"trap_$1"} ? 1 : 0);
397 # use a call instead of just setting the $variable to check argument
398 $class->$key($set_args->{$key});
402 # now return actual configuration
406 lib_version => ${"${CALC}::VERSION"},
408 trap_nan => ${"${class}::_trap_nan"},
409 trap_inf => ${"${class}::_trap_inf"},
410 version => ${"${class}::VERSION"},
413 upgrade downgrade precision accuracy round_mode div_scale
416 $cfg->{$key} = ${"${class}::$key"};
423 # select accuracy parameter based on precedence,
424 # used by bround() and bfround(), may return undef for scale (means no op)
425 my ($x,$scale,$mode) = @_;
427 $scale = $x->{_a} unless defined $scale;
432 $scale = ${ $class . '::accuracy' } unless defined $scale;
433 $mode = ${ $class . '::round_mode' } unless defined $mode;
440 # select precision parameter based on precedence,
441 # used by bround() and bfround(), may return undef for scale (means no op)
442 my ($x,$scale,$mode) = @_;
444 $scale = $x->{_p} unless defined $scale;
449 $scale = ${ $class . '::precision' } unless defined $scale;
450 $mode = ${ $class . '::round_mode' } unless defined $mode;
455 ##############################################################################
463 # if two arguments, the first one is the class to "swallow" subclasses
471 return unless ref($x); # only for objects
473 my $self = bless {}, $c;
475 $self->{sign} = $x->{sign};
476 $self->{value} = $CALC->_copy($x->{value});
477 $self->{_a} = $x->{_a} if defined $x->{_a};
478 $self->{_p} = $x->{_p} if defined $x->{_p};
484 # create a new BigInt object from a string or another BigInt object.
485 # see hash keys documented at top
487 # the argument could be an object, so avoid ||, && etc on it, this would
488 # cause costly overloaded code to be called. The only allowed ops are
491 my ($class,$wanted,$a,$p,$r) = @_;
493 # avoid numify-calls by not using || on $wanted!
494 return $class->bzero($a,$p) if !defined $wanted; # default to 0
495 return $class->copy($wanted,$a,$p,$r)
496 if ref($wanted) && $wanted->isa($class); # MBI or subclass
498 $class->import() if $IMPORT == 0; # make require work
500 my $self = bless {}, $class;
502 # shortcut for "normal" numbers
503 if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*\z/))
505 $self->{sign} = $1 || '+';
507 if ($wanted =~ /^[+-]/)
509 # remove sign without touching wanted to make it work with constants
510 my $t = $wanted; $t =~ s/^[+-]//;
511 $self->{value} = $CALC->_new($t);
515 $self->{value} = $CALC->_new($wanted);
518 if ( (defined $a) || (defined $p)
519 || (defined ${"${class}::precision"})
520 || (defined ${"${class}::accuracy"})
523 $self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p);
528 # handle '+inf', '-inf' first
529 if ($wanted =~ /^[+-]?inf$/)
531 $self->{value} = $CALC->_zero();
532 $self->{sign} = $wanted; $self->{sign} = '+inf' if $self->{sign} eq 'inf';
535 # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign
536 my ($mis,$miv,$mfv,$es,$ev) = _split($wanted);
541 require Carp; Carp::croak("$wanted is not a number in $class");
543 $self->{value} = $CALC->_zero();
544 $self->{sign} = $nan;
549 # _from_hex or _from_bin
550 $self->{value} = $mis->{value};
551 $self->{sign} = $mis->{sign};
552 return $self; # throw away $mis
554 # make integer from mantissa by adjusting exp, then convert to bigint
555 $self->{sign} = $$mis; # store sign
556 $self->{value} = $CALC->_zero(); # for all the NaN cases
557 my $e = int("$$es$$ev"); # exponent (avoid recursion)
560 my $diff = $e - CORE::length($$mfv);
561 if ($diff < 0) # Not integer
565 require Carp; Carp::croak("$wanted not an integer in $class");
568 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
569 $self->{sign} = $nan;
573 # adjust fraction and add it to value
574 #print "diff > 0 $$miv\n";
575 $$miv = $$miv . ($$mfv . '0' x $diff);
580 if ($$mfv ne '') # e <= 0
582 # fraction and negative/zero E => NOI
585 require Carp; Carp::croak("$wanted not an integer in $class");
587 #print "NOI 2 \$\$mfv '$$mfv'\n";
588 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
589 $self->{sign} = $nan;
593 # xE-y, and empty mfv
596 if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
600 require Carp; Carp::croak("$wanted not an integer in $class");
603 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
604 $self->{sign} = $nan;
608 $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
609 $self->{value} = $CALC->_new($$miv) if $self->{sign} =~ /^[+-]$/;
610 # if any of the globals is set, use them to round and store them inside $self
611 # do not round for new($x,undef,undef) since that is used by MBF to signal
613 $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p;
619 # create a bigint 'NaN', if given a BigInt, set it to 'NaN'
621 $self = $class if !defined $self;
624 my $c = $self; $self = {}; bless $self, $c;
627 if (${"${class}::_trap_nan"})
630 Carp::croak ("Tried to set $self to NaN in $class\::bnan()");
632 $self->import() if $IMPORT == 0; # make require work
633 return if $self->modify('bnan');
634 if ($self->can('_bnan'))
636 # use subclass to initialize
641 # otherwise do our own thing
642 $self->{value} = $CALC->_zero();
644 $self->{sign} = $nan;
645 delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly
651 # create a bigint '+-inf', if given a BigInt, set it to '+-inf'
652 # the sign is either '+', or if given, used from there
654 my $sign = shift; $sign = '+' if !defined $sign || $sign !~ /^-(inf)?$/;
655 $self = $class if !defined $self;
658 my $c = $self; $self = {}; bless $self, $c;
661 if (${"${class}::_trap_inf"})
664 Carp::croak ("Tried to set $self to +-inf in $class\::binfn()");
666 $self->import() if $IMPORT == 0; # make require work
667 return if $self->modify('binf');
668 if ($self->can('_binf'))
670 # use subclass to initialize
675 # otherwise do our own thing
676 $self->{value} = $CALC->_zero();
678 $sign = $sign . 'inf' if $sign !~ /inf$/; # - => -inf
679 $self->{sign} = $sign;
680 ($self->{_a},$self->{_p}) = @_; # take over requested rounding
686 # create a bigint '+0', if given a BigInt, set it to 0
688 $self = __PACKAGE__ if !defined $self;
692 my $c = $self; $self = {}; bless $self, $c;
694 $self->import() if $IMPORT == 0; # make require work
695 return if $self->modify('bzero');
697 if ($self->can('_bzero'))
699 # use subclass to initialize
704 # otherwise do our own thing
705 $self->{value} = $CALC->_zero();
712 # call like: $x->bzero($a,$p,$r,$y);
713 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
718 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
720 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
728 # create a bigint '+1' (or -1 if given sign '-'),
729 # if given a BigInt, set it to +1 or -1, respecively
731 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
732 $self = $class if !defined $self;
736 my $c = $self; $self = {}; bless $self, $c;
738 $self->import() if $IMPORT == 0; # make require work
739 return if $self->modify('bone');
741 if ($self->can('_bone'))
743 # use subclass to initialize
748 # otherwise do our own thing
749 $self->{value} = $CALC->_one();
751 $self->{sign} = $sign;
756 # call like: $x->bone($sign,$a,$p,$r,$y);
757 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
761 # call like: $x->bone($sign,$a,$p,$r);
763 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
765 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
771 ##############################################################################
772 # string conversation
776 # (ref to BFLOAT or num_str ) return num_str
777 # Convert number from internal format to scientific string format.
778 # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
779 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
781 if ($x->{sign} !~ /^[+-]$/)
783 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
786 my ($m,$e) = $x->parts();
787 #$m->bstr() . 'e+' . $e->bstr(); # e can only be positive in BigInt
788 # 'e+' because E can only be positive in BigInt
789 $m->bstr() . 'e+' . $CALC->_str($e->{value});
794 # make a string from bigint object
795 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
797 if ($x->{sign} !~ /^[+-]$/)
799 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
802 my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
803 $es.$CALC->_str($x->{value});
808 # Make a "normal" scalar from a BigInt object
809 my $x = shift; $x = $class->new($x) unless ref $x;
811 return $x->bstr() if $x->{sign} !~ /^[+-]$/;
812 my $num = $CALC->_num($x->{value});
813 return -$num if $x->{sign} eq '-';
817 ##############################################################################
818 # public stuff (usually prefixed with "b")
822 # return the sign of the number: +/-/-inf/+inf/NaN
823 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
828 sub _find_round_parameters
830 # After any operation or when calling round(), the result is rounded by
831 # regarding the A & P from arguments, local parameters, or globals.
833 # !!!!!!! If you change this, remember to change round(), too! !!!!!!!!!!
835 # This procedure finds the round parameters, but it is for speed reasons
836 # duplicated in round. Otherwise, it is tested by the testsuite and used
839 # returns ($self) or ($self,$a,$p,$r) - sets $self to NaN of both A and P
840 # were requested/defined (locally or globally or both)
842 my ($self,$a,$p,$r,@args) = @_;
843 # $a accuracy, if given by caller
844 # $p precision, if given by caller
845 # $r round_mode, if given by caller
846 # @args all 'other' arguments (0 for unary, 1 for binary ops)
848 my $c = ref($self); # find out class of argument(s)
851 # now pick $a or $p, but only if we have got "arguments"
854 foreach ($self,@args)
856 # take the defined one, or if both defined, the one that is smaller
857 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
862 # even if $a is defined, take $p, to signal error for both defined
863 foreach ($self,@args)
865 # take the defined one, or if both defined, the one that is bigger
867 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
870 # if still none defined, use globals (#2)
871 $a = ${"$c\::accuracy"} unless defined $a;
872 $p = ${"$c\::precision"} unless defined $p;
874 # A == 0 is useless, so undef it to signal no rounding
875 $a = undef if defined $a && $a == 0;
878 return ($self) unless defined $a || defined $p; # early out
880 # set A and set P is an fatal error
881 return ($self->bnan()) if defined $a && defined $p; # error
883 $r = ${"$c\::round_mode"} unless defined $r;
884 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
886 require Carp; Carp::croak ("Unknown round mode '$r'");
894 # Round $self according to given parameters, or given second argument's
895 # parameters or global defaults
897 # for speed reasons, _find_round_parameters is embeded here:
899 my ($self,$a,$p,$r,@args) = @_;
900 # $a accuracy, if given by caller
901 # $p precision, if given by caller
902 # $r round_mode, if given by caller
903 # @args all 'other' arguments (0 for unary, 1 for binary ops)
905 my $c = ref($self); # find out class of argument(s)
908 # now pick $a or $p, but only if we have got "arguments"
911 foreach ($self,@args)
913 # take the defined one, or if both defined, the one that is smaller
914 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
919 # even if $a is defined, take $p, to signal error for both defined
920 foreach ($self,@args)
922 # take the defined one, or if both defined, the one that is bigger
924 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
927 # if still none defined, use globals (#2)
928 $a = ${"$c\::accuracy"} unless defined $a;
929 $p = ${"$c\::precision"} unless defined $p;
931 # A == 0 is useless, so undef it to signal no rounding
932 $a = undef if defined $a && $a == 0;
935 return $self unless defined $a || defined $p; # early out
937 # set A and set P is an fatal error
938 return $self->bnan() if defined $a && defined $p;
940 $r = ${"$c\::round_mode"} unless defined $r;
941 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
943 require Carp; Carp::croak ("Unknown round mode '$r'");
946 # now round, by calling either fround or ffround:
949 $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a;
951 else # both can't be undefined due to early out
953 $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p;
955 # bround() or bfround() already callled bnorm() if necc.
961 # (numstr or BINT) return BINT
962 # Normalize number -- no-op here
963 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
969 # (BINT or num_str) return BINT
970 # make number absolute, or return absolute BINT from string
971 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
973 return $x if $x->modify('babs');
974 # post-normalized abs for internal use (does nothing for NaN)
975 $x->{sign} =~ s/^-/+/;
981 # (BINT or num_str) return BINT
982 # negate number or make a negated number from string
983 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
985 return $x if $x->modify('bneg');
987 # for +0 dont negate (to have always normalized +0). Does nothing for 'NaN'
988 $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $CALC->_is_zero($x->{value}));
994 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
995 # (BINT or num_str, BINT or num_str) return cond_code
998 my ($self,$x,$y) = (ref($_[0]),@_);
1000 # objectify is costly, so avoid it
1001 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1003 ($self,$x,$y) = objectify(2,@_);
1006 return $upgrade->bcmp($x,$y) if defined $upgrade &&
1007 ((!$x->isa($self)) || (!$y->isa($self)));
1009 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1011 # handle +-inf and NaN
1012 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1013 return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
1014 return +1 if $x->{sign} eq '+inf';
1015 return -1 if $x->{sign} eq '-inf';
1016 return -1 if $y->{sign} eq '+inf';
1019 # check sign for speed first
1020 return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
1021 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
1023 # have same sign, so compare absolute values. Don't make tests for zero here
1024 # because it's actually slower than testin in Calc (especially w/ Pari et al)
1026 # post-normalized compare for internal use (honors signs)
1027 if ($x->{sign} eq '+')
1029 # $x and $y both > 0
1030 return $CALC->_acmp($x->{value},$y->{value});
1034 $CALC->_acmp($y->{value},$x->{value}); # swaped acmp (lib returns 0,1,-1)
1039 # Compares 2 values, ignoring their signs.
1040 # Returns one of undef, <0, =0, >0. (suitable for sort)
1041 # (BINT, BINT) return cond_code
1044 my ($self,$x,$y) = (ref($_[0]),@_);
1045 # objectify is costly, so avoid it
1046 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1048 ($self,$x,$y) = objectify(2,@_);
1051 return $upgrade->bacmp($x,$y) if defined $upgrade &&
1052 ((!$x->isa($self)) || (!$y->isa($self)));
1054 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1056 # handle +-inf and NaN
1057 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1058 return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
1059 return 1 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} !~ /^[+-]inf$/;
1062 $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
1067 # add second arg (BINT or string) to first (BINT) (modifies first)
1068 # return result as BINT
1071 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1072 # objectify is costly, so avoid it
1073 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1075 ($self,$x,$y,@r) = objectify(2,@_);
1078 return $x if $x->modify('badd');
1079 return $upgrade->badd($upgrade->new($x),$upgrade->new($y),@r) if defined $upgrade &&
1080 ((!$x->isa($self)) || (!$y->isa($self)));
1082 $r[3] = $y; # no push!
1083 # inf and NaN handling
1084 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1087 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1089 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1091 # +inf++inf or -inf+-inf => same, rest is NaN
1092 return $x if $x->{sign} eq $y->{sign};
1095 # +-inf + something => +inf
1096 # something +-inf => +-inf
1097 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
1101 my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
1105 $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
1109 my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
1112 $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
1117 # speedup, if equal, set result to 0
1118 $x->{value} = $CALC->_zero();
1123 $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
1131 # (BINT or num_str, BINT or num_str) return BINT
1132 # subtract second arg from first, modify first
1135 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1136 # objectify is costly, so avoid it
1137 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1139 ($self,$x,$y,@r) = objectify(2,@_);
1142 return $x if $x->modify('bsub');
1144 return $upgrade->new($x)->bsub($upgrade->new($y),@r) if defined $upgrade &&
1145 ((!$x->isa($self)) || (!$y->isa($self)));
1147 return $x->round(@r) if $y->is_zero();
1149 require Scalar::Util;
1150 if (Scalar::Util::refaddr($x) == Scalar::Util::refaddr($y))
1152 # if we get the same variable twice, the result must be zero (the code
1153 # below fails in that case)
1154 return $x->bzero(@r) if $x->{sign} =~ /^[+-]$/;
1155 return $x->bnan(); # NaN, -inf, +inf
1157 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
1158 $x->badd($y,@r); # badd does not leave internal zeros
1159 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
1160 $x; # already rounded by badd() or no round necc.
1165 # increment arg by one
1166 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1167 return $x if $x->modify('binc');
1169 if ($x->{sign} eq '+')
1171 $x->{value} = $CALC->_inc($x->{value});
1172 return $x->round($a,$p,$r);
1174 elsif ($x->{sign} eq '-')
1176 $x->{value} = $CALC->_dec($x->{value});
1177 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1178 return $x->round($a,$p,$r);
1180 # inf, nan handling etc
1181 $x->badd($self->bone(),$a,$p,$r); # badd does round
1186 # decrement arg by one
1187 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1188 return $x if $x->modify('bdec');
1190 if ($x->{sign} eq '-')
1193 $x->{value} = $CALC->_inc($x->{value});
1197 return $x->badd($self->bone('-'),@r) unless $x->{sign} eq '+'; # inf or NaN
1199 if ($CALC->_is_zero($x->{value}))
1202 $x->{value} = $CALC->_one(); $x->{sign} = '-'; # 0 => -1
1207 $x->{value} = $CALC->_dec($x->{value});
1215 # calculate $x = $a ** $base + $b and return $a (e.g. the log() to base
1219 my ($self,$x,$base,@r) = (undef,@_);
1220 # objectify is costly, so avoid it
1221 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1223 ($self,$x,$base,@r) = objectify(1,ref($x),@_);
1226 return $x if $x->modify('blog');
1228 # inf, -inf, NaN, <0 => NaN
1230 if $x->{sign} ne '+' || (defined $base && $base->{sign} ne '+');
1232 return $upgrade->blog($upgrade->new($x),$base,@r) if
1235 my ($rc,$exact) = $CALC->_log_int($x->{value},$base->{value});
1236 return $x->bnan() unless defined $rc; # not possible to take log?
1243 # (BINT or num_str, BINT or num_str) return BINT
1244 # does not modify arguments, but returns new object
1245 # Lowest Common Multiplicator
1247 my $y = shift; my ($x);
1254 $x = $class->new($y);
1259 my $y = shift; $y = $self->new($y) if !ref ($y);
1267 # (BINT or num_str, BINT or num_str) return BINT
1268 # does not modify arguments, but returns new object
1269 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1272 $y = $class->new($y) if !ref($y);
1274 my $x = $y->copy()->babs(); # keep arguments
1275 return $x->bnan() if $x->{sign} !~ /^[+-]$/; # x NaN?
1279 $y = shift; $y = $self->new($y) if !ref($y);
1280 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1281 $x->{value} = $CALC->_gcd($x->{value},$y->{value});
1282 last if $CALC->_is_one($x->{value});
1289 # (num_str or BINT) return BINT
1290 # represent ~x as twos-complement number
1291 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1292 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1294 return $x if $x->modify('bnot');
1295 $x->binc()->bneg(); # binc already does round
1298 ##############################################################################
1299 # is_foo test routines
1300 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1304 # return true if arg (BINT or num_str) is zero (array '+', '0')
1305 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1307 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1308 $CALC->_is_zero($x->{value});
1313 # return true if arg (BINT or num_str) is NaN
1314 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1316 $x->{sign} eq $nan ? 1 : 0;
1321 # return true if arg (BINT or num_str) is +-inf
1322 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1326 $sign = '[+-]inf' if $sign eq ''; # +- doesn't matter, only that's inf
1327 $sign = "[$1]inf" if $sign =~ /^([+-])(inf)?$/; # extract '+' or '-'
1328 return $x->{sign} =~ /^$sign$/ ? 1 : 0;
1330 $x->{sign} =~ /^[+-]inf$/ ? 1 : 0; # only +-inf is infinity
1335 # return true if arg (BINT or num_str) is +1, or -1 if sign is given
1336 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1338 $sign = '+' if !defined $sign || $sign ne '-';
1340 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1341 $CALC->_is_one($x->{value});
1346 # return true when arg (BINT or num_str) is odd, false for even
1347 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1349 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1350 $CALC->_is_odd($x->{value});
1355 # return true when arg (BINT or num_str) is even, false for odd
1356 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1358 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1359 $CALC->_is_even($x->{value});
1364 # return true when arg (BINT or num_str) is positive (>= 0)
1365 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1367 return 1 if $x->{sign} eq '+inf'; # +inf is positive
1369 # 0+ is neither positive nor negative
1370 ($x->{sign} eq '+' && !$x->is_zero()) ? 1 : 0;
1375 # return true when arg (BINT or num_str) is negative (< 0)
1376 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1378 $x->{sign} =~ /^-/ ? 1 : 0; # -inf is negative, but NaN is not
1383 # return true when arg (BINT or num_str) is an integer
1384 # always true for BigInt, but different for BigFloats
1385 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1387 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1390 ###############################################################################
1394 # multiply two numbers -- stolen from Knuth Vol 2 pg 233
1395 # (BINT or num_str, BINT or num_str) return BINT
1398 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1399 # objectify is costly, so avoid it
1400 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1402 ($self,$x,$y,@r) = objectify(2,@_);
1405 return $x if $x->modify('bmul');
1407 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1410 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1412 return $x->bnan() if $x->is_zero() || $y->is_zero();
1413 # result will always be +-inf:
1414 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1415 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1416 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1417 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1418 return $x->binf('-');
1421 return $upgrade->bmul($x,$upgrade->new($y),@r)
1422 if defined $upgrade && !$y->isa($self);
1424 $r[3] = $y; # no push here
1426 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1428 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1429 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1436 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1437 my ($self,$x,$y) = @_;
1439 # NaN if x == NaN or y == NaN or x==y==0
1440 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1441 if (($x->is_nan() || $y->is_nan()) ||
1442 ($x->is_zero() && $y->is_zero()));
1444 # +-inf / +-inf == NaN, reminder also NaN
1445 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1447 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1449 # x / +-inf => 0, remainder x (works even if x == 0)
1450 if ($y->{sign} =~ /^[+-]inf$/)
1452 my $t = $x->copy(); # bzero clobbers up $x
1453 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1456 # 5 / 0 => +inf, -6 / 0 => -inf
1457 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1458 # exception: -8 / 0 has remainder -8, not 8
1459 # exception: -inf / 0 has remainder -inf, not inf
1462 # +-inf / 0 => special case for -inf
1463 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1464 if (!$x->is_zero() && !$x->is_inf())
1466 my $t = $x->copy(); # binf clobbers up $x
1468 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1472 # last case: +-inf / ordinary number
1474 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1476 return wantarray ? ($x,$self->bzero()) : $x;
1481 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1482 # (BINT,BINT) (quo,rem) or BINT (only rem)
1485 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1486 # objectify is costly, so avoid it
1487 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1489 ($self,$x,$y,@r) = objectify(2,@_);
1492 return $x if $x->modify('bdiv');
1494 return $self->_div_inf($x,$y)
1495 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1497 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1498 if defined $upgrade;
1500 $r[3] = $y; # no push!
1502 # calc new sign and in case $y == +/- 1, return $x
1503 my $xsign = $x->{sign}; # keep
1504 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1508 my $rem = $self->bzero();
1509 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1510 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1511 $rem->{_a} = $x->{_a};
1512 $rem->{_p} = $x->{_p};
1514 if (! $CALC->_is_zero($rem->{value}))
1516 $rem->{sign} = $y->{sign};
1517 $rem = $y->copy()->bsub($rem) if $xsign ne $y->{sign}; # one of them '-'
1521 $rem->{sign} = '+'; # dont leave -0
1527 $x->{value} = $CALC->_div($x->{value},$y->{value});
1528 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1533 ###############################################################################
1538 # modulus (or remainder)
1539 # (BINT or num_str, BINT or num_str) return BINT
1542 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1543 # objectify is costly, so avoid it
1544 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1546 ($self,$x,$y,@r) = objectify(2,@_);
1549 return $x if $x->modify('bmod');
1550 $r[3] = $y; # no push!
1551 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1553 my ($d,$r) = $self->_div_inf($x,$y);
1554 $x->{sign} = $r->{sign};
1555 $x->{value} = $r->{value};
1556 return $x->round(@r);
1559 # calc new sign and in case $y == +/- 1, return $x
1560 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1561 if (!$CALC->_is_zero($x->{value}))
1563 $x->{value} = $CALC->_sub($y->{value},$x->{value},1) # $y-$x
1564 if ($x->{sign} ne $y->{sign});
1565 $x->{sign} = $y->{sign};
1569 $x->{sign} = '+'; # dont leave -0
1576 # Modular inverse. given a number which is (hopefully) relatively
1577 # prime to the modulus, calculate its inverse using Euclid's
1578 # alogrithm. If the number is not relatively prime to the modulus
1579 # (i.e. their gcd is not one) then NaN is returned.
1582 my ($self,$x,$y,@r) = (undef,@_);
1583 # objectify is costly, so avoid it
1584 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1586 ($self,$x,$y,@r) = objectify(2,@_);
1589 return $x if $x->modify('bmodinv');
1592 if ($y->{sign} ne '+' # -, NaN, +inf, -inf
1593 || $x->is_zero() # or num == 0
1594 || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1597 # put least residue into $x if $x was negative, and thus make it positive
1598 $x->bmod($y) if $x->{sign} eq '-';
1601 ($x->{value},$sign) = $CALC->_modinv($x->{value},$y->{value});
1602 return $x->bnan() if !defined $x->{value}; # in case no GCD found
1603 return $x if !defined $sign; # already real result
1604 $x->{sign} = $sign; # flip/flop see below
1605 $x->bmod($y); # calc real result
1611 # takes a very large number to a very large exponent in a given very
1612 # large modulus, quickly, thanks to binary exponentation. supports
1613 # negative exponents.
1614 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1616 return $num if $num->modify('bmodpow');
1618 # check modulus for valid values
1619 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1620 || $mod->is_zero());
1622 # check exponent for valid values
1623 if ($exp->{sign} =~ /\w/)
1625 # i.e., if it's NaN, +inf, or -inf...
1626 return $num->bnan();
1629 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
1631 # check num for valid values (also NaN if there was no inverse but $exp < 0)
1632 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1634 # $mod is positive, sign on $exp is ignored, result also positive
1635 $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
1639 ###############################################################################
1643 # (BINT or num_str, BINT or num_str) return BINT
1644 # compute factorial number from $x, modify $x in place
1645 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1647 return $x if $x->modify('bfac') || $x->{sign} eq '+inf'; # inf => inf
1648 return $x->bnan() if $x->{sign} ne '+'; # NaN, <0 etc => NaN
1650 $x->{value} = $CALC->_fac($x->{value});
1656 # (BINT or num_str, BINT or num_str) return BINT
1657 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1658 # modifies first argument
1661 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1662 # objectify is costly, so avoid it
1663 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1665 ($self,$x,$y,@r) = objectify(2,@_);
1668 return $x if $x->modify('bpow');
1670 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1673 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1675 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1681 if ($x->{sign} =~ /^[+-]inf/)
1684 return $x->bnan() if $y->is_zero();
1685 # -inf ** -1 => 1/inf => 0
1686 return $x->bzero() if $y->is_one('-') && $x->is_negative();
1689 return $x if $x->{sign} eq '+inf';
1691 # -inf ** Y => -inf if Y is odd
1692 return $x if $y->is_odd();
1698 return $x if $x->is_one();
1701 return $x if $x->is_zero() && $y->{sign} =~ /^[+]/;
1704 return $x->binf() if $x->is_zero();
1707 return $x->bnan() if $x->is_one('-') && $y->{sign} =~ /^[-]/;
1710 return $x->bzero() if $x->{sign} eq '-' && $y->{sign} =~ /^[-]/;
1713 return $x->bnan() if $x->{sign} eq '-';
1716 return $x->binf() if $y->{sign} =~ /^[+]/;
1721 return $upgrade->bpow($upgrade->new($x),$y,@r)
1722 if defined $upgrade && !$y->isa($self);
1724 $r[3] = $y; # no push!
1726 # cases 0 ** Y, X ** 0, X ** 1, 1 ** Y are handled by Calc or Emu
1729 $new_sign = $y->is_odd() ? '-' : '+' if ($x->{sign} ne '+');
1731 # 0 ** -7 => ( 1 / (0 ** 7)) => 1 / 0 => +inf
1733 if $y->{sign} eq '-' && $x->{sign} eq '+' && $CALC->_is_zero($x->{value});
1734 # 1 ** -y => 1 / (1 ** |y|)
1735 # so do test for negative $y after above's clause
1736 return $x->bnan() if $y->{sign} eq '-' && !$CALC->_is_one($x->{value});
1738 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1739 $x->{sign} = $new_sign;
1740 $x->{sign} = '+' if $CALC->_is_zero($y->{value});
1746 # (BINT or num_str, BINT or num_str) return BINT
1747 # compute x << y, base n, y >= 0
1750 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1751 # objectify is costly, so avoid it
1752 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1754 ($self,$x,$y,$n,@r) = objectify(2,@_);
1757 return $x if $x->modify('blsft');
1758 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1759 return $x->round(@r) if $y->is_zero();
1761 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1763 $x->{value} = $CALC->_lsft($x->{value},$y->{value},$n);
1769 # (BINT or num_str, BINT or num_str) return BINT
1770 # compute x >> y, base n, y >= 0
1773 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1774 # objectify is costly, so avoid it
1775 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1777 ($self,$x,$y,$n,@r) = objectify(2,@_);
1780 return $x if $x->modify('brsft');
1781 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1782 return $x->round(@r) if $y->is_zero();
1783 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
1785 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1787 # this only works for negative numbers when shifting in base 2
1788 if (($x->{sign} eq '-') && ($n == 2))
1790 return $x->round(@r) if $x->is_one('-'); # -1 => -1
1793 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1794 # but perhaps there is a better emulation for two's complement shift...
1795 # if $y != 1, we must simulate it by doing:
1796 # convert to bin, flip all bits, shift, and be done
1797 $x->binc(); # -3 => -2
1798 my $bin = $x->as_bin();
1799 $bin =~ s/^-0b//; # strip '-0b' prefix
1800 $bin =~ tr/10/01/; # flip bits
1802 if (CORE::length($bin) <= $y)
1804 $bin = '0'; # shifting to far right creates -1
1805 # 0, because later increment makes
1806 # that 1, attached '-' makes it '-1'
1807 # because -1 >> x == -1 !
1811 $bin =~ s/.{$y}$//; # cut off at the right side
1812 $bin = '1' . $bin; # extend left side by one dummy '1'
1813 $bin =~ tr/10/01/; # flip bits back
1815 my $res = $self->new('0b'.$bin); # add prefix and convert back
1816 $res->binc(); # remember to increment
1817 $x->{value} = $res->{value}; # take over value
1818 return $x->round(@r); # we are done now, magic, isn't?
1820 # x < 0, n == 2, y == 1
1821 $x->bdec(); # n == 2, but $y == 1: this fixes it
1824 $x->{value} = $CALC->_rsft($x->{value},$y->{value},$n);
1830 #(BINT or num_str, BINT or num_str) return BINT
1834 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1835 # objectify is costly, so avoid it
1836 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1838 ($self,$x,$y,@r) = objectify(2,@_);
1841 return $x if $x->modify('band');
1843 $r[3] = $y; # no push!
1845 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1847 my $sx = $x->{sign} eq '+' ? 1 : -1;
1848 my $sy = $y->{sign} eq '+' ? 1 : -1;
1850 if ($sx == 1 && $sy == 1)
1852 $x->{value} = $CALC->_and($x->{value},$y->{value});
1853 return $x->round(@r);
1856 if ($CAN{signed_and})
1858 $x->{value} = $CALC->_signed_and($x->{value},$y->{value},$sx,$sy);
1859 return $x->round(@r);
1863 __emu_band($self,$x,$y,$sx,$sy,@r);
1868 #(BINT or num_str, BINT or num_str) return BINT
1872 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1873 # objectify is costly, so avoid it
1874 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1876 ($self,$x,$y,@r) = objectify(2,@_);
1879 return $x if $x->modify('bior');
1880 $r[3] = $y; # no push!
1882 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1884 my $sx = $x->{sign} eq '+' ? 1 : -1;
1885 my $sy = $y->{sign} eq '+' ? 1 : -1;
1887 # the sign of X follows the sign of X, e.g. sign of Y irrelevant for bior()
1889 # don't use lib for negative values
1890 if ($sx == 1 && $sy == 1)
1892 $x->{value} = $CALC->_or($x->{value},$y->{value});
1893 return $x->round(@r);
1896 # if lib can do negative values, let it handle this
1897 if ($CAN{signed_or})
1899 $x->{value} = $CALC->_signed_or($x->{value},$y->{value},$sx,$sy);
1900 return $x->round(@r);
1904 __emu_bior($self,$x,$y,$sx,$sy,@r);
1909 #(BINT or num_str, BINT or num_str) return BINT
1913 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1914 # objectify is costly, so avoid it
1915 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1917 ($self,$x,$y,@r) = objectify(2,@_);
1920 return $x if $x->modify('bxor');
1921 $r[3] = $y; # no push!
1923 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1925 my $sx = $x->{sign} eq '+' ? 1 : -1;
1926 my $sy = $y->{sign} eq '+' ? 1 : -1;
1928 # don't use lib for negative values
1929 if ($sx == 1 && $sy == 1)
1931 $x->{value} = $CALC->_xor($x->{value},$y->{value});
1932 return $x->round(@r);
1935 # if lib can do negative values, let it handle this
1936 if ($CAN{signed_xor})
1938 $x->{value} = $CALC->_signed_xor($x->{value},$y->{value},$sx,$sy);
1939 return $x->round(@r);
1943 __emu_bxor($self,$x,$y,$sx,$sy,@r);
1948 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1950 my $e = $CALC->_len($x->{value});
1951 wantarray ? ($e,0) : $e;
1956 # return the nth decimal digit, negative values count backward, 0 is right
1957 my ($self,$x,$n) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1959 $n = $n->numify() if ref($n);
1960 $CALC->_digit($x->{value},$n||0);
1965 # return the amount of trailing zeros in $x (as scalar)
1967 $x = $class->new($x) unless ref $x;
1969 return 0 if $x->{sign} !~ /^[+-]$/; # NaN, inf, -inf etc
1971 $CALC->_zeros($x->{value}); # must handle odd values, 0 etc
1976 # calculate square root of $x
1977 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1979 return $x if $x->modify('bsqrt');
1981 return $x->bnan() if $x->{sign} !~ /^\+/; # -x or -inf or NaN => NaN
1982 return $x if $x->{sign} eq '+inf'; # sqrt(+inf) == inf
1984 return $upgrade->bsqrt($x,@r) if defined $upgrade;
1986 $x->{value} = $CALC->_sqrt($x->{value});
1992 # calculate $y'th root of $x
1995 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1997 $y = $self->new(2) unless defined $y;
1999 # objectify is costly, so avoid it
2000 if ((!ref($x)) || (ref($x) ne ref($y)))
2002 ($self,$x,$y,@r) = objectify(2,$self || $class,@_);
2005 return $x if $x->modify('broot');
2007 # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
2008 return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
2009 $y->{sign} !~ /^\+$/;
2011 return $x->round(@r)
2012 if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
2014 return $upgrade->new($x)->broot($upgrade->new($y),@r) if defined $upgrade;
2016 $x->{value} = $CALC->_root($x->{value},$y->{value});
2022 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
2023 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2025 if ($x->{sign} !~ /^[+-]$/)
2027 my $s = $x->{sign}; $s =~ s/^[+-]//; # NaN, -inf,+inf => NaN or inf
2028 return $self->new($s);
2030 return $self->bone() if $x->is_zero();
2032 $self->new($x->_trailing_zeros());
2037 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
2038 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2040 if ($x->{sign} !~ /^[+-]$/)
2042 # for NaN, +inf, -inf: keep the sign
2043 return $self->new($x->{sign});
2045 my $m = $x->copy(); delete $m->{_p}; delete $m->{_a};
2046 # that's a bit inefficient:
2047 my $zeros = $m->_trailing_zeros();
2048 $m->brsft($zeros,10) if $zeros != 0;
2054 # return a copy of both the exponent and the mantissa
2055 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2057 ($x->mantissa(),$x->exponent());
2060 ##############################################################################
2061 # rounding functions
2065 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
2066 # $n == 0 || $n == 1 => round to integer
2067 my $x = shift; my $self = ref($x) || $x; $x = $self->new($x) unless ref $x;
2069 my ($scale,$mode) = $x->_scale_p(@_);
2071 return $x if !defined $scale || $x->modify('bfround'); # no-op
2073 # no-op for BigInts if $n <= 0
2074 $x->bround( $x->length()-$scale, $mode) if $scale > 0;
2076 delete $x->{_a}; # delete to save memory
2077 $x->{_p} = $scale; # store new _p
2081 sub _scan_for_nonzero
2083 # internal, used by bround() to scan for non-zeros after a '5'
2084 my ($x,$pad,$xs,$len) = @_;
2086 return 0 if $len == 1; # "5" is trailed by invisible zeros
2087 my $follow = $pad - 1;
2088 return 0 if $follow > $len || $follow < 1;
2090 # use the string form to check whether only '0's follow or not
2091 substr ($xs,-$follow) =~ /[^0]/ ? 1 : 0;
2096 # Exists to make life easier for switch between MBF and MBI (should we
2097 # autoload fxxx() like MBF does for bxxx()?)
2098 my $x = shift; $x = $class->new($x) unless ref $x;
2104 # accuracy: +$n preserve $n digits from left,
2105 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
2107 # and overwrite the rest with 0's, return normalized number
2108 # do not return $x->bnorm(), but $x
2110 my $x = shift; $x = $class->new($x) unless ref $x;
2111 my ($scale,$mode) = $x->_scale_a(@_);
2112 return $x if !defined $scale || $x->modify('bround'); # no-op
2114 if ($x->is_zero() || $scale == 0)
2116 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2119 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2121 # we have fewer digits than we want to scale to
2122 my $len = $x->length();
2123 # convert $scale to a scalar in case it is an object (put's a limit on the
2124 # number length, but this would already limited by memory constraints), makes
2126 $scale = $scale->numify() if ref ($scale);
2128 # scale < 0, but > -len (not >=!)
2129 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2131 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2135 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2136 my ($pad,$digit_round,$digit_after);
2137 $pad = $len - $scale;
2138 $pad = abs($scale-1) if $scale < 0;
2140 # do not use digit(), it is very costly for binary => decimal
2141 # getting the entire string is also costly, but we need to do it only once
2142 my $xs = $CALC->_str($x->{value});
2145 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2146 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2147 $digit_round = '0'; $digit_round = substr($xs,$pl,1) if $pad <= $len;
2148 $pl++; $pl ++ if $pad >= $len;
2149 $digit_after = '0'; $digit_after = substr($xs,$pl,1) if $pad > 0;
2151 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2152 # closer at the remaining digits of the original $x, remember decision
2153 my $round_up = 1; # default round up
2155 ($mode eq 'trunc') || # trunc by round down
2156 ($digit_after =~ /[01234]/) || # round down anyway,
2158 ($digit_after eq '5') && # not 5000...0000
2159 ($x->_scan_for_nonzero($pad,$xs,$len) == 0) &&
2161 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2162 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2163 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2164 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2165 ($mode eq 'zero') # round down if zero, sign adjusted below
2167 my $put_back = 0; # not yet modified
2169 if (($pad > 0) && ($pad <= $len))
2171 substr($xs,-$pad,$pad) = '0' x $pad; # replace with '00...'
2172 $put_back = 1; # need to put back
2176 $x->bzero(); # round to '0'
2179 if ($round_up) # what gave test above?
2181 $put_back = 1; # need to put back
2182 $pad = $len, $xs = '0' x $pad if $scale < 0; # tlr: whack 0.51=>1.0
2184 # we modify directly the string variant instead of creating a number and
2185 # adding it, since that is faster (we already have the string)
2186 my $c = 0; $pad ++; # for $pad == $len case
2187 while ($pad <= $len)
2189 $c = substr($xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2190 substr($xs,-$pad,1) = $c; $pad++;
2191 last if $c != 0; # no overflow => early out
2193 $xs = '1'.$xs if $c == 0;
2196 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back, if needed
2198 $x->{_a} = $scale if $scale >= 0;
2201 $x->{_a} = $len+$scale;
2202 $x->{_a} = 0 if $scale < -$len;
2209 # return integer less or equal then number; no-op since it's already integer
2210 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2217 # return integer greater or equal then number; no-op since it's already int
2218 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2225 # An object might be asked to return itself as bigint on certain overloaded
2226 # operations, this does exactly this, so that sub classes can simple inherit
2227 # it or override with their own integer conversion routine.
2233 # return as hex string, with prefixed 0x
2234 my $x = shift; $x = $class->new($x) if !ref($x);
2236 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2239 $s = $x->{sign} if $x->{sign} eq '-';
2240 $s . $CALC->_as_hex($x->{value});
2245 # return as binary string, with prefixed 0b
2246 my $x = shift; $x = $class->new($x) if !ref($x);
2248 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2250 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2251 return $s . $CALC->_as_bin($x->{value});
2254 ##############################################################################
2255 # private stuff (internal use only)
2259 # check for strings, if yes, return objects instead
2261 # the first argument is number of args objectify() should look at it will
2262 # return $count+1 elements, the first will be a classname. This is because
2263 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2264 # useless objects beeing created and thrown away. So we cannot simple loop
2265 # over @_. If the given count is 0, all arguments will be used.
2267 # If the second arg is a ref, use it as class.
2268 # If not, try to use it as classname, unless undef, then use $class
2269 # (aka Math::BigInt). The latter shouldn't happen,though.
2272 # $x->badd(1); => ref x, scalar y
2273 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2274 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2275 # Math::BigInt::badd(1,2); => scalar x, scalar y
2276 # In the last case we check number of arguments to turn it silently into
2277 # $class,1,2. (We can not take '1' as class ;o)
2278 # badd($class,1) is not supported (it should, eventually, try to add undef)
2279 # currently it tries 'Math::BigInt' + 1, which will not work.
2281 # some shortcut for the common cases
2283 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2285 my $count = abs(shift || 0);
2287 my (@a,$k,$d); # resulting array, temp, and downgrade
2290 # okay, got object as first
2295 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2297 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2301 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2302 if (defined ${"$a[0]::downgrade"})
2304 $d = ${"$a[0]::downgrade"};
2305 ${"$a[0]::downgrade"} = undef;
2308 my $up = ${"$a[0]::upgrade"};
2309 #print "Now in objectify, my class is today $a[0], count = $count\n";
2317 $k = $a[0]->new($k);
2319 elsif (!defined $up && ref($k) ne $a[0])
2321 # foreign object, try to convert to integer
2322 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2335 $k = $a[0]->new($k);
2337 elsif (!defined $up && ref($k) ne $a[0])
2339 # foreign object, try to convert to integer
2340 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2344 push @a,@_; # return other params, too
2348 require Carp; Carp::croak ("$class objectify needs list context");
2350 ${"$a[0]::downgrade"} = $d;
2354 sub _register_callback
2356 my ($class,$callback) = @_;
2358 if (ref($callback) ne 'CODE')
2361 Carp::croak ("$callback is not a coderef");
2363 $CALLBACKS{$class} = $callback;
2370 $IMPORT++; # remember we did import()
2371 my @a; my $l = scalar @_;
2372 for ( my $i = 0; $i < $l ; $i++ )
2374 if ($_[$i] eq ':constant')
2376 # this causes overlord er load to step in
2378 integer => sub { $self->new(shift) },
2379 binary => sub { $self->new(shift) };
2381 elsif ($_[$i] eq 'upgrade')
2383 # this causes upgrading
2384 $upgrade = $_[$i+1]; # or undef to disable
2387 elsif ($_[$i] =~ /^lib$/i)
2389 # this causes a different low lib to take care...
2390 $CALC = $_[$i+1] || '';
2398 # any non :constant stuff is handled by our parent, Exporter
2403 $self->SUPER::import(@a); # need it for subclasses
2404 $self->export_to_level(1,$self,@a); # need it for MBF
2407 # try to load core math lib
2408 my @c = split /\s*,\s*/,$CALC;
2411 $_ =~ tr/a-zA-Z0-9://cd; # limit to sane characters
2413 push @c,'Calc'; # if all fail, try this
2414 $CALC = ''; # signal error
2415 foreach my $lib (@c)
2417 next if ($lib || '') eq '';
2418 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2422 # Perl < 5.6.0 dies with "out of memory!" when eval("") and ':constant' is
2423 # used in the same script, or eval("") inside import().
2424 my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
2425 my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
2427 $file = File::Spec->catfile (@parts, $file);
2428 eval { require "$file"; $lib->import( @c ); }
2432 eval "use $lib qw/@c/;";
2437 # loaded it ok, see if the api_version() is high enough
2438 if ($lib->can('api_version') && $lib->api_version() >= 1.0)
2441 # api_version matches, check if it really provides anything we need
2445 add mul div sub dec inc
2446 acmp len digit is_one is_zero is_even is_odd
2448 new copy check from_hex from_bin as_hex as_bin zeros
2449 rsft lsft xor and or
2450 mod sqrt root fac pow modinv modpow log_int gcd
2453 if (!$lib->can("_$method"))
2455 if (($WARN{$lib}||0) < 2)
2458 Carp::carp ("$lib is missing method '_$method'");
2459 $WARN{$lib} = 1; # still warn about the lib
2468 last; # found a usable one, break
2472 if (($WARN{$lib}||0) < 2)
2474 my $ver = eval "\$$lib\::VERSION";
2476 Carp::carp ("Cannot load outdated $lib v$ver, please upgrade");
2477 $WARN{$lib} = 2; # never warn again
2485 Carp::croak ("Couldn't load any math lib, not even 'Calc.pm'");
2489 foreach my $class (keys %CALLBACKS)
2491 &{$CALLBACKS{$class}}($CALC);
2494 # Fill $CAN with the results of $CALC->can(...) for emulating lower math lib
2498 for my $method (qw/ signed_and signed_or signed_xor /)
2500 $CAN{$method} = $CALC->can("_$method") ? 1 : 0;
2509 # convert a (ref to) big hex string to BigInt, return undef for error
2512 my $x = Math::BigInt->bzero();
2515 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2516 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2518 return $x->bnan() if $hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2520 my $sign = '+'; $sign = '-' if $hs =~ /^-/;
2522 $hs =~ s/^[+-]//; # strip sign
2523 $x->{value} = $CALC->_from_hex($hs);
2524 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2531 # convert a (ref to) big binary string to BigInt, return undef for error
2534 my $x = Math::BigInt->bzero();
2536 $bs =~ s/([01])_([01])/$1$2/g;
2537 $bs =~ s/([01])_([01])/$1$2/g;
2538 return $x->bnan() if $bs !~ /^[+-]?0b[01]+$/;
2540 my $sign = '+'; $sign = '-' if $bs =~ /^\-/;
2541 $bs =~ s/^[+-]//; # strip sign
2543 $x->{value} = $CALC->_from_bin($bs);
2544 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2550 # input: num_str; output: undef for invalid or
2551 # (\$mantissa_sign,\$mantissa_value,\$mantissa_fraction,\$exp_sign,\$exp_value)
2552 # Internal, take apart a string and return the pieces.
2553 # Strip leading/trailing whitespace, leading zeros, underscore and reject
2557 # strip white space at front, also extranous leading zeros
2558 $x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2559 $x =~ s/^\s+//; # but this will
2560 $x =~ s/\s+$//g; # strip white space at end
2562 # shortcut, if nothing to split, return early
2563 if ($x =~ /^[+-]?\d+\z/)
2565 $x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2566 return (\$sign, \$x, \'', \'', \0);
2569 # invalid starting char?
2570 return if $x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2572 return __from_hex($x) if $x =~ /^[\-\+]?0x/; # hex string
2573 return __from_bin($x) if $x =~ /^[\-\+]?0b/; # binary string
2575 # strip underscores between digits
2576 $x =~ s/(\d)_(\d)/$1$2/g;
2577 $x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3
2579 # some possible inputs:
2580 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2581 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 # 0e999
2583 my ($m,$e,$last) = split /[Ee]/,$x;
2584 return if defined $last; # last defined => 1e2E3 or others
2585 $e = '0' if !defined $e || $e eq "";
2587 # sign,value for exponent,mantint,mantfrac
2588 my ($es,$ev,$mis,$miv,$mfv);
2590 if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2594 return if $m eq '.' || $m eq '';
2595 my ($mi,$mf,$lastf) = split /\./,$m;
2596 return if defined $lastf; # lastf defined => 1.2.3 or others
2597 $mi = '0' if !defined $mi;
2598 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2599 $mf = '0' if !defined $mf || $mf eq '';
2600 if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2602 $mis = $1||'+'; $miv = $2;
2603 return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
2605 # handle the 0e999 case here
2606 $ev = 0 if $miv eq '0' && $mfv eq '';
2607 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2610 return; # NaN, not a number
2613 ##############################################################################
2614 # internal calculation routines (others are in Math::BigInt::Calc etc)
2618 # (BINT or num_str, BINT or num_str) return BINT
2619 # does modify first argument
2623 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2624 my $method = ref($x) . '::bgcd';
2626 $x * $ty / &$method($x,$ty);
2629 ###############################################################################
2630 # this method returns 0 if the object can be modified, or 1 if not.
2631 # We use a fast constant sub() here, to avoid costly calls. Subclasses
2632 # may override it with special code (f.i. Math::BigInt::Constant does so)
2634 sub modify () { 0; }
2641 Math::BigInt - Arbitrary size integer math package
2647 # or make it faster: install (optional) Math::BigInt::GMP
2648 # and always use (it will fall back to pure Perl if the
2649 # GMP library is not installed):
2651 use Math::BigInt lib => 'GMP';
2653 my $str = '1234567890';
2654 my @values = (64,74,18);
2655 my $n = 1; my $sign = '-';
2658 $x = Math::BigInt->new($str); # defaults to 0
2659 $y = $x->copy(); # make a true copy
2660 $nan = Math::BigInt->bnan(); # create a NotANumber
2661 $zero = Math::BigInt->bzero(); # create a +0
2662 $inf = Math::BigInt->binf(); # create a +inf
2663 $inf = Math::BigInt->binf('-'); # create a -inf
2664 $one = Math::BigInt->bone(); # create a +1
2665 $one = Math::BigInt->bone('-'); # create a -1
2667 # Testing (don't modify their arguments)
2668 # (return true if the condition is met, otherwise false)
2670 $x->is_zero(); # if $x is +0
2671 $x->is_nan(); # if $x is NaN
2672 $x->is_one(); # if $x is +1
2673 $x->is_one('-'); # if $x is -1
2674 $x->is_odd(); # if $x is odd
2675 $x->is_even(); # if $x is even
2676 $x->is_pos(); # if $x >= 0
2677 $x->is_neg(); # if $x < 0
2678 $x->is_inf($sign); # if $x is +inf, or -inf (sign is default '+')
2679 $x->is_int(); # if $x is an integer (not a float)
2681 # comparing and digit/sign extration
2682 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2683 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2684 $x->sign(); # return the sign, either +,- or NaN
2685 $x->digit($n); # return the nth digit, counting from right
2686 $x->digit(-$n); # return the nth digit, counting from left
2688 # The following all modify their first argument. If you want to preserve
2689 # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
2690 # neccessary when mixing $a = $b assigments with non-overloaded math.
2692 $x->bzero(); # set $x to 0
2693 $x->bnan(); # set $x to NaN
2694 $x->bone(); # set $x to +1
2695 $x->bone('-'); # set $x to -1
2696 $x->binf(); # set $x to inf
2697 $x->binf('-'); # set $x to -inf
2699 $x->bneg(); # negation
2700 $x->babs(); # absolute value
2701 $x->bnorm(); # normalize (no-op in BigInt)
2702 $x->bnot(); # two's complement (bit wise not)
2703 $x->binc(); # increment $x by 1
2704 $x->bdec(); # decrement $x by 1
2706 $x->badd($y); # addition (add $y to $x)
2707 $x->bsub($y); # subtraction (subtract $y from $x)
2708 $x->bmul($y); # multiplication (multiply $x by $y)
2709 $x->bdiv($y); # divide, set $x to quotient
2710 # return (quo,rem) or quo if scalar
2712 $x->bmod($y); # modulus (x % y)
2713 $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
2714 $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
2716 $x->bpow($y); # power of arguments (x ** y)
2717 $x->blsft($y); # left shift
2718 $x->brsft($y); # right shift
2719 $x->blsft($y,$n); # left shift, by base $n (like 10)
2720 $x->brsft($y,$n); # right shift, by base $n (like 10)
2722 $x->band($y); # bitwise and
2723 $x->bior($y); # bitwise inclusive or
2724 $x->bxor($y); # bitwise exclusive or
2725 $x->bnot(); # bitwise not (two's complement)
2727 $x->bsqrt(); # calculate square-root
2728 $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
2729 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2731 $x->round($A,$P,$mode); # round to accuracy or precision using mode $mode
2732 $x->bround($n); # accuracy: preserve $n digits
2733 $x->bfround($n); # round to $nth digit, no-op for BigInts
2735 # The following do not modify their arguments in BigInt (are no-ops),
2736 # but do so in BigFloat:
2738 $x->bfloor(); # return integer less or equal than $x
2739 $x->bceil(); # return integer greater or equal than $x
2741 # The following do not modify their arguments:
2743 # greatest common divisor (no OO style)
2744 my $gcd = Math::BigInt::bgcd(@values);
2745 # lowest common multiplicator (no OO style)
2746 my $lcm = Math::BigInt::blcm(@values);
2748 $x->length(); # return number of digits in number
2749 ($xl,$f) = $x->length(); # length of number and length of fraction part,
2750 # latter is always 0 digits long for BigInts
2752 $x->exponent(); # return exponent as BigInt
2753 $x->mantissa(); # return (signed) mantissa as BigInt
2754 $x->parts(); # return (mantissa,exponent) as BigInt
2755 $x->copy(); # make a true copy of $x (unlike $y = $x;)
2756 $x->as_int(); # return as BigInt (in BigInt: same as copy())
2757 $x->numify(); # return as scalar (might overflow!)
2759 # conversation to string (do not modify their argument)
2760 $x->bstr(); # normalized string (e.g. '3')
2761 $x->bsstr(); # norm. string in scientific notation (e.g. '3E0')
2762 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
2763 $x->as_bin(); # as signed binary string with prefixed 0b
2766 # precision and accuracy (see section about rounding for more)
2767 $x->precision(); # return P of $x (or global, if P of $x undef)
2768 $x->precision($n); # set P of $x to $n
2769 $x->accuracy(); # return A of $x (or global, if A of $x undef)
2770 $x->accuracy($n); # set A $x to $n
2773 Math::BigInt->precision(); # get/set global P for all BigInt objects
2774 Math::BigInt->accuracy(); # get/set global A for all BigInt objects
2775 Math::BigInt->round_mode(); # get/set global round mode, one of
2776 # 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
2777 Math::BigInt->config(); # return hash containing configuration
2781 All operators (inlcuding basic math operations) are overloaded if you
2782 declare your big integers as
2784 $i = new Math::BigInt '123_456_789_123_456_789';
2786 Operations with overloaded operators preserve the arguments which is
2787 exactly what you expect.
2793 Input values to these routines may be any string, that looks like a number
2794 and results in an integer, including hexadecimal and binary numbers.
2796 Scalars holding numbers may also be passed, but note that non-integer numbers
2797 may already have lost precision due to the conversation to float. Quote
2798 your input if you want BigInt to see all the digits:
2800 $x = Math::BigInt->new(12345678890123456789); # bad
2801 $x = Math::BigInt->new('12345678901234567890'); # good
2803 You can include one underscore between any two digits.
2805 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
2806 Non-integer values result in NaN.
2808 Currently, Math::BigInt::new() defaults to 0, while Math::BigInt::new('')
2809 results in 'NaN'. This might change in the future, so use always the following
2810 explicit forms to get a zero or NaN:
2812 $zero = Math::BigInt->bzero();
2813 $nan = Math::BigInt->bnan();
2815 C<bnorm()> on a BigInt object is now effectively a no-op, since the numbers
2816 are always stored in normalized form. If passed a string, creates a BigInt
2817 object from the input.
2821 Output values are BigInt objects (normalized), except for the methods which
2822 return a string (see L<SYNOPSIS>).
2824 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
2825 C<is_nan()>, etc.) return true or false, while others (C<bcmp()>, C<bacmp()>)
2826 return either undef (if NaN is involved), <0, 0 or >0 and are suited for sort.
2832 Each of the methods below (except config(), accuracy() and precision())
2833 accepts three additional parameters. These arguments C<$A>, C<$P> and C<$R>
2834 are C<accuracy>, C<precision> and C<round_mode>. Please see the section about
2835 L<ACCURACY and PRECISION> for more information.
2841 print Dumper ( Math::BigInt->config() );
2842 print Math::BigInt->config()->{lib},"\n";
2844 Returns a hash containing the configuration, e.g. the version number, lib
2845 loaded etc. The following hash keys are currently filled in with the
2846 appropriate information.
2850 ============================================================
2851 lib Name of the low-level math library
2853 lib_version Version of low-level math library (see 'lib')
2855 class The class name of config() you just called
2857 upgrade To which class math operations might be upgraded
2859 downgrade To which class math operations might be downgraded
2861 precision Global precision
2863 accuracy Global accuracy
2865 round_mode Global round mode
2867 version version number of the class you used
2869 div_scale Fallback acccuracy for div
2871 trap_nan If true, traps creation of NaN via croak()
2873 trap_inf If true, traps creation of +inf/-inf via croak()
2876 The following values can be set by passing C<config()> a reference to a hash:
2879 upgrade downgrade precision accuracy round_mode div_scale
2883 $new_cfg = Math::BigInt->config( { trap_inf => 1, precision => 5 } );
2887 $x->accuracy(5); # local for $x
2888 CLASS->accuracy(5); # global for all members of CLASS
2889 $A = $x->accuracy(); # read out
2890 $A = CLASS->accuracy(); # read out
2892 Set or get the global or local accuracy, aka how many significant digits the
2895 Please see the section about L<ACCURACY AND PRECISION> for further details.
2897 Value must be greater than zero. Pass an undef value to disable it:
2899 $x->accuracy(undef);
2900 Math::BigInt->accuracy(undef);
2902 Returns the current accuracy. For C<$x->accuracy()> it will return either the
2903 local accuracy, or if not defined, the global. This means the return value
2904 represents the accuracy that will be in effect for $x:
2906 $y = Math::BigInt->new(1234567); # unrounded
2907 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
2908 $x = Math::BigInt->new(123456); # will be automatically rounded
2909 print "$x $y\n"; # '123500 1234567'
2910 print $x->accuracy(),"\n"; # will be 4
2911 print $y->accuracy(),"\n"; # also 4, since global is 4
2912 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
2913 print $x->accuracy(),"\n"; # still 4
2914 print $y->accuracy(),"\n"; # 5, since global is 5
2916 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
2917 globals separated from Math::BigInt, but it is possible to subclass
2918 Math::BigInt and make the globals of the subclass aliases to the ones from
2923 $x->precision(-2); # local for $x, round right of the dot
2924 $x->precision(2); # ditto, but round left of the dot
2925 CLASS->accuracy(5); # global for all members of CLASS
2926 CLASS->precision(-5); # ditto
2927 $P = CLASS->precision(); # read out
2928 $P = $x->precision(); # read out
2930 Set or get the global or local precision, aka how many digits the result has
2931 after the dot (or where to round it when passing a positive number). In
2932 Math::BigInt, passing a negative number precision has no effect since no
2933 numbers have digits after the dot.
2935 Please see the section about L<ACCURACY AND PRECISION> for further details.
2937 Value must be greater than zero. Pass an undef value to disable it:
2939 $x->precision(undef);
2940 Math::BigInt->precision(undef);
2942 Returns the current precision. For C<$x->precision()> it will return either the
2943 local precision of $x, or if not defined, the global. This means the return
2944 value represents the accuracy that will be in effect for $x:
2946 $y = Math::BigInt->new(1234567); # unrounded
2947 print Math::BigInt->precision(4),"\n"; # set 4, print 4
2948 $x = Math::BigInt->new(123456); # will be automatically rounded
2950 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
2951 globals separated from Math::BigInt, but it is possible to subclass
2952 Math::BigInt and make the globals of the subclass aliases to the ones from
2959 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
2960 2, but others work, too.
2962 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
2966 $x = Math::BigInt->new(10);
2967 $x->brsft(1); # same as $x >> 1: 5
2968 $x = Math::BigInt->new(1234);
2969 $x->brsft(2,10); # result 12
2971 There is one exception, and that is base 2 with negative $x:
2974 $x = Math::BigInt->new(-5);
2977 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
2982 $x = Math::BigInt->new($str,$A,$P,$R);
2984 Creates a new BigInt object from a scalar or another BigInt object. The
2985 input is accepted as decimal, hex (with leading '0x') or binary (with leading
2988 See L<Input> for more info on accepted input formats.
2992 $x = Math::BigInt->bnan();
2994 Creates a new BigInt object representing NaN (Not A Number).
2995 If used on an object, it will set it to NaN:
3001 $x = Math::BigInt->bzero();
3003 Creates a new BigInt object representing zero.
3004 If used on an object, it will set it to zero:
3010 $x = Math::BigInt->binf($sign);
3012 Creates a new BigInt object representing infinity. The optional argument is
3013 either '-' or '+', indicating whether you want infinity or minus infinity.
3014 If used on an object, it will set it to infinity:
3021 $x = Math::BigInt->binf($sign);
3023 Creates a new BigInt object representing one. The optional argument is
3024 either '-' or '+', indicating whether you want one or minus one.
3025 If used on an object, it will set it to one:
3030 =head2 is_one()/is_zero()/is_nan()/is_inf()
3033 $x->is_zero(); # true if arg is +0
3034 $x->is_nan(); # true if arg is NaN
3035 $x->is_one(); # true if arg is +1
3036 $x->is_one('-'); # true if arg is -1
3037 $x->is_inf(); # true if +inf
3038 $x->is_inf('-'); # true if -inf (sign is default '+')
3040 These methods all test the BigInt for beeing one specific value and return
3041 true or false depending on the input. These are faster than doing something
3046 =head2 is_pos()/is_neg()
3048 $x->is_pos(); # true if > 0
3049 $x->is_neg(); # true if < 0
3051 The methods return true if the argument is positive or negative, respectively.
3052 C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
3053 C<-inf> is negative. A C<zero> is neither positive nor negative.
3055 These methods are only testing the sign, and not the value.
3057 C<is_positive()> and C<is_negative()> are aliase to C<is_pos()> and
3058 C<is_neg()>, respectively. C<is_positive()> and C<is_negative()> were
3059 introduced in v1.36, while C<is_pos()> and C<is_neg()> were only introduced
3062 =head2 is_odd()/is_even()/is_int()
3064 $x->is_odd(); # true if odd, false for even
3065 $x->is_even(); # true if even, false for odd
3066 $x->is_int(); # true if $x is an integer
3068 The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
3069 C<-inf> are not integers and are neither odd nor even.
3071 In BigInt, all numbers except C<NaN>, C<+inf> and C<-inf> are integers.
3077 Compares $x with $y and takes the sign into account.
3078 Returns -1, 0, 1 or undef.
3084 Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
3090 Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
3092 If you want $x to have a certain sign, use one of the following methods:
3095 $x->babs()->bneg(); # '-'
3097 $x->binf(); # '+inf'
3098 $x->binf('-'); # '-inf'
3102 $x->digit($n); # return the nth digit, counting from right
3104 If C<$n> is negative, returns the digit counting from left.
3110 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
3111 and '-inf', respectively. Does nothing for NaN or zero.
3117 Set the number to it's absolute value, e.g. change the sign from '-' to '+'
3118 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
3123 $x->bnorm(); # normalize (no-op)
3129 Two's complement (bit wise not). This is equivalent to
3137 $x->binc(); # increment x by 1
3141 $x->bdec(); # decrement x by 1
3145 $x->badd($y); # addition (add $y to $x)
3149 $x->bsub($y); # subtraction (subtract $y from $x)
3153 $x->bmul($y); # multiplication (multiply $x by $y)
3157 $x->bdiv($y); # divide, set $x to quotient
3158 # return (quo,rem) or quo if scalar
3162 $x->bmod($y); # modulus (x % y)
3166 num->bmodinv($mod); # modular inverse
3168 Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
3169 returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
3170 C<bgcd($num, $mod)==1>.
3174 $num->bmodpow($exp,$mod); # modular exponentation
3175 # ($num**$exp % $mod)
3177 Returns the value of C<$num> taken to the power C<$exp> in the modulus
3178 C<$mod> using binary exponentation. C<bmodpow> is far superior to
3183 because it is much faster - it reduces internal variables into
3184 the modulus whenever possible, so it operates on smaller numbers.
3186 C<bmodpow> also supports negative exponents.
3188 bmodpow($num, -1, $mod)
3190 is exactly equivalent to
3196 $x->bpow($y); # power of arguments (x ** y)
3200 $x->blsft($y); # left shift
3201 $x->blsft($y,$n); # left shift, in base $n (like 10)
3205 $x->brsft($y); # right shift
3206 $x->brsft($y,$n); # right shift, in base $n (like 10)
3210 $x->band($y); # bitwise and
3214 $x->bior($y); # bitwise inclusive or
3218 $x->bxor($y); # bitwise exclusive or
3222 $x->bnot(); # bitwise not (two's complement)
3226 $x->bsqrt(); # calculate square-root
3230 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3234 $x->round($A,$P,$round_mode);
3236 Round $x to accuracy C<$A> or precision C<$P> using the round mode
3241 $x->bround($N); # accuracy: preserve $N digits
3245 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3251 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3252 does change $x in BigFloat.
3258 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3259 does change $x in BigFloat.
3263 bgcd(@values); # greatest common divisor (no OO style)
3267 blcm(@values); # lowest common multiplicator (no OO style)
3272 ($xl,$fl) = $x->length();
3274 Returns the number of digits in the decimal representation of the number.
3275 In list context, returns the length of the integer and fraction part. For
3276 BigInt's, the length of the fraction part will always be 0.
3282 Return the exponent of $x as BigInt.
3288 Return the signed mantissa of $x as BigInt.
3292 $x->parts(); # return (mantissa,exponent) as BigInt
3296 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3302 Returns $x as a BigInt (truncated towards zero). In BigInt this is the same as
3305 C<as_number()> is an alias to this method. C<as_number> was introduced in
3306 v1.22, while C<as_int()> was only introduced in v1.68.
3312 Returns a normalized string represantation of C<$x>.
3316 $x->bsstr(); # normalized string in scientific notation
3320 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3324 $x->as_bin(); # as signed binary string with prefixed 0b
3326 =head1 ACCURACY and PRECISION
3328 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3329 accuracy and precision based rounding, both automatically after every
3330 operation, as well as manually.
3332 This section describes the accuracy/precision handling in Math::Big* as it
3333 used to be and as it is now, complete with an explanation of all terms and
3336 Not yet implemented things (but with correct description) are marked with '!',
3337 things that need to be answered are marked with '?'.
3339 In the next paragraph follows a short description of terms used here (because
3340 these may differ from terms used by others people or documentation).
3342 During the rest of this document, the shortcuts A (for accuracy), P (for
3343 precision), F (fallback) and R (rounding mode) will be used.
3347 A fixed number of digits before (positive) or after (negative)
3348 the decimal point. For example, 123.45 has a precision of -2. 0 means an
3349 integer like 123 (or 120). A precision of 2 means two digits to the left
3350 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3351 numbers with zeros before the decimal point may have different precisions,
3352 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3353 was). It could also have p < 0, when the digits after the decimal point
3356 The string output (of floating point numbers) will be padded with zeros:
3358 Initial value P A Result String
3359 ------------------------------------------------------------
3360 1234.01 -3 1000 1000
3363 1234.001 1 1234 1234.0
3365 1234.01 2 1234.01 1234.01
3366 1234.01 5 1234.01 1234.01000
3368 For BigInts, no padding occurs.
3372 Number of significant digits. Leading zeros are not counted. A
3373 number may have an accuracy greater than the non-zero digits
3374 when there are zeros in it or trailing zeros. For example, 123.456 has
3375 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3377 The string output (of floating point numbers) will be padded with zeros:
3379 Initial value P A Result String
3380 ------------------------------------------------------------
3382 1234.01 6 1234.01 1234.01
3383 1234.1 8 1234.1 1234.1000
3385 For BigInts, no padding occurs.
3389 When both A and P are undefined, this is used as a fallback accuracy when
3392 =head2 Rounding mode R
3394 When rounding a number, different 'styles' or 'kinds'
3395 of rounding are possible. (Note that random rounding, as in
3396 Math::Round, is not implemented.)
3402 truncation invariably removes all digits following the
3403 rounding place, replacing them with zeros. Thus, 987.65 rounded
3404 to tens (P=1) becomes 980, and rounded to the fourth sigdig
3405 becomes 987.6 (A=4). 123.456 rounded to the second place after the
3406 decimal point (P=-2) becomes 123.46.
3408 All other implemented styles of rounding attempt to round to the
3409 "nearest digit." If the digit D immediately to the right of the
3410 rounding place (skipping the decimal point) is greater than 5, the
3411 number is incremented at the rounding place (possibly causing a
3412 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3413 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3414 truncated at the rounding place: e.g. when rounding to units, 0.4
3415 rounds to 0, and -19.4 rounds to -19.
3417 However the results of other styles of rounding differ if the
3418 digit immediately to the right of the rounding place (skipping the
3419 decimal point) is 5 and if there are no digits, or no digits other
3420 than 0, after that 5. In such cases:
3424 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3425 if it is not already. E.g., when rounding to the first sigdig, 0.45
3426 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3430 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3431 it is not already. E.g., when rounding to the first sigdig, 0.45
3432 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3436 round to plus infinity, i.e. always round up. E.g., when
3437 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3438 and 0.4501 also becomes 0.5.
3442 round to minus infinity, i.e. always round down. E.g., when
3443 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3444 but 0.4501 becomes 0.5.
3448 round to zero, i.e. positive numbers down, negative ones up.
3449 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3450 becomes -0.5, but 0.4501 becomes 0.5.
3454 The handling of A & P in MBI/MBF (the old core code shipped with Perl
3455 versions <= 5.7.2) is like this:
3461 * ffround($p) is able to round to $p number of digits after the decimal
3463 * otherwise P is unused
3465 =item Accuracy (significant digits)
3467 * fround($a) rounds to $a significant digits
3468 * only fdiv() and fsqrt() take A as (optional) paramater
3469 + other operations simply create the same number (fneg etc), or more (fmul)
3471 + rounding/truncating is only done when explicitly calling one of fround
3472 or ffround, and never for BigInt (not implemented)
3473 * fsqrt() simply hands its accuracy argument over to fdiv.
3474 * the documentation and the comment in the code indicate two different ways
3475 on how fdiv() determines the maximum number of digits it should calculate,
3476 and the actual code does yet another thing
3478 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3480 result has at most max(scale, length(dividend), length(divisor)) digits
3482 scale = max(scale, length(dividend)-1,length(divisor)-1);
3483 scale += length(divisior) - length(dividend);
3484 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3485 Actually, the 'difference' added to the scale is calculated from the
3486 number of "significant digits" in dividend and divisor, which is derived
3487 by looking at the length of the mantissa. Which is wrong, since it includes
3488 the + sign (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops
3489 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3490 assumption that 124 has 3 significant digits, while 120/7 will get you
3491 '17', not '17.1' since 120 is thought to have 2 significant digits.
3492 The rounding after the division then uses the remainder and $y to determine
3493 wether it must round up or down.
3494 ? I have no idea which is the right way. That's why I used a slightly more
3495 ? simple scheme and tweaked the few failing testcases to match it.
3499 This is how it works now:
3503 =item Setting/Accessing
3505 * You can set the A global via C<< Math::BigInt->accuracy() >> or
3506 C<< Math::BigFloat->accuracy() >> or whatever class you are using.
3507 * You can also set P globally by using C<< Math::SomeClass->precision() >>
3509 * Globals are classwide, and not inherited by subclasses.
3510 * to undefine A, use C<< Math::SomeCLass->accuracy(undef); >>
3511 * to undefine P, use C<< Math::SomeClass->precision(undef); >>
3512 * Setting C<< Math::SomeClass->accuracy() >> clears automatically
3513 C<< Math::SomeClass->precision() >>, and vice versa.
3514 * To be valid, A must be > 0, P can have any value.
3515 * If P is negative, this means round to the P'th place to the right of the
3516 decimal point; positive values mean to the left of the decimal point.
3517 P of 0 means round to integer.
3518 * to find out the current global A, use C<< Math::SomeClass->accuracy() >>
3519 * to find out the current global P, use C<< Math::SomeClass->precision() >>
3520 * use C<< $x->accuracy() >> respective C<< $x->precision() >> for the local
3521 setting of C<< $x >>.
3522 * Please note that C<< $x->accuracy() >> respecive C<< $x->precision() >>
3523 return eventually defined global A or P, when C<< $x >>'s A or P is not
3526 =item Creating numbers
3528 * When you create a number, you can give it's desired A or P via:
3529 $x = Math::BigInt->new($number,$A,$P);
3530 * Only one of A or P can be defined, otherwise the result is NaN
3531 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
3532 globals (if set) will be used. Thus changing the global defaults later on
3533 will not change the A or P of previously created numbers (i.e., A and P of
3534 $x will be what was in effect when $x was created)
3535 * If given undef for A and P, B<no> rounding will occur, and the globals will
3536 B<not> be used. This is used by subclasses to create numbers without
3537 suffering rounding in the parent. Thus a subclass is able to have it's own
3538 globals enforced upon creation of a number by using
3539 C<< $x = Math::BigInt->new($number,undef,undef) >>:
3541 use Math::BigInt::SomeSubclass;
3544 Math::BigInt->accuracy(2);
3545 Math::BigInt::SomeSubClass->accuracy(3);
3546 $x = Math::BigInt::SomeSubClass->new(1234);
3548 $x is now 1230, and not 1200. A subclass might choose to implement
3549 this otherwise, e.g. falling back to the parent's A and P.
3553 * If A or P are enabled/defined, they are used to round the result of each
3554 operation according to the rules below
3555 * Negative P is ignored in Math::BigInt, since BigInts never have digits
3556 after the decimal point
3557 * Math::BigFloat uses Math::BigInt internally, but setting A or P inside
3558 Math::BigInt as globals does not tamper with the parts of a BigFloat.
3559 A flag is used to mark all Math::BigFloat numbers as 'never round'.
3563 * It only makes sense that a number has only one of A or P at a time.
3564 If you set either A or P on one object, or globally, the other one will
3565 be automatically cleared.
3566 * If two objects are involved in an operation, and one of them has A in
3567 effect, and the other P, this results in an error (NaN).
3568 * A takes precendence over P (Hint: A comes before P).
3569 If neither of them is defined, nothing is used, i.e. the result will have
3570 as many digits as it can (with an exception for fdiv/fsqrt) and will not
3572 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
3573 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
3574 If either the dividend's or the divisor's mantissa has more digits than
3575 the value of F, the higher value will be used instead of F.
3576 This is to limit the digits (A) of the result (just consider what would
3577 happen with unlimited A and P in the case of 1/3 :-)
3578 * fdiv will calculate (at least) 4 more digits than required (determined by
3579 A, P or F), and, if F is not used, round the result
3580 (this will still fail in the case of a result like 0.12345000000001 with A
3581 or P of 5, but this can not be helped - or can it?)
3582 * Thus you can have the math done by on Math::Big* class in two modi:
3583 + never round (this is the default):
3584 This is done by setting A and P to undef. No math operation
3585 will round the result, with fdiv() and fsqrt() as exceptions to guard
3586 against overflows. You must explicitely call bround(), bfround() or
3587 round() (the latter with parameters).
3588 Note: Once you have rounded a number, the settings will 'stick' on it
3589 and 'infect' all other numbers engaged in math operations with it, since
3590 local settings have the highest precedence. So, to get SaferRound[tm],
3591 use a copy() before rounding like this:
3593 $x = Math::BigFloat->new(12.34);
3594 $y = Math::BigFloat->new(98.76);
3595 $z = $x * $y; # 1218.6984
3596 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
3597 $z = $x * $y; # still 1218.6984, without
3598 # copy would have been 1210!
3600 + round after each op:
3601 After each single operation (except for testing like is_zero()), the
3602 method round() is called and the result is rounded appropriately. By
3603 setting proper values for A and P, you can have all-the-same-A or
3604 all-the-same-P modes. For example, Math::Currency might set A to undef,
3605 and P to -2, globally.
3607 ?Maybe an extra option that forbids local A & P settings would be in order,
3608 ?so that intermediate rounding does not 'poison' further math?
3610 =item Overriding globals
3612 * you will be able to give A, P and R as an argument to all the calculation
3613 routines; the second parameter is A, the third one is P, and the fourth is
3614 R (shift right by one for binary operations like badd). P is used only if
3615 the first parameter (A) is undefined. These three parameters override the
3616 globals in the order detailed as follows, i.e. the first defined value
3618 (local: per object, global: global default, parameter: argument to sub)
3621 + local A (if defined on both of the operands: smaller one is taken)
3622 + local P (if defined on both of the operands: bigger one is taken)
3626 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
3627 arguments (A and P) instead of one
3629 =item Local settings
3631 * You can set A or P locally by using C<< $x->accuracy() >> or
3632 C<< $x->precision() >>
3633 and thus force different A and P for different objects/numbers.
3634 * Setting A or P this way immediately rounds $x to the new value.
3635 * C<< $x->accuracy() >> clears C<< $x->precision() >>, and vice versa.
3639 * the rounding routines will use the respective global or local settings.
3640 fround()/bround() is for accuracy rounding, while ffround()/bfround()
3642 * the two rounding functions take as the second parameter one of the
3643 following rounding modes (R):
3644 'even', 'odd', '+inf', '-inf', 'zero', 'trunc'
3645 * you can set/get the global R by using C<< Math::SomeClass->round_mode() >>
3646 or by setting C<< $Math::SomeClass::round_mode >>
3647 * after each operation, C<< $result->round() >> is called, and the result may
3648 eventually be rounded (that is, if A or P were set either locally,
3649 globally or as parameter to the operation)
3650 * to manually round a number, call C<< $x->round($A,$P,$round_mode); >>
3651 this will round the number by using the appropriate rounding function
3652 and then normalize it.
3653 * rounding modifies the local settings of the number:
3655 $x = Math::BigFloat->new(123.456);
3659 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
3660 will be 4 from now on.
3662 =item Default values
3671 * The defaults are set up so that the new code gives the same results as
3672 the old code (except in a few cases on fdiv):
3673 + Both A and P are undefined and thus will not be used for rounding
3674 after each operation.
3675 + round() is thus a no-op, unless given extra parameters A and P
3679 =head1 Infinity and Not a Number
3681 While BigInt has extensive handling of inf and NaN, certain quirks remain.
3687 These perl routines currently (as of Perl v.5.8.6) cannot handle passed
3690 te@linux:~> perl -wle 'print 2 ** 3333'
3692 te@linux:~> perl -wle 'print 2 ** 3333 == 2 ** 3333'
3694 te@linux:~> perl -wle 'print oct(2 ** 3333)'
3696 te@linux:~> perl -wle 'print hex(2 ** 3333)'
3697 Illegal hexadecimal digit 'i' ignored at -e line 1.
3700 The same problems occur if you pass them Math::BigInt->binf() objects. Since
3701 overloading these routines is not possible, this cannot be fixed from BigInt.
3703 =item ==, !=, <, >, <=, >= with NaNs
3705 BigInt's bcmp() routine currently returns undef to signal that a NaN was
3706 involved in a comparisation. However, the overload code turns that into
3707 either 1 or '' and thus operations like C<< NaN != NaN >> might return
3712 C<< log(-inf) >> is highly weird. Since log(-x)=pi*i+log(x), then
3713 log(-inf)=pi*i+inf. However, since the imaginary part is finite, the real
3714 infinity "overshadows" it, so the number might as well just be infinity.
3715 However, the result is a complex number, and since BigInt/BigFloat can only
3716 have real numbers as results, the result is NaN.
3718 =item exp(), cos(), sin(), atan2()
3720 These all might have problems handling infinity right.
3726 The actual numbers are stored as unsigned big integers (with seperate sign).
3728 You should neither care about nor depend on the internal representation; it
3729 might change without notice. Use B<ONLY> method calls like C<< $x->sign(); >>
3730 instead relying on the internal representation.
3734 Math with the numbers is done (by default) by a module called
3735 C<Math::BigInt::Calc>. This is equivalent to saying:
3737 use Math::BigInt lib => 'Calc';
3739 You can change this by using:
3741 use Math::BigInt lib => 'BitVect';
3743 The following would first try to find Math::BigInt::Foo, then
3744 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
3746 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
3748 Since Math::BigInt::GMP is in almost all cases faster than Calc (especially in
3749 math involving really big numbers, where it is B<much> faster), and there is
3750 no penalty if Math::BigInt::GMP is not installed, it is a good idea to always
3753 use Math::BigInt lib => 'GMP';
3755 Different low-level libraries use different formats to store the
3756 numbers. You should B<NOT> depend on the number having a specific format
3759 See the respective math library module documentation for further details.
3763 The sign is either '+', '-', 'NaN', '+inf' or '-inf'.
3765 A sign of 'NaN' is used to represent the result when input arguments are not
3766 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
3767 minus infinity. You will get '+inf' when dividing a positive number by 0, and
3768 '-inf' when dividing any negative number by 0.
3770 =head2 mantissa(), exponent() and parts()
3772 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
3775 $m = $x->mantissa();
3776 $e = $x->exponent();
3777 $y = $m * ( 10 ** $e );
3778 print "ok\n" if $x == $y;
3780 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
3781 in one go. Both the returned mantissa and exponent have a sign.
3783 Currently, for BigInts C<$e> is always 0, except for NaN, +inf and -inf,
3784 where it is C<NaN>; and for C<$x == 0>, where it is C<1> (to be compatible
3785 with Math::BigFloat's internal representation of a zero as C<0E1>).
3787 C<$m> is currently just a copy of the original number. The relation between
3788 C<$e> and C<$m> will stay always the same, though their real values might
3795 sub bint { Math::BigInt->new(shift); }
3797 $x = Math::BigInt->bstr("1234") # string "1234"
3798 $x = "$x"; # same as bstr()
3799 $x = Math::BigInt->bneg("1234"); # BigInt "-1234"
3800 $x = Math::BigInt->babs("-12345"); # BigInt "12345"
3801 $x = Math::BigInt->bnorm("-0 00"); # BigInt "0"
3802 $x = bint(1) + bint(2); # BigInt "3"
3803 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
3804 $x = bint(1); # BigInt "1"
3805 $x = $x + 5 / 2; # BigInt "3"
3806 $x = $x ** 3; # BigInt "27"
3807 $x *= 2; # BigInt "54"
3808 $x = Math::BigInt->new(0); # BigInt "0"
3810 $x = Math::BigInt->badd(4,5) # BigInt "9"
3811 print $x->bsstr(); # 9e+0
3813 Examples for rounding:
3818 $x = Math::BigFloat->new(123.4567);
3819 $y = Math::BigFloat->new(123.456789);
3820 Math::BigFloat->accuracy(4); # no more A than 4
3822 ok ($x->copy()->fround(),123.4); # even rounding
3823 print $x->copy()->fround(),"\n"; # 123.4
3824 Math::BigFloat->round_mode('odd'); # round to odd
3825 print $x->copy()->fround(),"\n"; # 123.5
3826 Math::BigFloat->accuracy(5); # no more A than 5
3827 Math::BigFloat->round_mode('odd'); # round to odd
3828 print $x->copy()->fround(),"\n"; # 123.46
3829 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
3830 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
3832 Math::BigFloat->accuracy(undef); # A not important now
3833 Math::BigFloat->precision(2); # P important
3834 print $x->copy()->bnorm(),"\n"; # 123.46
3835 print $x->copy()->fround(),"\n"; # 123.46
3837 Examples for converting:
3839 my $x = Math::BigInt->new('0b1'.'01' x 123);
3840 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
3842 =head1 Autocreating constants
3844 After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
3845 and binary constants in the given scope are converted to C<Math::BigInt>.
3846 This conversion happens at compile time.
3850 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
3852 prints the integer value of C<2**100>. Note that without conversion of
3853 constants the expression 2**100 will be calculated as perl scalar.
3855 Please note that strings and floating point constants are not affected,
3858 use Math::BigInt qw/:constant/;
3860 $x = 1234567890123456789012345678901234567890
3861 + 123456789123456789;
3862 $y = '1234567890123456789012345678901234567890'
3863 + '123456789123456789';
3865 do not work. You need an explicit Math::BigInt->new() around one of the
3866 operands. You should also quote large constants to protect loss of precision:
3870 $x = Math::BigInt->new('1234567889123456789123456789123456789');
3872 Without the quotes Perl would convert the large number to a floating point
3873 constant at compile time and then hand the result to BigInt, which results in
3874 an truncated result or a NaN.
3876 This also applies to integers that look like floating point constants:
3878 use Math::BigInt ':constant';
3880 print ref(123e2),"\n";
3881 print ref(123.2e2),"\n";
3883 will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
3884 to get this to work.
3888 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
3889 must be made in the second case. For long numbers, the copy can eat up to 20%
3890 of the work (in the case of addition/subtraction, less for
3891 multiplication/division). If $y is very small compared to $x, the form
3892 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
3893 more time then the actual addition.
3895 With a technique called copy-on-write, the cost of copying with overload could
3896 be minimized or even completely avoided. A test implementation of COW did show
3897 performance gains for overloaded math, but introduced a performance loss due
3898 to a constant overhead for all other operatons. So Math::BigInt does currently
3901 The rewritten version of this module (vs. v0.01) is slower on certain
3902 operations, like C<new()>, C<bstr()> and C<numify()>. The reason are that it
3903 does now more work and handles much more cases. The time spent in these
3904 operations is usually gained in the other math operations so that code on
3905 the average should get (much) faster. If they don't, please contact the author.
3907 Some operations may be slower for small numbers, but are significantly faster
3908 for big numbers. Other operations are now constant (O(1), like C<bneg()>,
3909 C<babs()> etc), instead of O(N) and thus nearly always take much less time.
3910 These optimizations were done on purpose.
3912 If you find the Calc module to slow, try to install any of the replacement
3913 modules and see if they help you.
3915 =head2 Alternative math libraries
3917 You can use an alternative library to drive Math::BigInt via:
3919 use Math::BigInt lib => 'Module';
3921 See L<MATH LIBRARY> for more information.
3923 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
3927 =head1 Subclassing Math::BigInt
3929 The basic design of Math::BigInt allows simple subclasses with very little
3930 work, as long as a few simple rules are followed:
3936 The public API must remain consistent, i.e. if a sub-class is overloading
3937 addition, the sub-class must use the same name, in this case badd(). The
3938 reason for this is that Math::BigInt is optimized to call the object methods
3943 The private object hash keys like C<$x->{sign}> may not be changed, but
3944 additional keys can be added, like C<$x->{_custom}>.
3948 Accessor functions are available for all existing object hash keys and should
3949 be used instead of directly accessing the internal hash keys. The reason for
3950 this is that Math::BigInt itself has a pluggable interface which permits it
3951 to support different storage methods.
3955 More complex sub-classes may have to replicate more of the logic internal of
3956 Math::BigInt if they need to change more basic behaviors. A subclass that
3957 needs to merely change the output only needs to overload C<bstr()>.
3959 All other object methods and overloaded functions can be directly inherited
3960 from the parent class.
3962 At the very minimum, any subclass will need to provide it's own C<new()> and can
3963 store additional hash keys in the object. There are also some package globals
3964 that must be defined, e.g.:
3968 $precision = -2; # round to 2 decimal places
3969 $round_mode = 'even';
3972 Additionally, you might want to provide the following two globals to allow
3973 auto-upgrading and auto-downgrading to work correctly:
3978 This allows Math::BigInt to correctly retrieve package globals from the
3979 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
3980 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
3986 in your subclass to automatically inherit the overloading from the parent. If
3987 you like, you can change part of the overloading, look at Math::String for an
3992 When used like this:
3994 use Math::BigInt upgrade => 'Foo::Bar';
3996 certain operations will 'upgrade' their calculation and thus the result to
3997 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
3999 use Math::BigInt upgrade => 'Math::BigFloat';
4001 As a shortcut, you can use the module C<bignum>:
4005 Also good for oneliners:
4007 perl -Mbignum -le 'print 2 ** 255'
4009 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
4010 as well es preserve accuracy (as in sqrt(3)).
4012 Beware: This feature is not fully implemented yet.
4016 The following methods upgrade themselves unconditionally; that is if upgrade
4017 is in effect, they will always hand up their work:
4029 Beware: This list is not complete.
4031 All other methods upgrade themselves only when one (or all) of their
4032 arguments are of the class mentioned in $upgrade (This might change in later
4033 versions to a more sophisticated scheme):
4039 =item broot() does not work
4041 The broot() function in BigInt may only work for small values. This will be
4042 fixed in a later version.
4044 =item Out of Memory!
4046 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
4047 C<eval()> in your code will crash with "Out of memory". This is probably an
4048 overload/exporter bug. You can workaround by not having C<eval()>
4049 and ':constant' at the same time or upgrade your Perl to a newer version.
4051 =item Fails to load Calc on Perl prior 5.6.0
4053 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
4054 will fall back to eval { require ... } when loading the math lib on Perls
4055 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
4056 filesystems using a different seperator.
4062 Some things might not work as you expect them. Below is documented what is
4063 known to be troublesome:
4067 =item bstr(), bsstr() and 'cmp'
4069 Both C<bstr()> and C<bsstr()> as well as automated stringify via overload now
4070 drop the leading '+'. The old code would return '+3', the new returns '3'.
4071 This is to be consistent with Perl and to make C<cmp> (especially with
4072 overloading) to work as you expect. It also solves problems with C<Test.pm>,
4073 because it's C<ok()> uses 'eq' internally.
4075 Mark Biggar said, when asked about to drop the '+' altogether, or make only
4078 I agree (with the first alternative), don't add the '+' on positive
4079 numbers. It's not as important anymore with the new internal
4080 form for numbers. It made doing things like abs and neg easier,
4081 but those have to be done differently now anyway.
4083 So, the following examples will now work all as expected:
4086 BEGIN { plan tests => 1 }
4089 my $x = new Math::BigInt 3*3;
4090 my $y = new Math::BigInt 3*3;
4093 print "$x eq 9" if $x eq $y;
4094 print "$x eq 9" if $x eq '9';
4095 print "$x eq 9" if $x eq 3*3;
4097 Additionally, the following still works:
4099 print "$x == 9" if $x == $y;
4100 print "$x == 9" if $x == 9;
4101 print "$x == 9" if $x == 3*3;
4103 There is now a C<bsstr()> method to get the string in scientific notation aka
4104 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
4105 for comparisation, but Perl will represent some numbers as 100 and others
4106 as 1e+308. If in doubt, convert both arguments to Math::BigInt before
4107 comparing them as strings:
4110 BEGIN { plan tests => 3 }
4113 $x = Math::BigInt->new('1e56'); $y = 1e56;
4114 ok ($x,$y); # will fail
4115 ok ($x->bsstr(),$y); # okay
4116 $y = Math::BigInt->new($y);
4119 Alternatively, simple use C<< <=> >> for comparisations, this will get it
4120 always right. There is not yet a way to get a number automatically represented
4121 as a string that matches exactly the way Perl represents it.
4123 See also the section about L<Infinity and Not a Number> for problems in
4128 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
4131 $x = Math::BigInt->new(123);
4132 $y = int($x); # BigInt 123
4133 $x = Math::BigFloat->new(123.45);
4134 $y = int($x); # BigInt 123
4136 In all Perl versions you can use C<as_number()> or C<as_int> for the same
4139 $x = Math::BigFloat->new(123.45);
4140 $y = $x->as_number(); # BigInt 123
4141 $y = $x->as_int(); # ditto
4143 This also works for other subclasses, like Math::String.
4145 It is yet unlcear whether overloaded int() should return a scalar or a BigInt.
4147 If you want a real Perl scalar, use C<numify()>:
4149 $y = $x->numify(); # 123 as scalar
4151 This is seldom necessary, though, because this is done automatically, like
4152 when you access an array:
4154 $z = $array[$x]; # does work automatically
4158 The following will probably not do what you expect:
4160 $c = Math::BigInt->new(123);
4161 print $c->length(),"\n"; # prints 30
4163 It prints both the number of digits in the number and in the fraction part
4164 since print calls C<length()> in list context. Use something like:
4166 print scalar $c->length(),"\n"; # prints 3
4170 The following will probably not do what you expect:
4172 print $c->bdiv(10000),"\n";
4174 It prints both quotient and remainder since print calls C<bdiv()> in list
4175 context. Also, C<bdiv()> will modify $c, so be carefull. You probably want
4178 print $c / 10000,"\n";
4179 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
4183 The quotient is always the greatest integer less than or equal to the
4184 real-valued quotient of the two operands, and the remainder (when it is
4185 nonzero) always has the same sign as the second operand; so, for
4195 As a consequence, the behavior of the operator % agrees with the
4196 behavior of Perl's built-in % operator (as documented in the perlop
4197 manpage), and the equation
4199 $x == ($x / $y) * $y + ($x % $y)
4201 holds true for any $x and $y, which justifies calling the two return
4202 values of bdiv() the quotient and remainder. The only exception to this rule
4203 are when $y == 0 and $x is negative, then the remainder will also be
4204 negative. See below under "infinity handling" for the reasoning behing this.
4206 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
4207 not change BigInt's way to do things. This is because under 'use integer' Perl
4208 will do what the underlying C thinks is right and this is different for each
4209 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
4210 the author to implement it ;)
4212 =item infinity handling
4214 Here are some examples that explain the reasons why certain results occur while
4217 The following table shows the result of the division and the remainder, so that
4218 the equation above holds true. Some "ordinary" cases are strewn in to show more
4219 clearly the reasoning:
4221 A / B = C, R so that C * B + R = A
4222 =========================================================
4223 5 / 8 = 0, 5 0 * 8 + 5 = 5
4224 0 / 8 = 0, 0 0 * 8 + 0 = 0
4225 0 / inf = 0, 0 0 * inf + 0 = 0
4226 0 /-inf = 0, 0 0 * -inf + 0 = 0
4227 5 / inf = 0, 5 0 * inf + 5 = 5
4228 5 /-inf = 0, 5 0 * -inf + 5 = 5
4229 -5/ inf = 0, -5 0 * inf + -5 = -5
4230 -5/-inf = 0, -5 0 * -inf + -5 = -5
4231 inf/ 5 = inf, 0 inf * 5 + 0 = inf
4232 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
4233 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
4234 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
4235 5/ 5 = 1, 0 1 * 5 + 0 = 5
4236 -5/ -5 = 1, 0 1 * -5 + 0 = -5
4237 inf/ inf = 1, 0 1 * inf + 0 = inf
4238 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
4239 inf/-inf = -1, 0 -1 * -inf + 0 = inf
4240 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
4241 8/ 0 = inf, 8 inf * 0 + 8 = 8
4242 inf/ 0 = inf, inf inf * 0 + inf = inf
4245 These cases below violate the "remainder has the sign of the second of the two
4246 arguments", since they wouldn't match up otherwise.
4248 A / B = C, R so that C * B + R = A
4249 ========================================================
4250 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
4251 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
4253 =item Modifying and =
4257 $x = Math::BigFloat->new(5);
4260 It will not do what you think, e.g. making a copy of $x. Instead it just makes
4261 a second reference to the B<same> object and stores it in $y. Thus anything
4262 that modifies $x (except overloaded operators) will modify $y, and vice versa.
4263 Or in other words, C<=> is only safe if you modify your BigInts only via
4264 overloaded math. As soon as you use a method call it breaks:
4267 print "$x, $y\n"; # prints '10, 10'
4269 If you want a true copy of $x, use:
4273 You can also chain the calls like this, this will make first a copy and then
4276 $y = $x->copy()->bmul(2);
4278 See also the documentation for overload.pm regarding C<=>.
4282 C<bpow()> (and the rounding functions) now modifies the first argument and
4283 returns it, unlike the old code which left it alone and only returned the
4284 result. This is to be consistent with C<badd()> etc. The first three will
4285 modify $x, the last one won't:
4287 print bpow($x,$i),"\n"; # modify $x
4288 print $x->bpow($i),"\n"; # ditto
4289 print $x **= $i,"\n"; # the same
4290 print $x ** $i,"\n"; # leave $x alone
4292 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4294 =item Overloading -$x
4304 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4305 needs to preserve $x since it does not know that it later will get overwritten.
4306 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4308 =item Mixing different object types
4310 In Perl you will get a floating point value if you do one of the following:
4316 With overloaded math, only the first two variants will result in a BigFloat:
4321 $mbf = Math::BigFloat->new(5);
4322 $mbi2 = Math::BigInteger->new(5);
4323 $mbi = Math::BigInteger->new(2);
4325 # what actually gets called:
4326 $float = $mbf + $mbi; # $mbf->badd()
4327 $float = $mbf / $mbi; # $mbf->bdiv()
4328 $integer = $mbi + $mbf; # $mbi->badd()
4329 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4330 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4332 This is because math with overloaded operators follows the first (dominating)
4333 operand, and the operation of that is called and returns thus the result. So,
4334 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4335 the result should be a Math::BigFloat or the second operant is one.
4337 To get a Math::BigFloat you either need to call the operation manually,
4338 make sure the operands are already of the proper type or casted to that type
4339 via Math::BigFloat->new():
4341 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4343 Beware of simple "casting" the entire expression, this would only convert
4344 the already computed result:
4346 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4348 Beware also of the order of more complicated expressions like:
4350 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4351 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4353 If in doubt, break the expression into simpler terms, or cast all operands
4354 to the desired resulting type.
4356 Scalar values are a bit different, since:
4361 will both result in the proper type due to the way the overloaded math works.
4363 This section also applies to other overloaded math packages, like Math::String.
4365 One solution to you problem might be autoupgrading|upgrading. See the
4366 pragmas L<bignum>, L<bigint> and L<bigrat> for an easy way to do this.
4370 C<bsqrt()> works only good if the result is a big integer, e.g. the square
4371 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4372 mode. The reason is that the result is always truncated to an integer.
4374 If you want a better approximation of the square root, then use:
4376 $x = Math::BigFloat->new(12);
4377 Math::BigFloat->precision(0);
4378 Math::BigFloat->round_mode('even');
4379 print $x->copy->bsqrt(),"\n"; # 4
4381 Math::BigFloat->precision(2);
4382 print $x->bsqrt(),"\n"; # 3.46
4383 print $x->bsqrt(3),"\n"; # 3.464
4387 For negative numbers in base see also L<brsft|brsft>.
4393 This program is free software; you may redistribute it and/or modify it under
4394 the same terms as Perl itself.
4398 L<Math::BigFloat>, L<Math::BigRat> and L<Math::Big> as well as
4399 L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4401 The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest
4402 because they solve the autoupgrading/downgrading issue, at least partly.
4405 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4406 more documentation including a full version history, testcases, empty
4407 subclass files and benchmarks.
4411 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4412 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001 - 2004
4413 and still at it in 2005.
4415 Many people contributed in one or more ways to the final beast, see the file
4416 CREDITS for an (uncomplete) list. If you miss your name, please drop me a