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 $class = "Math::BigInt";
23 @ISA = qw( Exporter );
24 @EXPORT_OK = qw( objectify bgcd blcm);
25 # _trap_inf and _trap_nan are internal and should never be accessed from the
27 use vars qw/$round_mode $accuracy $precision $div_scale $rnd_mode
28 $upgrade $downgrade $_trap_nan $_trap_inf/;
31 # Inside overload, the first arg is always an object. If the original code had
32 # it reversed (like $x = 2 * $y), then the third paramater is true.
33 # In some cases (like add, $x = $x + 2 is the same as $x = 2 + $x) this makes
34 # no difference, but in some cases it does.
36 # For overloaded ops with only one argument we simple use $_[0]->copy() to
37 # preserve the argument.
39 # Thus inheritance of overload operators becomes possible and transparent for
40 # our subclasses without the need to repeat the entire overload section there.
43 '=' => sub { $_[0]->copy(); },
45 # some shortcuts for speed (assumes that reversed order of arguments is routed
46 # to normal '+' and we thus can always modify first arg. If this is changed,
47 # this breaks and must be adjusted.)
48 '+=' => sub { $_[0]->badd($_[1]); },
49 '-=' => sub { $_[0]->bsub($_[1]); },
50 '*=' => sub { $_[0]->bmul($_[1]); },
51 '/=' => sub { scalar $_[0]->bdiv($_[1]); },
52 '%=' => sub { $_[0]->bmod($_[1]); },
53 '^=' => sub { $_[0]->bxor($_[1]); },
54 '&=' => sub { $_[0]->band($_[1]); },
55 '|=' => sub { $_[0]->bior($_[1]); },
56 '**=' => sub { $_[0]->bpow($_[1]); },
58 # not supported by Perl yet
59 '..' => \&_pointpoint,
61 '<=>' => sub { $_[2] ?
62 ref($_[0])->bcmp($_[1],$_[0]) :
66 "$_[1]" cmp $_[0]->bstr() :
67 $_[0]->bstr() cmp "$_[1]" },
69 # make cos()/sin()/exp() "work" with BigInt's or subclasses
70 'cos' => sub { cos($_[0]->numify()) },
71 'sin' => sub { sin($_[0]->numify()) },
72 'exp' => sub { exp($_[0]->numify()) },
73 'atan2' => sub { atan2($_[0]->numify(),$_[1]) },
75 'log' => sub { $_[0]->copy()->blog($_[1]); },
76 'int' => sub { $_[0]->copy(); },
77 'neg' => sub { $_[0]->copy()->bneg(); },
78 'abs' => sub { $_[0]->copy()->babs(); },
79 'sqrt' => sub { $_[0]->copy()->bsqrt(); },
80 '~' => sub { $_[0]->copy()->bnot(); },
82 # for sub it is a bit tricky to keep b: b-a => -a+b
83 '-' => sub { my $c = $_[0]->copy; $_[2] ?
84 $c->bneg()->badd($_[1]) :
86 '+' => sub { $_[0]->copy()->badd($_[1]); },
87 '*' => sub { $_[0]->copy()->bmul($_[1]); },
90 $_[2] ? ref($_[0])->new($_[1])->bdiv($_[0]) : $_[0]->copy->bdiv($_[1]);
93 $_[2] ? ref($_[0])->new($_[1])->bmod($_[0]) : $_[0]->copy->bmod($_[1]);
96 $_[2] ? ref($_[0])->new($_[1])->bpow($_[0]) : $_[0]->copy->bpow($_[1]);
99 $_[2] ? ref($_[0])->new($_[1])->blsft($_[0]) : $_[0]->copy->blsft($_[1]);
102 $_[2] ? ref($_[0])->new($_[1])->brsft($_[0]) : $_[0]->copy->brsft($_[1]);
105 $_[2] ? ref($_[0])->new($_[1])->band($_[0]) : $_[0]->copy->band($_[1]);
108 $_[2] ? ref($_[0])->new($_[1])->bior($_[0]) : $_[0]->copy->bior($_[1]);
111 $_[2] ? ref($_[0])->new($_[1])->bxor($_[0]) : $_[0]->copy->bxor($_[1]);
114 # can modify arg of ++ and --, so avoid a copy() for speed, but don't
115 # use $_[0]->bone(), it would modify $_[0] to be 1!
116 '++' => sub { $_[0]->binc() },
117 '--' => sub { $_[0]->bdec() },
119 # if overloaded, O(1) instead of O(N) and twice as fast for small numbers
121 # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/
122 # v5.6.1 dumps on this: return !$_[0]->is_zero() || undef; :-(
124 $t = 1 if !$_[0]->is_zero();
128 # the original qw() does not work with the TIESCALAR below, why?
129 # Order of arguments unsignificant
130 '""' => sub { $_[0]->bstr(); },
131 '0+' => sub { $_[0]->numify(); }
134 ##############################################################################
135 # global constants, flags and accessory
137 # these are public, but their usage is not recommended, use the accessor
140 $round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
145 $upgrade = undef; # default is no upgrade
146 $downgrade = undef; # default is no downgrade
148 # these are internally, and not to be used from the outside
150 sub MB_NEVER_ROUND () { 0x0001; }
152 $_trap_nan = 0; # are NaNs ok? set w/ config()
153 $_trap_inf = 0; # are infs ok? set w/ config()
154 my $nan = 'NaN'; # constants for easier life
156 my $CALC = 'Math::BigInt::Calc'; # module to do the low level math
158 my $IMPORT = 0; # was import() called yet?
159 # used to make require work
160 my %WARN; # warn only once for low-level libs
161 my %CAN; # cache for $CALC->can(...)
162 my $EMU_LIB = 'Math/BigInt/CalcEmu.pm'; # emulate low-level math
164 ##############################################################################
165 # the old code had $rnd_mode, so we need to support it, too
168 sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; }
169 sub FETCH { return $round_mode; }
170 sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
174 # tie to enable $rnd_mode to work transparently
175 tie $rnd_mode, 'Math::BigInt';
177 # set up some handy alias names
178 *as_int = \&as_number;
179 *is_pos = \&is_positive;
180 *is_neg = \&is_negative;
183 ##############################################################################
188 # make Class->round_mode() work
190 my $class = ref($self) || $self || __PACKAGE__;
194 if ($m !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
196 require Carp; Carp::croak ("Unknown round mode '$m'");
198 return ${"${class}::round_mode"} = $m;
200 ${"${class}::round_mode"};
206 # make Class->upgrade() work
208 my $class = ref($self) || $self || __PACKAGE__;
209 # need to set new value?
213 return ${"${class}::upgrade"} = $u;
215 ${"${class}::upgrade"};
221 # make Class->downgrade() work
223 my $class = ref($self) || $self || __PACKAGE__;
224 # need to set new value?
228 return ${"${class}::downgrade"} = $u;
230 ${"${class}::downgrade"};
236 # make Class->div_scale() work
238 my $class = ref($self) || $self || __PACKAGE__;
243 require Carp; Carp::croak ('div_scale must be greater than zero');
245 ${"${class}::div_scale"} = shift;
247 ${"${class}::div_scale"};
252 # $x->accuracy($a); ref($x) $a
253 # $x->accuracy(); ref($x)
254 # Class->accuracy(); class
255 # Class->accuracy($a); class $a
258 my $class = ref($x) || $x || __PACKAGE__;
261 # need to set new value?
265 # convert objects to scalars to avoid deep recursion. If object doesn't
266 # have numify(), then hopefully it will have overloading for int() and
267 # boolean test without wandering into a deep recursion path...
268 $a = $a->numify() if ref($a) && $a->can('numify');
272 # also croak on non-numerical
276 Carp::croak ('Argument to accuracy must be greater than zero');
280 require Carp; Carp::croak ('Argument to accuracy must be an integer');
285 # $object->accuracy() or fallback to global
286 $x->bround($a) if $a; # not for undef, 0
287 $x->{_a} = $a; # set/overwrite, even if not rounded
288 delete $x->{_p}; # clear P
289 $a = ${"${class}::accuracy"} unless defined $a; # proper return value
293 ${"${class}::accuracy"} = $a; # set global A
294 ${"${class}::precision"} = undef; # clear global P
296 return $a; # shortcut
300 # $object->accuracy() or fallback to global
301 $r = $x->{_a} if ref($x);
302 # but don't return global undef, when $x's accuracy is 0!
303 $r = ${"${class}::accuracy"} if !defined $r;
309 # $x->precision($p); ref($x) $p
310 # $x->precision(); ref($x)
311 # Class->precision(); class
312 # Class->precision($p); class $p
315 my $class = ref($x) || $x || __PACKAGE__;
321 # convert objects to scalars to avoid deep recursion. If object doesn't
322 # have numify(), then hopefully it will have overloading for int() and
323 # boolean test without wandering into a deep recursion path...
324 $p = $p->numify() if ref($p) && $p->can('numify');
325 if ((defined $p) && (int($p) != $p))
327 require Carp; Carp::croak ('Argument to precision must be an integer');
331 # $object->precision() or fallback to global
332 $x->bfround($p) if $p; # not for undef, 0
333 $x->{_p} = $p; # set/overwrite, even if not rounded
334 delete $x->{_a}; # clear A
335 $p = ${"${class}::precision"} unless defined $p; # proper return value
339 ${"${class}::precision"} = $p; # set global P
340 ${"${class}::accuracy"} = undef; # clear global A
342 return $p; # shortcut
346 # $object->precision() or fallback to global
347 $r = $x->{_p} if ref($x);
348 # but don't return global undef, when $x's precision is 0!
349 $r = ${"${class}::precision"} if !defined $r;
355 # return (or set) configuration data as hash ref
356 my $class = shift || 'Math::BigInt';
361 # try to set given options as arguments from hash
364 if (ref($args) ne 'HASH')
368 # these values can be "set"
372 upgrade downgrade precision accuracy round_mode div_scale/
375 $set_args->{$key} = $args->{$key} if exists $args->{$key};
376 delete $args->{$key};
381 Carp::croak ("Illegal key(s) '",
382 join("','",keys %$args),"' passed to $class\->config()");
384 foreach my $key (keys %$set_args)
386 if ($key =~ /^trap_(inf|nan)\z/)
388 ${"${class}::_trap_$1"} = ($set_args->{"trap_$1"} ? 1 : 0);
391 # use a call instead of just setting the $variable to check argument
392 $class->$key($set_args->{$key});
396 # now return actual configuration
400 lib_version => ${"${CALC}::VERSION"},
402 trap_nan => ${"${class}::_trap_nan"},
403 trap_inf => ${"${class}::_trap_inf"},
404 version => ${"${class}::VERSION"},
407 upgrade downgrade precision accuracy round_mode div_scale
410 $cfg->{$key} = ${"${class}::$key"};
417 # select accuracy parameter based on precedence,
418 # used by bround() and bfround(), may return undef for scale (means no op)
419 my ($x,$s,$m,$scale,$mode) = @_;
420 $scale = $x->{_a} if !defined $scale;
421 $scale = $s if (!defined $scale);
422 $mode = $m if !defined $mode;
423 return ($scale,$mode);
428 # select precision parameter based on precedence,
429 # used by bround() and bfround(), may return undef for scale (means no op)
430 my ($x,$s,$m,$scale,$mode) = @_;
431 $scale = $x->{_p} if !defined $scale;
432 $scale = $s if (!defined $scale);
433 $mode = $m if !defined $mode;
434 return ($scale,$mode);
437 ##############################################################################
445 # if two arguments, the first one is the class to "swallow" subclasses
453 return unless ref($x); # only for objects
455 my $self = {}; bless $self,$c;
457 $self->{sign} = $x->{sign};
458 $self->{value} = $CALC->_copy($x->{value});
459 $self->{_a} = $x->{_a} if defined $x->{_a};
460 $self->{_p} = $x->{_p} if defined $x->{_p};
466 # create a new BigInt object from a string or another BigInt object.
467 # see hash keys documented at top
469 # the argument could be an object, so avoid ||, && etc on it, this would
470 # cause costly overloaded code to be called. The only allowed ops are
473 my ($class,$wanted,$a,$p,$r) = @_;
475 # avoid numify-calls by not using || on $wanted!
476 return $class->bzero($a,$p) if !defined $wanted; # default to 0
477 return $class->copy($wanted,$a,$p,$r)
478 if ref($wanted) && $wanted->isa($class); # MBI or subclass
480 $class->import() if $IMPORT == 0; # make require work
482 my $self = bless {}, $class;
484 # shortcut for "normal" numbers
485 if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*\z/))
487 $self->{sign} = $1 || '+';
489 if ($wanted =~ /^[+-]/)
491 # remove sign without touching wanted to make it work with constants
492 my $t = $wanted; $t =~ s/^[+-]//;
493 $self->{value} = $CALC->_new($t);
497 $self->{value} = $CALC->_new($wanted);
500 if ( (defined $a) || (defined $p)
501 || (defined ${"${class}::precision"})
502 || (defined ${"${class}::accuracy"})
505 $self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p);
510 # handle '+inf', '-inf' first
511 if ($wanted =~ /^[+-]?inf$/)
513 $self->{value} = $CALC->_zero();
514 $self->{sign} = $wanted; $self->{sign} = '+inf' if $self->{sign} eq 'inf';
517 # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign
518 my ($mis,$miv,$mfv,$es,$ev) = _split($wanted);
523 require Carp; Carp::croak("$wanted is not a number in $class");
525 $self->{value} = $CALC->_zero();
526 $self->{sign} = $nan;
531 # _from_hex or _from_bin
532 $self->{value} = $mis->{value};
533 $self->{sign} = $mis->{sign};
534 return $self; # throw away $mis
536 # make integer from mantissa by adjusting exp, then convert to bigint
537 $self->{sign} = $$mis; # store sign
538 $self->{value} = $CALC->_zero(); # for all the NaN cases
539 my $e = int("$$es$$ev"); # exponent (avoid recursion)
542 my $diff = $e - CORE::length($$mfv);
543 if ($diff < 0) # Not integer
547 require Carp; Carp::croak("$wanted not an integer in $class");
550 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
551 $self->{sign} = $nan;
555 # adjust fraction and add it to value
556 #print "diff > 0 $$miv\n";
557 $$miv = $$miv . ($$mfv . '0' x $diff);
562 if ($$mfv ne '') # e <= 0
564 # fraction and negative/zero E => NOI
567 require Carp; Carp::croak("$wanted not an integer in $class");
569 #print "NOI 2 \$\$mfv '$$mfv'\n";
570 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
571 $self->{sign} = $nan;
575 # xE-y, and empty mfv
578 if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
582 require Carp; Carp::croak("$wanted not an integer in $class");
585 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
586 $self->{sign} = $nan;
590 $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
591 $self->{value} = $CALC->_new($$miv) if $self->{sign} =~ /^[+-]$/;
592 # if any of the globals is set, use them to round and store them inside $self
593 # do not round for new($x,undef,undef) since that is used by MBF to signal
595 $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p;
601 # create a bigint 'NaN', if given a BigInt, set it to 'NaN'
603 $self = $class if !defined $self;
606 my $c = $self; $self = {}; bless $self, $c;
609 if (${"${class}::_trap_nan"})
612 Carp::croak ("Tried to set $self to NaN in $class\::bnan()");
614 $self->import() if $IMPORT == 0; # make require work
615 return if $self->modify('bnan');
616 if ($self->can('_bnan'))
618 # use subclass to initialize
623 # otherwise do our own thing
624 $self->{value} = $CALC->_zero();
626 $self->{sign} = $nan;
627 delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly
633 # create a bigint '+-inf', if given a BigInt, set it to '+-inf'
634 # the sign is either '+', or if given, used from there
636 my $sign = shift; $sign = '+' if !defined $sign || $sign !~ /^-(inf)?$/;
637 $self = $class if !defined $self;
640 my $c = $self; $self = {}; bless $self, $c;
643 if (${"${class}::_trap_inf"})
646 Carp::croak ("Tried to set $self to +-inf in $class\::binfn()");
648 $self->import() if $IMPORT == 0; # make require work
649 return if $self->modify('binf');
650 if ($self->can('_binf'))
652 # use subclass to initialize
657 # otherwise do our own thing
658 $self->{value} = $CALC->_zero();
660 $sign = $sign . 'inf' if $sign !~ /inf$/; # - => -inf
661 $self->{sign} = $sign;
662 ($self->{_a},$self->{_p}) = @_; # take over requested rounding
668 # create a bigint '+0', if given a BigInt, set it to 0
670 $self = $class if !defined $self;
674 my $c = $self; $self = {}; bless $self, $c;
676 $self->import() if $IMPORT == 0; # make require work
677 return if $self->modify('bzero');
679 if ($self->can('_bzero'))
681 # use subclass to initialize
686 # otherwise do our own thing
687 $self->{value} = $CALC->_zero();
694 # call like: $x->bzero($a,$p,$r,$y);
695 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
700 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
702 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
710 # create a bigint '+1' (or -1 if given sign '-'),
711 # if given a BigInt, set it to +1 or -1, respecively
713 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
714 $self = $class if !defined $self;
718 my $c = $self; $self = {}; bless $self, $c;
720 $self->import() if $IMPORT == 0; # make require work
721 return if $self->modify('bone');
723 if ($self->can('_bone'))
725 # use subclass to initialize
730 # otherwise do our own thing
731 $self->{value} = $CALC->_one();
733 $self->{sign} = $sign;
738 # call like: $x->bone($sign,$a,$p,$r,$y);
739 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
743 # call like: $x->bone($sign,$a,$p,$r);
745 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
747 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
753 ##############################################################################
754 # string conversation
758 # (ref to BFLOAT or num_str ) return num_str
759 # Convert number from internal format to scientific string format.
760 # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
761 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
762 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
764 if ($x->{sign} !~ /^[+-]$/)
766 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
769 my ($m,$e) = $x->parts();
770 #$m->bstr() . 'e+' . $e->bstr(); # e can only be positive in BigInt
771 # 'e+' because E can only be positive in BigInt
772 $m->bstr() . 'e+' . $CALC->_str($e->{value});
777 # make a string from bigint object
778 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
779 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
781 if ($x->{sign} !~ /^[+-]$/)
783 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
786 my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
787 $es.$CALC->_str($x->{value});
792 # Make a "normal" scalar from a BigInt object
793 my $x = shift; $x = $class->new($x) unless ref $x;
795 return $x->bstr() if $x->{sign} !~ /^[+-]$/;
796 my $num = $CALC->_num($x->{value});
797 return -$num if $x->{sign} eq '-';
801 ##############################################################################
802 # public stuff (usually prefixed with "b")
806 # return the sign of the number: +/-/-inf/+inf/NaN
807 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
812 sub _find_round_parameters
814 # After any operation or when calling round(), the result is rounded by
815 # regarding the A & P from arguments, local parameters, or globals.
817 # !!!!!!! If you change this, remember to change round(), too! !!!!!!!!!!
819 # This procedure finds the round parameters, but it is for speed reasons
820 # duplicated in round. Otherwise, it is tested by the testsuite and used
823 # returns ($self) or ($self,$a,$p,$r) - sets $self to NaN of both A and P
824 # were requested/defined (locally or globally or both)
826 my ($self,$a,$p,$r,@args) = @_;
827 # $a accuracy, if given by caller
828 # $p precision, if given by caller
829 # $r round_mode, if given by caller
830 # @args all 'other' arguments (0 for unary, 1 for binary ops)
832 # leave bigfloat parts alone
833 return ($self) if exists $self->{_f} && ($self->{_f} & MB_NEVER_ROUND) != 0;
835 my $c = ref($self); # find out class of argument(s)
838 # now pick $a or $p, but only if we have got "arguments"
841 foreach ($self,@args)
843 # take the defined one, or if both defined, the one that is smaller
844 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
849 # even if $a is defined, take $p, to signal error for both defined
850 foreach ($self,@args)
852 # take the defined one, or if both defined, the one that is bigger
854 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
857 # if still none defined, use globals (#2)
858 $a = ${"$c\::accuracy"} unless defined $a;
859 $p = ${"$c\::precision"} unless defined $p;
861 # A == 0 is useless, so undef it to signal no rounding
862 $a = undef if defined $a && $a == 0;
865 return ($self) unless defined $a || defined $p; # early out
867 # set A and set P is an fatal error
868 return ($self->bnan()) if defined $a && defined $p; # error
870 $r = ${"$c\::round_mode"} unless defined $r;
871 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
873 require Carp; Carp::croak ("Unknown round mode '$r'");
881 # Round $self according to given parameters, or given second argument's
882 # parameters or global defaults
884 # for speed reasons, _find_round_parameters is embeded here:
886 my ($self,$a,$p,$r,@args) = @_;
887 # $a accuracy, if given by caller
888 # $p precision, if given by caller
889 # $r round_mode, if given by caller
890 # @args all 'other' arguments (0 for unary, 1 for binary ops)
892 # leave bigfloat parts alone
893 return ($self) if exists $self->{_f} && ($self->{_f} & MB_NEVER_ROUND) != 0;
895 my $c = ref($self); # find out class of argument(s)
898 # now pick $a or $p, but only if we have got "arguments"
901 foreach ($self,@args)
903 # take the defined one, or if both defined, the one that is smaller
904 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
909 # even if $a is defined, take $p, to signal error for both defined
910 foreach ($self,@args)
912 # take the defined one, or if both defined, the one that is bigger
914 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
917 # if still none defined, use globals (#2)
918 $a = ${"$c\::accuracy"} unless defined $a;
919 $p = ${"$c\::precision"} unless defined $p;
921 # A == 0 is useless, so undef it to signal no rounding
922 $a = undef if defined $a && $a == 0;
925 return $self unless defined $a || defined $p; # early out
927 # set A and set P is an fatal error
928 return $self->bnan() if defined $a && defined $p;
930 $r = ${"$c\::round_mode"} unless defined $r;
931 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
933 require Carp; Carp::croak ("Unknown round mode '$r'");
936 # now round, by calling either fround or ffround:
939 $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a;
941 else # both can't be undefined due to early out
943 $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p;
945 $self->bnorm(); # after round, normalize
950 # (numstr or BINT) return BINT
951 # Normalize number -- no-op here
952 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
958 # (BINT or num_str) return BINT
959 # make number absolute, or return absolute BINT from string
960 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
962 return $x if $x->modify('babs');
963 # post-normalized abs for internal use (does nothing for NaN)
964 $x->{sign} =~ s/^-/+/;
970 # (BINT or num_str) return BINT
971 # negate number or make a negated number from string
972 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
974 return $x if $x->modify('bneg');
976 # for +0 dont negate (to have always normalized)
977 $x->{sign} =~ tr/+-/-+/ if !$x->is_zero(); # does nothing for NaN
983 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
984 # (BINT or num_str, BINT or num_str) return cond_code
987 my ($self,$x,$y) = (ref($_[0]),@_);
989 # objectify is costly, so avoid it
990 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
992 ($self,$x,$y) = objectify(2,@_);
995 return $upgrade->bcmp($x,$y) if defined $upgrade &&
996 ((!$x->isa($self)) || (!$y->isa($self)));
998 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1000 # handle +-inf and NaN
1001 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1002 return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
1003 return +1 if $x->{sign} eq '+inf';
1004 return -1 if $x->{sign} eq '-inf';
1005 return -1 if $y->{sign} eq '+inf';
1008 # check sign for speed first
1009 return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
1010 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
1012 # have same sign, so compare absolute values. Don't make tests for zero here
1013 # because it's actually slower than testin in Calc (especially w/ Pari et al)
1015 # post-normalized compare for internal use (honors signs)
1016 if ($x->{sign} eq '+')
1018 # $x and $y both > 0
1019 return $CALC->_acmp($x->{value},$y->{value});
1023 $CALC->_acmp($y->{value},$x->{value}); # swaped acmp (lib returns 0,1,-1)
1028 # Compares 2 values, ignoring their signs.
1029 # Returns one of undef, <0, =0, >0. (suitable for sort)
1030 # (BINT, BINT) return cond_code
1033 my ($self,$x,$y) = (ref($_[0]),@_);
1034 # objectify is costly, so avoid it
1035 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1037 ($self,$x,$y) = objectify(2,@_);
1040 return $upgrade->bacmp($x,$y) if defined $upgrade &&
1041 ((!$x->isa($self)) || (!$y->isa($self)));
1043 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1045 # handle +-inf and NaN
1046 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1047 return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
1048 return 1 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} !~ /^[+-]inf$/;
1051 $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
1056 # add second arg (BINT or string) to first (BINT) (modifies first)
1057 # return result as BINT
1060 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1061 # objectify is costly, so avoid it
1062 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1064 ($self,$x,$y,@r) = objectify(2,@_);
1067 return $x if $x->modify('badd');
1068 return $upgrade->badd($upgrade->new($x),$upgrade->new($y),@r) if defined $upgrade &&
1069 ((!$x->isa($self)) || (!$y->isa($self)));
1071 $r[3] = $y; # no push!
1072 # inf and NaN handling
1073 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1076 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1078 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1080 # +inf++inf or -inf+-inf => same, rest is NaN
1081 return $x if $x->{sign} eq $y->{sign};
1084 # +-inf + something => +inf
1085 # something +-inf => +-inf
1086 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
1090 my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
1094 $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
1098 my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
1101 $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
1106 # speedup, if equal, set result to 0
1107 $x->{value} = $CALC->_zero();
1112 $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
1115 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1121 # (BINT or num_str, BINT or num_str) return BINT
1122 # subtract second arg from first, modify first
1125 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1126 # objectify is costly, so avoid it
1127 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1129 ($self,$x,$y,@r) = objectify(2,@_);
1132 return $x if $x->modify('bsub');
1134 return $upgrade->new($x)->bsub($upgrade->new($y),@r) if defined $upgrade &&
1135 ((!$x->isa($self)) || (!$y->isa($self)));
1139 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1143 require Scalar::Util;
1144 if (Scalar::Util::refaddr($x) == Scalar::Util::refaddr($y))
1146 # if we get the same variable twice, the result must be zero (the code
1147 # below fails in that case)
1148 return $x->bzero(@r) if $x->{sign} =~ /^[+-]$/;
1149 return $x->bnan(); # NaN, -inf, +inf
1151 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
1152 $x->badd($y,@r); # badd does not leave internal zeros
1153 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
1154 $x; # already rounded by badd() or no round necc.
1159 # increment arg by one
1160 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1161 return $x if $x->modify('binc');
1163 if ($x->{sign} eq '+')
1165 $x->{value} = $CALC->_inc($x->{value});
1166 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1169 elsif ($x->{sign} eq '-')
1171 $x->{value} = $CALC->_dec($x->{value});
1172 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1173 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1176 # inf, nan handling etc
1177 $x->badd($self->bone(),$a,$p,$r); # badd does round
1182 # decrement arg by one
1183 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1184 return $x if $x->modify('bdec');
1186 if ($x->{sign} eq '-')
1189 $x->{value} = $CALC->_inc($x->{value});
1193 return $x->badd($self->bone('-'),@r) unless $x->{sign} eq '+'; # inf/NaN
1195 if ($CALC->_is_zero($x->{value}))
1198 $x->{value} = $CALC->_one(); $x->{sign} = '-'; # 0 => -1
1203 $x->{value} = $CALC->_dec($x->{value});
1206 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1212 # calculate $x = $a ** $base + $b and return $a (e.g. the log() to base
1216 my ($self,$x,$base,@r) = (ref($_[0]),@_);
1217 # objectify is costly, so avoid it
1218 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1220 ($self,$x,$base,@r) = objectify(1,$class,@_);
1223 return $x if $x->modify('blog');
1225 # inf, -inf, NaN, <0 => NaN
1227 if $x->{sign} ne '+' || (defined $base && $base->{sign} ne '+');
1229 return $upgrade->blog($upgrade->new($x),$base,@r) if
1232 my ($rc,$exact) = $CALC->_log_int($x->{value},$base->{value});
1233 return $x->bnan() unless defined $rc; # not possible to take log?
1240 # (BINT or num_str, BINT or num_str) return BINT
1241 # does not modify arguments, but returns new object
1242 # Lowest Common Multiplicator
1244 my $y = shift; my ($x);
1251 $x = __PACKAGE__->new($y);
1256 my $y = shift; $y = $self->new($y) if !ref ($y);
1264 # (BINT or num_str, BINT or num_str) return BINT
1265 # does not modify arguments, but returns new object
1266 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1269 $y = __PACKAGE__->new($y) if !ref($y);
1271 my $x = $y->copy()->babs(); # keep arguments
1272 return $x->bnan() if $x->{sign} !~ /^[+-]$/; # x NaN?
1276 $y = shift; $y = $self->new($y) if !ref($y);
1277 next if $y->is_zero();
1278 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1279 $x->{value} = $CALC->_gcd($x->{value},$y->{value}); last if $x->is_one();
1286 # (num_str or BINT) return BINT
1287 # represent ~x as twos-complement number
1288 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1289 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1291 return $x if $x->modify('bnot');
1292 $x->binc()->bneg(); # binc already does round
1295 ##############################################################################
1296 # is_foo test routines
1297 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1301 # return true if arg (BINT or num_str) is zero (array '+', '0')
1302 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1304 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1305 $CALC->_is_zero($x->{value});
1310 # return true if arg (BINT or num_str) is NaN
1311 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1313 $x->{sign} eq $nan ? 1 : 0;
1318 # return true if arg (BINT or num_str) is +-inf
1319 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1323 $sign = '[+-]inf' if $sign eq ''; # +- doesn't matter, only that's inf
1324 $sign = "[$1]inf" if $sign =~ /^([+-])(inf)?$/; # extract '+' or '-'
1325 return $x->{sign} =~ /^$sign$/ ? 1 : 0;
1327 $x->{sign} =~ /^[+-]inf$/ ? 1 : 0; # only +-inf is infinity
1332 # return true if arg (BINT or num_str) is +1, or -1 if sign is given
1333 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1335 $sign = '+' if !defined $sign || $sign ne '-';
1337 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1338 $CALC->_is_one($x->{value});
1343 # return true when arg (BINT or num_str) is odd, false for even
1344 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1346 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1347 $CALC->_is_odd($x->{value});
1352 # return true when arg (BINT or num_str) is even, false for odd
1353 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1355 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1356 $CALC->_is_even($x->{value});
1361 # return true when arg (BINT or num_str) is positive (>= 0)
1362 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1364 $x->{sign} =~ /^\+/ ? 1 : 0; # +inf is also positive, but NaN not
1369 # return true when arg (BINT or num_str) is negative (< 0)
1370 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1372 $x->{sign} =~ /^-/ ? 1 : 0; # -inf is also negative, but NaN not
1377 # return true when arg (BINT or num_str) is an integer
1378 # always true for BigInt, but different for BigFloats
1379 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1381 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1384 ###############################################################################
1388 # multiply two numbers -- stolen from Knuth Vol 2 pg 233
1389 # (BINT or num_str, BINT or num_str) return BINT
1392 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1393 # objectify is costly, so avoid it
1394 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1396 ($self,$x,$y,@r) = objectify(2,@_);
1399 return $x if $x->modify('bmul');
1401 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1404 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1406 return $x->bnan() if $x->is_zero() || $y->is_zero();
1407 # result will always be +-inf:
1408 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1409 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1410 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1411 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1412 return $x->binf('-');
1415 return $upgrade->bmul($x,$upgrade->new($y),@r)
1416 if defined $upgrade && !$y->isa($self);
1418 $r[3] = $y; # no push here
1420 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1422 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1423 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1425 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1431 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1432 my ($self,$x,$y) = @_;
1434 # NaN if x == NaN or y == NaN or x==y==0
1435 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1436 if (($x->is_nan() || $y->is_nan()) ||
1437 ($x->is_zero() && $y->is_zero()));
1439 # +-inf / +-inf == NaN, reminder also NaN
1440 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1442 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1444 # x / +-inf => 0, remainder x (works even if x == 0)
1445 if ($y->{sign} =~ /^[+-]inf$/)
1447 my $t = $x->copy(); # bzero clobbers up $x
1448 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1451 # 5 / 0 => +inf, -6 / 0 => -inf
1452 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1453 # exception: -8 / 0 has remainder -8, not 8
1454 # exception: -inf / 0 has remainder -inf, not inf
1457 # +-inf / 0 => special case for -inf
1458 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1459 if (!$x->is_zero() && !$x->is_inf())
1461 my $t = $x->copy(); # binf clobbers up $x
1463 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1467 # last case: +-inf / ordinary number
1469 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1471 return wantarray ? ($x,$self->bzero()) : $x;
1476 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1477 # (BINT,BINT) (quo,rem) or BINT (only rem)
1480 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1481 # objectify is costly, so avoid it
1482 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1484 ($self,$x,$y,@r) = objectify(2,@_);
1487 return $x if $x->modify('bdiv');
1489 return $self->_div_inf($x,$y)
1490 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1492 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1493 if defined $upgrade;
1495 $r[3] = $y; # no push!
1497 # calc new sign and in case $y == +/- 1, return $x
1498 my $xsign = $x->{sign}; # keep
1499 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1503 my $rem = $self->bzero();
1504 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1505 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1506 $rem->{_a} = $x->{_a};
1507 $rem->{_p} = $x->{_p};
1508 $x->round(@r) if !exists $x->{_f} || ($x->{_f} & MB_NEVER_ROUND) == 0;
1509 if (! $CALC->_is_zero($rem->{value}))
1511 $rem->{sign} = $y->{sign};
1512 $rem = $y->copy()->bsub($rem) if $xsign ne $y->{sign}; # one of them '-'
1516 $rem->{sign} = '+'; # dont leave -0
1518 $rem->round(@r) if !exists $rem->{_f} || ($rem->{_f} & MB_NEVER_ROUND) == 0;
1522 $x->{value} = $CALC->_div($x->{value},$y->{value});
1523 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1525 $x->round(@r) if !exists $x->{_f} || ($x->{_f} & MB_NEVER_ROUND) == 0;
1529 ###############################################################################
1534 # modulus (or remainder)
1535 # (BINT or num_str, BINT or num_str) return BINT
1538 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1539 # objectify is costly, so avoid it
1540 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1542 ($self,$x,$y,@r) = objectify(2,@_);
1545 return $x if $x->modify('bmod');
1546 $r[3] = $y; # no push!
1547 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1549 my ($d,$r) = $self->_div_inf($x,$y);
1550 $x->{sign} = $r->{sign};
1551 $x->{value} = $r->{value};
1552 return $x->round(@r);
1555 # calc new sign and in case $y == +/- 1, return $x
1556 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1557 if (!$CALC->_is_zero($x->{value}))
1559 my $xsign = $x->{sign};
1560 $x->{sign} = $y->{sign};
1561 if ($xsign ne $y->{sign})
1563 my $t = $CALC->_copy($x->{value}); # copy $x
1564 $x->{value} = $CALC->_sub($y->{value},$t,1); # $y-$x
1569 $x->{sign} = '+'; # dont leave -0
1571 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1577 # Modular inverse. given a number which is (hopefully) relatively
1578 # prime to the modulus, calculate its inverse using Euclid's
1579 # alogrithm. If the number is not relatively prime to the modulus
1580 # (i.e. their gcd is not one) then NaN is returned.
1583 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1584 # objectify is costly, so avoid it
1585 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1587 ($self,$x,$y,@r) = objectify(2,@_);
1590 return $x if $x->modify('bmodinv');
1593 if ($y->{sign} ne '+' # -, NaN, +inf, -inf
1594 || $x->is_zero() # or num == 0
1595 || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1598 # put least residue into $x if $x was negative, and thus make it positive
1599 $x->bmod($y) if $x->{sign} eq '-';
1602 ($x->{value},$sign) = $CALC->_modinv($x->{value},$y->{value});
1603 return $x->bnan() if !defined $x->{value}; # in case no GCD found
1604 return $x if !defined $sign; # already real result
1605 $x->{sign} = $sign; # flip/flop see below
1606 $x->bmod($y); # calc real result
1612 # takes a very large number to a very large exponent in a given very
1613 # large modulus, quickly, thanks to binary exponentation. supports
1614 # negative exponents.
1615 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1617 return $num if $num->modify('bmodpow');
1619 # check modulus for valid values
1620 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1621 || $mod->is_zero());
1623 # check exponent for valid values
1624 if ($exp->{sign} =~ /\w/)
1626 # i.e., if it's NaN, +inf, or -inf...
1627 return $num->bnan();
1630 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
1632 # check num for valid values (also NaN if there was no inverse but $exp < 0)
1633 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1635 # $mod is positive, sign on $exp is ignored, result also positive
1636 $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
1640 ###############################################################################
1644 # (BINT or num_str, BINT or num_str) return BINT
1645 # compute factorial number from $x, modify $x in place
1646 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1648 return $x if $x->modify('bfac');
1650 return $x if $x->{sign} eq '+inf'; # inf => inf
1651 return $x->bnan() if $x->{sign} ne '+'; # NaN, <0 etc => NaN
1653 $x->{value} = $CALC->_fac($x->{value});
1659 # (BINT or num_str, BINT or num_str) return BINT
1660 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1661 # modifies first argument
1664 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1665 # objectify is costly, so avoid it
1666 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1668 ($self,$x,$y,@r) = objectify(2,@_);
1671 return $x if $x->modify('bpow');
1673 return $upgrade->bpow($upgrade->new($x),$y,@r)
1674 if defined $upgrade && !$y->isa($self);
1676 $r[3] = $y; # no push!
1677 return $x if $x->{sign} =~ /^[+-]inf$/; # -inf/+inf ** x
1678 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1680 # cases 0 ** Y, X ** 0, X ** 1, 1 ** Y are handled by Calc or Emu
1683 $new_sign = $y->is_odd() ? '-' : '+' if ($x->{sign} ne '+');
1685 # 0 ** -7 => ( 1 / (0 ** 7)) => 1 / 0 => +inf
1687 if $y->{sign} eq '-' && $x->{sign} eq '+' && $CALC->_is_zero($x->{value});
1688 # 1 ** -y => 1 / (1 ** |y|)
1689 # so do test for negative $y after above's clause
1690 return $x->bnan() if $y->{sign} eq '-' && !$CALC->_is_one($x->{value});
1692 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1693 $x->{sign} = $new_sign;
1694 $x->{sign} = '+' if $CALC->_is_zero($y->{value});
1695 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1701 # (BINT or num_str, BINT or num_str) return BINT
1702 # compute x << y, base n, y >= 0
1705 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1706 # objectify is costly, so avoid it
1707 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1709 ($self,$x,$y,$n,@r) = objectify(2,@_);
1712 return $x if $x->modify('blsft');
1713 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1714 return $x->round(@r) if $y->is_zero();
1716 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1718 $x->{value} = $CALC->_lsft($x->{value},$y->{value},$n);
1724 # (BINT or num_str, BINT or num_str) return BINT
1725 # compute x >> y, base n, y >= 0
1728 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1729 # objectify is costly, so avoid it
1730 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1732 ($self,$x,$y,$n,@r) = objectify(2,@_);
1735 return $x if $x->modify('brsft');
1736 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1737 return $x->round(@r) if $y->is_zero();
1738 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
1740 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1742 # this only works for negative numbers when shifting in base 2
1743 if (($x->{sign} eq '-') && ($n == 2))
1745 return $x->round(@r) if $x->is_one('-'); # -1 => -1
1748 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1749 # but perhaps there is a better emulation for two's complement shift...
1750 # if $y != 1, we must simulate it by doing:
1751 # convert to bin, flip all bits, shift, and be done
1752 $x->binc(); # -3 => -2
1753 my $bin = $x->as_bin();
1754 $bin =~ s/^-0b//; # strip '-0b' prefix
1755 $bin =~ tr/10/01/; # flip bits
1757 if (CORE::length($bin) <= $y)
1759 $bin = '0'; # shifting to far right creates -1
1760 # 0, because later increment makes
1761 # that 1, attached '-' makes it '-1'
1762 # because -1 >> x == -1 !
1766 $bin =~ s/.{$y}$//; # cut off at the right side
1767 $bin = '1' . $bin; # extend left side by one dummy '1'
1768 $bin =~ tr/10/01/; # flip bits back
1770 my $res = $self->new('0b'.$bin); # add prefix and convert back
1771 $res->binc(); # remember to increment
1772 $x->{value} = $res->{value}; # take over value
1773 return $x->round(@r); # we are done now, magic, isn't?
1775 # x < 0, n == 2, y == 1
1776 $x->bdec(); # n == 2, but $y == 1: this fixes it
1779 $x->{value} = $CALC->_rsft($x->{value},$y->{value},$n);
1785 #(BINT or num_str, BINT or num_str) return BINT
1789 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1790 # objectify is costly, so avoid it
1791 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1793 ($self,$x,$y,@r) = objectify(2,@_);
1796 return $x if $x->modify('band');
1798 $r[3] = $y; # no push!
1800 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1802 my $sx = $x->{sign} eq '+' ? 1 : -1;
1803 my $sy = $y->{sign} eq '+' ? 1 : -1;
1805 if ($sx == 1 && $sy == 1)
1807 $x->{value} = $CALC->_and($x->{value},$y->{value});
1808 return $x->round(@r);
1811 if ($CAN{signed_and})
1813 $x->{value} = $CALC->_signed_and($x->{value},$y->{value},$sx,$sy);
1814 return $x->round(@r);
1818 __emu_band($self,$x,$y,$sx,$sy,@r);
1823 #(BINT or num_str, BINT or num_str) return BINT
1827 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1828 # objectify is costly, so avoid it
1829 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1831 ($self,$x,$y,@r) = objectify(2,@_);
1834 return $x if $x->modify('bior');
1835 $r[3] = $y; # no push!
1837 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1839 my $sx = $x->{sign} eq '+' ? 1 : -1;
1840 my $sy = $y->{sign} eq '+' ? 1 : -1;
1842 # the sign of X follows the sign of X, e.g. sign of Y irrelevant for bior()
1844 # don't use lib for negative values
1845 if ($sx == 1 && $sy == 1)
1847 $x->{value} = $CALC->_or($x->{value},$y->{value});
1848 return $x->round(@r);
1851 # if lib can do negative values, let it handle this
1852 if ($CAN{signed_or})
1854 $x->{value} = $CALC->_signed_or($x->{value},$y->{value},$sx,$sy);
1855 return $x->round(@r);
1859 __emu_bior($self,$x,$y,$sx,$sy,@r);
1864 #(BINT or num_str, BINT or num_str) return BINT
1868 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1869 # objectify is costly, so avoid it
1870 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1872 ($self,$x,$y,@r) = objectify(2,@_);
1875 return $x if $x->modify('bxor');
1876 $r[3] = $y; # no push!
1878 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1880 my $sx = $x->{sign} eq '+' ? 1 : -1;
1881 my $sy = $y->{sign} eq '+' ? 1 : -1;
1883 # don't use lib for negative values
1884 if ($sx == 1 && $sy == 1)
1886 $x->{value} = $CALC->_xor($x->{value},$y->{value});
1887 return $x->round(@r);
1890 # if lib can do negative values, let it handle this
1891 if ($CAN{signed_xor})
1893 $x->{value} = $CALC->_signed_xor($x->{value},$y->{value},$sx,$sy);
1894 return $x->round(@r);
1898 __emu_bxor($self,$x,$y,$sx,$sy,@r);
1903 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1905 my $e = $CALC->_len($x->{value});
1906 wantarray ? ($e,0) : $e;
1911 # return the nth decimal digit, negative values count backward, 0 is right
1912 my ($self,$x,$n) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1914 $n = $n->numify() if ref($n);
1915 $CALC->_digit($x->{value},$n||0);
1920 # return the amount of trailing zeros in $x (as scalar)
1922 $x = $class->new($x) unless ref $x;
1924 return 0 if $x->{sign} !~ /^[+-]$/; # NaN, inf, -inf etc
1926 $CALC->_zeros($x->{value}); # must handle odd values, 0 etc
1931 # calculate square root of $x
1932 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1934 return $x if $x->modify('bsqrt');
1936 return $x->bnan() if $x->{sign} !~ /^\+/; # -x or -inf or NaN => NaN
1937 return $x if $x->{sign} eq '+inf'; # sqrt(+inf) == inf
1939 return $upgrade->bsqrt($x,@r) if defined $upgrade;
1941 $x->{value} = $CALC->_sqrt($x->{value});
1947 # calculate $y'th root of $x
1950 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1952 $y = $self->new(2) unless defined $y;
1954 # objectify is costly, so avoid it
1955 if ((!ref($x)) || (ref($x) ne ref($y)))
1957 ($self,$x,$y,@r) = objectify(2,$self || $class,@_);
1960 return $x if $x->modify('broot');
1962 # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
1963 return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
1964 $y->{sign} !~ /^\+$/;
1966 return $x->round(@r)
1967 if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
1969 return $upgrade->new($x)->broot($upgrade->new($y),@r) if defined $upgrade;
1971 $x->{value} = $CALC->_root($x->{value},$y->{value});
1977 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
1978 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1980 if ($x->{sign} !~ /^[+-]$/)
1982 my $s = $x->{sign}; $s =~ s/^[+-]//; # NaN, -inf,+inf => NaN or inf
1983 return $self->new($s);
1985 return $self->bone() if $x->is_zero();
1987 $self->new($x->_trailing_zeros());
1992 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
1993 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1995 if ($x->{sign} !~ /^[+-]$/)
1997 # for NaN, +inf, -inf: keep the sign
1998 return $self->new($x->{sign});
2000 my $m = $x->copy(); delete $m->{_p}; delete $m->{_a};
2001 # that's a bit inefficient:
2002 my $zeros = $m->_trailing_zeros();
2003 $m->brsft($zeros,10) if $zeros != 0;
2009 # return a copy of both the exponent and the mantissa
2010 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2012 ($x->mantissa(),$x->exponent());
2015 ##############################################################################
2016 # rounding functions
2020 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
2021 # $n == 0 || $n == 1 => round to integer
2022 my $x = shift; my $self = ref($x) || $x; $x = $self->new($x) unless ref $x;
2024 my ($scale,$mode) = $x->_scale_p($x->precision(),$x->round_mode(),@_);
2026 return $x if !defined $scale || $x->modify('bfround'); # no-op
2028 # no-op for BigInts if $n <= 0
2029 $x->bround( $x->length()-$scale, $mode) if $scale > 0;
2031 delete $x->{_a}; # delete to save memory
2032 $x->{_p} = $scale; # store new _p
2036 sub _scan_for_nonzero
2038 # internal, used by bround() to scan for non-zeros after a '5'
2039 my ($x,$pad,$xs,$len) = @_;
2041 return 0 if $len == 1; # "5" is trailed by invisible zeros
2042 my $follow = $pad - 1;
2043 return 0 if $follow > $len || $follow < 1;
2045 # use the string form to check whether only '0's follow or not
2046 substr ($xs,-$follow) =~ /[^0]/ ? 1 : 0;
2051 # Exists to make life easier for switch between MBF and MBI (should we
2052 # autoload fxxx() like MBF does for bxxx()?)
2059 # accuracy: +$n preserve $n digits from left,
2060 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
2062 # and overwrite the rest with 0's, return normalized number
2063 # do not return $x->bnorm(), but $x
2065 my $x = shift; $x = $class->new($x) unless ref $x;
2066 my ($scale,$mode) = $x->_scale_a($x->accuracy(),$x->round_mode(),@_);
2067 return $x if !defined $scale; # no-op
2068 return $x if $x->modify('bround');
2070 if ($x->is_zero() || $scale == 0)
2072 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2075 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2077 # we have fewer digits than we want to scale to
2078 my $len = $x->length();
2079 # convert $scale to a scalar in case it is an object (put's a limit on the
2080 # number length, but this would already limited by memory constraints), makes
2082 $scale = $scale->numify() if ref ($scale);
2084 # scale < 0, but > -len (not >=!)
2085 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2087 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2091 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2092 my ($pad,$digit_round,$digit_after);
2093 $pad = $len - $scale;
2094 $pad = abs($scale-1) if $scale < 0;
2096 # do not use digit(), it is very costly for binary => decimal
2097 # getting the entire string is also costly, but we need to do it only once
2098 my $xs = $CALC->_str($x->{value});
2101 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2102 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2103 $digit_round = '0'; $digit_round = substr($xs,$pl,1) if $pad <= $len;
2104 $pl++; $pl ++ if $pad >= $len;
2105 $digit_after = '0'; $digit_after = substr($xs,$pl,1) if $pad > 0;
2107 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2108 # closer at the remaining digits of the original $x, remember decision
2109 my $round_up = 1; # default round up
2111 ($mode eq 'trunc') || # trunc by round down
2112 ($digit_after =~ /[01234]/) || # round down anyway,
2114 ($digit_after eq '5') && # not 5000...0000
2115 ($x->_scan_for_nonzero($pad,$xs,$len) == 0) &&
2117 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2118 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2119 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2120 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2121 ($mode eq 'zero') # round down if zero, sign adjusted below
2123 my $put_back = 0; # not yet modified
2125 if (($pad > 0) && ($pad <= $len))
2127 substr($xs,-$pad,$pad) = '0' x $pad; # replace with '00...'
2128 $put_back = 1; # need to put back
2132 $x->bzero(); # round to '0'
2135 if ($round_up) # what gave test above?
2137 $put_back = 1; # need to put back
2138 $pad = $len, $xs = '0' x $pad if $scale < 0; # tlr: whack 0.51=>1.0
2140 # we modify directly the string variant instead of creating a number and
2141 # adding it, since that is faster (we already have the string)
2142 my $c = 0; $pad ++; # for $pad == $len case
2143 while ($pad <= $len)
2145 $c = substr($xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2146 substr($xs,-$pad,1) = $c; $pad++;
2147 last if $c != 0; # no overflow => early out
2149 $xs = '1'.$xs if $c == 0;
2152 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back, if needed
2154 $x->{_a} = $scale if $scale >= 0;
2157 $x->{_a} = $len+$scale;
2158 $x->{_a} = 0 if $scale < -$len;
2165 # return integer less or equal then number; no-op since it's already integer
2166 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2173 # return integer greater or equal then number; no-op since it's already int
2174 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2181 # An object might be asked to return itself as bigint on certain overloaded
2182 # operations, this does exactly this, so that sub classes can simple inherit
2183 # it or override with their own integer conversion routine.
2189 # return as hex string, with prefixed 0x
2190 my $x = shift; $x = $class->new($x) if !ref($x);
2192 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2195 $s = $x->{sign} if $x->{sign} eq '-';
2196 $s . $CALC->_as_hex($x->{value});
2201 # return as binary string, with prefixed 0b
2202 my $x = shift; $x = $class->new($x) if !ref($x);
2204 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2206 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2207 return $s . $CALC->_as_bin($x->{value});
2210 ##############################################################################
2211 # private stuff (internal use only)
2215 # check for strings, if yes, return objects instead
2217 # the first argument is number of args objectify() should look at it will
2218 # return $count+1 elements, the first will be a classname. This is because
2219 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2220 # useless objects beeing created and thrown away. So we cannot simple loop
2221 # over @_. If the given count is 0, all arguments will be used.
2223 # If the second arg is a ref, use it as class.
2224 # If not, try to use it as classname, unless undef, then use $class
2225 # (aka Math::BigInt). The latter shouldn't happen,though.
2228 # $x->badd(1); => ref x, scalar y
2229 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2230 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2231 # Math::BigInt::badd(1,2); => scalar x, scalar y
2232 # In the last case we check number of arguments to turn it silently into
2233 # $class,1,2. (We can not take '1' as class ;o)
2234 # badd($class,1) is not supported (it should, eventually, try to add undef)
2235 # currently it tries 'Math::BigInt' + 1, which will not work.
2237 # some shortcut for the common cases
2239 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2241 my $count = abs(shift || 0);
2243 my (@a,$k,$d); # resulting array, temp, and downgrade
2246 # okay, got object as first
2251 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2253 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2257 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2258 if (defined ${"$a[0]::downgrade"})
2260 $d = ${"$a[0]::downgrade"};
2261 ${"$a[0]::downgrade"} = undef;
2264 my $up = ${"$a[0]::upgrade"};
2265 #print "Now in objectify, my class is today $a[0], count = $count\n";
2273 $k = $a[0]->new($k);
2275 elsif (!defined $up && ref($k) ne $a[0])
2277 # foreign object, try to convert to integer
2278 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2291 $k = $a[0]->new($k);
2293 elsif (!defined $up && ref($k) ne $a[0])
2295 # foreign object, try to convert to integer
2296 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2300 push @a,@_; # return other params, too
2304 require Carp; Carp::croak ("$class objectify needs list context");
2306 ${"$a[0]::downgrade"} = $d;
2314 $IMPORT++; # remember we did import()
2315 my @a; my $l = scalar @_;
2316 for ( my $i = 0; $i < $l ; $i++ )
2318 if ($_[$i] eq ':constant')
2320 # this causes overlord er load to step in
2322 integer => sub { $self->new(shift) },
2323 binary => sub { $self->new(shift) };
2325 elsif ($_[$i] eq 'upgrade')
2327 # this causes upgrading
2328 $upgrade = $_[$i+1]; # or undef to disable
2331 elsif ($_[$i] =~ /^lib$/i)
2333 # this causes a different low lib to take care...
2334 $CALC = $_[$i+1] || '';
2342 # any non :constant stuff is handled by our parent, Exporter
2343 # even if @_ is empty, to give it a chance
2344 $self->SUPER::import(@a); # need it for subclasses
2345 $self->export_to_level(1,$self,@a); # need it for MBF
2347 # try to load core math lib
2348 my @c = split /\s*,\s*/,$CALC;
2349 push @c,'Calc'; # if all fail, try this
2350 $CALC = ''; # signal error
2351 foreach my $lib (@c)
2353 next if ($lib || '') eq '';
2354 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2358 # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
2359 # used in the same script, or eval inside import().
2360 my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
2361 my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
2363 $file = File::Spec->catfile (@parts, $file);
2364 eval { require "$file"; $lib->import( @c ); }
2368 eval "use $lib qw/@c/;";
2373 # loaded it ok, see if the api_version() is high enough
2374 if ($lib->can('api_version') && $lib->api_version() >= 1.0)
2377 # api_version matches, check if it really provides anything we need
2381 add mul div sub dec inc
2382 acmp len digit is_one is_zero is_even is_odd
2384 new copy check from_hex from_bin as_hex as_bin zeros
2385 rsft lsft xor and or
2386 mod sqrt root fac pow modinv modpow log_int gcd
2389 if (!$lib->can("_$method"))
2391 if (($WARN{$lib}||0) < 2)
2394 Carp::carp ("$lib is missing method '_$method'");
2395 $WARN{$lib} = 1; # still warn about the lib
2404 last; # found a usable one, break
2408 if (($WARN{$lib}||0) < 2)
2410 my $ver = eval "\$$lib\::VERSION";
2412 Carp::carp ("Cannot load outdated $lib v$ver, please upgrade");
2413 $WARN{$lib} = 2; # never warn again
2421 Carp::croak ("Couldn't load any math lib, not even 'Calc.pm'");
2423 _fill_can_cache(); # for emulating lower math lib functions
2428 # fill $CAN with the results of $CALC->can(...)
2431 for my $method (qw/ signed_and or signed_or xor signed_xor /)
2433 $CAN{$method} = $CALC->can("_$method") ? 1 : 0;
2439 # convert a (ref to) big hex string to BigInt, return undef for error
2442 my $x = Math::BigInt->bzero();
2445 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2446 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2448 return $x->bnan() if $hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2450 my $sign = '+'; $sign = '-' if $hs =~ /^-/;
2452 $hs =~ s/^[+-]//; # strip sign
2453 $x->{value} = $CALC->_from_hex($hs);
2454 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2460 # convert a (ref to) big binary string to BigInt, return undef for error
2463 my $x = Math::BigInt->bzero();
2465 $bs =~ s/([01])_([01])/$1$2/g;
2466 $bs =~ s/([01])_([01])/$1$2/g;
2467 return $x->bnan() if $bs !~ /^[+-]?0b[01]+$/;
2469 my $sign = '+'; $sign = '-' if $bs =~ /^\-/;
2470 $bs =~ s/^[+-]//; # strip sign
2472 $x->{value} = $CALC->_from_bin($bs);
2473 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2479 # (ref to num_str) return num_str
2480 # internal, take apart a string and return the pieces
2481 # strip leading/trailing whitespace, leading zeros, underscore and reject
2485 # strip white space at front, also extranous leading zeros
2486 $x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2487 $x =~ s/^\s+//; # but this will
2488 $x =~ s/\s+$//g; # strip white space at end
2490 # shortcut, if nothing to split, return early
2491 if ($x =~ /^[+-]?\d+\z/)
2493 $x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2494 return (\$sign, \$x, \'', \'', \0);
2497 # invalid starting char?
2498 return if $x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2500 return __from_hex($x) if $x =~ /^[\-\+]?0x/; # hex string
2501 return __from_bin($x) if $x =~ /^[\-\+]?0b/; # binary string
2503 # strip underscores between digits
2504 $x =~ s/(\d)_(\d)/$1$2/g;
2505 $x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3
2507 # some possible inputs:
2508 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2509 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 # 0e999
2511 my ($m,$e,$last) = split /[Ee]/,$x;
2512 return if defined $last; # last defined => 1e2E3 or others
2513 $e = '0' if !defined $e || $e eq "";
2515 # sign,value for exponent,mantint,mantfrac
2516 my ($es,$ev,$mis,$miv,$mfv);
2518 if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2522 return if $m eq '.' || $m eq '';
2523 my ($mi,$mf,$lastf) = split /\./,$m;
2524 return if defined $lastf; # lastf defined => 1.2.3 or others
2525 $mi = '0' if !defined $mi;
2526 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2527 $mf = '0' if !defined $mf || $mf eq '';
2528 if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2530 $mis = $1||'+'; $miv = $2;
2531 return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
2533 # handle the 0e999 case here
2534 $ev = 0 if $miv eq '0' && $mfv eq '';
2535 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2538 return; # NaN, not a number
2541 ##############################################################################
2542 # internal calculation routines (others are in Math::BigInt::Calc etc)
2546 # (BINT or num_str, BINT or num_str) return BINT
2547 # does modify first argument
2550 my $x = shift; my $ty = shift;
2551 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2552 $x * $ty / bgcd($x,$ty);
2555 ###############################################################################
2556 # this method return 0 if the object can be modified, or 1 for not
2557 # We use a fast constant sub() here, to avoid costly calls. Subclasses
2558 # may override it with special code (f.i. Math::BigInt::Constant does so)
2560 sub modify () { 0; }
2567 Math::BigInt - Arbitrary size integer math package
2573 # or make it faster: install (optional) Math::BigInt::GMP
2574 # and always use (it will fall back to pure Perl if the
2575 # GMP library is not installed):
2577 use Math::BigInt lib => 'GMP';
2579 my $str = '1234567890';
2580 my @values = (64,74,18);
2581 my $n = 1; my $sign = '-';
2584 $x = Math::BigInt->new($str); # defaults to 0
2585 $y = $x->copy(); # make a true copy
2586 $nan = Math::BigInt->bnan(); # create a NotANumber
2587 $zero = Math::BigInt->bzero(); # create a +0
2588 $inf = Math::BigInt->binf(); # create a +inf
2589 $inf = Math::BigInt->binf('-'); # create a -inf
2590 $one = Math::BigInt->bone(); # create a +1
2591 $one = Math::BigInt->bone('-'); # create a -1
2593 # Testing (don't modify their arguments)
2594 # (return true if the condition is met, otherwise false)
2596 $x->is_zero(); # if $x is +0
2597 $x->is_nan(); # if $x is NaN
2598 $x->is_one(); # if $x is +1
2599 $x->is_one('-'); # if $x is -1
2600 $x->is_odd(); # if $x is odd
2601 $x->is_even(); # if $x is even
2602 $x->is_pos(); # if $x >= 0
2603 $x->is_neg(); # if $x < 0
2604 $x->is_inf($sign); # if $x is +inf, or -inf (sign is default '+')
2605 $x->is_int(); # if $x is an integer (not a float)
2607 # comparing and digit/sign extration
2608 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2609 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2610 $x->sign(); # return the sign, either +,- or NaN
2611 $x->digit($n); # return the nth digit, counting from right
2612 $x->digit(-$n); # return the nth digit, counting from left
2614 # The following all modify their first argument. If you want to preserve
2615 # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
2616 # neccessary when mixing $a = $b assigments with non-overloaded math.
2618 $x->bzero(); # set $x to 0
2619 $x->bnan(); # set $x to NaN
2620 $x->bone(); # set $x to +1
2621 $x->bone('-'); # set $x to -1
2622 $x->binf(); # set $x to inf
2623 $x->binf('-'); # set $x to -inf
2625 $x->bneg(); # negation
2626 $x->babs(); # absolute value
2627 $x->bnorm(); # normalize (no-op in BigInt)
2628 $x->bnot(); # two's complement (bit wise not)
2629 $x->binc(); # increment $x by 1
2630 $x->bdec(); # decrement $x by 1
2632 $x->badd($y); # addition (add $y to $x)
2633 $x->bsub($y); # subtraction (subtract $y from $x)
2634 $x->bmul($y); # multiplication (multiply $x by $y)
2635 $x->bdiv($y); # divide, set $x to quotient
2636 # return (quo,rem) or quo if scalar
2638 $x->bmod($y); # modulus (x % y)
2639 $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
2640 $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
2642 $x->bpow($y); # power of arguments (x ** y)
2643 $x->blsft($y); # left shift
2644 $x->brsft($y); # right shift
2645 $x->blsft($y,$n); # left shift, by base $n (like 10)
2646 $x->brsft($y,$n); # right shift, by base $n (like 10)
2648 $x->band($y); # bitwise and
2649 $x->bior($y); # bitwise inclusive or
2650 $x->bxor($y); # bitwise exclusive or
2651 $x->bnot(); # bitwise not (two's complement)
2653 $x->bsqrt(); # calculate square-root
2654 $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
2655 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2657 $x->round($A,$P,$mode); # round to accuracy or precision using mode $mode
2658 $x->bround($n); # accuracy: preserve $n digits
2659 $x->bfround($n); # round to $nth digit, no-op for BigInts
2661 # The following do not modify their arguments in BigInt (are no-ops),
2662 # but do so in BigFloat:
2664 $x->bfloor(); # return integer less or equal than $x
2665 $x->bceil(); # return integer greater or equal than $x
2667 # The following do not modify their arguments:
2669 # greatest common divisor (no OO style)
2670 my $gcd = Math::BigInt::bgcd(@values);
2671 # lowest common multiplicator (no OO style)
2672 my $lcm = Math::BigInt::blcm(@values);
2674 $x->length(); # return number of digits in number
2675 ($xl,$f) = $x->length(); # length of number and length of fraction part,
2676 # latter is always 0 digits long for BigInt's
2678 $x->exponent(); # return exponent as BigInt
2679 $x->mantissa(); # return (signed) mantissa as BigInt
2680 $x->parts(); # return (mantissa,exponent) as BigInt
2681 $x->copy(); # make a true copy of $x (unlike $y = $x;)
2682 $x->as_int(); # return as BigInt (in BigInt: same as copy())
2683 $x->numify(); # return as scalar (might overflow!)
2685 # conversation to string (do not modify their argument)
2686 $x->bstr(); # normalized string
2687 $x->bsstr(); # normalized string in scientific notation
2688 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
2689 $x->as_bin(); # as signed binary string with prefixed 0b
2692 # precision and accuracy (see section about rounding for more)
2693 $x->precision(); # return P of $x (or global, if P of $x undef)
2694 $x->precision($n); # set P of $x to $n
2695 $x->accuracy(); # return A of $x (or global, if A of $x undef)
2696 $x->accuracy($n); # set A $x to $n
2699 Math::BigInt->precision(); # get/set global P for all BigInt objects
2700 Math::BigInt->accuracy(); # get/set global A for all BigInt objects
2701 Math::BigInt->config(); # return hash containing configuration
2705 All operators (inlcuding basic math operations) are overloaded if you
2706 declare your big integers as
2708 $i = new Math::BigInt '123_456_789_123_456_789';
2710 Operations with overloaded operators preserve the arguments which is
2711 exactly what you expect.
2717 Input values to these routines may be any string, that looks like a number
2718 and results in an integer, including hexadecimal and binary numbers.
2720 Scalars holding numbers may also be passed, but note that non-integer numbers
2721 may already have lost precision due to the conversation to float. Quote
2722 your input if you want BigInt to see all the digits:
2724 $x = Math::BigInt->new(12345678890123456789); # bad
2725 $x = Math::BigInt->new('12345678901234567890'); # good
2727 You can include one underscore between any two digits.
2729 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
2730 Non-integer values result in NaN.
2732 Currently, Math::BigInt::new() defaults to 0, while Math::BigInt::new('')
2733 results in 'NaN'. This might change in the future, so use always the following
2734 explicit forms to get a zero or NaN:
2736 $zero = Math::BigInt->bzero();
2737 $nan = Math::BigInt->bnan();
2739 C<bnorm()> on a BigInt object is now effectively a no-op, since the numbers
2740 are always stored in normalized form. If passed a string, creates a BigInt
2741 object from the input.
2745 Output values are BigInt objects (normalized), except for bstr(), which
2746 returns a string in normalized form.
2747 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
2748 C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
2749 return either undef, <0, 0 or >0 and are suited for sort.
2755 Each of the methods below (except config(), accuracy() and precision())
2756 accepts three additional parameters. These arguments $A, $P and $R are
2757 accuracy, precision and round_mode. Please see the section about
2758 L<ACCURACY and PRECISION> for more information.
2764 print Dumper ( Math::BigInt->config() );
2765 print Math::BigInt->config()->{lib},"\n";
2767 Returns a hash containing the configuration, e.g. the version number, lib
2768 loaded etc. The following hash keys are currently filled in with the
2769 appropriate information.
2773 ============================================================
2774 lib Name of the low-level math library
2776 lib_version Version of low-level math library (see 'lib')
2778 class The class name of config() you just called
2780 upgrade To which class math operations might be upgraded
2782 downgrade To which class math operations might be downgraded
2784 precision Global precision
2786 accuracy Global accuracy
2788 round_mode Global round mode
2790 version version number of the class you used
2792 div_scale Fallback acccuracy for div
2794 trap_nan If true, traps creation of NaN via croak()
2796 trap_inf If true, traps creation of +inf/-inf via croak()
2799 The following values can be set by passing C<config()> a reference to a hash:
2802 upgrade downgrade precision accuracy round_mode div_scale
2806 $new_cfg = Math::BigInt->config( { trap_inf => 1, precision => 5 } );
2810 $x->accuracy(5); # local for $x
2811 CLASS->accuracy(5); # global for all members of CLASS
2812 $A = $x->accuracy(); # read out
2813 $A = CLASS->accuracy(); # read out
2815 Set or get the global or local accuracy, aka how many significant digits the
2818 Please see the section about L<ACCURACY AND PRECISION> for further details.
2820 Value must be greater than zero. Pass an undef value to disable it:
2822 $x->accuracy(undef);
2823 Math::BigInt->accuracy(undef);
2825 Returns the current accuracy. For C<$x->accuracy()> it will return either the
2826 local accuracy, or if not defined, the global. This means the return value
2827 represents the accuracy that will be in effect for $x:
2829 $y = Math::BigInt->new(1234567); # unrounded
2830 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
2831 $x = Math::BigInt->new(123456); # will be automatically rounded
2832 print "$x $y\n"; # '123500 1234567'
2833 print $x->accuracy(),"\n"; # will be 4
2834 print $y->accuracy(),"\n"; # also 4, since global is 4
2835 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
2836 print $x->accuracy(),"\n"; # still 4
2837 print $y->accuracy(),"\n"; # 5, since global is 5
2839 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
2840 globals separated from Math::BigInt, but it is possible to subclass
2841 Math::BigInt and make the globals of the subclass aliases to the ones from
2846 $x->precision(-2); # local for $x, round right of the dot
2847 $x->precision(2); # ditto, but round left of the dot
2848 CLASS->accuracy(5); # global for all members of CLASS
2849 CLASS->precision(-5); # ditto
2850 $P = CLASS->precision(); # read out
2851 $P = $x->precision(); # read out
2853 Set or get the global or local precision, aka how many digits the result has
2854 after the dot (or where to round it when passing a positive number). In
2855 Math::BigInt, passing a negative number precision has no effect since no
2856 numbers have digits after the dot.
2858 Please see the section about L<ACCURACY AND PRECISION> for further details.
2860 Value must be greater than zero. Pass an undef value to disable it:
2862 $x->precision(undef);
2863 Math::BigInt->precision(undef);
2865 Returns the current precision. For C<$x->precision()> it will return either the
2866 local precision of $x, or if not defined, the global. This means the return
2867 value represents the accuracy that will be in effect for $x:
2869 $y = Math::BigInt->new(1234567); # unrounded
2870 print Math::BigInt->precision(4),"\n"; # set 4, print 4
2871 $x = Math::BigInt->new(123456); # will be automatically rounded
2873 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
2874 globals separated from Math::BigInt, but it is possible to subclass
2875 Math::BigInt and make the globals of the subclass aliases to the ones from
2882 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
2883 2, but others work, too.
2885 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
2889 $x = Math::BigInt->new(10);
2890 $x->brsft(1); # same as $x >> 1: 5
2891 $x = Math::BigInt->new(1234);
2892 $x->brsft(2,10); # result 12
2894 There is one exception, and that is base 2 with negative $x:
2897 $x = Math::BigInt->new(-5);
2900 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
2905 $x = Math::BigInt->new($str,$A,$P,$R);
2907 Creates a new BigInt object from a scalar or another BigInt object. The
2908 input is accepted as decimal, hex (with leading '0x') or binary (with leading
2911 See L<Input> for more info on accepted input formats.
2915 $x = Math::BigInt->bnan();
2917 Creates a new BigInt object representing NaN (Not A Number).
2918 If used on an object, it will set it to NaN:
2924 $x = Math::BigInt->bzero();
2926 Creates a new BigInt object representing zero.
2927 If used on an object, it will set it to zero:
2933 $x = Math::BigInt->binf($sign);
2935 Creates a new BigInt object representing infinity. The optional argument is
2936 either '-' or '+', indicating whether you want infinity or minus infinity.
2937 If used on an object, it will set it to infinity:
2944 $x = Math::BigInt->binf($sign);
2946 Creates a new BigInt object representing one. The optional argument is
2947 either '-' or '+', indicating whether you want one or minus one.
2948 If used on an object, it will set it to one:
2953 =head2 is_one()/is_zero()/is_nan()/is_inf()
2956 $x->is_zero(); # true if arg is +0
2957 $x->is_nan(); # true if arg is NaN
2958 $x->is_one(); # true if arg is +1
2959 $x->is_one('-'); # true if arg is -1
2960 $x->is_inf(); # true if +inf
2961 $x->is_inf('-'); # true if -inf (sign is default '+')
2963 These methods all test the BigInt for beeing one specific value and return
2964 true or false depending on the input. These are faster than doing something
2969 =head2 is_pos()/is_neg()
2971 $x->is_pos(); # true if >= 0
2972 $x->is_neg(); # true if < 0
2974 The methods return true if the argument is positive or negative, respectively.
2975 C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
2976 C<-inf> is negative. A C<zero> is positive.
2978 These methods are only testing the sign, and not the value.
2980 C<is_positive()> and C<is_negative()> are aliase to C<is_pos()> and
2981 C<is_neg()>, respectively. C<is_positive()> and C<is_negative()> were
2982 introduced in v1.36, while C<is_pos()> and C<is_neg()> were only introduced
2985 =head2 is_odd()/is_even()/is_int()
2987 $x->is_odd(); # true if odd, false for even
2988 $x->is_even(); # true if even, false for odd
2989 $x->is_int(); # true if $x is an integer
2991 The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
2992 C<-inf> are not integers and are neither odd nor even.
2994 In BigInt, all numbers except C<NaN>, C<+inf> and C<-inf> are integers.
3000 Compares $x with $y and takes the sign into account.
3001 Returns -1, 0, 1 or undef.
3007 Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
3013 Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
3017 $x->digit($n); # return the nth digit, counting from right
3019 If C<$n> is negative, returns the digit counting from left.
3025 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
3026 and '-inf', respectively. Does nothing for NaN or zero.
3032 Set the number to it's absolute value, e.g. change the sign from '-' to '+'
3033 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
3038 $x->bnorm(); # normalize (no-op)
3044 Two's complement (bit wise not). This is equivalent to
3052 $x->binc(); # increment x by 1
3056 $x->bdec(); # decrement x by 1
3060 $x->badd($y); # addition (add $y to $x)
3064 $x->bsub($y); # subtraction (subtract $y from $x)
3068 $x->bmul($y); # multiplication (multiply $x by $y)
3072 $x->bdiv($y); # divide, set $x to quotient
3073 # return (quo,rem) or quo if scalar
3077 $x->bmod($y); # modulus (x % y)
3081 num->bmodinv($mod); # modular inverse
3083 Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
3084 returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
3085 C<bgcd($num, $mod)==1>.
3089 $num->bmodpow($exp,$mod); # modular exponentation
3090 # ($num**$exp % $mod)
3092 Returns the value of C<$num> taken to the power C<$exp> in the modulus
3093 C<$mod> using binary exponentation. C<bmodpow> is far superior to
3098 because it is much faster - it reduces internal variables into
3099 the modulus whenever possible, so it operates on smaller numbers.
3101 C<bmodpow> also supports negative exponents.
3103 bmodpow($num, -1, $mod)
3105 is exactly equivalent to
3111 $x->bpow($y); # power of arguments (x ** y)
3115 $x->blsft($y); # left shift
3116 $x->blsft($y,$n); # left shift, in base $n (like 10)
3120 $x->brsft($y); # right shift
3121 $x->brsft($y,$n); # right shift, in base $n (like 10)
3125 $x->band($y); # bitwise and
3129 $x->bior($y); # bitwise inclusive or
3133 $x->bxor($y); # bitwise exclusive or
3137 $x->bnot(); # bitwise not (two's complement)
3141 $x->bsqrt(); # calculate square-root
3145 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3149 $x->round($A,$P,$round_mode);
3151 Round $x to accuracy C<$A> or precision C<$P> using the round mode
3156 $x->bround($N); # accuracy: preserve $N digits
3160 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3166 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3167 does change $x in BigFloat.
3173 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3174 does change $x in BigFloat.
3178 bgcd(@values); # greatest common divisor (no OO style)
3182 blcm(@values); # lowest common multiplicator (no OO style)
3187 ($xl,$fl) = $x->length();
3189 Returns the number of digits in the decimal representation of the number.
3190 In list context, returns the length of the integer and fraction part. For
3191 BigInt's, the length of the fraction part will always be 0.
3197 Return the exponent of $x as BigInt.
3203 Return the signed mantissa of $x as BigInt.
3207 $x->parts(); # return (mantissa,exponent) as BigInt
3211 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3217 Returns $x as a BigInt (truncated towards zero). In BigInt this is the same as
3220 C<as_number()> is an alias to this method. C<as_number> was introduced in
3221 v1.22, while C<as_int()> was only introduced in v1.68.
3227 Returns a normalized string represantation of C<$x>.
3231 $x->bsstr(); # normalized string in scientific notation
3235 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3239 $x->as_bin(); # as signed binary string with prefixed 0b
3241 =head1 ACCURACY and PRECISION
3243 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3244 accuracy and precision based rounding, both automatically after every
3245 operation, as well as manually.
3247 This section describes the accuracy/precision handling in Math::Big* as it
3248 used to be and as it is now, complete with an explanation of all terms and
3251 Not yet implemented things (but with correct description) are marked with '!',
3252 things that need to be answered are marked with '?'.
3254 In the next paragraph follows a short description of terms used here (because
3255 these may differ from terms used by others people or documentation).
3257 During the rest of this document, the shortcuts A (for accuracy), P (for
3258 precision), F (fallback) and R (rounding mode) will be used.
3262 A fixed number of digits before (positive) or after (negative)
3263 the decimal point. For example, 123.45 has a precision of -2. 0 means an
3264 integer like 123 (or 120). A precision of 2 means two digits to the left
3265 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3266 numbers with zeros before the decimal point may have different precisions,
3267 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3268 was). It could also have p < 0, when the digits after the decimal point
3271 The string output (of floating point numbers) will be padded with zeros:
3273 Initial value P A Result String
3274 ------------------------------------------------------------
3275 1234.01 -3 1000 1000
3278 1234.001 1 1234 1234.0
3280 1234.01 2 1234.01 1234.01
3281 1234.01 5 1234.01 1234.01000
3283 For BigInts, no padding occurs.
3287 Number of significant digits. Leading zeros are not counted. A
3288 number may have an accuracy greater than the non-zero digits
3289 when there are zeros in it or trailing zeros. For example, 123.456 has
3290 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3292 The string output (of floating point numbers) will be padded with zeros:
3294 Initial value P A Result String
3295 ------------------------------------------------------------
3297 1234.01 6 1234.01 1234.01
3298 1234.1 8 1234.1 1234.1000
3300 For BigInts, no padding occurs.
3304 When both A and P are undefined, this is used as a fallback accuracy when
3307 =head2 Rounding mode R
3309 When rounding a number, different 'styles' or 'kinds'
3310 of rounding are possible. (Note that random rounding, as in
3311 Math::Round, is not implemented.)
3317 truncation invariably removes all digits following the
3318 rounding place, replacing them with zeros. Thus, 987.65 rounded
3319 to tens (P=1) becomes 980, and rounded to the fourth sigdig
3320 becomes 987.6 (A=4). 123.456 rounded to the second place after the
3321 decimal point (P=-2) becomes 123.46.
3323 All other implemented styles of rounding attempt to round to the
3324 "nearest digit." If the digit D immediately to the right of the
3325 rounding place (skipping the decimal point) is greater than 5, the
3326 number is incremented at the rounding place (possibly causing a
3327 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3328 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3329 truncated at the rounding place: e.g. when rounding to units, 0.4
3330 rounds to 0, and -19.4 rounds to -19.
3332 However the results of other styles of rounding differ if the
3333 digit immediately to the right of the rounding place (skipping the
3334 decimal point) is 5 and if there are no digits, or no digits other
3335 than 0, after that 5. In such cases:
3339 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3340 if it is not already. E.g., when rounding to the first sigdig, 0.45
3341 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3345 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3346 it is not already. E.g., when rounding to the first sigdig, 0.45
3347 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3351 round to plus infinity, i.e. always round up. E.g., when
3352 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3353 and 0.4501 also becomes 0.5.
3357 round to minus infinity, i.e. always round down. E.g., when
3358 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3359 but 0.4501 becomes 0.5.
3363 round to zero, i.e. positive numbers down, negative ones up.
3364 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3365 becomes -0.5, but 0.4501 becomes 0.5.
3369 The handling of A & P in MBI/MBF (the old core code shipped with Perl
3370 versions <= 5.7.2) is like this:
3376 * ffround($p) is able to round to $p number of digits after the decimal
3378 * otherwise P is unused
3380 =item Accuracy (significant digits)
3382 * fround($a) rounds to $a significant digits
3383 * only fdiv() and fsqrt() take A as (optional) paramater
3384 + other operations simply create the same number (fneg etc), or more (fmul)
3386 + rounding/truncating is only done when explicitly calling one of fround
3387 or ffround, and never for BigInt (not implemented)
3388 * fsqrt() simply hands its accuracy argument over to fdiv.
3389 * the documentation and the comment in the code indicate two different ways
3390 on how fdiv() determines the maximum number of digits it should calculate,
3391 and the actual code does yet another thing
3393 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3395 result has at most max(scale, length(dividend), length(divisor)) digits
3397 scale = max(scale, length(dividend)-1,length(divisor)-1);
3398 scale += length(divisior) - length(dividend);
3399 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3400 Actually, the 'difference' added to the scale is calculated from the
3401 number of "significant digits" in dividend and divisor, which is derived
3402 by looking at the length of the mantissa. Which is wrong, since it includes
3403 the + sign (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops
3404 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3405 assumption that 124 has 3 significant digits, while 120/7 will get you
3406 '17', not '17.1' since 120 is thought to have 2 significant digits.
3407 The rounding after the division then uses the remainder and $y to determine
3408 wether it must round up or down.
3409 ? I have no idea which is the right way. That's why I used a slightly more
3410 ? simple scheme and tweaked the few failing testcases to match it.
3414 This is how it works now:
3418 =item Setting/Accessing
3420 * You can set the A global via C<< Math::BigInt->accuracy() >> or
3421 C<< Math::BigFloat->accuracy() >> or whatever class you are using.
3422 * You can also set P globally by using C<< Math::SomeClass->precision() >>
3424 * Globals are classwide, and not inherited by subclasses.
3425 * to undefine A, use C<< Math::SomeCLass->accuracy(undef); >>
3426 * to undefine P, use C<< Math::SomeClass->precision(undef); >>
3427 * Setting C<< Math::SomeClass->accuracy() >> clears automatically
3428 C<< Math::SomeClass->precision() >>, and vice versa.
3429 * To be valid, A must be > 0, P can have any value.
3430 * If P is negative, this means round to the P'th place to the right of the
3431 decimal point; positive values mean to the left of the decimal point.
3432 P of 0 means round to integer.
3433 * to find out the current global A, use C<< Math::SomeClass->accuracy() >>
3434 * to find out the current global P, use C<< Math::SomeClass->precision() >>
3435 * use C<< $x->accuracy() >> respective C<< $x->precision() >> for the local
3436 setting of C<< $x >>.
3437 * Please note that C<< $x->accuracy() >> respecive C<< $x->precision() >>
3438 return eventually defined global A or P, when C<< $x >>'s A or P is not
3441 =item Creating numbers
3443 * When you create a number, you can give it's desired A or P via:
3444 $x = Math::BigInt->new($number,$A,$P);
3445 * Only one of A or P can be defined, otherwise the result is NaN
3446 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
3447 globals (if set) will be used. Thus changing the global defaults later on
3448 will not change the A or P of previously created numbers (i.e., A and P of
3449 $x will be what was in effect when $x was created)
3450 * If given undef for A and P, B<no> rounding will occur, and the globals will
3451 B<not> be used. This is used by subclasses to create numbers without
3452 suffering rounding in the parent. Thus a subclass is able to have it's own
3453 globals enforced upon creation of a number by using
3454 C<< $x = Math::BigInt->new($number,undef,undef) >>:
3456 use Math::BigInt::SomeSubclass;
3459 Math::BigInt->accuracy(2);
3460 Math::BigInt::SomeSubClass->accuracy(3);
3461 $x = Math::BigInt::SomeSubClass->new(1234);
3463 $x is now 1230, and not 1200. A subclass might choose to implement
3464 this otherwise, e.g. falling back to the parent's A and P.
3468 * If A or P are enabled/defined, they are used to round the result of each
3469 operation according to the rules below
3470 * Negative P is ignored in Math::BigInt, since BigInts never have digits
3471 after the decimal point
3472 * Math::BigFloat uses Math::BigInt internally, but setting A or P inside
3473 Math::BigInt as globals does not tamper with the parts of a BigFloat.
3474 A flag is used to mark all Math::BigFloat numbers as 'never round'.
3478 * It only makes sense that a number has only one of A or P at a time.
3479 If you set either A or P on one object, or globally, the other one will
3480 be automatically cleared.
3481 * If two objects are involved in an operation, and one of them has A in
3482 effect, and the other P, this results in an error (NaN).
3483 * A takes precendence over P (Hint: A comes before P).
3484 If neither of them is defined, nothing is used, i.e. the result will have
3485 as many digits as it can (with an exception for fdiv/fsqrt) and will not
3487 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
3488 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
3489 If either the dividend's or the divisor's mantissa has more digits than
3490 the value of F, the higher value will be used instead of F.
3491 This is to limit the digits (A) of the result (just consider what would
3492 happen with unlimited A and P in the case of 1/3 :-)
3493 * fdiv will calculate (at least) 4 more digits than required (determined by
3494 A, P or F), and, if F is not used, round the result
3495 (this will still fail in the case of a result like 0.12345000000001 with A
3496 or P of 5, but this can not be helped - or can it?)
3497 * Thus you can have the math done by on Math::Big* class in two modi:
3498 + never round (this is the default):
3499 This is done by setting A and P to undef. No math operation
3500 will round the result, with fdiv() and fsqrt() as exceptions to guard
3501 against overflows. You must explicitely call bround(), bfround() or
3502 round() (the latter with parameters).
3503 Note: Once you have rounded a number, the settings will 'stick' on it
3504 and 'infect' all other numbers engaged in math operations with it, since
3505 local settings have the highest precedence. So, to get SaferRound[tm],
3506 use a copy() before rounding like this:
3508 $x = Math::BigFloat->new(12.34);
3509 $y = Math::BigFloat->new(98.76);
3510 $z = $x * $y; # 1218.6984
3511 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
3512 $z = $x * $y; # still 1218.6984, without
3513 # copy would have been 1210!
3515 + round after each op:
3516 After each single operation (except for testing like is_zero()), the
3517 method round() is called and the result is rounded appropriately. By
3518 setting proper values for A and P, you can have all-the-same-A or
3519 all-the-same-P modes. For example, Math::Currency might set A to undef,
3520 and P to -2, globally.
3522 ?Maybe an extra option that forbids local A & P settings would be in order,
3523 ?so that intermediate rounding does not 'poison' further math?
3525 =item Overriding globals
3527 * you will be able to give A, P and R as an argument to all the calculation
3528 routines; the second parameter is A, the third one is P, and the fourth is
3529 R (shift right by one for binary operations like badd). P is used only if
3530 the first parameter (A) is undefined. These three parameters override the
3531 globals in the order detailed as follows, i.e. the first defined value
3533 (local: per object, global: global default, parameter: argument to sub)
3536 + local A (if defined on both of the operands: smaller one is taken)
3537 + local P (if defined on both of the operands: bigger one is taken)
3541 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
3542 arguments (A and P) instead of one
3544 =item Local settings
3546 * You can set A or P locally by using C<< $x->accuracy() >> or
3547 C<< $x->precision() >>
3548 and thus force different A and P for different objects/numbers.
3549 * Setting A or P this way immediately rounds $x to the new value.
3550 * C<< $x->accuracy() >> clears C<< $x->precision() >>, and vice versa.
3554 * the rounding routines will use the respective global or local settings.
3555 fround()/bround() is for accuracy rounding, while ffround()/bfround()
3557 * the two rounding functions take as the second parameter one of the
3558 following rounding modes (R):
3559 'even', 'odd', '+inf', '-inf', 'zero', 'trunc'
3560 * you can set/get the global R by using C<< Math::SomeClass->round_mode() >>
3561 or by setting C<< $Math::SomeClass::round_mode >>
3562 * after each operation, C<< $result->round() >> is called, and the result may
3563 eventually be rounded (that is, if A or P were set either locally,
3564 globally or as parameter to the operation)
3565 * to manually round a number, call C<< $x->round($A,$P,$round_mode); >>
3566 this will round the number by using the appropriate rounding function
3567 and then normalize it.
3568 * rounding modifies the local settings of the number:
3570 $x = Math::BigFloat->new(123.456);
3574 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
3575 will be 4 from now on.
3577 =item Default values
3586 * The defaults are set up so that the new code gives the same results as
3587 the old code (except in a few cases on fdiv):
3588 + Both A and P are undefined and thus will not be used for rounding
3589 after each operation.
3590 + round() is thus a no-op, unless given extra parameters A and P
3596 The actual numbers are stored as unsigned big integers (with seperate sign).
3597 You should neither care about nor depend on the internal representation; it
3598 might change without notice. Use only method calls like C<< $x->sign(); >>
3599 instead relying on the internal hash keys like in C<< $x->{sign}; >>.
3603 Math with the numbers is done (by default) by a module called
3604 C<Math::BigInt::Calc>. This is equivalent to saying:
3606 use Math::BigInt lib => 'Calc';
3608 You can change this by using:
3610 use Math::BigInt lib => 'BitVect';
3612 The following would first try to find Math::BigInt::Foo, then
3613 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
3615 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
3617 Since Math::BigInt::GMP is in almost all cases faster than Calc (especially in
3618 cases involving really big numbers, where it is B<much> faster), and there is
3619 no penalty if Math::BigInt::GMP is not installed, it is a good idea to always
3622 use Math::BigInt lib => 'GMP';
3624 Different low-level libraries use different formats to store the
3625 numbers. You should not depend on the number having a specific format.
3627 See the respective math library module documentation for further details.
3631 The sign is either '+', '-', 'NaN', '+inf' or '-inf' and stored seperately.
3633 A sign of 'NaN' is used to represent the result when input arguments are not
3634 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
3635 minus infinity. You will get '+inf' when dividing a positive number by 0, and
3636 '-inf' when dividing any negative number by 0.
3638 =head2 mantissa(), exponent() and parts()
3640 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
3643 $m = $x->mantissa();
3644 $e = $x->exponent();
3645 $y = $m * ( 10 ** $e );
3646 print "ok\n" if $x == $y;
3648 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
3649 in one go. Both the returned mantissa and exponent have a sign.
3651 Currently, for BigInts C<$e> is always 0, except for NaN, +inf and -inf,
3652 where it is C<NaN>; and for C<$x == 0>, where it is C<1> (to be compatible
3653 with Math::BigFloat's internal representation of a zero as C<0E1>).
3655 C<$m> is currently just a copy of the original number. The relation between
3656 C<$e> and C<$m> will stay always the same, though their real values might
3663 sub bint { Math::BigInt->new(shift); }
3665 $x = Math::BigInt->bstr("1234") # string "1234"
3666 $x = "$x"; # same as bstr()
3667 $x = Math::BigInt->bneg("1234"); # BigInt "-1234"
3668 $x = Math::BigInt->babs("-12345"); # BigInt "12345"
3669 $x = Math::BigInt->bnorm("-0 00"); # BigInt "0"
3670 $x = bint(1) + bint(2); # BigInt "3"
3671 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
3672 $x = bint(1); # BigInt "1"
3673 $x = $x + 5 / 2; # BigInt "3"
3674 $x = $x ** 3; # BigInt "27"
3675 $x *= 2; # BigInt "54"
3676 $x = Math::BigInt->new(0); # BigInt "0"
3678 $x = Math::BigInt->badd(4,5) # BigInt "9"
3679 print $x->bsstr(); # 9e+0
3681 Examples for rounding:
3686 $x = Math::BigFloat->new(123.4567);
3687 $y = Math::BigFloat->new(123.456789);
3688 Math::BigFloat->accuracy(4); # no more A than 4
3690 ok ($x->copy()->fround(),123.4); # even rounding
3691 print $x->copy()->fround(),"\n"; # 123.4
3692 Math::BigFloat->round_mode('odd'); # round to odd
3693 print $x->copy()->fround(),"\n"; # 123.5
3694 Math::BigFloat->accuracy(5); # no more A than 5
3695 Math::BigFloat->round_mode('odd'); # round to odd
3696 print $x->copy()->fround(),"\n"; # 123.46
3697 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
3698 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
3700 Math::BigFloat->accuracy(undef); # A not important now
3701 Math::BigFloat->precision(2); # P important
3702 print $x->copy()->bnorm(),"\n"; # 123.46
3703 print $x->copy()->fround(),"\n"; # 123.46
3705 Examples for converting:
3707 my $x = Math::BigInt->new('0b1'.'01' x 123);
3708 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
3710 =head1 Autocreating constants
3712 After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
3713 and binary constants in the given scope are converted to C<Math::BigInt>.
3714 This conversion happens at compile time.
3718 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
3720 prints the integer value of C<2**100>. Note that without conversion of
3721 constants the expression 2**100 will be calculated as perl scalar.
3723 Please note that strings and floating point constants are not affected,
3726 use Math::BigInt qw/:constant/;
3728 $x = 1234567890123456789012345678901234567890
3729 + 123456789123456789;
3730 $y = '1234567890123456789012345678901234567890'
3731 + '123456789123456789';
3733 do not work. You need an explicit Math::BigInt->new() around one of the
3734 operands. You should also quote large constants to protect loss of precision:
3738 $x = Math::BigInt->new('1234567889123456789123456789123456789');
3740 Without the quotes Perl would convert the large number to a floating point
3741 constant at compile time and then hand the result to BigInt, which results in
3742 an truncated result or a NaN.
3744 This also applies to integers that look like floating point constants:
3746 use Math::BigInt ':constant';
3748 print ref(123e2),"\n";
3749 print ref(123.2e2),"\n";
3751 will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
3752 to get this to work.
3756 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
3757 must be made in the second case. For long numbers, the copy can eat up to 20%
3758 of the work (in the case of addition/subtraction, less for
3759 multiplication/division). If $y is very small compared to $x, the form
3760 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
3761 more time then the actual addition.
3763 With a technique called copy-on-write, the cost of copying with overload could
3764 be minimized or even completely avoided. A test implementation of COW did show
3765 performance gains for overloaded math, but introduced a performance loss due
3766 to a constant overhead for all other operatons. So Math::BigInt does currently
3769 The rewritten version of this module (vs. v0.01) is slower on certain
3770 operations, like C<new()>, C<bstr()> and C<numify()>. The reason are that it
3771 does now more work and handles much more cases. The time spent in these
3772 operations is usually gained in the other math operations so that code on
3773 the average should get (much) faster. If they don't, please contact the author.
3775 Some operations may be slower for small numbers, but are significantly faster
3776 for big numbers. Other operations are now constant (O(1), like C<bneg()>,
3777 C<babs()> etc), instead of O(N) and thus nearly always take much less time.
3778 These optimizations were done on purpose.
3780 If you find the Calc module to slow, try to install any of the replacement
3781 modules and see if they help you.
3783 =head2 Alternative math libraries
3785 You can use an alternative library to drive Math::BigInt via:
3787 use Math::BigInt lib => 'Module';
3789 See L<MATH LIBRARY> for more information.
3791 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
3795 =head1 Subclassing Math::BigInt
3797 The basic design of Math::BigInt allows simple subclasses with very little
3798 work, as long as a few simple rules are followed:
3804 The public API must remain consistent, i.e. if a sub-class is overloading
3805 addition, the sub-class must use the same name, in this case badd(). The
3806 reason for this is that Math::BigInt is optimized to call the object methods
3811 The private object hash keys like C<$x->{sign}> may not be changed, but
3812 additional keys can be added, like C<$x->{_custom}>.
3816 Accessor functions are available for all existing object hash keys and should
3817 be used instead of directly accessing the internal hash keys. The reason for
3818 this is that Math::BigInt itself has a pluggable interface which permits it
3819 to support different storage methods.
3823 More complex sub-classes may have to replicate more of the logic internal of
3824 Math::BigInt if they need to change more basic behaviors. A subclass that
3825 needs to merely change the output only needs to overload C<bstr()>.
3827 All other object methods and overloaded functions can be directly inherited
3828 from the parent class.
3830 At the very minimum, any subclass will need to provide it's own C<new()> and can
3831 store additional hash keys in the object. There are also some package globals
3832 that must be defined, e.g.:
3836 $precision = -2; # round to 2 decimal places
3837 $round_mode = 'even';
3840 Additionally, you might want to provide the following two globals to allow
3841 auto-upgrading and auto-downgrading to work correctly:
3846 This allows Math::BigInt to correctly retrieve package globals from the
3847 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
3848 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
3854 in your subclass to automatically inherit the overloading from the parent. If
3855 you like, you can change part of the overloading, look at Math::String for an
3860 When used like this:
3862 use Math::BigInt upgrade => 'Foo::Bar';
3864 certain operations will 'upgrade' their calculation and thus the result to
3865 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
3867 use Math::BigInt upgrade => 'Math::BigFloat';
3869 As a shortcut, you can use the module C<bignum>:
3873 Also good for oneliners:
3875 perl -Mbignum -le 'print 2 ** 255'
3877 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
3878 as well es preserve accuracy (as in sqrt(3)).
3880 Beware: This feature is not fully implemented yet.
3884 The following methods upgrade themselves unconditionally; that is if upgrade
3885 is in effect, they will always hand up their work:
3897 Beware: This list is not complete.
3899 All other methods upgrade themselves only when one (or all) of their
3900 arguments are of the class mentioned in $upgrade (This might change in later
3901 versions to a more sophisticated scheme):
3907 =item broot() does not work
3909 The broot() function in BigInt may only work for small values. This will be
3910 fixed in a later version.
3912 =item Out of Memory!
3914 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
3915 C<eval()> in your code will crash with "Out of memory". This is probably an
3916 overload/exporter bug. You can workaround by not having C<eval()>
3917 and ':constant' at the same time or upgrade your Perl to a newer version.
3919 =item Fails to load Calc on Perl prior 5.6.0
3921 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
3922 will fall back to eval { require ... } when loading the math lib on Perls
3923 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
3924 filesystems using a different seperator.
3930 Some things might not work as you expect them. Below is documented what is
3931 known to be troublesome:
3935 =item bstr(), bsstr() and 'cmp'
3937 Both C<bstr()> and C<bsstr()> as well as automated stringify via overload now
3938 drop the leading '+'. The old code would return '+3', the new returns '3'.
3939 This is to be consistent with Perl and to make C<cmp> (especially with
3940 overloading) to work as you expect. It also solves problems with C<Test.pm>,
3941 because it's C<ok()> uses 'eq' internally.
3943 Mark Biggar said, when asked about to drop the '+' altogether, or make only
3946 I agree (with the first alternative), don't add the '+' on positive
3947 numbers. It's not as important anymore with the new internal
3948 form for numbers. It made doing things like abs and neg easier,
3949 but those have to be done differently now anyway.
3951 So, the following examples will now work all as expected:
3954 BEGIN { plan tests => 1 }
3957 my $x = new Math::BigInt 3*3;
3958 my $y = new Math::BigInt 3*3;
3961 print "$x eq 9" if $x eq $y;
3962 print "$x eq 9" if $x eq '9';
3963 print "$x eq 9" if $x eq 3*3;
3965 Additionally, the following still works:
3967 print "$x == 9" if $x == $y;
3968 print "$x == 9" if $x == 9;
3969 print "$x == 9" if $x == 3*3;
3971 There is now a C<bsstr()> method to get the string in scientific notation aka
3972 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
3973 for comparisation, but Perl will represent some numbers as 100 and others
3974 as 1e+308. If in doubt, convert both arguments to Math::BigInt before
3975 comparing them as strings:
3978 BEGIN { plan tests => 3 }
3981 $x = Math::BigInt->new('1e56'); $y = 1e56;
3982 ok ($x,$y); # will fail
3983 ok ($x->bsstr(),$y); # okay
3984 $y = Math::BigInt->new($y);
3987 Alternatively, simple use C<< <=> >> for comparisations, this will get it
3988 always right. There is not yet a way to get a number automatically represented
3989 as a string that matches exactly the way Perl represents it.
3993 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
3996 $x = Math::BigInt->new(123);
3997 $y = int($x); # BigInt 123
3998 $x = Math::BigFloat->new(123.45);
3999 $y = int($x); # BigInt 123
4001 In all Perl versions you can use C<as_number()> for the same effect:
4003 $x = Math::BigFloat->new(123.45);
4004 $y = $x->as_number(); # BigInt 123
4006 This also works for other subclasses, like Math::String.
4008 It is yet unlcear whether overloaded int() should return a scalar or a BigInt.
4012 The following will probably not do what you expect:
4014 $c = Math::BigInt->new(123);
4015 print $c->length(),"\n"; # prints 30
4017 It prints both the number of digits in the number and in the fraction part
4018 since print calls C<length()> in list context. Use something like:
4020 print scalar $c->length(),"\n"; # prints 3
4024 The following will probably not do what you expect:
4026 print $c->bdiv(10000),"\n";
4028 It prints both quotient and remainder since print calls C<bdiv()> in list
4029 context. Also, C<bdiv()> will modify $c, so be carefull. You probably want
4032 print $c / 10000,"\n";
4033 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
4037 The quotient is always the greatest integer less than or equal to the
4038 real-valued quotient of the two operands, and the remainder (when it is
4039 nonzero) always has the same sign as the second operand; so, for
4049 As a consequence, the behavior of the operator % agrees with the
4050 behavior of Perl's built-in % operator (as documented in the perlop
4051 manpage), and the equation
4053 $x == ($x / $y) * $y + ($x % $y)
4055 holds true for any $x and $y, which justifies calling the two return
4056 values of bdiv() the quotient and remainder. The only exception to this rule
4057 are when $y == 0 and $x is negative, then the remainder will also be
4058 negative. See below under "infinity handling" for the reasoning behing this.
4060 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
4061 not change BigInt's way to do things. This is because under 'use integer' Perl
4062 will do what the underlying C thinks is right and this is different for each
4063 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
4064 the author to implement it ;)
4066 =item infinity handling
4068 Here are some examples that explain the reasons why certain results occur while
4071 The following table shows the result of the division and the remainder, so that
4072 the equation above holds true. Some "ordinary" cases are strewn in to show more
4073 clearly the reasoning:
4075 A / B = C, R so that C * B + R = A
4076 =========================================================
4077 5 / 8 = 0, 5 0 * 8 + 5 = 5
4078 0 / 8 = 0, 0 0 * 8 + 0 = 0
4079 0 / inf = 0, 0 0 * inf + 0 = 0
4080 0 /-inf = 0, 0 0 * -inf + 0 = 0
4081 5 / inf = 0, 5 0 * inf + 5 = 5
4082 5 /-inf = 0, 5 0 * -inf + 5 = 5
4083 -5/ inf = 0, -5 0 * inf + -5 = -5
4084 -5/-inf = 0, -5 0 * -inf + -5 = -5
4085 inf/ 5 = inf, 0 inf * 5 + 0 = inf
4086 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
4087 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
4088 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
4089 5/ 5 = 1, 0 1 * 5 + 0 = 5
4090 -5/ -5 = 1, 0 1 * -5 + 0 = -5
4091 inf/ inf = 1, 0 1 * inf + 0 = inf
4092 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
4093 inf/-inf = -1, 0 -1 * -inf + 0 = inf
4094 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
4095 8/ 0 = inf, 8 inf * 0 + 8 = 8
4096 inf/ 0 = inf, inf inf * 0 + inf = inf
4099 These cases below violate the "remainder has the sign of the second of the two
4100 arguments", since they wouldn't match up otherwise.
4102 A / B = C, R so that C * B + R = A
4103 ========================================================
4104 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
4105 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
4107 =item Modifying and =
4111 $x = Math::BigFloat->new(5);
4114 It will not do what you think, e.g. making a copy of $x. Instead it just makes
4115 a second reference to the B<same> object and stores it in $y. Thus anything
4116 that modifies $x (except overloaded operators) will modify $y, and vice versa.
4117 Or in other words, C<=> is only safe if you modify your BigInts only via
4118 overloaded math. As soon as you use a method call it breaks:
4121 print "$x, $y\n"; # prints '10, 10'
4123 If you want a true copy of $x, use:
4127 You can also chain the calls like this, this will make first a copy and then
4130 $y = $x->copy()->bmul(2);
4132 See also the documentation for overload.pm regarding C<=>.
4136 C<bpow()> (and the rounding functions) now modifies the first argument and
4137 returns it, unlike the old code which left it alone and only returned the
4138 result. This is to be consistent with C<badd()> etc. The first three will
4139 modify $x, the last one won't:
4141 print bpow($x,$i),"\n"; # modify $x
4142 print $x->bpow($i),"\n"; # ditto
4143 print $x **= $i,"\n"; # the same
4144 print $x ** $i,"\n"; # leave $x alone
4146 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4148 =item Overloading -$x
4158 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4159 needs to preserve $x since it does not know that it later will get overwritten.
4160 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4162 With Copy-On-Write, this issue would be gone, but C-o-W is not implemented
4163 since it is slower for all other things.
4165 =item Mixing different object types
4167 In Perl you will get a floating point value if you do one of the following:
4173 With overloaded math, only the first two variants will result in a BigFloat:
4178 $mbf = Math::BigFloat->new(5);
4179 $mbi2 = Math::BigInteger->new(5);
4180 $mbi = Math::BigInteger->new(2);
4182 # what actually gets called:
4183 $float = $mbf + $mbi; # $mbf->badd()
4184 $float = $mbf / $mbi; # $mbf->bdiv()
4185 $integer = $mbi + $mbf; # $mbi->badd()
4186 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4187 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4189 This is because math with overloaded operators follows the first (dominating)
4190 operand, and the operation of that is called and returns thus the result. So,
4191 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4192 the result should be a Math::BigFloat or the second operant is one.
4194 To get a Math::BigFloat you either need to call the operation manually,
4195 make sure the operands are already of the proper type or casted to that type
4196 via Math::BigFloat->new():
4198 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4200 Beware of simple "casting" the entire expression, this would only convert
4201 the already computed result:
4203 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4205 Beware also of the order of more complicated expressions like:
4207 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4208 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4210 If in doubt, break the expression into simpler terms, or cast all operands
4211 to the desired resulting type.
4213 Scalar values are a bit different, since:
4218 will both result in the proper type due to the way the overloaded math works.
4220 This section also applies to other overloaded math packages, like Math::String.
4222 One solution to you problem might be autoupgrading|upgrading. See the
4223 pragmas L<bignum>, L<bigint> and L<bigrat> for an easy way to do this.
4227 C<bsqrt()> works only good if the result is a big integer, e.g. the square
4228 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4229 mode. The reason is that the result is always truncated to an integer.
4231 If you want a better approximation of the square root, then use:
4233 $x = Math::BigFloat->new(12);
4234 Math::BigFloat->precision(0);
4235 Math::BigFloat->round_mode('even');
4236 print $x->copy->bsqrt(),"\n"; # 4
4238 Math::BigFloat->precision(2);
4239 print $x->bsqrt(),"\n"; # 3.46
4240 print $x->bsqrt(3),"\n"; # 3.464
4244 For negative numbers in base see also L<brsft|brsft>.
4250 This program is free software; you may redistribute it and/or modify it under
4251 the same terms as Perl itself.
4255 L<Math::BigFloat>, L<Math::BigRat> and L<Math::Big> as well as
4256 L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4258 The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest
4259 because they solve the autoupgrading/downgrading issue, at least partly.
4262 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4263 more documentation including a full version history, testcases, empty
4264 subclass files and benchmarks.
4268 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4269 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001 - 2003
4270 and still at it in 2004.
4272 Many people contributed in one or more ways to the final beast, see the file
4273 CREDITS for an (uncomplete) list. If you miss your name, please drop me a