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 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
1144 $x->badd($y,@r); # badd does not leave internal zeros
1145 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
1146 $x; # already rounded by badd() or no round necc.
1151 # increment arg by one
1152 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1153 return $x if $x->modify('binc');
1155 if ($x->{sign} eq '+')
1157 $x->{value} = $CALC->_inc($x->{value});
1158 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1161 elsif ($x->{sign} eq '-')
1163 $x->{value} = $CALC->_dec($x->{value});
1164 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1165 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1168 # inf, nan handling etc
1169 $x->badd($self->bone(),$a,$p,$r); # badd does round
1174 # decrement arg by one
1175 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1176 return $x if $x->modify('bdec');
1178 if ($x->{sign} eq '-')
1181 $x->{value} = $CALC->_inc($x->{value});
1185 return $x->badd($self->bone('-'),@r) unless $x->{sign} eq '+'; # inf/NaN
1187 if ($CALC->_is_zero($x->{value}))
1190 $x->{value} = $CALC->_one(); $x->{sign} = '-'; # 0 => -1
1195 $x->{value} = $CALC->_dec($x->{value});
1198 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1204 # calculate $x = $a ** $base + $b and return $a (e.g. the log() to base
1208 my ($self,$x,$base,@r) = (ref($_[0]),@_);
1209 # objectify is costly, so avoid it
1210 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1212 ($self,$x,$base,@r) = objectify(1,$class,@_);
1215 return $x if $x->modify('blog');
1217 # inf, -inf, NaN, <0 => NaN
1219 if $x->{sign} ne '+' || (defined $base && $base->{sign} ne '+');
1221 return $upgrade->blog($upgrade->new($x),$base,@r) if
1224 my ($rc,$exact) = $CALC->_log_int($x->{value},$base->{value});
1225 return $x->bnan() unless defined $rc; # not possible to take log?
1232 # (BINT or num_str, BINT or num_str) return BINT
1233 # does not modify arguments, but returns new object
1234 # Lowest Common Multiplicator
1236 my $y = shift; my ($x);
1243 $x = __PACKAGE__->new($y);
1248 my $y = shift; $y = $self->new($y) if !ref ($y);
1256 # (BINT or num_str, BINT or num_str) return BINT
1257 # does not modify arguments, but returns new object
1258 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1261 $y = __PACKAGE__->new($y) if !ref($y);
1263 my $x = $y->copy()->babs(); # keep arguments
1264 return $x->bnan() if $x->{sign} !~ /^[+-]$/; # x NaN?
1268 $y = shift; $y = $self->new($y) if !ref($y);
1269 next if $y->is_zero();
1270 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1271 $x->{value} = $CALC->_gcd($x->{value},$y->{value}); last if $x->is_one();
1278 # (num_str or BINT) return BINT
1279 # represent ~x as twos-complement number
1280 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1281 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1283 return $x if $x->modify('bnot');
1284 $x->binc()->bneg(); # binc already does round
1287 ##############################################################################
1288 # is_foo test routines
1289 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1293 # return true if arg (BINT or num_str) is zero (array '+', '0')
1294 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1296 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1297 $CALC->_is_zero($x->{value});
1302 # return true if arg (BINT or num_str) is NaN
1303 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1305 $x->{sign} eq $nan ? 1 : 0;
1310 # return true if arg (BINT or num_str) is +-inf
1311 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1315 $sign = '[+-]inf' if $sign eq ''; # +- doesn't matter, only that's inf
1316 $sign = "[$1]inf" if $sign =~ /^([+-])(inf)?$/; # extract '+' or '-'
1317 return $x->{sign} =~ /^$sign$/ ? 1 : 0;
1319 $x->{sign} =~ /^[+-]inf$/ ? 1 : 0; # only +-inf is infinity
1324 # return true if arg (BINT or num_str) is +1, or -1 if sign is given
1325 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1327 $sign = '+' if !defined $sign || $sign ne '-';
1329 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1330 $CALC->_is_one($x->{value});
1335 # return true when arg (BINT or num_str) is odd, false for even
1336 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1338 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1339 $CALC->_is_odd($x->{value});
1344 # return true when arg (BINT or num_str) is even, false for odd
1345 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1347 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1348 $CALC->_is_even($x->{value});
1353 # return true when arg (BINT or num_str) is positive (>= 0)
1354 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1356 $x->{sign} =~ /^\+/ ? 1 : 0; # +inf is also positive, but NaN not
1361 # return true when arg (BINT or num_str) is negative (< 0)
1362 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1364 $x->{sign} =~ /^-/ ? 1 : 0; # -inf is also negative, but NaN not
1369 # return true when arg (BINT or num_str) is an integer
1370 # always true for BigInt, but different for BigFloats
1371 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1373 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1376 ###############################################################################
1380 # multiply two numbers -- stolen from Knuth Vol 2 pg 233
1381 # (BINT or num_str, BINT or num_str) return BINT
1384 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1385 # objectify is costly, so avoid it
1386 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1388 ($self,$x,$y,@r) = objectify(2,@_);
1391 return $x if $x->modify('bmul');
1393 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1396 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1398 return $x->bnan() if $x->is_zero() || $y->is_zero();
1399 # result will always be +-inf:
1400 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1401 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1402 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1403 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1404 return $x->binf('-');
1407 return $upgrade->bmul($x,$upgrade->new($y),@r)
1408 if defined $upgrade && !$y->isa($self);
1410 $r[3] = $y; # no push here
1412 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1414 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1415 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1417 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1423 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1424 my ($self,$x,$y) = @_;
1426 # NaN if x == NaN or y == NaN or x==y==0
1427 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1428 if (($x->is_nan() || $y->is_nan()) ||
1429 ($x->is_zero() && $y->is_zero()));
1431 # +-inf / +-inf == NaN, reminder also NaN
1432 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1434 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1436 # x / +-inf => 0, remainder x (works even if x == 0)
1437 if ($y->{sign} =~ /^[+-]inf$/)
1439 my $t = $x->copy(); # bzero clobbers up $x
1440 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1443 # 5 / 0 => +inf, -6 / 0 => -inf
1444 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1445 # exception: -8 / 0 has remainder -8, not 8
1446 # exception: -inf / 0 has remainder -inf, not inf
1449 # +-inf / 0 => special case for -inf
1450 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1451 if (!$x->is_zero() && !$x->is_inf())
1453 my $t = $x->copy(); # binf clobbers up $x
1455 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1459 # last case: +-inf / ordinary number
1461 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1463 return wantarray ? ($x,$self->bzero()) : $x;
1468 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1469 # (BINT,BINT) (quo,rem) or BINT (only rem)
1472 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1473 # objectify is costly, so avoid it
1474 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1476 ($self,$x,$y,@r) = objectify(2,@_);
1479 return $x if $x->modify('bdiv');
1481 return $self->_div_inf($x,$y)
1482 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1484 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1485 if defined $upgrade;
1487 $r[3] = $y; # no push!
1489 # calc new sign and in case $y == +/- 1, return $x
1490 my $xsign = $x->{sign}; # keep
1491 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1495 my $rem = $self->bzero();
1496 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1497 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1498 $rem->{_a} = $x->{_a};
1499 $rem->{_p} = $x->{_p};
1500 $x->round(@r) if !exists $x->{_f} || ($x->{_f} & MB_NEVER_ROUND) == 0;
1501 if (! $CALC->_is_zero($rem->{value}))
1503 $rem->{sign} = $y->{sign};
1504 $rem = $y->copy()->bsub($rem) if $xsign ne $y->{sign}; # one of them '-'
1508 $rem->{sign} = '+'; # dont leave -0
1510 $rem->round(@r) if !exists $rem->{_f} || ($rem->{_f} & MB_NEVER_ROUND) == 0;
1514 $x->{value} = $CALC->_div($x->{value},$y->{value});
1515 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1517 $x->round(@r) if !exists $x->{_f} || ($x->{_f} & MB_NEVER_ROUND) == 0;
1521 ###############################################################################
1526 # modulus (or remainder)
1527 # (BINT or num_str, BINT or num_str) return BINT
1530 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1531 # objectify is costly, so avoid it
1532 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1534 ($self,$x,$y,@r) = objectify(2,@_);
1537 return $x if $x->modify('bmod');
1538 $r[3] = $y; # no push!
1539 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1541 my ($d,$r) = $self->_div_inf($x,$y);
1542 $x->{sign} = $r->{sign};
1543 $x->{value} = $r->{value};
1544 return $x->round(@r);
1547 # calc new sign and in case $y == +/- 1, return $x
1548 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1549 if (!$CALC->_is_zero($x->{value}))
1551 my $xsign = $x->{sign};
1552 $x->{sign} = $y->{sign};
1553 if ($xsign ne $y->{sign})
1555 my $t = $CALC->_copy($x->{value}); # copy $x
1556 $x->{value} = $CALC->_sub($y->{value},$t,1); # $y-$x
1561 $x->{sign} = '+'; # dont leave -0
1563 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1569 # Modular inverse. given a number which is (hopefully) relatively
1570 # prime to the modulus, calculate its inverse using Euclid's
1571 # alogrithm. If the number is not relatively prime to the modulus
1572 # (i.e. their gcd is not one) then NaN is returned.
1575 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1576 # objectify is costly, so avoid it
1577 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1579 ($self,$x,$y,@r) = objectify(2,@_);
1582 return $x if $x->modify('bmodinv');
1585 if ($y->{sign} ne '+' # -, NaN, +inf, -inf
1586 || $x->is_zero() # or num == 0
1587 || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1590 # put least residue into $x if $x was negative, and thus make it positive
1591 $x->bmod($y) if $x->{sign} eq '-';
1594 ($x->{value},$sign) = $CALC->_modinv($x->{value},$y->{value});
1595 return $x->bnan() if !defined $x->{value}; # in case no GCD found
1596 return $x if !defined $sign; # already real result
1597 $x->{sign} = $sign; # flip/flop see below
1598 $x->bmod($y); # calc real result
1604 # takes a very large number to a very large exponent in a given very
1605 # large modulus, quickly, thanks to binary exponentation. supports
1606 # negative exponents.
1607 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1609 return $num if $num->modify('bmodpow');
1611 # check modulus for valid values
1612 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1613 || $mod->is_zero());
1615 # check exponent for valid values
1616 if ($exp->{sign} =~ /\w/)
1618 # i.e., if it's NaN, +inf, or -inf...
1619 return $num->bnan();
1622 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
1624 # check num for valid values (also NaN if there was no inverse but $exp < 0)
1625 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1627 # $mod is positive, sign on $exp is ignored, result also positive
1628 $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
1632 ###############################################################################
1636 # (BINT or num_str, BINT or num_str) return BINT
1637 # compute factorial number from $x, modify $x in place
1638 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1640 return $x if $x->modify('bfac');
1642 return $x if $x->{sign} eq '+inf'; # inf => inf
1643 return $x->bnan() if $x->{sign} ne '+'; # NaN, <0 etc => NaN
1645 $x->{value} = $CALC->_fac($x->{value});
1651 # (BINT or num_str, BINT or num_str) return BINT
1652 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1653 # modifies first argument
1656 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1657 # objectify is costly, so avoid it
1658 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1660 ($self,$x,$y,@r) = objectify(2,@_);
1663 return $x if $x->modify('bpow');
1665 return $upgrade->bpow($upgrade->new($x),$y,@r)
1666 if defined $upgrade && !$y->isa($self);
1668 $r[3] = $y; # no push!
1669 return $x if $x->{sign} =~ /^[+-]inf$/; # -inf/+inf ** x
1670 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1672 # cases 0 ** Y, X ** 0, X ** 1, 1 ** Y are handled by Calc or Emu
1675 $new_sign = $y->is_odd() ? '-' : '+' if ($x->{sign} ne '+');
1677 # 0 ** -7 => ( 1 / (0 ** 7)) => 1 / 0 => +inf
1679 if $y->{sign} eq '-' && $x->{sign} eq '+' && $CALC->_is_zero($x->{value});
1680 # 1 ** -y => 1 / (1 ** |y|)
1681 # so do test for negative $y after above's clause
1682 return $x->bnan() if $y->{sign} eq '-' && !$CALC->_is_one($x->{value});
1684 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1685 $x->{sign} = $new_sign;
1686 $x->{sign} = '+' if $CALC->_is_zero($y->{value});
1687 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1693 # (BINT or num_str, BINT or num_str) return BINT
1694 # compute x << y, base n, y >= 0
1697 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1698 # objectify is costly, so avoid it
1699 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1701 ($self,$x,$y,$n,@r) = objectify(2,@_);
1704 return $x if $x->modify('blsft');
1705 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1706 return $x->round(@r) if $y->is_zero();
1708 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1710 $x->{value} = $CALC->_lsft($x->{value},$y->{value},$n);
1716 # (BINT or num_str, BINT or num_str) return BINT
1717 # compute x >> y, base n, y >= 0
1720 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1721 # objectify is costly, so avoid it
1722 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1724 ($self,$x,$y,$n,@r) = objectify(2,@_);
1727 return $x if $x->modify('brsft');
1728 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1729 return $x->round(@r) if $y->is_zero();
1730 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
1732 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1734 # this only works for negative numbers when shifting in base 2
1735 if (($x->{sign} eq '-') && ($n == 2))
1737 return $x->round(@r) if $x->is_one('-'); # -1 => -1
1740 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1741 # but perhaps there is a better emulation for two's complement shift...
1742 # if $y != 1, we must simulate it by doing:
1743 # convert to bin, flip all bits, shift, and be done
1744 $x->binc(); # -3 => -2
1745 my $bin = $x->as_bin();
1746 $bin =~ s/^-0b//; # strip '-0b' prefix
1747 $bin =~ tr/10/01/; # flip bits
1749 if (CORE::length($bin) <= $y)
1751 $bin = '0'; # shifting to far right creates -1
1752 # 0, because later increment makes
1753 # that 1, attached '-' makes it '-1'
1754 # because -1 >> x == -1 !
1758 $bin =~ s/.{$y}$//; # cut off at the right side
1759 $bin = '1' . $bin; # extend left side by one dummy '1'
1760 $bin =~ tr/10/01/; # flip bits back
1762 my $res = $self->new('0b'.$bin); # add prefix and convert back
1763 $res->binc(); # remember to increment
1764 $x->{value} = $res->{value}; # take over value
1765 return $x->round(@r); # we are done now, magic, isn't?
1767 # x < 0, n == 2, y == 1
1768 $x->bdec(); # n == 2, but $y == 1: this fixes it
1771 $x->{value} = $CALC->_rsft($x->{value},$y->{value},$n);
1777 #(BINT or num_str, BINT or num_str) return BINT
1781 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1782 # objectify is costly, so avoid it
1783 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1785 ($self,$x,$y,@r) = objectify(2,@_);
1788 return $x if $x->modify('band');
1790 $r[3] = $y; # no push!
1792 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1794 my $sx = $x->{sign} eq '+' ? 1 : -1;
1795 my $sy = $y->{sign} eq '+' ? 1 : -1;
1797 if ($sx == 1 && $sy == 1)
1799 $x->{value} = $CALC->_and($x->{value},$y->{value});
1800 return $x->round(@r);
1803 if ($CAN{signed_and})
1805 $x->{value} = $CALC->_signed_and($x->{value},$y->{value},$sx,$sy);
1806 return $x->round(@r);
1810 __emu_band($self,$x,$y,$sx,$sy,@r);
1815 #(BINT or num_str, BINT or num_str) return BINT
1819 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1820 # objectify is costly, so avoid it
1821 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1823 ($self,$x,$y,@r) = objectify(2,@_);
1826 return $x if $x->modify('bior');
1827 $r[3] = $y; # no push!
1829 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1831 my $sx = $x->{sign} eq '+' ? 1 : -1;
1832 my $sy = $y->{sign} eq '+' ? 1 : -1;
1834 # the sign of X follows the sign of X, e.g. sign of Y irrelevant for bior()
1836 # don't use lib for negative values
1837 if ($sx == 1 && $sy == 1)
1839 $x->{value} = $CALC->_or($x->{value},$y->{value});
1840 return $x->round(@r);
1843 # if lib can do negative values, let it handle this
1844 if ($CAN{signed_or})
1846 $x->{value} = $CALC->_signed_or($x->{value},$y->{value},$sx,$sy);
1847 return $x->round(@r);
1851 __emu_bior($self,$x,$y,$sx,$sy,@r);
1856 #(BINT or num_str, BINT or num_str) return BINT
1860 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1861 # objectify is costly, so avoid it
1862 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1864 ($self,$x,$y,@r) = objectify(2,@_);
1867 return $x if $x->modify('bxor');
1868 $r[3] = $y; # no push!
1870 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1872 my $sx = $x->{sign} eq '+' ? 1 : -1;
1873 my $sy = $y->{sign} eq '+' ? 1 : -1;
1875 # don't use lib for negative values
1876 if ($sx == 1 && $sy == 1)
1878 $x->{value} = $CALC->_xor($x->{value},$y->{value});
1879 return $x->round(@r);
1882 # if lib can do negative values, let it handle this
1883 if ($CAN{signed_xor})
1885 $x->{value} = $CALC->_signed_xor($x->{value},$y->{value},$sx,$sy);
1886 return $x->round(@r);
1890 __emu_bxor($self,$x,$y,$sx,$sy,@r);
1895 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1897 my $e = $CALC->_len($x->{value});
1898 wantarray ? ($e,0) : $e;
1903 # return the nth decimal digit, negative values count backward, 0 is right
1904 my ($self,$x,$n) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1906 $n = $n->numify() if ref($n);
1907 $CALC->_digit($x->{value},$n||0);
1912 # return the amount of trailing zeros in $x (as scalar)
1914 $x = $class->new($x) unless ref $x;
1916 return 0 if $x->{sign} !~ /^[+-]$/; # NaN, inf, -inf etc
1918 $CALC->_zeros($x->{value}); # must handle odd values, 0 etc
1923 # calculate square root of $x
1924 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1926 return $x if $x->modify('bsqrt');
1928 return $x->bnan() if $x->{sign} !~ /^\+/; # -x or -inf or NaN => NaN
1929 return $x if $x->{sign} eq '+inf'; # sqrt(+inf) == inf
1931 return $upgrade->bsqrt($x,@r) if defined $upgrade;
1933 $x->{value} = $CALC->_sqrt($x->{value});
1939 # calculate $y'th root of $x
1942 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1944 $y = $self->new(2) unless defined $y;
1946 # objectify is costly, so avoid it
1947 if ((!ref($x)) || (ref($x) ne ref($y)))
1949 ($self,$x,$y,@r) = objectify(2,$self || $class,@_);
1952 return $x if $x->modify('broot');
1954 # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
1955 return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
1956 $y->{sign} !~ /^\+$/;
1958 return $x->round(@r)
1959 if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
1961 return $upgrade->new($x)->broot($upgrade->new($y),@r) if defined $upgrade;
1963 $x->{value} = $CALC->_root($x->{value},$y->{value});
1969 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
1970 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1972 if ($x->{sign} !~ /^[+-]$/)
1974 my $s = $x->{sign}; $s =~ s/^[+-]//; # NaN, -inf,+inf => NaN or inf
1975 return $self->new($s);
1977 return $self->bone() if $x->is_zero();
1979 $self->new($x->_trailing_zeros());
1984 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
1985 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1987 if ($x->{sign} !~ /^[+-]$/)
1989 # for NaN, +inf, -inf: keep the sign
1990 return $self->new($x->{sign});
1992 my $m = $x->copy(); delete $m->{_p}; delete $m->{_a};
1993 # that's a bit inefficient:
1994 my $zeros = $m->_trailing_zeros();
1995 $m->brsft($zeros,10) if $zeros != 0;
2001 # return a copy of both the exponent and the mantissa
2002 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2004 ($x->mantissa(),$x->exponent());
2007 ##############################################################################
2008 # rounding functions
2012 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
2013 # $n == 0 || $n == 1 => round to integer
2014 my $x = shift; my $self = ref($x) || $x; $x = $self->new($x) unless ref $x;
2016 my ($scale,$mode) = $x->_scale_p($x->precision(),$x->round_mode(),@_);
2018 return $x if !defined $scale || $x->modify('bfround'); # no-op
2020 # no-op for BigInts if $n <= 0
2021 $x->bround( $x->length()-$scale, $mode) if $scale > 0;
2023 delete $x->{_a}; # delete to save memory
2024 $x->{_p} = $scale; # store new _p
2028 sub _scan_for_nonzero
2030 # internal, used by bround()
2031 my ($x,$pad,$xs) = @_;
2033 my $len = $x->length();
2034 return 0 if $len == 1; # '5' is trailed by invisible zeros
2035 my $follow = $pad - 1;
2036 return 0 if $follow > $len || $follow < 1;
2038 # since we do not know underlying represention of $x, use decimal string
2039 my $r = substr ("$x",-$follow);
2040 $r =~ /[^0]/ ? 1 : 0;
2045 # Exists to make life easier for switch between MBF and MBI (should we
2046 # autoload fxxx() like MBF does for bxxx()?)
2053 # accuracy: +$n preserve $n digits from left,
2054 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
2056 # and overwrite the rest with 0's, return normalized number
2057 # do not return $x->bnorm(), but $x
2059 my $x = shift; $x = $class->new($x) unless ref $x;
2060 my ($scale,$mode) = $x->_scale_a($x->accuracy(),$x->round_mode(),@_);
2061 return $x if !defined $scale; # no-op
2062 return $x if $x->modify('bround');
2064 if ($x->is_zero() || $scale == 0)
2066 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2069 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2071 # we have fewer digits than we want to scale to
2072 my $len = $x->length();
2073 # convert $scale to a scalar in case it is an object (put's a limit on the
2074 # number length, but this would already limited by memory constraints), makes
2076 $scale = $scale->numify() if ref ($scale);
2078 # scale < 0, but > -len (not >=!)
2079 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2081 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2085 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2086 my ($pad,$digit_round,$digit_after);
2087 $pad = $len - $scale;
2088 $pad = abs($scale-1) if $scale < 0;
2090 # do not use digit(), it is costly for binary => decimal
2092 my $xs = $CALC->_str($x->{value});
2095 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2096 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2097 $digit_round = '0'; $digit_round = substr($xs,$pl,1) if $pad <= $len;
2098 $pl++; $pl ++ if $pad >= $len;
2099 $digit_after = '0'; $digit_after = substr($xs,$pl,1) if $pad > 0;
2101 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2102 # closer at the remaining digits of the original $x, remember decision
2103 my $round_up = 1; # default round up
2105 ($mode eq 'trunc') || # trunc by round down
2106 ($digit_after =~ /[01234]/) || # round down anyway,
2108 ($digit_after eq '5') && # not 5000...0000
2109 ($x->_scan_for_nonzero($pad,$xs) == 0) &&
2111 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2112 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2113 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2114 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2115 ($mode eq 'zero') # round down if zero, sign adjusted below
2117 my $put_back = 0; # not yet modified
2119 if (($pad > 0) && ($pad <= $len))
2121 substr($xs,-$pad,$pad) = '0' x $pad;
2126 $x->bzero(); # round to '0'
2129 if ($round_up) # what gave test above?
2132 $pad = $len, $xs = '0' x $pad if $scale < 0; # tlr: whack 0.51=>1.0
2134 # we modify directly the string variant instead of creating a number and
2135 # adding it, since that is faster (we already have the string)
2136 my $c = 0; $pad ++; # for $pad == $len case
2137 while ($pad <= $len)
2139 $c = substr($xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2140 substr($xs,-$pad,1) = $c; $pad++;
2141 last if $c != 0; # no overflow => early out
2143 $xs = '1'.$xs if $c == 0;
2146 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back in if needed
2148 $x->{_a} = $scale if $scale >= 0;
2151 $x->{_a} = $len+$scale;
2152 $x->{_a} = 0 if $scale < -$len;
2159 # return integer less or equal then number; no-op since it's already integer
2160 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2167 # return integer greater or equal then number; no-op since it's already int
2168 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2175 # An object might be asked to return itself as bigint on certain overloaded
2176 # operations, this does exactly this, so that sub classes can simple inherit
2177 # it or override with their own integer conversion routine.
2183 # return as hex string, with prefixed 0x
2184 my $x = shift; $x = $class->new($x) if !ref($x);
2186 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2189 $s = $x->{sign} if $x->{sign} eq '-';
2190 $s . $CALC->_as_hex($x->{value});
2195 # return as binary string, with prefixed 0b
2196 my $x = shift; $x = $class->new($x) if !ref($x);
2198 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2200 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2201 return $s . $CALC->_as_bin($x->{value});
2204 ##############################################################################
2205 # private stuff (internal use only)
2209 # check for strings, if yes, return objects instead
2211 # the first argument is number of args objectify() should look at it will
2212 # return $count+1 elements, the first will be a classname. This is because
2213 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2214 # useless objects beeing created and thrown away. So we cannot simple loop
2215 # over @_. If the given count is 0, all arguments will be used.
2217 # If the second arg is a ref, use it as class.
2218 # If not, try to use it as classname, unless undef, then use $class
2219 # (aka Math::BigInt). The latter shouldn't happen,though.
2222 # $x->badd(1); => ref x, scalar y
2223 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2224 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2225 # Math::BigInt::badd(1,2); => scalar x, scalar y
2226 # In the last case we check number of arguments to turn it silently into
2227 # $class,1,2. (We can not take '1' as class ;o)
2228 # badd($class,1) is not supported (it should, eventually, try to add undef)
2229 # currently it tries 'Math::BigInt' + 1, which will not work.
2231 # some shortcut for the common cases
2233 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2235 my $count = abs(shift || 0);
2237 my (@a,$k,$d); # resulting array, temp, and downgrade
2240 # okay, got object as first
2245 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2247 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2251 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2252 if (defined ${"$a[0]::downgrade"})
2254 $d = ${"$a[0]::downgrade"};
2255 ${"$a[0]::downgrade"} = undef;
2258 my $up = ${"$a[0]::upgrade"};
2259 #print "Now in objectify, my class is today $a[0], count = $count\n";
2267 $k = $a[0]->new($k);
2269 elsif (!defined $up && ref($k) ne $a[0])
2271 # foreign object, try to convert to integer
2272 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2285 $k = $a[0]->new($k);
2287 elsif (!defined $up && ref($k) ne $a[0])
2289 # foreign object, try to convert to integer
2290 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2294 push @a,@_; # return other params, too
2298 require Carp; Carp::croak ("$class objectify needs list context");
2300 ${"$a[0]::downgrade"} = $d;
2308 $IMPORT++; # remember we did import()
2309 my @a; my $l = scalar @_;
2310 for ( my $i = 0; $i < $l ; $i++ )
2312 if ($_[$i] eq ':constant')
2314 # this causes overlord er load to step in
2316 integer => sub { $self->new(shift) },
2317 binary => sub { $self->new(shift) };
2319 elsif ($_[$i] eq 'upgrade')
2321 # this causes upgrading
2322 $upgrade = $_[$i+1]; # or undef to disable
2325 elsif ($_[$i] =~ /^lib$/i)
2327 # this causes a different low lib to take care...
2328 $CALC = $_[$i+1] || '';
2336 # any non :constant stuff is handled by our parent, Exporter
2337 # even if @_ is empty, to give it a chance
2338 $self->SUPER::import(@a); # need it for subclasses
2339 $self->export_to_level(1,$self,@a); # need it for MBF
2341 # try to load core math lib
2342 my @c = split /\s*,\s*/,$CALC;
2343 push @c,'Calc'; # if all fail, try this
2344 $CALC = ''; # signal error
2345 foreach my $lib (@c)
2347 next if ($lib || '') eq '';
2348 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2352 # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
2353 # used in the same script, or eval inside import().
2354 my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
2355 my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
2357 $file = File::Spec->catfile (@parts, $file);
2358 eval { require "$file"; $lib->import( @c ); }
2362 eval "use $lib qw/@c/;";
2367 # loaded it ok, see if the api_version() is high enough
2368 if ($lib->can('api_version') && $lib->api_version() >= 1.0)
2371 # api_version matches, check if it really provides anything we need
2375 add mul div sub dec inc
2376 acmp len digit is_one is_zero is_even is_odd
2378 new copy check from_hex from_bin as_hex as_bin zeros
2379 rsft lsft xor and or
2380 mod sqrt root fac pow modinv modpow log_int gcd
2383 if (!$lib->can("_$method"))
2385 if (($WARN{$lib}||0) < 2)
2388 Carp::carp ("$lib is missing method '_$method'");
2389 $WARN{$lib} = 1; # still warn about the lib
2398 last; # found a usable one, break
2402 if (($WARN{$lib}||0) < 2)
2404 my $ver = eval "\$$lib\::VERSION";
2406 Carp::carp ("Cannot load outdated $lib v$ver, please upgrade");
2407 $WARN{$lib} = 2; # never warn again
2415 Carp::croak ("Couldn't load any math lib, not even 'Calc.pm'");
2417 _fill_can_cache(); # for emulating lower math lib functions
2422 # fill $CAN with the results of $CALC->can(...)
2425 for my $method (qw/ signed_and or signed_or xor signed_xor /)
2427 $CAN{$method} = $CALC->can("_$method") ? 1 : 0;
2433 # convert a (ref to) big hex string to BigInt, return undef for error
2436 my $x = Math::BigInt->bzero();
2439 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2440 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2442 return $x->bnan() if $hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2444 my $sign = '+'; $sign = '-' if $hs =~ /^-/;
2446 $hs =~ s/^[+-]//; # strip sign
2447 $x->{value} = $CALC->_from_hex($hs);
2448 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2454 # convert a (ref to) big binary string to BigInt, return undef for error
2457 my $x = Math::BigInt->bzero();
2459 $bs =~ s/([01])_([01])/$1$2/g;
2460 $bs =~ s/([01])_([01])/$1$2/g;
2461 return $x->bnan() if $bs !~ /^[+-]?0b[01]+$/;
2463 my $sign = '+'; $sign = '-' if $bs =~ /^\-/;
2464 $bs =~ s/^[+-]//; # strip sign
2466 $x->{value} = $CALC->_from_bin($bs);
2467 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2473 # (ref to num_str) return num_str
2474 # internal, take apart a string and return the pieces
2475 # strip leading/trailing whitespace, leading zeros, underscore and reject
2479 # strip white space at front, also extranous leading zeros
2480 $x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2481 $x =~ s/^\s+//; # but this will
2482 $x =~ s/\s+$//g; # strip white space at end
2484 # shortcut, if nothing to split, return early
2485 if ($x =~ /^[+-]?\d+\z/)
2487 $x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2488 return (\$sign, \$x, \'', \'', \0);
2491 # invalid starting char?
2492 return if $x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2494 return __from_hex($x) if $x =~ /^[\-\+]?0x/; # hex string
2495 return __from_bin($x) if $x =~ /^[\-\+]?0b/; # binary string
2497 # strip underscores between digits
2498 $x =~ s/(\d)_(\d)/$1$2/g;
2499 $x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3
2501 # some possible inputs:
2502 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2503 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 # 0e999
2505 my ($m,$e,$last) = split /[Ee]/,$x;
2506 return if defined $last; # last defined => 1e2E3 or others
2507 $e = '0' if !defined $e || $e eq "";
2509 # sign,value for exponent,mantint,mantfrac
2510 my ($es,$ev,$mis,$miv,$mfv);
2512 if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2516 return if $m eq '.' || $m eq '';
2517 my ($mi,$mf,$lastf) = split /\./,$m;
2518 return if defined $lastf; # lastf defined => 1.2.3 or others
2519 $mi = '0' if !defined $mi;
2520 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2521 $mf = '0' if !defined $mf || $mf eq '';
2522 if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2524 $mis = $1||'+'; $miv = $2;
2525 return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
2527 # handle the 0e999 case here
2528 $ev = 0 if $miv eq '0' && $mfv eq '';
2529 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2532 return; # NaN, not a number
2535 ##############################################################################
2536 # internal calculation routines (others are in Math::BigInt::Calc etc)
2540 # (BINT or num_str, BINT or num_str) return BINT
2541 # does modify first argument
2544 my $x = shift; my $ty = shift;
2545 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2546 $x * $ty / bgcd($x,$ty);
2549 ###############################################################################
2550 # this method return 0 if the object can be modified, or 1 for not
2551 # We use a fast constant sub() here, to avoid costly calls. Subclasses
2552 # may override it with special code (f.i. Math::BigInt::Constant does so)
2554 sub modify () { 0; }
2561 Math::BigInt - Arbitrary size integer math package
2567 # or make it faster: install (optional) Math::BigInt::GMP
2568 # and always use (it will fall back to pure Perl if the
2569 # GMP library is not installed):
2571 use Math::BigInt lib => 'GMP';
2573 my $str = '1234567890';
2574 my @values = (64,74,18);
2575 my $n = 1; my $sign = '-';
2578 $x = Math::BigInt->new($str); # defaults to 0
2579 $y = $x->copy(); # make a true copy
2580 $nan = Math::BigInt->bnan(); # create a NotANumber
2581 $zero = Math::BigInt->bzero(); # create a +0
2582 $inf = Math::BigInt->binf(); # create a +inf
2583 $inf = Math::BigInt->binf('-'); # create a -inf
2584 $one = Math::BigInt->bone(); # create a +1
2585 $one = Math::BigInt->bone('-'); # create a -1
2587 # Testing (don't modify their arguments)
2588 # (return true if the condition is met, otherwise false)
2590 $x->is_zero(); # if $x is +0
2591 $x->is_nan(); # if $x is NaN
2592 $x->is_one(); # if $x is +1
2593 $x->is_one('-'); # if $x is -1
2594 $x->is_odd(); # if $x is odd
2595 $x->is_even(); # if $x is even
2596 $x->is_pos(); # if $x >= 0
2597 $x->is_neg(); # if $x < 0
2598 $x->is_inf($sign); # if $x is +inf, or -inf (sign is default '+')
2599 $x->is_int(); # if $x is an integer (not a float)
2601 # comparing and digit/sign extration
2602 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2603 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2604 $x->sign(); # return the sign, either +,- or NaN
2605 $x->digit($n); # return the nth digit, counting from right
2606 $x->digit(-$n); # return the nth digit, counting from left
2608 # The following all modify their first argument. If you want to preserve
2609 # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
2610 # neccessary when mixing $a = $b assigments with non-overloaded math.
2612 $x->bzero(); # set $x to 0
2613 $x->bnan(); # set $x to NaN
2614 $x->bone(); # set $x to +1
2615 $x->bone('-'); # set $x to -1
2616 $x->binf(); # set $x to inf
2617 $x->binf('-'); # set $x to -inf
2619 $x->bneg(); # negation
2620 $x->babs(); # absolute value
2621 $x->bnorm(); # normalize (no-op in BigInt)
2622 $x->bnot(); # two's complement (bit wise not)
2623 $x->binc(); # increment $x by 1
2624 $x->bdec(); # decrement $x by 1
2626 $x->badd($y); # addition (add $y to $x)
2627 $x->bsub($y); # subtraction (subtract $y from $x)
2628 $x->bmul($y); # multiplication (multiply $x by $y)
2629 $x->bdiv($y); # divide, set $x to quotient
2630 # return (quo,rem) or quo if scalar
2632 $x->bmod($y); # modulus (x % y)
2633 $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
2634 $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
2636 $x->bpow($y); # power of arguments (x ** y)
2637 $x->blsft($y); # left shift
2638 $x->brsft($y); # right shift
2639 $x->blsft($y,$n); # left shift, by base $n (like 10)
2640 $x->brsft($y,$n); # right shift, by base $n (like 10)
2642 $x->band($y); # bitwise and
2643 $x->bior($y); # bitwise inclusive or
2644 $x->bxor($y); # bitwise exclusive or
2645 $x->bnot(); # bitwise not (two's complement)
2647 $x->bsqrt(); # calculate square-root
2648 $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
2649 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2651 $x->round($A,$P,$mode); # round to accuracy or precision using mode $mode
2652 $x->bround($n); # accuracy: preserve $n digits
2653 $x->bfround($n); # round to $nth digit, no-op for BigInts
2655 # The following do not modify their arguments in BigInt (are no-ops),
2656 # but do so in BigFloat:
2658 $x->bfloor(); # return integer less or equal than $x
2659 $x->bceil(); # return integer greater or equal than $x
2661 # The following do not modify their arguments:
2663 # greatest common divisor (no OO style)
2664 my $gcd = Math::BigInt::bgcd(@values);
2665 # lowest common multiplicator (no OO style)
2666 my $lcm = Math::BigInt::blcm(@values);
2668 $x->length(); # return number of digits in number
2669 ($xl,$f) = $x->length(); # length of number and length of fraction part,
2670 # latter is always 0 digits long for BigInt's
2672 $x->exponent(); # return exponent as BigInt
2673 $x->mantissa(); # return (signed) mantissa as BigInt
2674 $x->parts(); # return (mantissa,exponent) as BigInt
2675 $x->copy(); # make a true copy of $x (unlike $y = $x;)
2676 $x->as_int(); # return as BigInt (in BigInt: same as copy())
2677 $x->numify(); # return as scalar (might overflow!)
2679 # conversation to string (do not modify their argument)
2680 $x->bstr(); # normalized string
2681 $x->bsstr(); # normalized string in scientific notation
2682 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
2683 $x->as_bin(); # as signed binary string with prefixed 0b
2686 # precision and accuracy (see section about rounding for more)
2687 $x->precision(); # return P of $x (or global, if P of $x undef)
2688 $x->precision($n); # set P of $x to $n
2689 $x->accuracy(); # return A of $x (or global, if A of $x undef)
2690 $x->accuracy($n); # set A $x to $n
2693 Math::BigInt->precision(); # get/set global P for all BigInt objects
2694 Math::BigInt->accuracy(); # get/set global A for all BigInt objects
2695 Math::BigInt->config(); # return hash containing configuration
2699 All operators (inlcuding basic math operations) are overloaded if you
2700 declare your big integers as
2702 $i = new Math::BigInt '123_456_789_123_456_789';
2704 Operations with overloaded operators preserve the arguments which is
2705 exactly what you expect.
2711 Input values to these routines may be any string, that looks like a number
2712 and results in an integer, including hexadecimal and binary numbers.
2714 Scalars holding numbers may also be passed, but note that non-integer numbers
2715 may already have lost precision due to the conversation to float. Quote
2716 your input if you want BigInt to see all the digits:
2718 $x = Math::BigInt->new(12345678890123456789); # bad
2719 $x = Math::BigInt->new('12345678901234567890'); # good
2721 You can include one underscore between any two digits.
2723 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
2724 Non-integer values result in NaN.
2726 Currently, Math::BigInt::new() defaults to 0, while Math::BigInt::new('')
2727 results in 'NaN'. This might change in the future, so use always the following
2728 explicit forms to get a zero or NaN:
2730 $zero = Math::BigInt->bzero();
2731 $nan = Math::BigInt->bnan();
2733 C<bnorm()> on a BigInt object is now effectively a no-op, since the numbers
2734 are always stored in normalized form. If passed a string, creates a BigInt
2735 object from the input.
2739 Output values are BigInt objects (normalized), except for bstr(), which
2740 returns a string in normalized form.
2741 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
2742 C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
2743 return either undef, <0, 0 or >0 and are suited for sort.
2749 Each of the methods below (except config(), accuracy() and precision())
2750 accepts three additional parameters. These arguments $A, $P and $R are
2751 accuracy, precision and round_mode. Please see the section about
2752 L<ACCURACY and PRECISION> for more information.
2758 print Dumper ( Math::BigInt->config() );
2759 print Math::BigInt->config()->{lib},"\n";
2761 Returns a hash containing the configuration, e.g. the version number, lib
2762 loaded etc. The following hash keys are currently filled in with the
2763 appropriate information.
2767 ============================================================
2768 lib Name of the low-level math library
2770 lib_version Version of low-level math library (see 'lib')
2772 class The class name of config() you just called
2774 upgrade To which class math operations might be upgraded
2776 downgrade To which class math operations might be downgraded
2778 precision Global precision
2780 accuracy Global accuracy
2782 round_mode Global round mode
2784 version version number of the class you used
2786 div_scale Fallback acccuracy for div
2788 trap_nan If true, traps creation of NaN via croak()
2790 trap_inf If true, traps creation of +inf/-inf via croak()
2793 The following values can be set by passing C<config()> a reference to a hash:
2796 upgrade downgrade precision accuracy round_mode div_scale
2800 $new_cfg = Math::BigInt->config( { trap_inf => 1, precision => 5 } );
2804 $x->accuracy(5); # local for $x
2805 CLASS->accuracy(5); # global for all members of CLASS
2806 $A = $x->accuracy(); # read out
2807 $A = CLASS->accuracy(); # read out
2809 Set or get the global or local accuracy, aka how many significant digits the
2812 Please see the section about L<ACCURACY AND PRECISION> for further details.
2814 Value must be greater than zero. Pass an undef value to disable it:
2816 $x->accuracy(undef);
2817 Math::BigInt->accuracy(undef);
2819 Returns the current accuracy. For C<$x->accuracy()> it will return either the
2820 local accuracy, or if not defined, the global. This means the return value
2821 represents the accuracy that will be in effect for $x:
2823 $y = Math::BigInt->new(1234567); # unrounded
2824 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
2825 $x = Math::BigInt->new(123456); # will be automatically rounded
2826 print "$x $y\n"; # '123500 1234567'
2827 print $x->accuracy(),"\n"; # will be 4
2828 print $y->accuracy(),"\n"; # also 4, since global is 4
2829 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
2830 print $x->accuracy(),"\n"; # still 4
2831 print $y->accuracy(),"\n"; # 5, since global is 5
2833 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
2834 globals separated from Math::BigInt, but it is possible to subclass
2835 Math::BigInt and make the globals of the subclass aliases to the ones from
2840 $x->precision(-2); # local for $x, round right of the dot
2841 $x->precision(2); # ditto, but round left of the dot
2842 CLASS->accuracy(5); # global for all members of CLASS
2843 CLASS->precision(-5); # ditto
2844 $P = CLASS->precision(); # read out
2845 $P = $x->precision(); # read out
2847 Set or get the global or local precision, aka how many digits the result has
2848 after the dot (or where to round it when passing a positive number). In
2849 Math::BigInt, passing a negative number precision has no effect since no
2850 numbers have digits after the dot.
2852 Please see the section about L<ACCURACY AND PRECISION> for further details.
2854 Value must be greater than zero. Pass an undef value to disable it:
2856 $x->precision(undef);
2857 Math::BigInt->precision(undef);
2859 Returns the current precision. For C<$x->precision()> it will return either the
2860 local precision of $x, or if not defined, the global. This means the return
2861 value represents the accuracy that will be in effect for $x:
2863 $y = Math::BigInt->new(1234567); # unrounded
2864 print Math::BigInt->precision(4),"\n"; # set 4, print 4
2865 $x = Math::BigInt->new(123456); # will be automatically rounded
2867 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
2868 globals separated from Math::BigInt, but it is possible to subclass
2869 Math::BigInt and make the globals of the subclass aliases to the ones from
2876 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
2877 2, but others work, too.
2879 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
2883 $x = Math::BigInt->new(10);
2884 $x->brsft(1); # same as $x >> 1: 5
2885 $x = Math::BigInt->new(1234);
2886 $x->brsft(2,10); # result 12
2888 There is one exception, and that is base 2 with negative $x:
2891 $x = Math::BigInt->new(-5);
2894 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
2899 $x = Math::BigInt->new($str,$A,$P,$R);
2901 Creates a new BigInt object from a scalar or another BigInt object. The
2902 input is accepted as decimal, hex (with leading '0x') or binary (with leading
2905 See L<Input> for more info on accepted input formats.
2909 $x = Math::BigInt->bnan();
2911 Creates a new BigInt object representing NaN (Not A Number).
2912 If used on an object, it will set it to NaN:
2918 $x = Math::BigInt->bzero();
2920 Creates a new BigInt object representing zero.
2921 If used on an object, it will set it to zero:
2927 $x = Math::BigInt->binf($sign);
2929 Creates a new BigInt object representing infinity. The optional argument is
2930 either '-' or '+', indicating whether you want infinity or minus infinity.
2931 If used on an object, it will set it to infinity:
2938 $x = Math::BigInt->binf($sign);
2940 Creates a new BigInt object representing one. The optional argument is
2941 either '-' or '+', indicating whether you want one or minus one.
2942 If used on an object, it will set it to one:
2947 =head2 is_one()/is_zero()/is_nan()/is_inf()
2950 $x->is_zero(); # true if arg is +0
2951 $x->is_nan(); # true if arg is NaN
2952 $x->is_one(); # true if arg is +1
2953 $x->is_one('-'); # true if arg is -1
2954 $x->is_inf(); # true if +inf
2955 $x->is_inf('-'); # true if -inf (sign is default '+')
2957 These methods all test the BigInt for beeing one specific value and return
2958 true or false depending on the input. These are faster than doing something
2963 =head2 is_pos()/is_neg()
2965 $x->is_pos(); # true if >= 0
2966 $x->is_neg(); # true if < 0
2968 The methods return true if the argument is positive or negative, respectively.
2969 C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
2970 C<-inf> is negative. A C<zero> is positive.
2972 These methods are only testing the sign, and not the value.
2974 C<is_positive()> and C<is_negative()> are aliase to C<is_pos()> and
2975 C<is_neg()>, respectively. C<is_positive()> and C<is_negative()> were
2976 introduced in v1.36, while C<is_pos()> and C<is_neg()> were only introduced
2979 =head2 is_odd()/is_even()/is_int()
2981 $x->is_odd(); # true if odd, false for even
2982 $x->is_even(); # true if even, false for odd
2983 $x->is_int(); # true if $x is an integer
2985 The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
2986 C<-inf> are not integers and are neither odd nor even.
2988 In BigInt, all numbers except C<NaN>, C<+inf> and C<-inf> are integers.
2994 Compares $x with $y and takes the sign into account.
2995 Returns -1, 0, 1 or undef.
3001 Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
3007 Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
3011 $x->digit($n); # return the nth digit, counting from right
3013 If C<$n> is negative, returns the digit counting from left.
3019 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
3020 and '-inf', respectively. Does nothing for NaN or zero.
3026 Set the number to it's absolute value, e.g. change the sign from '-' to '+'
3027 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
3032 $x->bnorm(); # normalize (no-op)
3038 Two's complement (bit wise not). This is equivalent to
3046 $x->binc(); # increment x by 1
3050 $x->bdec(); # decrement x by 1
3054 $x->badd($y); # addition (add $y to $x)
3058 $x->bsub($y); # subtraction (subtract $y from $x)
3062 $x->bmul($y); # multiplication (multiply $x by $y)
3066 $x->bdiv($y); # divide, set $x to quotient
3067 # return (quo,rem) or quo if scalar
3071 $x->bmod($y); # modulus (x % y)
3075 num->bmodinv($mod); # modular inverse
3077 Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
3078 returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
3079 C<bgcd($num, $mod)==1>.
3083 $num->bmodpow($exp,$mod); # modular exponentation
3084 # ($num**$exp % $mod)
3086 Returns the value of C<$num> taken to the power C<$exp> in the modulus
3087 C<$mod> using binary exponentation. C<bmodpow> is far superior to
3092 because it is much faster - it reduces internal variables into
3093 the modulus whenever possible, so it operates on smaller numbers.
3095 C<bmodpow> also supports negative exponents.
3097 bmodpow($num, -1, $mod)
3099 is exactly equivalent to
3105 $x->bpow($y); # power of arguments (x ** y)
3109 $x->blsft($y); # left shift
3110 $x->blsft($y,$n); # left shift, in base $n (like 10)
3114 $x->brsft($y); # right shift
3115 $x->brsft($y,$n); # right shift, in base $n (like 10)
3119 $x->band($y); # bitwise and
3123 $x->bior($y); # bitwise inclusive or
3127 $x->bxor($y); # bitwise exclusive or
3131 $x->bnot(); # bitwise not (two's complement)
3135 $x->bsqrt(); # calculate square-root
3139 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3143 $x->round($A,$P,$round_mode);
3145 Round $x to accuracy C<$A> or precision C<$P> using the round mode
3150 $x->bround($N); # accuracy: preserve $N digits
3154 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3160 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3161 does change $x in BigFloat.
3167 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3168 does change $x in BigFloat.
3172 bgcd(@values); # greatest common divisor (no OO style)
3176 blcm(@values); # lowest common multiplicator (no OO style)
3181 ($xl,$fl) = $x->length();
3183 Returns the number of digits in the decimal representation of the number.
3184 In list context, returns the length of the integer and fraction part. For
3185 BigInt's, the length of the fraction part will always be 0.
3191 Return the exponent of $x as BigInt.
3197 Return the signed mantissa of $x as BigInt.
3201 $x->parts(); # return (mantissa,exponent) as BigInt
3205 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3211 Returns $x as a BigInt (truncated towards zero). In BigInt this is the same as
3214 C<as_number()> is an alias to this method. C<as_number> was introduced in
3215 v1.22, while C<as_int()> was only introduced in v1.68.
3221 Returns a normalized string represantation of C<$x>.
3225 $x->bsstr(); # normalized string in scientific notation
3229 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3233 $x->as_bin(); # as signed binary string with prefixed 0b
3235 =head1 ACCURACY and PRECISION
3237 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3238 accuracy and precision based rounding, both automatically after every
3239 operation, as well as manually.
3241 This section describes the accuracy/precision handling in Math::Big* as it
3242 used to be and as it is now, complete with an explanation of all terms and
3245 Not yet implemented things (but with correct description) are marked with '!',
3246 things that need to be answered are marked with '?'.
3248 In the next paragraph follows a short description of terms used here (because
3249 these may differ from terms used by others people or documentation).
3251 During the rest of this document, the shortcuts A (for accuracy), P (for
3252 precision), F (fallback) and R (rounding mode) will be used.
3256 A fixed number of digits before (positive) or after (negative)
3257 the decimal point. For example, 123.45 has a precision of -2. 0 means an
3258 integer like 123 (or 120). A precision of 2 means two digits to the left
3259 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3260 numbers with zeros before the decimal point may have different precisions,
3261 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3262 was). It could also have p < 0, when the digits after the decimal point
3265 The string output (of floating point numbers) will be padded with zeros:
3267 Initial value P A Result String
3268 ------------------------------------------------------------
3269 1234.01 -3 1000 1000
3272 1234.001 1 1234 1234.0
3274 1234.01 2 1234.01 1234.01
3275 1234.01 5 1234.01 1234.01000
3277 For BigInts, no padding occurs.
3281 Number of significant digits. Leading zeros are not counted. A
3282 number may have an accuracy greater than the non-zero digits
3283 when there are zeros in it or trailing zeros. For example, 123.456 has
3284 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3286 The string output (of floating point numbers) will be padded with zeros:
3288 Initial value P A Result String
3289 ------------------------------------------------------------
3291 1234.01 6 1234.01 1234.01
3292 1234.1 8 1234.1 1234.1000
3294 For BigInts, no padding occurs.
3298 When both A and P are undefined, this is used as a fallback accuracy when
3301 =head2 Rounding mode R
3303 When rounding a number, different 'styles' or 'kinds'
3304 of rounding are possible. (Note that random rounding, as in
3305 Math::Round, is not implemented.)
3311 truncation invariably removes all digits following the
3312 rounding place, replacing them with zeros. Thus, 987.65 rounded
3313 to tens (P=1) becomes 980, and rounded to the fourth sigdig
3314 becomes 987.6 (A=4). 123.456 rounded to the second place after the
3315 decimal point (P=-2) becomes 123.46.
3317 All other implemented styles of rounding attempt to round to the
3318 "nearest digit." If the digit D immediately to the right of the
3319 rounding place (skipping the decimal point) is greater than 5, the
3320 number is incremented at the rounding place (possibly causing a
3321 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3322 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3323 truncated at the rounding place: e.g. when rounding to units, 0.4
3324 rounds to 0, and -19.4 rounds to -19.
3326 However the results of other styles of rounding differ if the
3327 digit immediately to the right of the rounding place (skipping the
3328 decimal point) is 5 and if there are no digits, or no digits other
3329 than 0, after that 5. In such cases:
3333 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3334 if it is not already. E.g., when rounding to the first sigdig, 0.45
3335 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3339 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3340 it is not already. E.g., when rounding to the first sigdig, 0.45
3341 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3345 round to plus infinity, i.e. always round up. E.g., when
3346 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3347 and 0.4501 also becomes 0.5.
3351 round to minus infinity, i.e. always round down. E.g., when
3352 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3353 but 0.4501 becomes 0.5.
3357 round to zero, i.e. positive numbers down, negative ones up.
3358 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3359 becomes -0.5, but 0.4501 becomes 0.5.
3363 The handling of A & P in MBI/MBF (the old core code shipped with Perl
3364 versions <= 5.7.2) is like this:
3370 * ffround($p) is able to round to $p number of digits after the decimal
3372 * otherwise P is unused
3374 =item Accuracy (significant digits)
3376 * fround($a) rounds to $a significant digits
3377 * only fdiv() and fsqrt() take A as (optional) paramater
3378 + other operations simply create the same number (fneg etc), or more (fmul)
3380 + rounding/truncating is only done when explicitly calling one of fround
3381 or ffround, and never for BigInt (not implemented)
3382 * fsqrt() simply hands its accuracy argument over to fdiv.
3383 * the documentation and the comment in the code indicate two different ways
3384 on how fdiv() determines the maximum number of digits it should calculate,
3385 and the actual code does yet another thing
3387 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3389 result has at most max(scale, length(dividend), length(divisor)) digits
3391 scale = max(scale, length(dividend)-1,length(divisor)-1);
3392 scale += length(divisior) - length(dividend);
3393 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3394 Actually, the 'difference' added to the scale is calculated from the
3395 number of "significant digits" in dividend and divisor, which is derived
3396 by looking at the length of the mantissa. Which is wrong, since it includes
3397 the + sign (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops
3398 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3399 assumption that 124 has 3 significant digits, while 120/7 will get you
3400 '17', not '17.1' since 120 is thought to have 2 significant digits.
3401 The rounding after the division then uses the remainder and $y to determine
3402 wether it must round up or down.
3403 ? I have no idea which is the right way. That's why I used a slightly more
3404 ? simple scheme and tweaked the few failing testcases to match it.
3408 This is how it works now:
3412 =item Setting/Accessing
3414 * You can set the A global via C<< Math::BigInt->accuracy() >> or
3415 C<< Math::BigFloat->accuracy() >> or whatever class you are using.
3416 * You can also set P globally by using C<< Math::SomeClass->precision() >>
3418 * Globals are classwide, and not inherited by subclasses.
3419 * to undefine A, use C<< Math::SomeCLass->accuracy(undef); >>
3420 * to undefine P, use C<< Math::SomeClass->precision(undef); >>
3421 * Setting C<< Math::SomeClass->accuracy() >> clears automatically
3422 C<< Math::SomeClass->precision() >>, and vice versa.
3423 * To be valid, A must be > 0, P can have any value.
3424 * If P is negative, this means round to the P'th place to the right of the
3425 decimal point; positive values mean to the left of the decimal point.
3426 P of 0 means round to integer.
3427 * to find out the current global A, use C<< Math::SomeClass->accuracy() >>
3428 * to find out the current global P, use C<< Math::SomeClass->precision() >>
3429 * use C<< $x->accuracy() >> respective C<< $x->precision() >> for the local
3430 setting of C<< $x >>.
3431 * Please note that C<< $x->accuracy() >> respecive C<< $x->precision() >>
3432 return eventually defined global A or P, when C<< $x >>'s A or P is not
3435 =item Creating numbers
3437 * When you create a number, you can give it's desired A or P via:
3438 $x = Math::BigInt->new($number,$A,$P);
3439 * Only one of A or P can be defined, otherwise the result is NaN
3440 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
3441 globals (if set) will be used. Thus changing the global defaults later on
3442 will not change the A or P of previously created numbers (i.e., A and P of
3443 $x will be what was in effect when $x was created)
3444 * If given undef for A and P, B<no> rounding will occur, and the globals will
3445 B<not> be used. This is used by subclasses to create numbers without
3446 suffering rounding in the parent. Thus a subclass is able to have it's own
3447 globals enforced upon creation of a number by using
3448 C<< $x = Math::BigInt->new($number,undef,undef) >>:
3450 use Math::BigInt::SomeSubclass;
3453 Math::BigInt->accuracy(2);
3454 Math::BigInt::SomeSubClass->accuracy(3);
3455 $x = Math::BigInt::SomeSubClass->new(1234);
3457 $x is now 1230, and not 1200. A subclass might choose to implement
3458 this otherwise, e.g. falling back to the parent's A and P.
3462 * If A or P are enabled/defined, they are used to round the result of each
3463 operation according to the rules below
3464 * Negative P is ignored in Math::BigInt, since BigInts never have digits
3465 after the decimal point
3466 * Math::BigFloat uses Math::BigInt internally, but setting A or P inside
3467 Math::BigInt as globals does not tamper with the parts of a BigFloat.
3468 A flag is used to mark all Math::BigFloat numbers as 'never round'.
3472 * It only makes sense that a number has only one of A or P at a time.
3473 If you set either A or P on one object, or globally, the other one will
3474 be automatically cleared.
3475 * If two objects are involved in an operation, and one of them has A in
3476 effect, and the other P, this results in an error (NaN).
3477 * A takes precendence over P (Hint: A comes before P).
3478 If neither of them is defined, nothing is used, i.e. the result will have
3479 as many digits as it can (with an exception for fdiv/fsqrt) and will not
3481 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
3482 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
3483 If either the dividend's or the divisor's mantissa has more digits than
3484 the value of F, the higher value will be used instead of F.
3485 This is to limit the digits (A) of the result (just consider what would
3486 happen with unlimited A and P in the case of 1/3 :-)
3487 * fdiv will calculate (at least) 4 more digits than required (determined by
3488 A, P or F), and, if F is not used, round the result
3489 (this will still fail in the case of a result like 0.12345000000001 with A
3490 or P of 5, but this can not be helped - or can it?)
3491 * Thus you can have the math done by on Math::Big* class in two modi:
3492 + never round (this is the default):
3493 This is done by setting A and P to undef. No math operation
3494 will round the result, with fdiv() and fsqrt() as exceptions to guard
3495 against overflows. You must explicitely call bround(), bfround() or
3496 round() (the latter with parameters).
3497 Note: Once you have rounded a number, the settings will 'stick' on it
3498 and 'infect' all other numbers engaged in math operations with it, since
3499 local settings have the highest precedence. So, to get SaferRound[tm],
3500 use a copy() before rounding like this:
3502 $x = Math::BigFloat->new(12.34);
3503 $y = Math::BigFloat->new(98.76);
3504 $z = $x * $y; # 1218.6984
3505 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
3506 $z = $x * $y; # still 1218.6984, without
3507 # copy would have been 1210!
3509 + round after each op:
3510 After each single operation (except for testing like is_zero()), the
3511 method round() is called and the result is rounded appropriately. By
3512 setting proper values for A and P, you can have all-the-same-A or
3513 all-the-same-P modes. For example, Math::Currency might set A to undef,
3514 and P to -2, globally.
3516 ?Maybe an extra option that forbids local A & P settings would be in order,
3517 ?so that intermediate rounding does not 'poison' further math?
3519 =item Overriding globals
3521 * you will be able to give A, P and R as an argument to all the calculation
3522 routines; the second parameter is A, the third one is P, and the fourth is
3523 R (shift right by one for binary operations like badd). P is used only if
3524 the first parameter (A) is undefined. These three parameters override the
3525 globals in the order detailed as follows, i.e. the first defined value
3527 (local: per object, global: global default, parameter: argument to sub)
3530 + local A (if defined on both of the operands: smaller one is taken)
3531 + local P (if defined on both of the operands: bigger one is taken)
3535 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
3536 arguments (A and P) instead of one
3538 =item Local settings
3540 * You can set A or P locally by using C<< $x->accuracy() >> or
3541 C<< $x->precision() >>
3542 and thus force different A and P for different objects/numbers.
3543 * Setting A or P this way immediately rounds $x to the new value.
3544 * C<< $x->accuracy() >> clears C<< $x->precision() >>, and vice versa.
3548 * the rounding routines will use the respective global or local settings.
3549 fround()/bround() is for accuracy rounding, while ffround()/bfround()
3551 * the two rounding functions take as the second parameter one of the
3552 following rounding modes (R):
3553 'even', 'odd', '+inf', '-inf', 'zero', 'trunc'
3554 * you can set/get the global R by using C<< Math::SomeClass->round_mode() >>
3555 or by setting C<< $Math::SomeClass::round_mode >>
3556 * after each operation, C<< $result->round() >> is called, and the result may
3557 eventually be rounded (that is, if A or P were set either locally,
3558 globally or as parameter to the operation)
3559 * to manually round a number, call C<< $x->round($A,$P,$round_mode); >>
3560 this will round the number by using the appropriate rounding function
3561 and then normalize it.
3562 * rounding modifies the local settings of the number:
3564 $x = Math::BigFloat->new(123.456);
3568 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
3569 will be 4 from now on.
3571 =item Default values
3580 * The defaults are set up so that the new code gives the same results as
3581 the old code (except in a few cases on fdiv):
3582 + Both A and P are undefined and thus will not be used for rounding
3583 after each operation.
3584 + round() is thus a no-op, unless given extra parameters A and P
3590 The actual numbers are stored as unsigned big integers (with seperate sign).
3591 You should neither care about nor depend on the internal representation; it
3592 might change without notice. Use only method calls like C<< $x->sign(); >>
3593 instead relying on the internal hash keys like in C<< $x->{sign}; >>.
3597 Math with the numbers is done (by default) by a module called
3598 C<Math::BigInt::Calc>. This is equivalent to saying:
3600 use Math::BigInt lib => 'Calc';
3602 You can change this by using:
3604 use Math::BigInt lib => 'BitVect';
3606 The following would first try to find Math::BigInt::Foo, then
3607 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
3609 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
3611 Since Math::BigInt::GMP is in almost all cases faster than Calc (especially in
3612 cases involving really big numbers, where it is B<much> faster), and there is
3613 no penalty if Math::BigInt::GMP is not installed, it is a good idea to always
3616 use Math::BigInt lib => 'GMP';
3618 Different low-level libraries use different formats to store the
3619 numbers. You should not depend on the number having a specific format.
3621 See the respective math library module documentation for further details.
3625 The sign is either '+', '-', 'NaN', '+inf' or '-inf' and stored seperately.
3627 A sign of 'NaN' is used to represent the result when input arguments are not
3628 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
3629 minus infinity. You will get '+inf' when dividing a positive number by 0, and
3630 '-inf' when dividing any negative number by 0.
3632 =head2 mantissa(), exponent() and parts()
3634 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
3637 $m = $x->mantissa();
3638 $e = $x->exponent();
3639 $y = $m * ( 10 ** $e );
3640 print "ok\n" if $x == $y;
3642 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
3643 in one go. Both the returned mantissa and exponent have a sign.
3645 Currently, for BigInts C<$e> is always 0, except for NaN, +inf and -inf,
3646 where it is C<NaN>; and for C<$x == 0>, where it is C<1> (to be compatible
3647 with Math::BigFloat's internal representation of a zero as C<0E1>).
3649 C<$m> is currently just a copy of the original number. The relation between
3650 C<$e> and C<$m> will stay always the same, though their real values might
3657 sub bint { Math::BigInt->new(shift); }
3659 $x = Math::BigInt->bstr("1234") # string "1234"
3660 $x = "$x"; # same as bstr()
3661 $x = Math::BigInt->bneg("1234"); # BigInt "-1234"
3662 $x = Math::BigInt->babs("-12345"); # BigInt "12345"
3663 $x = Math::BigInt->bnorm("-0 00"); # BigInt "0"
3664 $x = bint(1) + bint(2); # BigInt "3"
3665 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
3666 $x = bint(1); # BigInt "1"
3667 $x = $x + 5 / 2; # BigInt "3"
3668 $x = $x ** 3; # BigInt "27"
3669 $x *= 2; # BigInt "54"
3670 $x = Math::BigInt->new(0); # BigInt "0"
3672 $x = Math::BigInt->badd(4,5) # BigInt "9"
3673 print $x->bsstr(); # 9e+0
3675 Examples for rounding:
3680 $x = Math::BigFloat->new(123.4567);
3681 $y = Math::BigFloat->new(123.456789);
3682 Math::BigFloat->accuracy(4); # no more A than 4
3684 ok ($x->copy()->fround(),123.4); # even rounding
3685 print $x->copy()->fround(),"\n"; # 123.4
3686 Math::BigFloat->round_mode('odd'); # round to odd
3687 print $x->copy()->fround(),"\n"; # 123.5
3688 Math::BigFloat->accuracy(5); # no more A than 5
3689 Math::BigFloat->round_mode('odd'); # round to odd
3690 print $x->copy()->fround(),"\n"; # 123.46
3691 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
3692 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
3694 Math::BigFloat->accuracy(undef); # A not important now
3695 Math::BigFloat->precision(2); # P important
3696 print $x->copy()->bnorm(),"\n"; # 123.46
3697 print $x->copy()->fround(),"\n"; # 123.46
3699 Examples for converting:
3701 my $x = Math::BigInt->new('0b1'.'01' x 123);
3702 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
3704 =head1 Autocreating constants
3706 After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
3707 and binary constants in the given scope are converted to C<Math::BigInt>.
3708 This conversion happens at compile time.
3712 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
3714 prints the integer value of C<2**100>. Note that without conversion of
3715 constants the expression 2**100 will be calculated as perl scalar.
3717 Please note that strings and floating point constants are not affected,
3720 use Math::BigInt qw/:constant/;
3722 $x = 1234567890123456789012345678901234567890
3723 + 123456789123456789;
3724 $y = '1234567890123456789012345678901234567890'
3725 + '123456789123456789';
3727 do not work. You need an explicit Math::BigInt->new() around one of the
3728 operands. You should also quote large constants to protect loss of precision:
3732 $x = Math::BigInt->new('1234567889123456789123456789123456789');
3734 Without the quotes Perl would convert the large number to a floating point
3735 constant at compile time and then hand the result to BigInt, which results in
3736 an truncated result or a NaN.
3738 This also applies to integers that look like floating point constants:
3740 use Math::BigInt ':constant';
3742 print ref(123e2),"\n";
3743 print ref(123.2e2),"\n";
3745 will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
3746 to get this to work.
3750 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
3751 must be made in the second case. For long numbers, the copy can eat up to 20%
3752 of the work (in the case of addition/subtraction, less for
3753 multiplication/division). If $y is very small compared to $x, the form
3754 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
3755 more time then the actual addition.
3757 With a technique called copy-on-write, the cost of copying with overload could
3758 be minimized or even completely avoided. A test implementation of COW did show
3759 performance gains for overloaded math, but introduced a performance loss due
3760 to a constant overhead for all other operatons. So Math::BigInt does currently
3763 The rewritten version of this module (vs. v0.01) is slower on certain
3764 operations, like C<new()>, C<bstr()> and C<numify()>. The reason are that it
3765 does now more work and handles much more cases. The time spent in these
3766 operations is usually gained in the other math operations so that code on
3767 the average should get (much) faster. If they don't, please contact the author.
3769 Some operations may be slower for small numbers, but are significantly faster
3770 for big numbers. Other operations are now constant (O(1), like C<bneg()>,
3771 C<babs()> etc), instead of O(N) and thus nearly always take much less time.
3772 These optimizations were done on purpose.
3774 If you find the Calc module to slow, try to install any of the replacement
3775 modules and see if they help you.
3777 =head2 Alternative math libraries
3779 You can use an alternative library to drive Math::BigInt via:
3781 use Math::BigInt lib => 'Module';
3783 See L<MATH LIBRARY> for more information.
3785 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
3789 =head1 Subclassing Math::BigInt
3791 The basic design of Math::BigInt allows simple subclasses with very little
3792 work, as long as a few simple rules are followed:
3798 The public API must remain consistent, i.e. if a sub-class is overloading
3799 addition, the sub-class must use the same name, in this case badd(). The
3800 reason for this is that Math::BigInt is optimized to call the object methods
3805 The private object hash keys like C<$x->{sign}> may not be changed, but
3806 additional keys can be added, like C<$x->{_custom}>.
3810 Accessor functions are available for all existing object hash keys and should
3811 be used instead of directly accessing the internal hash keys. The reason for
3812 this is that Math::BigInt itself has a pluggable interface which permits it
3813 to support different storage methods.
3817 More complex sub-classes may have to replicate more of the logic internal of
3818 Math::BigInt if they need to change more basic behaviors. A subclass that
3819 needs to merely change the output only needs to overload C<bstr()>.
3821 All other object methods and overloaded functions can be directly inherited
3822 from the parent class.
3824 At the very minimum, any subclass will need to provide it's own C<new()> and can
3825 store additional hash keys in the object. There are also some package globals
3826 that must be defined, e.g.:
3830 $precision = -2; # round to 2 decimal places
3831 $round_mode = 'even';
3834 Additionally, you might want to provide the following two globals to allow
3835 auto-upgrading and auto-downgrading to work correctly:
3840 This allows Math::BigInt to correctly retrieve package globals from the
3841 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
3842 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
3848 in your subclass to automatically inherit the overloading from the parent. If
3849 you like, you can change part of the overloading, look at Math::String for an
3854 When used like this:
3856 use Math::BigInt upgrade => 'Foo::Bar';
3858 certain operations will 'upgrade' their calculation and thus the result to
3859 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
3861 use Math::BigInt upgrade => 'Math::BigFloat';
3863 As a shortcut, you can use the module C<bignum>:
3867 Also good for oneliners:
3869 perl -Mbignum -le 'print 2 ** 255'
3871 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
3872 as well es preserve accuracy (as in sqrt(3)).
3874 Beware: This feature is not fully implemented yet.
3878 The following methods upgrade themselves unconditionally; that is if upgrade
3879 is in effect, they will always hand up their work:
3891 Beware: This list is not complete.
3893 All other methods upgrade themselves only when one (or all) of their
3894 arguments are of the class mentioned in $upgrade (This might change in later
3895 versions to a more sophisticated scheme):
3901 =item broot() does not work
3903 The broot() function in BigInt may only work for small values. This will be
3904 fixed in a later version.
3906 =item Out of Memory!
3908 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
3909 C<eval()> in your code will crash with "Out of memory". This is probably an
3910 overload/exporter bug. You can workaround by not having C<eval()>
3911 and ':constant' at the same time or upgrade your Perl to a newer version.
3913 =item Fails to load Calc on Perl prior 5.6.0
3915 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
3916 will fall back to eval { require ... } when loading the math lib on Perls
3917 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
3918 filesystems using a different seperator.
3924 Some things might not work as you expect them. Below is documented what is
3925 known to be troublesome:
3929 =item bstr(), bsstr() and 'cmp'
3931 Both C<bstr()> and C<bsstr()> as well as automated stringify via overload now
3932 drop the leading '+'. The old code would return '+3', the new returns '3'.
3933 This is to be consistent with Perl and to make C<cmp> (especially with
3934 overloading) to work as you expect. It also solves problems with C<Test.pm>,
3935 because it's C<ok()> uses 'eq' internally.
3937 Mark Biggar said, when asked about to drop the '+' altogether, or make only
3940 I agree (with the first alternative), don't add the '+' on positive
3941 numbers. It's not as important anymore with the new internal
3942 form for numbers. It made doing things like abs and neg easier,
3943 but those have to be done differently now anyway.
3945 So, the following examples will now work all as expected:
3948 BEGIN { plan tests => 1 }
3951 my $x = new Math::BigInt 3*3;
3952 my $y = new Math::BigInt 3*3;
3955 print "$x eq 9" if $x eq $y;
3956 print "$x eq 9" if $x eq '9';
3957 print "$x eq 9" if $x eq 3*3;
3959 Additionally, the following still works:
3961 print "$x == 9" if $x == $y;
3962 print "$x == 9" if $x == 9;
3963 print "$x == 9" if $x == 3*3;
3965 There is now a C<bsstr()> method to get the string in scientific notation aka
3966 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
3967 for comparisation, but Perl will represent some numbers as 100 and others
3968 as 1e+308. If in doubt, convert both arguments to Math::BigInt before
3969 comparing them as strings:
3972 BEGIN { plan tests => 3 }
3975 $x = Math::BigInt->new('1e56'); $y = 1e56;
3976 ok ($x,$y); # will fail
3977 ok ($x->bsstr(),$y); # okay
3978 $y = Math::BigInt->new($y);
3981 Alternatively, simple use C<< <=> >> for comparisations, this will get it
3982 always right. There is not yet a way to get a number automatically represented
3983 as a string that matches exactly the way Perl represents it.
3987 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
3990 $x = Math::BigInt->new(123);
3991 $y = int($x); # BigInt 123
3992 $x = Math::BigFloat->new(123.45);
3993 $y = int($x); # BigInt 123
3995 In all Perl versions you can use C<as_number()> for the same effect:
3997 $x = Math::BigFloat->new(123.45);
3998 $y = $x->as_number(); # BigInt 123
4000 This also works for other subclasses, like Math::String.
4002 It is yet unlcear whether overloaded int() should return a scalar or a BigInt.
4006 The following will probably not do what you expect:
4008 $c = Math::BigInt->new(123);
4009 print $c->length(),"\n"; # prints 30
4011 It prints both the number of digits in the number and in the fraction part
4012 since print calls C<length()> in list context. Use something like:
4014 print scalar $c->length(),"\n"; # prints 3
4018 The following will probably not do what you expect:
4020 print $c->bdiv(10000),"\n";
4022 It prints both quotient and remainder since print calls C<bdiv()> in list
4023 context. Also, C<bdiv()> will modify $c, so be carefull. You probably want
4026 print $c / 10000,"\n";
4027 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
4031 The quotient is always the greatest integer less than or equal to the
4032 real-valued quotient of the two operands, and the remainder (when it is
4033 nonzero) always has the same sign as the second operand; so, for
4043 As a consequence, the behavior of the operator % agrees with the
4044 behavior of Perl's built-in % operator (as documented in the perlop
4045 manpage), and the equation
4047 $x == ($x / $y) * $y + ($x % $y)
4049 holds true for any $x and $y, which justifies calling the two return
4050 values of bdiv() the quotient and remainder. The only exception to this rule
4051 are when $y == 0 and $x is negative, then the remainder will also be
4052 negative. See below under "infinity handling" for the reasoning behing this.
4054 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
4055 not change BigInt's way to do things. This is because under 'use integer' Perl
4056 will do what the underlying C thinks is right and this is different for each
4057 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
4058 the author to implement it ;)
4060 =item infinity handling
4062 Here are some examples that explain the reasons why certain results occur while
4065 The following table shows the result of the division and the remainder, so that
4066 the equation above holds true. Some "ordinary" cases are strewn in to show more
4067 clearly the reasoning:
4069 A / B = C, R so that C * B + R = A
4070 =========================================================
4071 5 / 8 = 0, 5 0 * 8 + 5 = 5
4072 0 / 8 = 0, 0 0 * 8 + 0 = 0
4073 0 / inf = 0, 0 0 * inf + 0 = 0
4074 0 /-inf = 0, 0 0 * -inf + 0 = 0
4075 5 / inf = 0, 5 0 * inf + 5 = 5
4076 5 /-inf = 0, 5 0 * -inf + 5 = 5
4077 -5/ inf = 0, -5 0 * inf + -5 = -5
4078 -5/-inf = 0, -5 0 * -inf + -5 = -5
4079 inf/ 5 = inf, 0 inf * 5 + 0 = inf
4080 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
4081 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
4082 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
4083 5/ 5 = 1, 0 1 * 5 + 0 = 5
4084 -5/ -5 = 1, 0 1 * -5 + 0 = -5
4085 inf/ inf = 1, 0 1 * inf + 0 = inf
4086 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
4087 inf/-inf = -1, 0 -1 * -inf + 0 = inf
4088 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
4089 8/ 0 = inf, 8 inf * 0 + 8 = 8
4090 inf/ 0 = inf, inf inf * 0 + inf = inf
4093 These cases below violate the "remainder has the sign of the second of the two
4094 arguments", since they wouldn't match up otherwise.
4096 A / B = C, R so that C * B + R = A
4097 ========================================================
4098 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
4099 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
4101 =item Modifying and =
4105 $x = Math::BigFloat->new(5);
4108 It will not do what you think, e.g. making a copy of $x. Instead it just makes
4109 a second reference to the B<same> object and stores it in $y. Thus anything
4110 that modifies $x (except overloaded operators) will modify $y, and vice versa.
4111 Or in other words, C<=> is only safe if you modify your BigInts only via
4112 overloaded math. As soon as you use a method call it breaks:
4115 print "$x, $y\n"; # prints '10, 10'
4117 If you want a true copy of $x, use:
4121 You can also chain the calls like this, this will make first a copy and then
4124 $y = $x->copy()->bmul(2);
4126 See also the documentation for overload.pm regarding C<=>.
4130 C<bpow()> (and the rounding functions) now modifies the first argument and
4131 returns it, unlike the old code which left it alone and only returned the
4132 result. This is to be consistent with C<badd()> etc. The first three will
4133 modify $x, the last one won't:
4135 print bpow($x,$i),"\n"; # modify $x
4136 print $x->bpow($i),"\n"; # ditto
4137 print $x **= $i,"\n"; # the same
4138 print $x ** $i,"\n"; # leave $x alone
4140 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4142 =item Overloading -$x
4152 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4153 needs to preserve $x since it does not know that it later will get overwritten.
4154 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4156 With Copy-On-Write, this issue would be gone, but C-o-W is not implemented
4157 since it is slower for all other things.
4159 =item Mixing different object types
4161 In Perl you will get a floating point value if you do one of the following:
4167 With overloaded math, only the first two variants will result in a BigFloat:
4172 $mbf = Math::BigFloat->new(5);
4173 $mbi2 = Math::BigInteger->new(5);
4174 $mbi = Math::BigInteger->new(2);
4176 # what actually gets called:
4177 $float = $mbf + $mbi; # $mbf->badd()
4178 $float = $mbf / $mbi; # $mbf->bdiv()
4179 $integer = $mbi + $mbf; # $mbi->badd()
4180 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4181 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4183 This is because math with overloaded operators follows the first (dominating)
4184 operand, and the operation of that is called and returns thus the result. So,
4185 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4186 the result should be a Math::BigFloat or the second operant is one.
4188 To get a Math::BigFloat you either need to call the operation manually,
4189 make sure the operands are already of the proper type or casted to that type
4190 via Math::BigFloat->new():
4192 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4194 Beware of simple "casting" the entire expression, this would only convert
4195 the already computed result:
4197 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4199 Beware also of the order of more complicated expressions like:
4201 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4202 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4204 If in doubt, break the expression into simpler terms, or cast all operands
4205 to the desired resulting type.
4207 Scalar values are a bit different, since:
4212 will both result in the proper type due to the way the overloaded math works.
4214 This section also applies to other overloaded math packages, like Math::String.
4216 One solution to you problem might be autoupgrading|upgrading. See the
4217 pragmas L<bignum>, L<bigint> and L<bigrat> for an easy way to do this.
4221 C<bsqrt()> works only good if the result is a big integer, e.g. the square
4222 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4223 mode. The reason is that the result is always truncated to an integer.
4225 If you want a better approximation of the square root, then use:
4227 $x = Math::BigFloat->new(12);
4228 Math::BigFloat->precision(0);
4229 Math::BigFloat->round_mode('even');
4230 print $x->copy->bsqrt(),"\n"; # 4
4232 Math::BigFloat->precision(2);
4233 print $x->bsqrt(),"\n"; # 3.46
4234 print $x->bsqrt(3),"\n"; # 3.464
4238 For negative numbers in base see also L<brsft|brsft>.
4244 This program is free software; you may redistribute it and/or modify it under
4245 the same terms as Perl itself.
4249 L<Math::BigFloat>, L<Math::BigRat> and L<Math::Big> as well as
4250 L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4252 The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest
4253 because they solve the autoupgrading/downgrading issue, at least partly.
4256 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4257 more documentation including a full version history, testcases, empty
4258 subclass files and benchmarks.
4262 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4263 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001 - 2003
4264 and still at it in 2004.
4266 Many people contributed in one or more ways to the final beast, see the file
4267 CREDITS for an (uncomplete) list. If you miss your name, please drop me a