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 $class = "Math::BigInt";
23 @ISA = qw( Exporter );
24 @EXPORT_OK = qw( objectify _swap bgcd blcm);
25 use vars qw/$round_mode $accuracy $precision $div_scale $rnd_mode/;
26 use vars qw/$upgrade $downgrade/;
29 # Inside overload, the first arg is always an object. If the original code had
30 # it reversed (like $x = 2 * $y), then the third paramater indicates this
31 # swapping. To make it work, we use a helper routine which not only reswaps the
32 # params, but also makes a new object in this case. See _swap() for details,
33 # especially the cases of operators with different classes.
35 # For overloaded ops with only one argument we simple use $_[0]->copy() to
36 # preserve the argument.
38 # Thus inheritance of overload operators becomes possible and transparent for
39 # our subclasses without the need to repeat the entire overload section there.
42 '=' => sub { $_[0]->copy(); },
44 # '+' and '-' do not use _swap, since it is a triffle slower. If you want to
45 # override _swap (if ever), then override overload of '+' and '-', too!
46 # for sub it is a bit tricky to keep b: b-a => -a+b
47 '-' => sub { my $c = $_[0]->copy; $_[2] ?
48 $c->bneg()->badd($_[1]) :
50 '+' => sub { $_[0]->copy()->badd($_[1]); },
52 # some shortcuts for speed (assumes that reversed order of arguments is routed
53 # to normal '+' and we thus can always modify first arg. If this is changed,
54 # this breaks and must be adjusted.)
55 '+=' => sub { $_[0]->badd($_[1]); },
56 '-=' => sub { $_[0]->bsub($_[1]); },
57 '*=' => sub { $_[0]->bmul($_[1]); },
58 '/=' => sub { scalar $_[0]->bdiv($_[1]); },
59 '%=' => sub { $_[0]->bmod($_[1]); },
60 '^=' => sub { $_[0]->bxor($_[1]); },
61 '&=' => sub { $_[0]->band($_[1]); },
62 '|=' => sub { $_[0]->bior($_[1]); },
63 '**=' => sub { $_[0]->bpow($_[1]); },
65 # not supported by Perl yet
66 '..' => \&_pointpoint,
68 '<=>' => sub { $_[2] ?
69 ref($_[0])->bcmp($_[1],$_[0]) :
73 "$_[1]" cmp $_[0]->bstr() :
74 $_[0]->bstr() cmp "$_[1]" },
76 'log' => sub { $_[0]->copy()->blog(); },
77 'int' => sub { $_[0]->copy(); },
78 'neg' => sub { $_[0]->copy()->bneg(); },
79 'abs' => sub { $_[0]->copy()->babs(); },
80 'sqrt' => sub { $_[0]->copy()->bsqrt(); },
81 '~' => sub { $_[0]->copy()->bnot(); },
83 '*' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bmul($a[1]); },
84 '/' => sub { my @a = ref($_[0])->_swap(@_);scalar $a[0]->bdiv($a[1]);},
85 '%' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bmod($a[1]); },
86 '**' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bpow($a[1]); },
87 '<<' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->blsft($a[1]); },
88 '>>' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->brsft($a[1]); },
90 '&' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->band($a[1]); },
91 '|' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bior($a[1]); },
92 '^' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bxor($a[1]); },
94 # can modify arg of ++ and --, so avoid a new-copy for speed, but don't
95 # use $_[0]->__one(), it modifies $_[0] to be 1!
96 '++' => sub { $_[0]->binc() },
97 '--' => sub { $_[0]->bdec() },
99 # if overloaded, O(1) instead of O(N) and twice as fast for small numbers
101 # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/
102 # v5.6.1 dumps on that: return !$_[0]->is_zero() || undef; :-(
103 my $t = !$_[0]->is_zero();
108 # the original qw() does not work with the TIESCALAR below, why?
109 # Order of arguments unsignificant
110 '""' => sub { $_[0]->bstr(); },
111 '0+' => sub { $_[0]->numify(); }
114 ##############################################################################
115 # global constants, flags and accessory
117 use constant MB_NEVER_ROUND => 0x0001;
119 my $NaNOK=1; # are NaNs ok?
120 my $nan = 'NaN'; # constants for easier life
122 my $CALC = 'Math::BigInt::Calc'; # module to do low level math
123 my $IMPORT = 0; # did import() yet?
125 $round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
130 $upgrade = undef; # default is no upgrade
131 $downgrade = undef; # default is no downgrade
133 ##############################################################################
134 # the old code had $rnd_mode, so we need to support it, too
137 sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; }
138 sub FETCH { return $round_mode; }
139 sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
141 BEGIN { tie $rnd_mode, 'Math::BigInt'; }
143 ##############################################################################
148 # make Class->round_mode() work
150 my $class = ref($self) || $self || __PACKAGE__;
154 die "Unknown round mode $m"
155 if $m !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/;
156 return ${"${class}::round_mode"} = $m;
158 return ${"${class}::round_mode"};
164 # make Class->upgrade() work
166 my $class = ref($self) || $self || __PACKAGE__;
167 # need to set new value?
171 return ${"${class}::upgrade"} = $u;
173 return ${"${class}::upgrade"};
179 # make Class->downgrade() work
181 my $class = ref($self) || $self || __PACKAGE__;
182 # need to set new value?
186 return ${"${class}::downgrade"} = $u;
188 return ${"${class}::downgrade"};
194 # make Class->round_mode() work
196 my $class = ref($self) || $self || __PACKAGE__;
199 die ('div_scale must be greater than zero') if $_[0] < 0;
200 ${"${class}::div_scale"} = shift;
202 return ${"${class}::div_scale"};
207 # $x->accuracy($a); ref($x) $a
208 # $x->accuracy(); ref($x)
209 # Class->accuracy(); class
210 # Class->accuracy($a); class $a
213 my $class = ref($x) || $x || __PACKAGE__;
216 # need to set new value?
220 die ('accuracy must not be zero') if defined $a && $a == 0;
223 # $object->accuracy() or fallback to global
224 $x->bround($a) if defined $a;
225 $x->{_a} = $a; # set/overwrite, even if not rounded
226 $x->{_p} = undef; # clear P
231 ${"${class}::accuracy"} = $a;
232 ${"${class}::precision"} = undef; # clear P
234 return $a; # shortcut
238 # $object->accuracy() or fallback to global
239 $r = $x->{_a} if ref($x);
240 # but don't return global undef, when $x's accuracy is 0!
241 $r = ${"${class}::accuracy"} if !defined $r;
247 # $x->precision($p); ref($x) $p
248 # $x->precision(); ref($x)
249 # Class->precision(); class
250 # Class->precision($p); class $p
253 my $class = ref($x) || $x || __PACKAGE__;
256 # need to set new value?
262 # $object->precision() or fallback to global
263 $x->bfround($p) if defined $p;
264 $x->{_p} = $p; # set/overwrite, even if not rounded
265 $x->{_a} = undef; # clear A
270 ${"${class}::precision"} = $p;
271 ${"${class}::accuracy"} = undef; # clear A
273 return $p; # shortcut
277 # $object->precision() or fallback to global
278 $r = $x->{_p} if ref($x);
279 # but don't return global undef, when $x's precision is 0!
280 $r = ${"${class}::precision"} if !defined $r;
286 # return (later set?) configuration data as hash ref
287 my $class = shift || 'Math::BigInt';
293 lib_version => ${"${lib}::VERSION"},
297 qw/upgrade downgrade precision accuracy round_mode VERSION div_scale/)
299 $cfg->{lc($_)} = ${"${class}::$_"};
306 # select accuracy parameter based on precedence,
307 # used by bround() and bfround(), may return undef for scale (means no op)
308 my ($x,$s,$m,$scale,$mode) = @_;
309 $scale = $x->{_a} if !defined $scale;
310 $scale = $s if (!defined $scale);
311 $mode = $m if !defined $mode;
312 return ($scale,$mode);
317 # select precision parameter based on precedence,
318 # used by bround() and bfround(), may return undef for scale (means no op)
319 my ($x,$s,$m,$scale,$mode) = @_;
320 $scale = $x->{_p} if !defined $scale;
321 $scale = $s if (!defined $scale);
322 $mode = $m if !defined $mode;
323 return ($scale,$mode);
326 ##############################################################################
334 # if two arguments, the first one is the class to "swallow" subclasses
342 return unless ref($x); # only for objects
344 my $self = {}; bless $self,$c;
346 foreach my $k (keys %$x)
350 $self->{value} = $CALC->_copy($x->{value}); next;
352 if (!($r = ref($x->{$k})))
354 $self->{$k} = $x->{$k}; next;
358 $self->{$k} = \${$x->{$k}};
360 elsif ($r eq 'ARRAY')
362 $self->{$k} = [ @{$x->{$k}} ];
366 # only one level deep!
367 foreach my $h (keys %{$x->{$k}})
369 $self->{$k}->{$h} = $x->{$k}->{$h};
375 if ($xk->can('copy'))
377 $self->{$k} = $xk->copy();
381 $self->{$k} = $xk->new($xk);
390 # create a new BigInt object from a string or another BigInt object.
391 # see hash keys documented at top
393 # the argument could be an object, so avoid ||, && etc on it, this would
394 # cause costly overloaded code to be called. The only allowed ops are
397 my ($class,$wanted,$a,$p,$r) = @_;
399 # avoid numify-calls by not using || on $wanted!
400 return $class->bzero($a,$p) if !defined $wanted; # default to 0
401 return $class->copy($wanted,$a,$p,$r)
402 if ref($wanted) && $wanted->isa($class); # MBI or subclass
404 $class->import() if $IMPORT == 0; # make require work
406 my $self = bless {}, $class;
408 # shortcut for "normal" numbers
409 if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*\z/))
411 $self->{sign} = $1 || '+';
413 if ($wanted =~ /^[+-]/)
415 # remove sign without touching wanted to make it work with constants
416 my $t = $wanted; $t =~ s/^[+-]//; $ref = \$t;
418 $self->{value} = $CALC->_new($ref);
420 if ( (defined $a) || (defined $p)
421 || (defined ${"${class}::precision"})
422 || (defined ${"${class}::accuracy"})
425 $self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p);
430 # handle '+inf', '-inf' first
431 if ($wanted =~ /^[+-]?inf$/)
433 $self->{value} = $CALC->_zero();
434 $self->{sign} = $wanted; $self->{sign} = '+inf' if $self->{sign} eq 'inf';
437 # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign
438 my ($mis,$miv,$mfv,$es,$ev) = _split(\$wanted);
441 die "$wanted is not a number initialized to $class" if !$NaNOK;
443 $self->{value} = $CALC->_zero();
444 $self->{sign} = $nan;
449 # _from_hex or _from_bin
450 $self->{value} = $mis->{value};
451 $self->{sign} = $mis->{sign};
452 return $self; # throw away $mis
454 # make integer from mantissa by adjusting exp, then convert to bigint
455 $self->{sign} = $$mis; # store sign
456 $self->{value} = $CALC->_zero(); # for all the NaN cases
457 my $e = int("$$es$$ev"); # exponent (avoid recursion)
460 my $diff = $e - CORE::length($$mfv);
461 if ($diff < 0) # Not integer
464 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
465 $self->{sign} = $nan;
469 # adjust fraction and add it to value
470 # print "diff > 0 $$miv\n";
471 $$miv = $$miv . ($$mfv . '0' x $diff);
476 if ($$mfv ne '') # e <= 0
478 # fraction and negative/zero E => NOI
479 #print "NOI 2 \$\$mfv '$$mfv'\n";
480 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
481 $self->{sign} = $nan;
485 # xE-y, and empty mfv
488 if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
491 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
492 $self->{sign} = $nan;
496 $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
497 $self->{value} = $CALC->_new($miv) if $self->{sign} =~ /^[+-]$/;
498 # if any of the globals is set, use them to round and store them inside $self
499 # do not round for new($x,undef,undef) since that is used by MBF to signal
501 $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p;
507 # create a bigint 'NaN', if given a BigInt, set it to 'NaN'
509 $self = $class if !defined $self;
512 my $c = $self; $self = {}; bless $self, $c;
514 $self->import() if $IMPORT == 0; # make require work
515 return if $self->modify('bnan');
517 if ($self->can('_bnan'))
519 # use subclass to initialize
524 # otherwise do our own thing
525 $self->{value} = $CALC->_zero();
527 $self->{sign} = $nan;
528 delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly
534 # create a bigint '+-inf', if given a BigInt, set it to '+-inf'
535 # the sign is either '+', or if given, used from there
537 my $sign = shift; $sign = '+' if !defined $sign || $sign !~ /^-(inf)?$/;
538 $self = $class if !defined $self;
541 my $c = $self; $self = {}; bless $self, $c;
543 $self->import() if $IMPORT == 0; # make require work
544 return if $self->modify('binf');
546 if ($self->can('_binf'))
548 # use subclass to initialize
553 # otherwise do our own thing
554 $self->{value} = $CALC->_zero();
556 $sign = $sign . 'inf' if $sign !~ /inf$/; # - => -inf
557 $self->{sign} = $sign;
558 ($self->{_a},$self->{_p}) = @_; # take over requested rounding
564 # create a bigint '+0', if given a BigInt, set it to 0
566 $self = $class if !defined $self;
570 my $c = $self; $self = {}; bless $self, $c;
572 $self->import() if $IMPORT == 0; # make require work
573 return if $self->modify('bzero');
575 if ($self->can('_bzero'))
577 # use subclass to initialize
582 # otherwise do our own thing
583 $self->{value} = $CALC->_zero();
590 # call like: $x->bzero($a,$p,$r,$y);
591 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
596 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
598 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
606 # create a bigint '+1' (or -1 if given sign '-'),
607 # if given a BigInt, set it to +1 or -1, respecively
609 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
610 $self = $class if !defined $self;
614 my $c = $self; $self = {}; bless $self, $c;
616 $self->import() if $IMPORT == 0; # make require work
617 return if $self->modify('bone');
619 if ($self->can('_bone'))
621 # use subclass to initialize
626 # otherwise do our own thing
627 $self->{value} = $CALC->_one();
629 $self->{sign} = $sign;
634 # call like: $x->bone($sign,$a,$p,$r,$y);
635 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
640 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
642 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
648 ##############################################################################
649 # string conversation
653 # (ref to BFLOAT or num_str ) return num_str
654 # Convert number from internal format to scientific string format.
655 # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
656 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
657 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
659 if ($x->{sign} !~ /^[+-]$/)
661 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
664 my ($m,$e) = $x->parts();
665 my $sign = 'e+'; # e can only be positive
666 return $m->bstr().$sign.$e->bstr();
671 # make a string from bigint object
672 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
673 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
675 if ($x->{sign} !~ /^[+-]$/)
677 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
680 my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
681 return $es.${$CALC->_str($x->{value})};
686 # Make a "normal" scalar from a BigInt object
687 my $x = shift; $x = $class->new($x) unless ref $x;
689 return $x->bstr() if $x->{sign} !~ /^[+-]$/;
690 my $num = $CALC->_num($x->{value});
691 return -$num if $x->{sign} eq '-';
695 ##############################################################################
696 # public stuff (usually prefixed with "b")
700 # return the sign of the number: +/-/-inf/+inf/NaN
701 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
706 sub _find_round_parameters
708 # After any operation or when calling round(), the result is rounded by
709 # regarding the A & P from arguments, local parameters, or globals.
711 # This procedure finds the round parameters, but it is for speed reasons
712 # duplicated in round. Otherwise, it is tested by the testsuite and used
715 my ($self,$a,$p,$r,@args) = @_;
716 # $a accuracy, if given by caller
717 # $p precision, if given by caller
718 # $r round_mode, if given by caller
719 # @args all 'other' arguments (0 for unary, 1 for binary ops)
721 # leave bigfloat parts alone
722 return ($self) if exists $self->{_f} && $self->{_f} & MB_NEVER_ROUND != 0;
724 my $c = ref($self); # find out class of argument(s)
727 # now pick $a or $p, but only if we have got "arguments"
730 foreach ($self,@args)
732 # take the defined one, or if both defined, the one that is smaller
733 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
738 # even if $a is defined, take $p, to signal error for both defined
739 foreach ($self,@args)
741 # take the defined one, or if both defined, the one that is bigger
743 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
746 # if still none defined, use globals (#2)
747 $a = ${"$c\::accuracy"} unless defined $a;
748 $p = ${"$c\::precision"} unless defined $p;
751 return ($self) unless defined $a || defined $p; # early out
753 # set A and set P is an fatal error
754 return ($self->bnan()) if defined $a && defined $p;
756 $r = ${"$c\::round_mode"} unless defined $r;
757 die "Unknown round mode '$r'" if $r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/;
759 return ($self,$a,$p,$r);
764 # Round $self according to given parameters, or given second argument's
765 # parameters or global defaults
767 # for speed reasons, _find_round_parameters is embeded here:
769 my ($self,$a,$p,$r,@args) = @_;
770 # $a accuracy, if given by caller
771 # $p precision, if given by caller
772 # $r round_mode, if given by caller
773 # @args all 'other' arguments (0 for unary, 1 for binary ops)
775 # leave bigfloat parts alone
776 return ($self) if exists $self->{_f} && $self->{_f} & MB_NEVER_ROUND != 0;
778 my $c = ref($self); # find out class of argument(s)
781 # now pick $a or $p, but only if we have got "arguments"
784 foreach ($self,@args)
786 # take the defined one, or if both defined, the one that is smaller
787 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
792 # even if $a is defined, take $p, to signal error for both defined
793 foreach ($self,@args)
795 # take the defined one, or if both defined, the one that is bigger
797 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
800 # if still none defined, use globals (#2)
801 $a = ${"$c\::accuracy"} unless defined $a;
802 $p = ${"$c\::precision"} unless defined $p;
805 return $self unless defined $a || defined $p; # early out
807 # set A and set P is an fatal error
808 return $self->bnan() if defined $a && defined $p;
810 $r = ${"$c\::round_mode"} unless defined $r;
811 die "Unknown round mode '$r'" if $r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/;
813 # now round, by calling either fround or ffround:
816 $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a;
818 else # both can't be undefined due to early out
820 $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p;
822 $self->bnorm(); # after round, normalize
827 # (numstr or BINT) return BINT
828 # Normalize number -- no-op here
829 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
835 # (BINT or num_str) return BINT
836 # make number absolute, or return absolute BINT from string
837 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
839 return $x if $x->modify('babs');
840 # post-normalized abs for internal use (does nothing for NaN)
841 $x->{sign} =~ s/^-/+/;
847 # (BINT or num_str) return BINT
848 # negate number or make a negated number from string
849 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
851 return $x if $x->modify('bneg');
853 # for +0 dont negate (to have always normalized)
854 $x->{sign} =~ tr/+-/-+/ if !$x->is_zero(); # does nothing for NaN
860 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
861 # (BINT or num_str, BINT or num_str) return cond_code
864 my ($self,$x,$y) = (ref($_[0]),@_);
866 # objectify is costly, so avoid it
867 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
869 ($self,$x,$y) = objectify(2,@_);
872 return $upgrade->bcmp($x,$y) if defined $upgrade &&
873 ((!$x->isa($self)) || (!$y->isa($self)));
875 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
877 # handle +-inf and NaN
878 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
879 return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
880 return +1 if $x->{sign} eq '+inf';
881 return -1 if $x->{sign} eq '-inf';
882 return -1 if $y->{sign} eq '+inf';
885 # check sign for speed first
886 return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
887 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
889 # have same sign, so compare absolute values. Don't make tests for zero here
890 # because it's actually slower than testin in Calc (especially w/ Pari et al)
892 # post-normalized compare for internal use (honors signs)
893 if ($x->{sign} eq '+')
896 return $CALC->_acmp($x->{value},$y->{value});
900 $CALC->_acmp($y->{value},$x->{value}); # swaped (lib returns 0,1,-1)
905 # Compares 2 values, ignoring their signs.
906 # Returns one of undef, <0, =0, >0. (suitable for sort)
907 # (BINT, BINT) return cond_code
910 my ($self,$x,$y) = (ref($_[0]),@_);
911 # objectify is costly, so avoid it
912 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
914 ($self,$x,$y) = objectify(2,@_);
917 return $upgrade->bacmp($x,$y) if defined $upgrade &&
918 ((!$x->isa($self)) || (!$y->isa($self)));
920 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
922 # handle +-inf and NaN
923 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
924 return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
925 return +1; # inf is always bigger
927 $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
932 # add second arg (BINT or string) to first (BINT) (modifies first)
933 # return result as BINT
936 my ($self,$x,$y,@r) = (ref($_[0]),@_);
937 # objectify is costly, so avoid it
938 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
940 ($self,$x,$y,@r) = objectify(2,@_);
943 return $x if $x->modify('badd');
944 return $upgrade->badd($x,$y,@r) if defined $upgrade &&
945 ((!$x->isa($self)) || (!$y->isa($self)));
947 $r[3] = $y; # no push!
948 # inf and NaN handling
949 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
952 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
954 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
956 # +inf++inf or -inf+-inf => same, rest is NaN
957 return $x if $x->{sign} eq $y->{sign};
960 # +-inf + something => +inf
961 # something +-inf => +-inf
962 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
966 my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
970 $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
975 my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
978 #print "swapped sub (a=$a)\n";
979 $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
984 # speedup, if equal, set result to 0
985 #print "equal sub, result = 0\n";
986 $x->{value} = $CALC->_zero();
991 #print "unswapped sub (a=$a)\n";
992 $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
996 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1002 # (BINT or num_str, BINT or num_str) return num_str
1003 # subtract second arg from first, modify first
1006 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1007 # objectify is costly, so avoid it
1008 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1010 ($self,$x,$y,@r) = objectify(2,@_);
1013 return $x if $x->modify('bsub');
1015 # upgrade done by badd():
1016 # return $upgrade->badd($x,$y,@r) if defined $upgrade &&
1017 # ((!$x->isa($self)) || (!$y->isa($self)));
1021 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1025 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
1026 $x->badd($y,@r); # badd does not leave internal zeros
1027 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
1028 $x; # already rounded by badd() or no round necc.
1033 # increment arg by one
1034 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1035 return $x if $x->modify('binc');
1037 if ($x->{sign} eq '+')
1039 $x->{value} = $CALC->_inc($x->{value});
1040 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1043 elsif ($x->{sign} eq '-')
1045 $x->{value} = $CALC->_dec($x->{value});
1046 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1047 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1050 # inf, nan handling etc
1051 $x->badd($self->__one(),$a,$p,$r); # badd does round
1056 # decrement arg by one
1057 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1058 return $x if $x->modify('bdec');
1060 my $zero = $CALC->_is_zero($x->{value}) && $x->{sign} eq '+';
1062 if (($x->{sign} eq '-') || $zero)
1064 $x->{value} = $CALC->_inc($x->{value});
1065 $x->{sign} = '-' if $zero; # 0 => 1 => -1
1066 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1067 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1071 elsif ($x->{sign} eq '+')
1073 $x->{value} = $CALC->_dec($x->{value});
1074 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1077 # inf, nan handling etc
1078 $x->badd($self->__one('-'),$a,$p,$r); # badd does round
1083 # not implemented yet
1084 my ($self,$x,$base,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1086 return $upgrade->blog($x,$base,$a,$p,$r) if defined $upgrade;
1093 # (BINT or num_str, BINT or num_str) return BINT
1094 # does not modify arguments, but returns new object
1095 # Lowest Common Multiplicator
1097 my $y = shift; my ($x);
1104 $x = $class->new($y);
1106 while (@_) { $x = __lcm($x,shift); }
1112 # (BINT or num_str, BINT or num_str) return BINT
1113 # does not modify arguments, but returns new object
1114 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1117 $y = __PACKAGE__->new($y) if !ref($y);
1119 my $x = $y->copy(); # keep arguments
1120 if ($CALC->can('_gcd'))
1124 $y = shift; $y = $self->new($y) if !ref($y);
1125 next if $y->is_zero();
1126 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1127 $x->{value} = $CALC->_gcd($x->{value},$y->{value}); last if $x->is_one();
1134 $y = shift; $y = $self->new($y) if !ref($y);
1135 $x = __gcd($x,$y->copy()); last if $x->is_one(); # _gcd handles NaN
1143 # (num_str or BINT) return BINT
1144 # represent ~x as twos-complement number
1145 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1146 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1148 return $x if $x->modify('bnot');
1149 $x->bneg()->bdec(); # bdec already does round
1152 # is_foo test routines
1156 # return true if arg (BINT or num_str) is zero (array '+', '0')
1157 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1158 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1160 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1161 $CALC->_is_zero($x->{value});
1166 # return true if arg (BINT or num_str) is NaN
1167 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1169 return 1 if $x->{sign} eq $nan;
1175 # return true if arg (BINT or num_str) is +-inf
1176 my ($self,$x,$sign) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1178 $sign = '' if !defined $sign;
1179 return 1 if $sign eq $x->{sign}; # match ("+inf" eq "+inf")
1180 return 0 if $sign !~ /^([+-]|)$/;
1184 return 1 if ($x->{sign} =~ /^[+-]inf$/);
1187 $sign = quotemeta($sign.'inf');
1188 return 1 if ($x->{sign} =~ /^$sign$/);
1194 # return true if arg (BINT or num_str) is +1
1195 # or -1 if sign is given
1196 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1197 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1199 $sign = '' if !defined $sign; $sign = '+' if $sign ne '-';
1201 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1202 $CALC->_is_one($x->{value});
1207 # return true when arg (BINT or num_str) is odd, false for even
1208 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1209 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1211 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1212 $CALC->_is_odd($x->{value});
1217 # return true when arg (BINT or num_str) is even, false for odd
1218 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1219 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1221 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1222 $CALC->_is_even($x->{value});
1227 # return true when arg (BINT or num_str) is positive (>= 0)
1228 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1229 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1231 return 1 if $x->{sign} =~ /^\+/;
1237 # return true when arg (BINT or num_str) is negative (< 0)
1238 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1239 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1241 return 1 if ($x->{sign} =~ /^-/);
1247 # return true when arg (BINT or num_str) is an integer
1248 # always true for BigInt, but different for Floats
1249 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1250 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1252 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1255 ###############################################################################
1259 # multiply two numbers -- stolen from Knuth Vol 2 pg 233
1260 # (BINT or num_str, BINT or num_str) return BINT
1263 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1264 # objectify is costly, so avoid it
1265 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1267 ($self,$x,$y,@r) = objectify(2,@_);
1270 return $x if $x->modify('bmul');
1272 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1275 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1277 return $x->bnan() if $x->is_zero() || $y->is_zero();
1278 # result will always be +-inf:
1279 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1280 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1281 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1282 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1283 return $x->binf('-');
1286 return $upgrade->bmul($x,$y,@r)
1287 if defined $upgrade && $y->isa($upgrade);
1289 $r[3] = $y; # no push here
1291 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1293 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1294 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1296 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1302 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1303 my ($self,$x,$y) = @_;
1305 # NaN if x == NaN or y == NaN or x==y==0
1306 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1307 if (($x->is_nan() || $y->is_nan()) ||
1308 ($x->is_zero() && $y->is_zero()));
1310 # +-inf / +-inf == NaN, reminder also NaN
1311 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1313 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1315 # x / +-inf => 0, remainder x (works even if x == 0)
1316 if ($y->{sign} =~ /^[+-]inf$/)
1318 my $t = $x->copy(); # bzero clobbers up $x
1319 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1322 # 5 / 0 => +inf, -6 / 0 => -inf
1323 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1324 # exception: -8 / 0 has remainder -8, not 8
1325 # exception: -inf / 0 has remainder -inf, not inf
1328 # +-inf / 0 => special case for -inf
1329 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1330 if (!$x->is_zero() && !$x->is_inf())
1332 my $t = $x->copy(); # binf clobbers up $x
1334 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1338 # last case: +-inf / ordinary number
1340 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1342 return wantarray ? ($x,$self->bzero()) : $x;
1347 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1348 # (BINT,BINT) (quo,rem) or BINT (only rem)
1351 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1352 # objectify is costly, so avoid it
1353 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1355 ($self,$x,$y,@r) = objectify(2,@_);
1358 return $x if $x->modify('bdiv');
1360 return $self->_div_inf($x,$y)
1361 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1363 return $upgrade->bdiv($upgrade->new($x),$y,@r)
1364 if defined $upgrade && !$y->isa($self);
1366 $r[3] = $y; # no push!
1370 wantarray ? ($x->round(@r),$self->bzero(@r)):$x->round(@r) if $x->is_zero();
1372 # Is $x in the interval [0, $y) (aka $x <= $y) ?
1373 my $cmp = $CALC->_acmp($x->{value},$y->{value});
1374 if (($cmp < 0) and (($x->{sign} eq $y->{sign}) or !wantarray))
1376 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1377 if defined $upgrade;
1379 return $x->bzero()->round(@r) unless wantarray;
1380 my $t = $x->copy(); # make copy first, because $x->bzero() clobbers $x
1381 return ($x->bzero()->round(@r),$t);
1385 # shortcut, both are the same, so set to +/- 1
1386 $x->__one( ($x->{sign} ne $y->{sign} ? '-' : '+') );
1387 return $x unless wantarray;
1388 return ($x->round(@r),$self->bzero(@r));
1390 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1391 if defined $upgrade;
1393 # calc new sign and in case $y == +/- 1, return $x
1394 my $xsign = $x->{sign}; # keep
1395 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1396 # check for / +-1 (cant use $y->is_one due to '-'
1397 if ($CALC->_is_one($y->{value}))
1399 return wantarray ? ($x->round(@r),$self->bzero(@r)) : $x->round(@r);
1404 my $rem = $self->bzero();
1405 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1406 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1407 $rem->{_a} = $x->{_a};
1408 $rem->{_p} = $x->{_p};
1410 if (! $CALC->_is_zero($rem->{value}))
1412 $rem->{sign} = $y->{sign};
1413 $rem = $y-$rem if $xsign ne $y->{sign}; # one of them '-'
1417 $rem->{sign} = '+'; # dont leave -0
1419 return ($x,$rem->round(@r));
1422 $x->{value} = $CALC->_div($x->{value},$y->{value});
1423 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1425 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1429 ###############################################################################
1434 # modulus (or remainder)
1435 # (BINT or num_str, BINT or num_str) return BINT
1438 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1439 # objectify is costly, so avoid it
1440 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1442 ($self,$x,$y,@r) = objectify(2,@_);
1445 return $x if $x->modify('bmod');
1446 $r[3] = $y; # no push!
1447 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1449 my ($d,$r) = $self->_div_inf($x,$y);
1450 $x->{sign} = $r->{sign};
1451 $x->{value} = $r->{value};
1452 return $x->round(@r);
1455 if ($CALC->can('_mod'))
1457 # calc new sign and in case $y == +/- 1, return $x
1458 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1459 if (!$CALC->_is_zero($x->{value}))
1461 my $xsign = $x->{sign};
1462 $x->{sign} = $y->{sign};
1463 if ($xsign ne $y->{sign})
1465 my $t = $CALC->_copy($x->{value}); # copy $x
1466 $x->{value} = $CALC->_copy($y->{value}); # copy $y to $x
1467 $x->{value} = $CALC->_sub($y->{value},$t,1); # $y-$x
1472 $x->{sign} = '+'; # dont leave -0
1474 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1477 my ($t,$rem) = $self->bdiv($x->copy(),$y,@r); # slow way (also rounds)
1479 foreach (qw/value sign _a _p/)
1481 $x->{$_} = $rem->{$_};
1488 # Modular inverse. given a number which is (hopefully) relatively
1489 # prime to the modulus, calculate its inverse using Euclid's
1490 # alogrithm. If the number is not relatively prime to the modulus
1491 # (i.e. their gcd is not one) then NaN is returned.
1494 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1495 # objectify is costly, so avoid it
1496 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1498 ($self,$x,$y,@r) = objectify(2,@_);
1501 return $x if $x->modify('bmodinv');
1504 if ($y->{sign} ne '+' # -, NaN, +inf, -inf
1505 || $x->is_zero() # or num == 0
1506 || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1509 # put least residue into $x if $x was negative, and thus make it positive
1510 $x->bmod($y) if $x->{sign} eq '-';
1512 if ($CALC->can('_modinv'))
1515 ($x->{value},$sign) = $CALC->_modinv($x->{value},$y->{value});
1516 $x->bnan() if !defined $x->{value}; # in case no GCD found
1517 return $x if !defined $sign; # already real result
1518 $x->{sign} = $sign; # flip/flop see below
1519 $x->bmod($y); # calc real result
1522 my ($u, $u1) = ($self->bzero(), $self->bone());
1523 my ($a, $b) = ($y->copy(), $x->copy());
1525 # first step need always be done since $num (and thus $b) is never 0
1526 # Note that the loop is aligned so that the check occurs between #2 and #1
1527 # thus saving us one step #2 at the loop end. Typical loop count is 1. Even
1528 # a case with 28 loops still gains about 3% with this layout.
1530 ($a, $q, $b) = ($b, $a->bdiv($b)); # step #1
1531 # Euclid's Algorithm (calculate GCD of ($a,$b) in $a and also calculate
1532 # two values in $u and $u1, we use only $u1 afterwards)
1533 my $sign = 1; # flip-flop
1534 while (!$b->is_zero()) # found GCD if $b == 0
1536 # the original algorithm had:
1537 # ($u, $u1) = ($u1, $u->bsub($u1->copy()->bmul($q))); # step #2
1538 # The following creates exact the same sequence of numbers in $u1,
1539 # except for the sign ($u1 is now always positive). Since formerly
1540 # the sign of $u1 was alternating between '-' and '+', the $sign
1541 # flip-flop will take care of that, so that at the end of the loop
1542 # we have the real sign of $u1. Keeping numbers positive gains us
1543 # speed since badd() is faster than bsub() and makes it possible
1544 # to have the algorithmn in Calc for even more speed.
1546 ($u, $u1) = ($u1, $u->badd($u1->copy()->bmul($q))); # step #2
1547 $sign = - $sign; # flip sign
1549 ($a, $q, $b) = ($b, $a->bdiv($b)); # step #1 again
1552 # If the gcd is not 1, then return NaN! It would be pointless to
1553 # have called bgcd to check this first, because we would then be
1554 # performing the same Euclidean Algorithm *twice*.
1555 return $x->bnan() unless $a->is_one();
1557 $u1->bneg() if $sign != 1; # need to flip?
1559 $u1->bmod($y); # calc result
1560 $x->{value} = $u1->{value}; # and copy over to $x
1561 $x->{sign} = $u1->{sign}; # to modify in place
1567 # takes a very large number to a very large exponent in a given very
1568 # large modulus, quickly, thanks to binary exponentation. supports
1569 # negative exponents.
1570 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1572 return $num if $num->modify('bmodpow');
1574 # check modulus for valid values
1575 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1576 || $mod->is_zero());
1578 # check exponent for valid values
1579 if ($exp->{sign} =~ /\w/)
1581 # i.e., if it's NaN, +inf, or -inf...
1582 return $num->bnan();
1585 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
1587 # check num for valid values (also NaN if there was no inverse but $exp < 0)
1588 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1590 if ($CALC->can('_modpow'))
1592 # $mod is positive, sign on $exp is ignored, result also positive
1593 $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
1597 # in the trivial case,
1598 return $num->bzero(@r) if $mod->is_one();
1599 return $num->bone('+',@r) if $num->is_zero() or $num->is_one();
1601 # $num->bmod($mod); # if $x is large, make it smaller first
1602 my $acc = $num->copy(); # but this is not really faster...
1604 $num->bone(); # keep ref to $num
1606 my $expbin = $exp->as_bin(); $expbin =~ s/^[-]?0b//; # ignore sign and prefix
1607 my $len = length($expbin);
1610 if( substr($expbin,$len,1) eq '1')
1612 $num->bmul($acc)->bmod($mod);
1614 $acc->bmul($acc)->bmod($mod);
1620 ###############################################################################
1624 # (BINT or num_str, BINT or num_str) return BINT
1625 # compute factorial numbers
1626 # modifies first argument
1627 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1629 return $x if $x->modify('bfac');
1631 return $x->bnan() if $x->{sign} ne '+'; # inf, NnN, <0 etc => NaN
1632 return $x->bone('+',@r) if $x->is_zero() || $x->is_one(); # 0 or 1 => 1
1634 if ($CALC->can('_fac'))
1636 $x->{value} = $CALC->_fac($x->{value});
1637 return $x->round(@r);
1642 # seems we need not to temp. clear A/P of $x since the result is the same
1643 my $f = $self->new(2);
1644 while ($f->bacmp($n) < 0)
1646 $x->bmul($f); $f->binc();
1648 $x->bmul($f,@r); # last step and also round
1653 # (BINT or num_str, BINT or num_str) return BINT
1654 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1655 # modifies first argument
1658 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1659 # objectify is costly, so avoid it
1660 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1662 ($self,$x,$y,@r) = objectify(2,@_);
1665 return $x if $x->modify('bpow');
1667 return $upgrade->bpow($upgrade->new($x),$y,@r)
1668 if defined $upgrade && !$y->isa($self);
1670 $r[3] = $y; # no push!
1671 return $x if $x->{sign} =~ /^[+-]inf$/; # -inf/+inf ** x
1672 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1673 return $x->bone('+',@r) if $y->is_zero();
1674 return $x->round(@r) if $x->is_one() || $y->is_one();
1675 if ($x->{sign} eq '-' && $CALC->_is_one($x->{value}))
1677 # if $x == -1 and odd/even y => +1/-1
1678 return $y->is_odd() ? $x->round(@r) : $x->babs()->round(@r);
1679 # my Casio FX-5500L has a bug here: -1 ** 2 is -1, but -1 * -1 is 1;
1681 # 1 ** -y => 1 / (1 ** |y|)
1682 # so do test for negative $y after above's clause
1683 return $x->bnan() if $y->{sign} eq '-';
1684 return $x->round(@r) if $x->is_zero(); # 0**y => 0 (if not y <= 0)
1686 if ($CALC->can('_pow'))
1688 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1689 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1693 # based on the assumption that shifting in base 10 is fast, and that mul
1694 # works faster if numbers are small: we count trailing zeros (this step is
1695 # O(1)..O(N), but in case of O(N) we save much more time due to this),
1696 # stripping them out of the multiplication, and add $count * $y zeros
1697 # afterwards like this:
1698 # 300 ** 3 == 300*300*300 == 3*3*3 . '0' x 2 * 3 == 27 . '0' x 6
1699 # creates deep recursion since brsft/blsft use bpow sometimes.
1700 # my $zeros = $x->_trailing_zeros();
1703 # $x->brsft($zeros,10); # remove zeros
1704 # $x->bpow($y); # recursion (will not branch into here again)
1705 # $zeros = $y * $zeros; # real number of zeros to add
1706 # $x->blsft($zeros,10);
1707 # return $x->round(@r);
1710 my $pow2 = $self->__one();
1711 my $y_bin = $y->as_bin(); $y_bin =~ s/^0b//;
1712 my $len = length($y_bin);
1715 $pow2->bmul($x) if substr($y_bin,$len,1) eq '1'; # is odd?
1719 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1725 # (BINT or num_str, BINT or num_str) return BINT
1726 # compute x << y, base n, y >= 0
1729 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1730 # objectify is costly, so avoid it
1731 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1733 ($self,$x,$y,$n,@r) = objectify(2,@_);
1736 return $x if $x->modify('blsft');
1737 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1738 return $x->round(@r) if $y->is_zero();
1740 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1742 my $t; $t = $CALC->_lsft($x->{value},$y->{value},$n) if $CALC->can('_lsft');
1745 $x->{value} = $t; return $x->round(@r);
1748 return $x->bmul( $self->bpow($n, $y, @r), @r );
1753 # (BINT or num_str, BINT or num_str) return BINT
1754 # compute x >> y, base n, y >= 0
1757 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1758 # objectify is costly, so avoid it
1759 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1761 ($self,$x,$y,$n,@r) = objectify(2,@_);
1764 return $x if $x->modify('brsft');
1765 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1766 return $x->round(@r) if $y->is_zero();
1767 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
1769 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1771 # this only works for negative numbers when shifting in base 2
1772 if (($x->{sign} eq '-') && ($n == 2))
1774 return $x->round(@r) if $x->is_one('-'); # -1 => -1
1777 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1778 # but perhaps there is a better emulation for two's complement shift...
1779 # if $y != 1, we must simulate it by doing:
1780 # convert to bin, flip all bits, shift, and be done
1781 $x->binc(); # -3 => -2
1782 my $bin = $x->as_bin();
1783 $bin =~ s/^-0b//; # strip '-0b' prefix
1784 $bin =~ tr/10/01/; # flip bits
1786 if (CORE::length($bin) <= $y)
1788 $bin = '0'; # shifting to far right creates -1
1789 # 0, because later increment makes
1790 # that 1, attached '-' makes it '-1'
1791 # because -1 >> x == -1 !
1795 $bin =~ s/.{$y}$//; # cut off at the right side
1796 $bin = '1' . $bin; # extend left side by one dummy '1'
1797 $bin =~ tr/10/01/; # flip bits back
1799 my $res = $self->new('0b'.$bin); # add prefix and convert back
1800 $res->binc(); # remember to increment
1801 $x->{value} = $res->{value}; # take over value
1802 return $x->round(@r); # we are done now, magic, isn't?
1804 $x->bdec(); # n == 2, but $y == 1: this fixes it
1807 my $t; $t = $CALC->_rsft($x->{value},$y->{value},$n) if $CALC->can('_rsft');
1811 return $x->round(@r);
1814 $x->bdiv($self->bpow($n,$y, @r), @r);
1820 #(BINT or num_str, BINT or num_str) return BINT
1824 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1825 # objectify is costly, so avoid it
1826 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1828 ($self,$x,$y,@r) = objectify(2,@_);
1831 return $x if $x->modify('band');
1833 $r[3] = $y; # no push!
1834 local $Math::BigInt::upgrade = undef;
1836 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1837 return $x->bzero(@r) if $y->is_zero() || $x->is_zero();
1839 my $sign = 0; # sign of result
1840 $sign = 1 if ($x->{sign} eq '-') && ($y->{sign} eq '-');
1841 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1842 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1844 if ($CALC->can('_and') && $sx == 1 && $sy == 1)
1846 $x->{value} = $CALC->_and($x->{value},$y->{value});
1847 return $x->round(@r);
1850 my $m = $self->bone(); my ($xr,$yr);
1851 my $x10000 = $self->new (0x1000);
1852 my $y1 = copy(ref($x),$y); # make copy
1853 $y1->babs(); # and positive
1854 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1855 use integer; # need this for negative bools
1856 while (!$x1->is_zero() && !$y1->is_zero())
1858 ($x1, $xr) = bdiv($x1, $x10000);
1859 ($y1, $yr) = bdiv($y1, $x10000);
1860 # make both op's numbers!
1861 $x->badd( bmul( $class->new(
1862 abs($sx*int($xr->numify()) & $sy*int($yr->numify()))),
1866 $x->bneg() if $sign;
1872 #(BINT or num_str, BINT or num_str) return BINT
1876 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1877 # objectify is costly, so avoid it
1878 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1880 ($self,$x,$y,@r) = objectify(2,@_);
1883 return $x if $x->modify('bior');
1884 $r[3] = $y; # no push!
1886 local $Math::BigInt::upgrade = undef;
1888 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1889 return $x->round(@r) if $y->is_zero();
1891 my $sign = 0; # sign of result
1892 $sign = 1 if ($x->{sign} eq '-') || ($y->{sign} eq '-');
1893 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1894 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1896 # don't use lib for negative values
1897 if ($CALC->can('_or') && $sx == 1 && $sy == 1)
1899 $x->{value} = $CALC->_or($x->{value},$y->{value});
1900 return $x->round(@r);
1903 my $m = $self->bone(); my ($xr,$yr);
1904 my $x10000 = $self->new(0x10000);
1905 my $y1 = copy(ref($x),$y); # make copy
1906 $y1->babs(); # and positive
1907 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1908 use integer; # need this for negative bools
1909 while (!$x1->is_zero() || !$y1->is_zero())
1911 ($x1, $xr) = bdiv($x1,$x10000);
1912 ($y1, $yr) = bdiv($y1,$x10000);
1913 # make both op's numbers!
1914 $x->badd( bmul( $class->new(
1915 abs($sx*int($xr->numify()) | $sy*int($yr->numify()))),
1919 $x->bneg() if $sign;
1925 #(BINT or num_str, BINT or num_str) return BINT
1929 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1930 # objectify is costly, so avoid it
1931 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1933 ($self,$x,$y,@r) = objectify(2,@_);
1936 return $x if $x->modify('bxor');
1937 $r[3] = $y; # no push!
1939 local $Math::BigInt::upgrade = undef;
1941 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1942 return $x->round(@r) if $y->is_zero();
1944 my $sign = 0; # sign of result
1945 $sign = 1 if $x->{sign} ne $y->{sign};
1946 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1947 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1949 # don't use lib for negative values
1950 if ($CALC->can('_xor') && $sx == 1 && $sy == 1)
1952 $x->{value} = $CALC->_xor($x->{value},$y->{value});
1953 return $x->round(@r);
1956 my $m = $self->bone(); my ($xr,$yr);
1957 my $x10000 = $self->new(0x10000);
1958 my $y1 = copy(ref($x),$y); # make copy
1959 $y1->babs(); # and positive
1960 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1961 use integer; # need this for negative bools
1962 while (!$x1->is_zero() || !$y1->is_zero())
1964 ($x1, $xr) = bdiv($x1, $x10000);
1965 ($y1, $yr) = bdiv($y1, $x10000);
1966 # make both op's numbers!
1967 $x->badd( bmul( $class->new(
1968 abs($sx*int($xr->numify()) ^ $sy*int($yr->numify()))),
1972 $x->bneg() if $sign;
1978 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1980 my $e = $CALC->_len($x->{value});
1981 return wantarray ? ($e,0) : $e;
1986 # return the nth decimal digit, negative values count backward, 0 is right
1987 my ($self,$x,$n) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1989 $CALC->_digit($x->{value},$n||0);
1994 # return the amount of trailing zeros in $x
1996 $x = $class->new($x) unless ref $x;
1998 return 0 if $x->is_zero() || $x->is_odd() || $x->{sign} !~ /^[+-]$/;
2000 return $CALC->_zeros($x->{value}) if $CALC->can('_zeros');
2002 # if not: since we do not know underlying internal representation:
2003 my $es = "$x"; $es =~ /([0]*)$/;
2004 return 0 if !defined $1; # no zeros
2005 CORE::length("$1"); # as string, not as +0!
2010 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2012 return $x if $x->modify('bsqrt');
2014 return $x->bnan() if $x->{sign} ne '+'; # -x or inf or NaN => NaN
2015 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
2016 return $x->round(@r) if $x->is_one(); # 1 => 1
2018 return $upgrade->bsqrt($x,@r) if defined $upgrade;
2020 if ($CALC->can('_sqrt'))
2022 $x->{value} = $CALC->_sqrt($x->{value});
2023 return $x->round(@r);
2026 return $x->bone('+',@r) if $x < 4; # 2,3 => 1
2028 my $l = int($x->length()/2);
2030 $x->bone(); # keep ref($x), but modify it
2033 my $last = $self->bzero();
2034 my $two = $self->new(2);
2035 my $lastlast = $x+$two;
2036 while ($last != $x && $lastlast != $x)
2038 $lastlast = $last; $last = $x->copy();
2042 $x->bdec() if $x * $x > $y; # overshot?
2048 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
2049 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2051 if ($x->{sign} !~ /^[+-]$/)
2053 my $s = $x->{sign}; $s =~ s/^[+-]//;
2054 return $self->new($s); # -inf,+inf => inf
2056 my $e = $class->bzero();
2057 return $e->binc() if $x->is_zero();
2058 $e += $x->_trailing_zeros();
2064 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
2065 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2067 if ($x->{sign} !~ /^[+-]$/)
2069 return $self->new($x->{sign}); # keep + or - sign
2072 # that's inefficient
2073 my $zeros = $m->_trailing_zeros();
2074 $m->brsft($zeros,10) if $zeros != 0;
2080 # return a copy of both the exponent and the mantissa
2081 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2083 return ($x->mantissa(),$x->exponent());
2086 ##############################################################################
2087 # rounding functions
2091 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
2092 # $n == 0 || $n == 1 => round to integer
2093 my $x = shift; $x = $class->new($x) unless ref $x;
2094 my ($scale,$mode) = $x->_scale_p($x->precision(),$x->round_mode(),@_);
2095 return $x if !defined $scale; # no-op
2096 return $x if $x->modify('bfround');
2098 # no-op for BigInts if $n <= 0
2101 $x->{_a} = undef; # clear an eventual set A
2102 $x->{_p} = $scale; return $x;
2105 $x->bround( $x->length()-$scale, $mode);
2106 $x->{_a} = undef; # bround sets {_a}
2107 $x->{_p} = $scale; # so correct it
2111 sub _scan_for_nonzero
2117 my $len = $x->length();
2118 return 0 if $len == 1; # '5' is trailed by invisible zeros
2119 my $follow = $pad - 1;
2120 return 0 if $follow > $len || $follow < 1;
2122 # since we do not know underlying represention of $x, use decimal string
2123 #my $r = substr ($$xs,-$follow);
2124 my $r = substr ("$x",-$follow);
2125 return 1 if $r =~ /[^0]/;
2131 # to make life easier for switch between MBF and MBI (autoload fxxx()
2132 # like MBF does for bxxx()?)
2134 return $x->bround(@_);
2139 # accuracy: +$n preserve $n digits from left,
2140 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
2142 # and overwrite the rest with 0's, return normalized number
2143 # do not return $x->bnorm(), but $x
2145 my $x = shift; $x = $class->new($x) unless ref $x;
2146 my ($scale,$mode) = $x->_scale_a($x->accuracy(),$x->round_mode(),@_);
2147 return $x if !defined $scale; # no-op
2148 return $x if $x->modify('bround');
2150 if ($x->is_zero() || $scale == 0)
2152 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2155 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2157 # we have fewer digits than we want to scale to
2158 my $len = $x->length();
2159 # convert $scale to a scalar in case it is an object (put's a limit on the
2160 # number length, but this would already limited by memory constraints), makes
2162 $scale = $scale->numify() if ref ($scale);
2164 # scale < 0, but > -len (not >=!)
2165 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2167 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2171 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2172 my ($pad,$digit_round,$digit_after);
2173 $pad = $len - $scale;
2174 $pad = abs($scale-1) if $scale < 0;
2176 # do not use digit(), it is costly for binary => decimal
2178 my $xs = $CALC->_str($x->{value});
2181 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2182 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2183 $digit_round = '0'; $digit_round = substr($$xs,$pl,1) if $pad <= $len;
2184 $pl++; $pl ++ if $pad >= $len;
2185 $digit_after = '0'; $digit_after = substr($$xs,$pl,1) if $pad > 0;
2187 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2188 # closer at the remaining digits of the original $x, remember decision
2189 my $round_up = 1; # default round up
2191 ($mode eq 'trunc') || # trunc by round down
2192 ($digit_after =~ /[01234]/) || # round down anyway,
2194 ($digit_after eq '5') && # not 5000...0000
2195 ($x->_scan_for_nonzero($pad,$xs) == 0) &&
2197 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2198 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2199 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2200 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2201 ($mode eq 'zero') # round down if zero, sign adjusted below
2203 my $put_back = 0; # not yet modified
2205 if (($pad > 0) && ($pad <= $len))
2207 substr($$xs,-$pad,$pad) = '0' x $pad;
2212 $x->bzero(); # round to '0'
2215 if ($round_up) # what gave test above?
2218 $pad = $len, $$xs = '0' x $pad if $scale < 0; # tlr: whack 0.51=>1.0
2220 # we modify directly the string variant instead of creating a number and
2221 # adding it, since that is faster (we already have the string)
2222 my $c = 0; $pad ++; # for $pad == $len case
2223 while ($pad <= $len)
2225 $c = substr($$xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2226 substr($$xs,-$pad,1) = $c; $pad++;
2227 last if $c != 0; # no overflow => early out
2229 $$xs = '1'.$$xs if $c == 0;
2232 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back in if needed
2234 $x->{_a} = $scale if $scale >= 0;
2237 $x->{_a} = $len+$scale;
2238 $x->{_a} = 0 if $scale < -$len;
2245 # return integer less or equal then number, since it is already integer,
2246 # always returns $self
2247 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2254 # return integer greater or equal then number, since it is already integer,
2255 # always returns $self
2256 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2261 ##############################################################################
2262 # private stuff (internal use only)
2266 # internal speedup, set argument to 1, or create a +/- 1
2268 my $x = $self->bone(); # $x->{value} = $CALC->_one();
2269 $x->{sign} = shift || '+';
2275 # Overload will swap params if first one is no object ref so that the first
2276 # one is always an object ref. In this case, third param is true.
2277 # This routine is to overcome the effect of scalar,$object creating an object
2278 # of the class of this package, instead of the second param $object. This
2279 # happens inside overload, when the overload section of this package is
2280 # inherited by sub classes.
2281 # For overload cases (and this is used only there), we need to preserve the
2282 # args, hence the copy().
2283 # You can override this method in a subclass, the overload section will call
2284 # $object->_swap() to make sure it arrives at the proper subclass, with some
2285 # exceptions like '+' and '-'. To make '+' and '-' work, you also need to
2286 # specify your own overload for them.
2288 # object, (object|scalar) => preserve first and make copy
2289 # scalar, object => swapped, re-swap and create new from first
2290 # (using class of second object, not $class!!)
2291 my $self = shift; # for override in subclass
2294 my $c = ref ($_[0]) || $class; # fallback $class should not happen
2295 return ( $c->new($_[1]), $_[0] );
2297 return ( $_[0]->copy(), $_[1] );
2302 # check for strings, if yes, return objects instead
2304 # the first argument is number of args objectify() should look at it will
2305 # return $count+1 elements, the first will be a classname. This is because
2306 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2307 # useless objects beeing created and thrown away. So we cannot simple loop
2308 # over @_. If the given count is 0, all arguments will be used.
2310 # If the second arg is a ref, use it as class.
2311 # If not, try to use it as classname, unless undef, then use $class
2312 # (aka Math::BigInt). The latter shouldn't happen,though.
2315 # $x->badd(1); => ref x, scalar y
2316 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2317 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2318 # Math::BigInt::badd(1,2); => scalar x, scalar y
2319 # In the last case we check number of arguments to turn it silently into
2320 # $class,1,2. (We can not take '1' as class ;o)
2321 # badd($class,1) is not supported (it should, eventually, try to add undef)
2322 # currently it tries 'Math::BigInt' + 1, which will not work.
2324 # some shortcut for the common cases
2326 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2328 my $count = abs(shift || 0);
2330 my (@a,$k,$d); # resulting array, temp, and downgrade
2333 # okay, got object as first
2338 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2340 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2344 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2345 if (defined ${"$a[0]::downgrade"})
2347 $d = ${"$a[0]::downgrade"};
2348 ${"$a[0]::downgrade"} = undef;
2351 my $up = ${"$a[0]::upgrade"};
2352 # print "Now in objectify, my class is today $a[0]\n";
2360 $k = $a[0]->new($k);
2362 elsif (!defined $up && ref($k) ne $a[0])
2364 # foreign object, try to convert to integer
2365 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2378 $k = $a[0]->new($k);
2380 elsif (!defined $up && ref($k) ne $a[0])
2382 # foreign object, try to convert to integer
2383 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2387 push @a,@_; # return other params, too
2389 die "$class objectify needs list context" unless wantarray;
2390 ${"$a[0]::downgrade"} = $d;
2399 my @a; my $l = scalar @_;
2400 for ( my $i = 0; $i < $l ; $i++ )
2402 if ($_[$i] eq ':constant')
2404 # this causes overlord er load to step in
2405 overload::constant integer => sub { $self->new(shift) };
2406 overload::constant binary => sub { $self->new(shift) };
2408 elsif ($_[$i] eq 'upgrade')
2410 # this causes upgrading
2411 $upgrade = $_[$i+1]; # or undef to disable
2414 elsif ($_[$i] =~ /^lib$/i)
2416 # this causes a different low lib to take care...
2417 $CALC = $_[$i+1] || '';
2425 # any non :constant stuff is handled by our parent, Exporter
2426 # even if @_ is empty, to give it a chance
2427 $self->SUPER::import(@a); # need it for subclasses
2428 $self->export_to_level(1,$self,@a); # need it for MBF
2430 # try to load core math lib
2431 my @c = split /\s*,\s*/,$CALC;
2432 push @c,'Calc'; # if all fail, try this
2433 $CALC = ''; # signal error
2434 foreach my $lib (@c)
2436 next if ($lib || '') eq '';
2437 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2441 # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
2442 # used in the same script, or eval inside import().
2443 my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
2444 my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
2446 $file = File::Spec->catfile (@parts, $file);
2447 eval { require "$file"; $lib->import( @c ); }
2451 eval "use $lib qw/@c/;";
2453 $CALC = $lib, last if $@ eq ''; # no error in loading lib?
2455 die "Couldn't load any math lib, not even the default" if $CALC eq '';
2460 # convert a (ref to) big hex string to BigInt, return undef for error
2463 my $x = Math::BigInt->bzero();
2466 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2467 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2469 return $x->bnan() if $$hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2471 my $sign = '+'; $sign = '-' if ($$hs =~ /^-/);
2473 $$hs =~ s/^[+-]//; # strip sign
2474 if ($CALC->can('_from_hex'))
2476 $x->{value} = $CALC->_from_hex($hs);
2480 # fallback to pure perl
2481 my $mul = Math::BigInt->bzero(); $mul++;
2482 my $x65536 = Math::BigInt->new(65536);
2483 my $len = CORE::length($$hs)-2;
2484 $len = int($len/4); # 4-digit parts, w/o '0x'
2485 my $val; my $i = -4;
2488 $val = substr($$hs,$i,4);
2489 $val =~ s/^[+-]?0x// if $len == 0; # for last part only because
2490 $val = hex($val); # hex does not like wrong chars
2492 $x += $mul * $val if $val != 0;
2493 $mul *= $x65536 if $len >= 0; # skip last mul
2496 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2502 # convert a (ref to) big binary string to BigInt, return undef for error
2505 my $x = Math::BigInt->bzero();
2507 $$bs =~ s/([01])_([01])/$1$2/g;
2508 $$bs =~ s/([01])_([01])/$1$2/g;
2509 return $x->bnan() if $$bs !~ /^[+-]?0b[01]+$/;
2511 my $sign = '+'; $sign = '-' if ($$bs =~ /^\-/);
2512 $$bs =~ s/^[+-]//; # strip sign
2513 if ($CALC->can('_from_bin'))
2515 $x->{value} = $CALC->_from_bin($bs);
2519 my $mul = Math::BigInt->bzero(); $mul++;
2520 my $x256 = Math::BigInt->new(256);
2521 my $len = CORE::length($$bs)-2;
2522 $len = int($len/8); # 8-digit parts, w/o '0b'
2523 my $val; my $i = -8;
2526 $val = substr($$bs,$i,8);
2527 $val =~ s/^[+-]?0b// if $len == 0; # for last part only
2528 #$val = oct('0b'.$val); # does not work on Perl prior to 5.6.0
2530 # $val = ('0' x (8-CORE::length($val))).$val if CORE::length($val) < 8;
2531 $val = ord(pack('B8',substr('00000000'.$val,-8,8)));
2533 $x += $mul * $val if $val != 0;
2534 $mul *= $x256 if $len >= 0; # skip last mul
2537 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2543 # (ref to num_str) return num_str
2544 # internal, take apart a string and return the pieces
2545 # strip leading/trailing whitespace, leading zeros, underscore and reject
2549 # strip white space at front, also extranous leading zeros
2550 $$x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2551 $$x =~ s/^\s+//; # but this will
2552 $$x =~ s/\s+$//g; # strip white space at end
2554 # shortcut, if nothing to split, return early
2555 if ($$x =~ /^[+-]?\d+\z/)
2557 $$x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2558 return (\$sign, $x, \'', \'', \0);
2561 # invalid starting char?
2562 return if $$x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2564 return __from_hex($x) if $$x =~ /^[\-\+]?0x/; # hex string
2565 return __from_bin($x) if $$x =~ /^[\-\+]?0b/; # binary string
2567 # strip underscores between digits
2568 $$x =~ s/(\d)_(\d)/$1$2/g;
2569 $$x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3
2571 # some possible inputs:
2572 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2573 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2
2575 #return if $$x =~ /[Ee].*[Ee]/; # more than one E => error
2577 my ($m,$e,$last) = split /[Ee]/,$$x;
2578 return if defined $last; # last defined => 1e2E3 or others
2579 $e = '0' if !defined $e || $e eq "";
2581 # sign,value for exponent,mantint,mantfrac
2582 my ($es,$ev,$mis,$miv,$mfv);
2584 if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2588 return if $m eq '.' || $m eq '';
2589 my ($mi,$mf,$lastf) = split /\./,$m;
2590 return if defined $lastf; # last defined => 1.2.3 or others
2591 $mi = '0' if !defined $mi;
2592 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2593 $mf = '0' if !defined $mf || $mf eq '';
2594 if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2596 $mis = $1||'+'; $miv = $2;
2597 return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
2599 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2602 return; # NaN, not a number
2607 # an object might be asked to return itself as bigint on certain overloaded
2608 # operations, this does exactly this, so that sub classes can simple inherit
2609 # it or override with their own integer conversion routine
2617 # return as hex string, with prefixed 0x
2618 my $x = shift; $x = $class->new($x) if !ref($x);
2620 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2621 return '0x0' if $x->is_zero();
2623 my $es = ''; my $s = '';
2624 $s = $x->{sign} if $x->{sign} eq '-';
2625 if ($CALC->can('_as_hex'))
2627 $es = ${$CALC->_as_hex($x->{value})};
2631 my $x1 = $x->copy()->babs(); my ($xr,$x10000,$h);
2634 $x10000 = Math::BigInt->new (0x10000); $h = 'h4';
2638 $x10000 = Math::BigInt->new (0x1000); $h = 'h3';
2640 while (!$x1->is_zero())
2642 ($x1, $xr) = bdiv($x1,$x10000);
2643 $es .= unpack($h,pack('v',$xr->numify()));
2646 $es =~ s/^[0]+//; # strip leading zeros
2654 # return as binary string, with prefixed 0b
2655 my $x = shift; $x = $class->new($x) if !ref($x);
2657 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2658 return '0b0' if $x->is_zero();
2660 my $es = ''; my $s = '';
2661 $s = $x->{sign} if $x->{sign} eq '-';
2662 if ($CALC->can('_as_bin'))
2664 $es = ${$CALC->_as_bin($x->{value})};
2668 my $x1 = $x->copy()->babs(); my ($xr,$x10000,$b);
2671 $x10000 = Math::BigInt->new (0x10000); $b = 'b16';
2675 $x10000 = Math::BigInt->new (0x1000); $b = 'b12';
2677 while (!$x1->is_zero())
2679 ($x1, $xr) = bdiv($x1,$x10000);
2680 $es .= unpack($b,pack('v',$xr->numify()));
2683 $es =~ s/^[0]+//; # strip leading zeros
2689 ##############################################################################
2690 # internal calculation routines (others are in Math::BigInt::Calc etc)
2694 # (BINT or num_str, BINT or num_str) return BINT
2695 # does modify first argument
2698 my $x = shift; my $ty = shift;
2699 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2700 return $x * $ty / bgcd($x,$ty);
2705 # (BINT or num_str, BINT or num_str) return BINT
2706 # does modify both arguments
2707 # GCD -- Euclids algorithm E, Knuth Vol 2 pg 296
2710 return $x->bnan() if $x->{sign} !~ /^[+-]$/ || $ty->{sign} !~ /^[+-]$/;
2712 while (!$ty->is_zero())
2714 ($x, $ty) = ($ty,bmod($x,$ty));
2719 ###############################################################################
2720 # this method return 0 if the object can be modified, or 1 for not
2721 # We use a fast use constant statement here, to avoid costly calls. Subclasses
2722 # may override it with special code (f.i. Math::BigInt::Constant does so)
2724 sub modify () { 0; }
2731 Math::BigInt - Arbitrary size integer math package
2738 $x = Math::BigInt->new($str); # defaults to 0
2739 $nan = Math::BigInt->bnan(); # create a NotANumber
2740 $zero = Math::BigInt->bzero(); # create a +0
2741 $inf = Math::BigInt->binf(); # create a +inf
2742 $inf = Math::BigInt->binf('-'); # create a -inf
2743 $one = Math::BigInt->bone(); # create a +1
2744 $one = Math::BigInt->bone('-'); # create a -1
2746 # Testing (don't modify their arguments)
2747 # (return true if the condition is met, otherwise false)
2749 $x->is_zero(); # if $x is +0
2750 $x->is_nan(); # if $x is NaN
2751 $x->is_one(); # if $x is +1
2752 $x->is_one('-'); # if $x is -1
2753 $x->is_odd(); # if $x is odd
2754 $x->is_even(); # if $x is even
2755 $x->is_positive(); # if $x >= 0
2756 $x->is_negative(); # if $x < 0
2757 $x->is_inf(sign); # if $x is +inf, or -inf (sign is default '+')
2758 $x->is_int(); # if $x is an integer (not a float)
2760 # comparing and digit/sign extration
2761 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2762 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2763 $x->sign(); # return the sign, either +,- or NaN
2764 $x->digit($n); # return the nth digit, counting from right
2765 $x->digit(-$n); # return the nth digit, counting from left
2767 # The following all modify their first argument:
2769 $x->bzero(); # set $x to 0
2770 $x->bnan(); # set $x to NaN
2771 $x->bone(); # set $x to +1
2772 $x->bone('-'); # set $x to -1
2773 $x->binf(); # set $x to inf
2774 $x->binf('-'); # set $x to -inf
2776 $x->bneg(); # negation
2777 $x->babs(); # absolute value
2778 $x->bnorm(); # normalize (no-op in BigInt)
2779 $x->bnot(); # two's complement (bit wise not)
2780 $x->binc(); # increment $x by 1
2781 $x->bdec(); # decrement $x by 1
2783 $x->badd($y); # addition (add $y to $x)
2784 $x->bsub($y); # subtraction (subtract $y from $x)
2785 $x->bmul($y); # multiplication (multiply $x by $y)
2786 $x->bdiv($y); # divide, set $x to quotient
2787 # return (quo,rem) or quo if scalar
2789 $x->bmod($y); # modulus (x % y)
2790 $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
2791 $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
2793 $x->bpow($y); # power of arguments (x ** y)
2794 $x->blsft($y); # left shift
2795 $x->brsft($y); # right shift
2796 $x->blsft($y,$n); # left shift, by base $n (like 10)
2797 $x->brsft($y,$n); # right shift, by base $n (like 10)
2799 $x->band($y); # bitwise and
2800 $x->bior($y); # bitwise inclusive or
2801 $x->bxor($y); # bitwise exclusive or
2802 $x->bnot(); # bitwise not (two's complement)
2804 $x->bsqrt(); # calculate square-root
2805 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2807 $x->round($A,$P,$mode); # round to accuracy or precision using mode $r
2808 $x->bround($N); # accuracy: preserve $N digits
2809 $x->bfround($N); # round to $Nth digit, no-op for BigInts
2811 # The following do not modify their arguments in BigInt,
2812 # but do so in BigFloat:
2814 $x->bfloor(); # return integer less or equal than $x
2815 $x->bceil(); # return integer greater or equal than $x
2817 # The following do not modify their arguments:
2819 bgcd(@values); # greatest common divisor (no OO style)
2820 blcm(@values); # lowest common multiplicator (no OO style)
2822 $x->length(); # return number of digits in number
2823 ($x,$f) = $x->length(); # length of number and length of fraction part,
2824 # latter is always 0 digits long for BigInt's
2826 $x->exponent(); # return exponent as BigInt
2827 $x->mantissa(); # return (signed) mantissa as BigInt
2828 $x->parts(); # return (mantissa,exponent) as BigInt
2829 $x->copy(); # make a true copy of $x (unlike $y = $x;)
2830 $x->as_number(); # return as BigInt (in BigInt: same as copy())
2832 # conversation to string (do not modify their argument)
2833 $x->bstr(); # normalized string
2834 $x->bsstr(); # normalized string in scientific notation
2835 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
2836 $x->as_bin(); # as signed binary string with prefixed 0b
2839 # precision and accuracy (see section about rounding for more)
2840 $x->precision(); # return P of $x (or global, if P of $x undef)
2841 $x->precision($n); # set P of $x to $n
2842 $x->accuracy(); # return A of $x (or global, if A of $x undef)
2843 $x->accuracy($n); # set A $x to $n
2846 Math::BigInt->precision(); # get/set global P for all BigInt objects
2847 Math::BigInt->accuracy(); # get/set global A for all BigInt objects
2848 Math::BigInt->config(); # return hash containing configuration
2852 All operators (inlcuding basic math operations) are overloaded if you
2853 declare your big integers as
2855 $i = new Math::BigInt '123_456_789_123_456_789';
2857 Operations with overloaded operators preserve the arguments which is
2858 exactly what you expect.
2862 =item Canonical notation
2864 Big integer values are strings of the form C</^[+-]\d+$/> with leading
2867 '-0' canonical value '-0', normalized '0'
2868 ' -123_123_123' canonical value '-123123123'
2869 '1_23_456_7890' canonical value '1234567890'
2873 Input values to these routines may be either Math::BigInt objects or
2874 strings of the form C</^\s*[+-]?[\d]+\.?[\d]*E?[+-]?[\d]*$/>.
2876 You can include one underscore between any two digits.
2878 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
2879 Non integer values result in NaN.
2881 Math::BigInt::new() defaults to 0, while Math::BigInt::new('') results
2884 bnorm() on a BigInt object is now effectively a no-op, since the numbers
2885 are always stored in normalized form. On a string, it creates a BigInt
2890 Output values are BigInt objects (normalized), except for bstr(), which
2891 returns a string in normalized form.
2892 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
2893 C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
2894 return either undef, <0, 0 or >0 and are suited for sort.
2900 Each of the methods below (except config(), accuracy() and precision())
2901 accepts three additional parameters. These arguments $A, $P and $R are
2902 accuracy, precision and round_mode. Please see the section about
2903 L<ACCURACY and PRECISION> for more information.
2909 print Dumper ( Math::BigInt->config() );
2910 print Math::BigInt->config()->{lib},"\n";
2912 Returns a hash containing the configuration, e.g. the version number, lib
2913 loaded etc. The following hash keys are currently filled in with the
2914 appropriate information.
2918 ============================================================
2919 lib Name of the Math library
2921 lib_version Version of 'lib'
2923 class The class of config you just called
2925 upgrade To which class numbers are upgraded
2927 downgrade To which class numbers are downgraded
2929 precision Global precision
2931 accuracy Global accuracy
2933 round_mode Global round mode
2935 version version number of the class you used
2937 div_scale Fallback acccuracy for div
2940 It is currently not supported to set the configuration parameters by passing
2941 a hash ref to C<config()>.
2945 $x->accuracy(5); # local for $x
2946 CLASS->accuracy(5); # global for all members of CLASS
2947 $A = $x->accuracy(); # read out
2948 $A = CLASS->accuracy(); # read out
2950 Set or get the global or local accuracy, aka how many significant digits the
2953 Please see the section about L<ACCURACY AND PRECISION> for further details.
2955 Value must be greater than zero. Pass an undef value to disable it:
2957 $x->accuracy(undef);
2958 Math::BigInt->accuracy(undef);
2960 Returns the current accuracy. For C<$x->accuracy()> it will return either the
2961 local accuracy, or if not defined, the global. This means the return value
2962 represents the accuracy that will be in effect for $x:
2964 $y = Math::BigInt->new(1234567); # unrounded
2965 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
2966 $x = Math::BigInt->new(123456); # will be automatically rounded
2967 print "$x $y\n"; # '123500 1234567'
2968 print $x->accuracy(),"\n"; # will be 4
2969 print $y->accuracy(),"\n"; # also 4, since global is 4
2970 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
2971 print $x->accuracy(),"\n"; # still 4
2972 print $y->accuracy(),"\n"; # 5, since global is 5
2974 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
2975 globals separated from Math::BigInt, but it is possible to subclass
2976 Math::BigInt and make the globals of the subclass aliases to the ones from
2981 $x->precision(-2); # local for $x, round right of the dot
2982 $x->precision(2); # ditto, but round left of the dot
2983 CLASS->accuracy(5); # global for all members of CLASS
2984 CLASS->precision(-5); # ditto
2985 $P = CLASS->precision(); # read out
2986 $P = $x->precision(); # read out
2988 Set or get the global or local precision, aka how many digits the result has
2989 after the dot (or where to round it when passing a positive number). In
2990 Math::BigInt, passing a negative number precision has no effect since no
2991 numbers have digits after the dot.
2993 Please see the section about L<ACCURACY AND PRECISION> for further details.
2995 Value must be greater than zero. Pass an undef value to disable it:
2997 $x->precision(undef);
2998 Math::BigInt->precision(undef);
3000 Returns the current precision. For C<$x->precision()> it will return either the
3001 local precision of $x, or if not defined, the global. This means the return
3002 value represents the accuracy that will be in effect for $x:
3004 $y = Math::BigInt->new(1234567); # unrounded
3005 print Math::BigInt->precision(4),"\n"; # set 4, print 4
3006 $x = Math::BigInt->new(123456); # will be automatically rounded
3008 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
3009 globals separated from Math::BigInt, but it is possible to subclass
3010 Math::BigInt and make the globals of the subclass aliases to the ones from
3017 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
3018 2, but others work, too.
3020 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
3024 $x = Math::BigInt->new(10);
3025 $x->brsft(1); # same as $x >> 1: 5
3026 $x = Math::BigInt->new(1234);
3027 $x->brsft(2,10); # result 12
3029 There is one exception, and that is base 2 with negative $x:
3032 $x = Math::BigInt->new(-5);
3035 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
3040 $x = Math::BigInt->new($str,$A,$P,$R);
3042 Creates a new BigInt object from a string or another BigInt object. The
3043 input is accepted as decimal, hex (with leading '0x') or binary (with leading
3048 $x = Math::BigInt->bnan();
3050 Creates a new BigInt object representing NaN (Not A Number).
3051 If used on an object, it will set it to NaN:
3057 $x = Math::BigInt->bzero();
3059 Creates a new BigInt object representing zero.
3060 If used on an object, it will set it to zero:
3066 $x = Math::BigInt->binf($sign);
3068 Creates a new BigInt object representing infinity. The optional argument is
3069 either '-' or '+', indicating whether you want infinity or minus infinity.
3070 If used on an object, it will set it to infinity:
3077 $x = Math::BigInt->binf($sign);
3079 Creates a new BigInt object representing one. The optional argument is
3080 either '-' or '+', indicating whether you want one or minus one.
3081 If used on an object, it will set it to one:
3086 =head2 is_one()/is_zero()/is_nan()/is_inf()
3089 $x->is_zero(); # true if arg is +0
3090 $x->is_nan(); # true if arg is NaN
3091 $x->is_one(); # true if arg is +1
3092 $x->is_one('-'); # true if arg is -1
3093 $x->is_inf(); # true if +inf
3094 $x->is_inf('-'); # true if -inf (sign is default '+')
3096 These methods all test the BigInt for beeing one specific value and return
3097 true or false depending on the input. These are faster than doing something
3102 =head2 is_positive()/is_negative()
3104 $x->is_positive(); # true if >= 0
3105 $x->is_negative(); # true if < 0
3107 The methods return true if the argument is positive or negative, respectively.
3108 C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
3109 C<-inf> is negative. A C<zero> is positive.
3111 These methods are only testing the sign, and not the value.
3113 =head2 is_odd()/is_even()/is_int()
3115 $x->is_odd(); # true if odd, false for even
3116 $x->is_even(); # true if even, false for odd
3117 $x->is_int(); # true if $x is an integer
3119 The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
3120 C<-inf> are not integers and are neither odd nor even.
3126 Compares $x with $y and takes the sign into account.
3127 Returns -1, 0, 1 or undef.
3133 Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
3139 Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
3143 $x->digit($n); # return the nth digit, counting from right
3149 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
3150 and '-inf', respectively. Does nothing for NaN or zero.
3156 Set the number to it's absolute value, e.g. change the sign from '-' to '+'
3157 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
3162 $x->bnorm(); # normalize (no-op)
3166 $x->bnot(); # two's complement (bit wise not)
3170 $x->binc(); # increment x by 1
3174 $x->bdec(); # decrement x by 1
3178 $x->badd($y); # addition (add $y to $x)
3182 $x->bsub($y); # subtraction (subtract $y from $x)
3186 $x->bmul($y); # multiplication (multiply $x by $y)
3190 $x->bdiv($y); # divide, set $x to quotient
3191 # return (quo,rem) or quo if scalar
3195 $x->bmod($y); # modulus (x % y)
3199 num->bmodinv($mod); # modular inverse
3201 Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
3202 returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
3203 C<bgcd($num, $mod)==1>.
3207 $num->bmodpow($exp,$mod); # modular exponentation
3208 # ($num**$exp % $mod)
3210 Returns the value of C<$num> taken to the power C<$exp> in the modulus
3211 C<$mod> using binary exponentation. C<bmodpow> is far superior to
3216 because C<bmodpow> is much faster--it reduces internal variables into
3217 the modulus whenever possible, so it operates on smaller numbers.
3219 C<bmodpow> also supports negative exponents.
3221 bmodpow($num, -1, $mod)
3223 is exactly equivalent to
3229 $x->bpow($y); # power of arguments (x ** y)
3233 $x->blsft($y); # left shift
3234 $x->blsft($y,$n); # left shift, in base $n (like 10)
3238 $x->brsft($y); # right shift
3239 $x->brsft($y,$n); # right shift, in base $n (like 10)
3243 $x->band($y); # bitwise and
3247 $x->bior($y); # bitwise inclusive or
3251 $x->bxor($y); # bitwise exclusive or
3255 $x->bnot(); # bitwise not (two's complement)
3259 $x->bsqrt(); # calculate square-root
3263 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3267 $x->round($A,$P,$round_mode);
3269 Round $x to accuracy C<$A> or precision C<$P> using the round mode
3274 $x->bround($N); # accuracy: preserve $N digits
3278 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3284 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3285 does change $x in BigFloat.
3291 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3292 does change $x in BigFloat.
3296 bgcd(@values); # greatest common divisor (no OO style)
3300 blcm(@values); # lowest common multiplicator (no OO style)
3305 ($xl,$fl) = $x->length();
3307 Returns the number of digits in the decimal representation of the number.
3308 In list context, returns the length of the integer and fraction part. For
3309 BigInt's, the length of the fraction part will always be 0.
3315 Return the exponent of $x as BigInt.
3321 Return the signed mantissa of $x as BigInt.
3325 $x->parts(); # return (mantissa,exponent) as BigInt
3329 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3333 $x->as_number(); # return as BigInt (in BigInt: same as copy())
3337 $x->bstr(); # return normalized string
3341 $x->bsstr(); # normalized string in scientific notation
3345 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3349 $x->as_bin(); # as signed binary string with prefixed 0b
3351 =head1 ACCURACY and PRECISION
3353 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3354 accuracy and precision based rounding, both automatically after every
3355 operation as well as manually.
3357 This section describes the accuracy/precision handling in Math::Big* as it
3358 used to be and as it is now, complete with an explanation of all terms and
3361 Not yet implemented things (but with correct description) are marked with '!',
3362 things that need to be answered are marked with '?'.
3364 In the next paragraph follows a short description of terms used here (because
3365 these may differ from terms used by others people or documentation).
3367 During the rest of this document, the shortcuts A (for accuracy), P (for
3368 precision), F (fallback) and R (rounding mode) will be used.
3372 A fixed number of digits before (positive) or after (negative)
3373 the decimal point. For example, 123.45 has a precision of -2. 0 means an
3374 integer like 123 (or 120). A precision of 2 means two digits to the left
3375 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3376 numbers with zeros before the decimal point may have different precisions,
3377 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3378 was). It could also have p < 0, when the digits after the decimal point
3381 The string output (of floating point numbers) will be padded with zeros:
3383 Initial value P A Result String
3384 ------------------------------------------------------------
3385 1234.01 -3 1000 1000
3388 1234.001 1 1234 1234.0
3390 1234.01 2 1234.01 1234.01
3391 1234.01 5 1234.01 1234.01000
3393 For BigInts, no padding occurs.
3397 Number of significant digits. Leading zeros are not counted. A
3398 number may have an accuracy greater than the non-zero digits
3399 when there are zeros in it or trailing zeros. For example, 123.456 has
3400 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3402 The string output (of floating point numbers) will be padded with zeros:
3404 Initial value P A Result String
3405 ------------------------------------------------------------
3407 1234.01 6 1234.01 1234.01
3408 1234.1 8 1234.1 1234.1000
3410 For BigInts, no padding occurs.
3414 When both A and P are undefined, this is used as a fallback accuracy when
3417 =head2 Rounding mode R
3419 When rounding a number, different 'styles' or 'kinds'
3420 of rounding are possible. (Note that random rounding, as in
3421 Math::Round, is not implemented.)
3427 truncation invariably removes all digits following the
3428 rounding place, replacing them with zeros. Thus, 987.65 rounded
3429 to tens (P=1) becomes 980, and rounded to the fourth sigdig
3430 becomes 987.6 (A=4). 123.456 rounded to the second place after the
3431 decimal point (P=-2) becomes 123.46.
3433 All other implemented styles of rounding attempt to round to the
3434 "nearest digit." If the digit D immediately to the right of the
3435 rounding place (skipping the decimal point) is greater than 5, the
3436 number is incremented at the rounding place (possibly causing a
3437 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3438 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3439 truncated at the rounding place: e.g. when rounding to units, 0.4
3440 rounds to 0, and -19.4 rounds to -19.
3442 However the results of other styles of rounding differ if the
3443 digit immediately to the right of the rounding place (skipping the
3444 decimal point) is 5 and if there are no digits, or no digits other
3445 than 0, after that 5. In such cases:
3449 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3450 if it is not already. E.g., when rounding to the first sigdig, 0.45
3451 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3455 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3456 it is not already. E.g., when rounding to the first sigdig, 0.45
3457 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3461 round to plus infinity, i.e. always round up. E.g., when
3462 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3463 and 0.4501 also becomes 0.5.
3467 round to minus infinity, i.e. always round down. E.g., when
3468 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3469 but 0.4501 becomes 0.5.
3473 round to zero, i.e. positive numbers down, negative ones up.
3474 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3475 becomes -0.5, but 0.4501 becomes 0.5.
3479 The handling of A & P in MBI/MBF (the old core code shipped with Perl
3480 versions <= 5.7.2) is like this:
3486 * ffround($p) is able to round to $p number of digits after the decimal
3488 * otherwise P is unused
3490 =item Accuracy (significant digits)
3492 * fround($a) rounds to $a significant digits
3493 * only fdiv() and fsqrt() take A as (optional) paramater
3494 + other operations simply create the same number (fneg etc), or more (fmul)
3496 + rounding/truncating is only done when explicitly calling one of fround
3497 or ffround, and never for BigInt (not implemented)
3498 * fsqrt() simply hands its accuracy argument over to fdiv.
3499 * the documentation and the comment in the code indicate two different ways
3500 on how fdiv() determines the maximum number of digits it should calculate,
3501 and the actual code does yet another thing
3503 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3505 result has at most max(scale, length(dividend), length(divisor)) digits
3507 scale = max(scale, length(dividend)-1,length(divisor)-1);
3508 scale += length(divisior) - length(dividend);
3509 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3510 Actually, the 'difference' added to the scale is calculated from the
3511 number of "significant digits" in dividend and divisor, which is derived
3512 by looking at the length of the mantissa. Which is wrong, since it includes
3513 the + sign (oups) and actually gets 2 for '+100' and 4 for '+101'. Oups
3514 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3515 assumption that 124 has 3 significant digits, while 120/7 will get you
3516 '17', not '17.1' since 120 is thought to have 2 significant digits.
3517 The rounding after the division then uses the remainder and $y to determine
3518 wether it must round up or down.
3519 ? I have no idea which is the right way. That's why I used a slightly more
3520 ? simple scheme and tweaked the few failing testcases to match it.
3524 This is how it works now:
3528 =item Setting/Accessing
3530 * You can set the A global via Math::BigInt->accuracy() or
3531 Math::BigFloat->accuracy() or whatever class you are using.
3532 * You can also set P globally by using Math::SomeClass->precision() likewise.
3533 * Globals are classwide, and not inherited by subclasses.
3534 * to undefine A, use Math::SomeCLass->accuracy(undef);
3535 * to undefine P, use Math::SomeClass->precision(undef);
3536 * Setting Math::SomeClass->accuracy() clears automatically
3537 Math::SomeClass->precision(), and vice versa.
3538 * To be valid, A must be > 0, P can have any value.
3539 * If P is negative, this means round to the P'th place to the right of the
3540 decimal point; positive values mean to the left of the decimal point.
3541 P of 0 means round to integer.
3542 * to find out the current global A, take Math::SomeClass->accuracy()
3543 * to find out the current global P, take Math::SomeClass->precision()
3544 * use $x->accuracy() respective $x->precision() for the local setting of $x.
3545 * Please note that $x->accuracy() respecive $x->precision() fall back to the
3546 defined globals, when $x's A or P is not set.
3548 =item Creating numbers
3550 * When you create a number, you can give it's desired A or P via:
3551 $x = Math::BigInt->new($number,$A,$P);
3552 * Only one of A or P can be defined, otherwise the result is NaN
3553 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
3554 globals (if set) will be used. Thus changing the global defaults later on
3555 will not change the A or P of previously created numbers (i.e., A and P of
3556 $x will be what was in effect when $x was created)
3557 * If given undef for A and P, B<no> rounding will occur, and the globals will
3558 B<not> be used. This is used by subclasses to create numbers without
3559 suffering rounding in the parent. Thus a subclass is able to have it's own
3560 globals enforced upon creation of a number by using
3561 $x = Math::BigInt->new($number,undef,undef):
3563 use Math::Bigint::SomeSubclass;
3566 Math::BigInt->accuracy(2);
3567 Math::BigInt::SomeSubClass->accuracy(3);
3568 $x = Math::BigInt::SomeSubClass->new(1234);
3570 $x is now 1230, and not 1200. A subclass might choose to implement
3571 this otherwise, e.g. falling back to the parent's A and P.
3575 * If A or P are enabled/defined, they are used to round the result of each
3576 operation according to the rules below
3577 * Negative P is ignored in Math::BigInt, since BigInts never have digits
3578 after the decimal point
3579 * Math::BigFloat uses Math::BigInts internally, but setting A or P inside
3580 Math::BigInt as globals should not tamper with the parts of a BigFloat.
3581 Thus a flag is used to mark all Math::BigFloat numbers as 'never round'
3585 * It only makes sense that a number has only one of A or P at a time.
3586 Since you can set/get both A and P, there is a rule that will practically
3587 enforce only A or P to be in effect at a time, even if both are set.
3588 This is called precedence.
3589 * If two objects are involved in an operation, and one of them has A in
3590 effect, and the other P, this results in an error (NaN).
3591 * A takes precendence over P (Hint: A comes before P). If A is defined, it
3592 is used, otherwise P is used. If neither of them is defined, nothing is
3593 used, i.e. the result will have as many digits as it can (with an
3594 exception for fdiv/fsqrt) and will not be rounded.
3595 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
3596 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
3597 If either the dividend's or the divisor's mantissa has more digits than
3598 the value of F, the higher value will be used instead of F.
3599 This is to limit the digits (A) of the result (just consider what would
3600 happen with unlimited A and P in the case of 1/3 :-)
3601 * fdiv will calculate (at least) 4 more digits than required (determined by
3602 A, P or F), and, if F is not used, round the result
3603 (this will still fail in the case of a result like 0.12345000000001 with A
3604 or P of 5, but this can not be helped - or can it?)
3605 * Thus you can have the math done by on Math::Big* class in three modes:
3606 + never round (this is the default):
3607 This is done by setting A and P to undef. No math operation
3608 will round the result, with fdiv() and fsqrt() as exceptions to guard
3609 against overflows. You must explicitely call bround(), bfround() or
3610 round() (the latter with parameters).
3611 Note: Once you have rounded a number, the settings will 'stick' on it
3612 and 'infect' all other numbers engaged in math operations with it, since
3613 local settings have the highest precedence. So, to get SaferRound[tm],
3614 use a copy() before rounding like this:
3616 $x = Math::BigFloat->new(12.34);
3617 $y = Math::BigFloat->new(98.76);
3618 $z = $x * $y; # 1218.6984
3619 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
3620 $z = $x * $y; # still 1218.6984, without
3621 # copy would have been 1210!
3623 + round after each op:
3624 After each single operation (except for testing like is_zero()), the
3625 method round() is called and the result is rounded appropriately. By
3626 setting proper values for A and P, you can have all-the-same-A or
3627 all-the-same-P modes. For example, Math::Currency might set A to undef,
3628 and P to -2, globally.
3630 ?Maybe an extra option that forbids local A & P settings would be in order,
3631 ?so that intermediate rounding does not 'poison' further math?
3633 =item Overriding globals
3635 * you will be able to give A, P and R as an argument to all the calculation
3636 routines; the second parameter is A, the third one is P, and the fourth is
3637 R (shift right by one for binary operations like badd). P is used only if
3638 the first parameter (A) is undefined. These three parameters override the
3639 globals in the order detailed as follows, i.e. the first defined value
3641 (local: per object, global: global default, parameter: argument to sub)
3644 + local A (if defined on both of the operands: smaller one is taken)
3645 + local P (if defined on both of the operands: bigger one is taken)
3649 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
3650 arguments (A and P) instead of one
3652 =item Local settings
3654 * You can set A and P locally by using $x->accuracy() and $x->precision()
3655 and thus force different A and P for different objects/numbers.
3656 * Setting A or P this way immediately rounds $x to the new value.
3657 * $x->accuracy() clears $x->precision(), and vice versa.
3661 * the rounding routines will use the respective global or local settings.
3662 fround()/bround() is for accuracy rounding, while ffround()/bfround()
3664 * the two rounding functions take as the second parameter one of the
3665 following rounding modes (R):
3666 'even', 'odd', '+inf', '-inf', 'zero', 'trunc'
3667 * you can set and get the global R by using Math::SomeClass->round_mode()
3668 or by setting $Math::SomeClass::round_mode
3669 * after each operation, $result->round() is called, and the result may
3670 eventually be rounded (that is, if A or P were set either locally,
3671 globally or as parameter to the operation)
3672 * to manually round a number, call $x->round($A,$P,$round_mode);
3673 this will round the number by using the appropriate rounding function
3674 and then normalize it.
3675 * rounding modifies the local settings of the number:
3677 $x = Math::BigFloat->new(123.456);
3681 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
3682 will be 4 from now on.
3684 =item Default values
3693 * The defaults are set up so that the new code gives the same results as
3694 the old code (except in a few cases on fdiv):
3695 + Both A and P are undefined and thus will not be used for rounding
3696 after each operation.
3697 + round() is thus a no-op, unless given extra parameters A and P
3703 The actual numbers are stored as unsigned big integers (with seperate sign).
3704 You should neither care about nor depend on the internal representation; it
3705 might change without notice. Use only method calls like C<< $x->sign(); >>
3706 instead relying on the internal hash keys like in C<< $x->{sign}; >>.
3710 Math with the numbers is done (by default) by a module called
3711 Math::BigInt::Calc. This is equivalent to saying:
3713 use Math::BigInt lib => 'Calc';
3715 You can change this by using:
3717 use Math::BigInt lib => 'BitVect';
3719 The following would first try to find Math::BigInt::Foo, then
3720 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
3722 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
3724 Calc.pm uses as internal format an array of elements of some decimal base
3725 (usually 1e5 or 1e7) with the least significant digit first, while BitVect.pm
3726 uses a bit vector of base 2, most significant bit first. Other modules might
3727 use even different means of representing the numbers. See the respective
3728 module documentation for further details.
3732 The sign is either '+', '-', 'NaN', '+inf' or '-inf' and stored seperately.
3734 A sign of 'NaN' is used to represent the result when input arguments are not
3735 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
3736 minus infinity. You will get '+inf' when dividing a positive number by 0, and
3737 '-inf' when dividing any negative number by 0.
3739 =head2 mantissa(), exponent() and parts()
3741 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
3744 $m = $x->mantissa();
3745 $e = $x->exponent();
3746 $y = $m * ( 10 ** $e );
3747 print "ok\n" if $x == $y;
3749 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
3750 in one go. Both the returned mantissa and exponent have a sign.
3752 Currently, for BigInts C<$e> will be always 0, except for NaN, +inf and -inf,
3753 where it will be NaN; and for $x == 0, where it will be 1
3754 (to be compatible with Math::BigFloat's internal representation of a zero as
3757 C<$m> will always be a copy of the original number. The relation between $e
3758 and $m might change in the future, but will always be equivalent in a
3759 numerical sense, e.g. $m might get minimized.
3765 sub bint { Math::BigInt->new(shift); }
3767 $x = Math::BigInt->bstr("1234") # string "1234"
3768 $x = "$x"; # same as bstr()
3769 $x = Math::BigInt->bneg("1234"); # Bigint "-1234"
3770 $x = Math::BigInt->babs("-12345"); # Bigint "12345"
3771 $x = Math::BigInt->bnorm("-0 00"); # BigInt "0"
3772 $x = bint(1) + bint(2); # BigInt "3"
3773 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
3774 $x = bint(1); # BigInt "1"
3775 $x = $x + 5 / 2; # BigInt "3"
3776 $x = $x ** 3; # BigInt "27"
3777 $x *= 2; # BigInt "54"
3778 $x = Math::BigInt->new(0); # BigInt "0"
3780 $x = Math::BigInt->badd(4,5) # BigInt "9"
3781 print $x->bsstr(); # 9e+0
3783 Examples for rounding:
3788 $x = Math::BigFloat->new(123.4567);
3789 $y = Math::BigFloat->new(123.456789);
3790 Math::BigFloat->accuracy(4); # no more A than 4
3792 ok ($x->copy()->fround(),123.4); # even rounding
3793 print $x->copy()->fround(),"\n"; # 123.4
3794 Math::BigFloat->round_mode('odd'); # round to odd
3795 print $x->copy()->fround(),"\n"; # 123.5
3796 Math::BigFloat->accuracy(5); # no more A than 5
3797 Math::BigFloat->round_mode('odd'); # round to odd
3798 print $x->copy()->fround(),"\n"; # 123.46
3799 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
3800 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
3802 Math::BigFloat->accuracy(undef); # A not important now
3803 Math::BigFloat->precision(2); # P important
3804 print $x->copy()->bnorm(),"\n"; # 123.46
3805 print $x->copy()->fround(),"\n"; # 123.46
3807 Examples for converting:
3809 my $x = Math::BigInt->new('0b1'.'01' x 123);
3810 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
3812 =head1 Autocreating constants
3814 After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
3815 and binary constants in the given scope are converted to C<Math::BigInt>.
3816 This conversion happens at compile time.
3820 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
3822 prints the integer value of C<2**100>. Note that without conversion of
3823 constants the expression 2**100 will be calculated as perl scalar.
3825 Please note that strings and floating point constants are not affected,
3828 use Math::BigInt qw/:constant/;
3830 $x = 1234567890123456789012345678901234567890
3831 + 123456789123456789;
3832 $y = '1234567890123456789012345678901234567890'
3833 + '123456789123456789';
3835 do not work. You need an explicit Math::BigInt->new() around one of the
3836 operands. You should also quote large constants to protect loss of precision:
3840 $x = Math::BigInt->new('1234567889123456789123456789123456789');
3842 Without the quotes Perl would convert the large number to a floating point
3843 constant at compile time and then hand the result to BigInt, which results in
3844 an truncated result or a NaN.
3846 This also applies to integers that look like floating point constants:
3848 use Math::BigInt ':constant';
3850 print ref(123e2),"\n";
3851 print ref(123.2e2),"\n";
3853 will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
3854 to get this to work.
3858 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
3859 must be made in the second case. For long numbers, the copy can eat up to 20%
3860 of the work (in the case of addition/subtraction, less for
3861 multiplication/division). If $y is very small compared to $x, the form
3862 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
3863 more time then the actual addition.
3865 With a technique called copy-on-write, the cost of copying with overload could
3866 be minimized or even completely avoided. A test implementation of COW did show
3867 performance gains for overloaded math, but introduced a performance loss due
3868 to a constant overhead for all other operatons.
3870 The rewritten version of this module is slower on certain operations, like
3871 new(), bstr() and numify(). The reason are that it does now more work and
3872 handles more cases. The time spent in these operations is usually gained in
3873 the other operations so that programs on the average should get faster. If
3874 they don't, please contect the author.
3876 Some operations may be slower for small numbers, but are significantly faster
3877 for big numbers. Other operations are now constant (O(1), like bneg(), babs()
3878 etc), instead of O(N) and thus nearly always take much less time. These
3879 optimizations were done on purpose.
3881 If you find the Calc module to slow, try to install any of the replacement
3882 modules and see if they help you.
3884 =head2 Alternative math libraries
3886 You can use an alternative library to drive Math::BigInt via:
3888 use Math::BigInt lib => 'Module';
3890 See L<MATH LIBRARY> for more information.
3892 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
3896 =head1 Subclassing Math::BigInt
3898 The basic design of Math::BigInt allows simple subclasses with very little
3899 work, as long as a few simple rules are followed:
3905 The public API must remain consistent, i.e. if a sub-class is overloading
3906 addition, the sub-class must use the same name, in this case badd(). The
3907 reason for this is that Math::BigInt is optimized to call the object methods
3912 The private object hash keys like C<$x->{sign}> may not be changed, but
3913 additional keys can be added, like C<$x->{_custom}>.
3917 Accessor functions are available for all existing object hash keys and should
3918 be used instead of directly accessing the internal hash keys. The reason for
3919 this is that Math::BigInt itself has a pluggable interface which permits it
3920 to support different storage methods.
3924 More complex sub-classes may have to replicate more of the logic internal of
3925 Math::BigInt if they need to change more basic behaviors. A subclass that
3926 needs to merely change the output only needs to overload C<bstr()>.
3928 All other object methods and overloaded functions can be directly inherited
3929 from the parent class.
3931 At the very minimum, any subclass will need to provide it's own C<new()> and can
3932 store additional hash keys in the object. There are also some package globals
3933 that must be defined, e.g.:
3937 $precision = -2; # round to 2 decimal places
3938 $round_mode = 'even';
3941 Additionally, you might want to provide the following two globals to allow
3942 auto-upgrading and auto-downgrading to work correctly:
3947 This allows Math::BigInt to correctly retrieve package globals from the
3948 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
3949 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
3955 in your subclass to automatically inherit the overloading from the parent. If
3956 you like, you can change part of the overloading, look at Math::String for an
3961 When used like this:
3963 use Math::BigInt upgrade => 'Foo::Bar';
3965 certain operations will 'upgrade' their calculation and thus the result to
3966 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
3968 use Math::BigInt upgrade => 'Math::BigFloat';
3970 As a shortcut, you can use the module C<bignum>:
3974 Also good for oneliners:
3976 perl -Mbignum -le 'print 2 ** 255'
3978 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
3979 as well es preserve accuracy (as in sqrt(3)).
3981 Beware: This feature is not fully implemented yet.
3985 The following methods upgrade themselves unconditionally; that is if upgrade
3986 is in effect, they will always hand up their work:
3998 Beware: This list is not complete.
4000 All other methods upgrade themselves only when one (or all) of their
4001 arguments are of the class mentioned in $upgrade (This might change in later
4002 versions to a more sophisticated scheme):
4008 =item Out of Memory!
4010 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
4011 C<eval()> in your code will crash with "Out of memory". This is probably an
4012 overload/exporter bug. You can workaround by not having C<eval()>
4013 and ':constant' at the same time or upgrade your Perl to a newer version.
4015 =item Fails to load Calc on Perl prior 5.6.0
4017 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
4018 will fall back to eval { require ... } when loading the math lib on Perls
4019 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
4020 filesystems using a different seperator.
4026 Some things might not work as you expect them. Below is documented what is
4027 known to be troublesome:
4031 =item stringify, bstr(), bsstr() and 'cmp'
4033 Both stringify and bstr() now drop the leading '+'. The old code would return
4034 '+3', the new returns '3'. This is to be consistent with Perl and to make
4035 cmp (especially with overloading) to work as you expect. It also solves
4036 problems with Test.pm, it's ok() uses 'eq' internally.
4038 Mark said, when asked about to drop the '+' altogether, or make only cmp work:
4040 I agree (with the first alternative), don't add the '+' on positive
4041 numbers. It's not as important anymore with the new internal
4042 form for numbers. It made doing things like abs and neg easier,
4043 but those have to be done differently now anyway.
4045 So, the following examples will now work all as expected:
4048 BEGIN { plan tests => 1 }
4051 my $x = new Math::BigInt 3*3;
4052 my $y = new Math::BigInt 3*3;
4055 print "$x eq 9" if $x eq $y;
4056 print "$x eq 9" if $x eq '9';
4057 print "$x eq 9" if $x eq 3*3;
4059 Additionally, the following still works:
4061 print "$x == 9" if $x == $y;
4062 print "$x == 9" if $x == 9;
4063 print "$x == 9" if $x == 3*3;
4065 There is now a C<bsstr()> method to get the string in scientific notation aka
4066 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
4067 for comparisation, but Perl will represent some numbers as 100 and others
4068 as 1e+308. If in doubt, convert both arguments to Math::BigInt before doing eq:
4071 BEGIN { plan tests => 3 }
4074 $x = Math::BigInt->new('1e56'); $y = 1e56;
4075 ok ($x,$y); # will fail
4076 ok ($x->bsstr(),$y); # okay
4077 $y = Math::BigInt->new($y);
4080 Alternatively, simple use <=> for comparisations, that will get it always
4081 right. There is not yet a way to get a number automatically represented as
4082 a string that matches exactly the way Perl represents it.
4086 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
4089 $x = Math::BigInt->new(123);
4090 $y = int($x); # BigInt 123
4091 $x = Math::BigFloat->new(123.45);
4092 $y = int($x); # BigInt 123
4094 In all Perl versions you can use C<as_number()> for the same effect:
4096 $x = Math::BigFloat->new(123.45);
4097 $y = $x->as_number(); # BigInt 123
4099 This also works for other subclasses, like Math::String.
4101 It is yet unlcear whether overloaded int() should return a scalar or a BigInt.
4105 The following will probably not do what you expect:
4107 $c = Math::BigInt->new(123);
4108 print $c->length(),"\n"; # prints 30
4110 It prints both the number of digits in the number and in the fraction part
4111 since print calls C<length()> in list context. Use something like:
4113 print scalar $c->length(),"\n"; # prints 3
4117 The following will probably not do what you expect:
4119 print $c->bdiv(10000),"\n";
4121 It prints both quotient and remainder since print calls C<bdiv()> in list
4122 context. Also, C<bdiv()> will modify $c, so be carefull. You probably want
4125 print $c / 10000,"\n";
4126 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
4130 The quotient is always the greatest integer less than or equal to the
4131 real-valued quotient of the two operands, and the remainder (when it is
4132 nonzero) always has the same sign as the second operand; so, for
4142 As a consequence, the behavior of the operator % agrees with the
4143 behavior of Perl's built-in % operator (as documented in the perlop
4144 manpage), and the equation
4146 $x == ($x / $y) * $y + ($x % $y)
4148 holds true for any $x and $y, which justifies calling the two return
4149 values of bdiv() the quotient and remainder. The only exception to this rule
4150 are when $y == 0 and $x is negative, then the remainder will also be
4151 negative. See below under "infinity handling" for the reasoning behing this.
4153 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
4154 not change BigInt's way to do things. This is because under 'use integer' Perl
4155 will do what the underlying C thinks is right and this is different for each
4156 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
4157 the author to implement it ;)
4159 =item infinity handling
4161 Here are some examples that explain the reasons why certain results occur while
4164 The following table shows the result of the division and the remainder, so that
4165 the equation above holds true. Some "ordinary" cases are strewn in to show more
4166 clearly the reasoning:
4168 A / B = C, R so that C * B + R = A
4169 =========================================================
4170 5 / 8 = 0, 5 0 * 8 + 5 = 5
4171 0 / 8 = 0, 0 0 * 8 + 0 = 0
4172 0 / inf = 0, 0 0 * inf + 0 = 0
4173 0 /-inf = 0, 0 0 * -inf + 0 = 0
4174 5 / inf = 0, 5 0 * inf + 5 = 5
4175 5 /-inf = 0, 5 0 * -inf + 5 = 5
4176 -5/ inf = 0, -5 0 * inf + -5 = -5
4177 -5/-inf = 0, -5 0 * -inf + -5 = -5
4178 inf/ 5 = inf, 0 inf * 5 + 0 = inf
4179 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
4180 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
4181 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
4182 5/ 5 = 1, 0 1 * 5 + 0 = 5
4183 -5/ -5 = 1, 0 1 * -5 + 0 = -5
4184 inf/ inf = 1, 0 1 * inf + 0 = inf
4185 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
4186 inf/-inf = -1, 0 -1 * -inf + 0 = inf
4187 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
4188 8/ 0 = inf, 8 inf * 0 + 8 = 8
4189 inf/ 0 = inf, inf inf * 0 + inf = inf
4192 These cases below violate the "remainder has the sign of the second of the two
4193 arguments", since they wouldn't match up otherwise.
4195 A / B = C, R so that C * B + R = A
4196 ========================================================
4197 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
4198 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
4200 =item Modifying and =
4204 $x = Math::BigFloat->new(5);
4207 It will not do what you think, e.g. making a copy of $x. Instead it just makes
4208 a second reference to the B<same> object and stores it in $y. Thus anything
4209 that modifies $x (except overloaded operators) will modify $y, and vice versa.
4210 Or in other words, C<=> is only safe if you modify your BigInts only via
4211 overloaded math. As soon as you use a method call it breaks:
4214 print "$x, $y\n"; # prints '10, 10'
4216 If you want a true copy of $x, use:
4220 You can also chain the calls like this, this will make first a copy and then
4223 $y = $x->copy()->bmul(2);
4225 See also the documentation for overload.pm regarding C<=>.
4229 C<bpow()> (and the rounding functions) now modifies the first argument and
4230 returns it, unlike the old code which left it alone and only returned the
4231 result. This is to be consistent with C<badd()> etc. The first three will
4232 modify $x, the last one won't:
4234 print bpow($x,$i),"\n"; # modify $x
4235 print $x->bpow($i),"\n"; # ditto
4236 print $x **= $i,"\n"; # the same
4237 print $x ** $i,"\n"; # leave $x alone
4239 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4241 =item Overloading -$x
4251 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4252 needs to preserve $x since it does not know that it later will get overwritten.
4253 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4255 With Copy-On-Write, this issue would be gone, but C-o-W is not implemented
4256 since it is slower for all other things.
4258 =item Mixing different object types
4260 In Perl you will get a floating point value if you do one of the following:
4266 With overloaded math, only the first two variants will result in a BigFloat:
4271 $mbf = Math::BigFloat->new(5);
4272 $mbi2 = Math::BigInteger->new(5);
4273 $mbi = Math::BigInteger->new(2);
4275 # what actually gets called:
4276 $float = $mbf + $mbi; # $mbf->badd()
4277 $float = $mbf / $mbi; # $mbf->bdiv()
4278 $integer = $mbi + $mbf; # $mbi->badd()
4279 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4280 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4282 This is because math with overloaded operators follows the first (dominating)
4283 operand, and the operation of that is called and returns thus the result. So,
4284 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4285 the result should be a Math::BigFloat or the second operant is one.
4287 To get a Math::BigFloat you either need to call the operation manually,
4288 make sure the operands are already of the proper type or casted to that type
4289 via Math::BigFloat->new():
4291 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4293 Beware of simple "casting" the entire expression, this would only convert
4294 the already computed result:
4296 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4298 Beware also of the order of more complicated expressions like:
4300 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4301 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4303 If in doubt, break the expression into simpler terms, or cast all operands
4304 to the desired resulting type.
4306 Scalar values are a bit different, since:
4311 will both result in the proper type due to the way the overloaded math works.
4313 This section also applies to other overloaded math packages, like Math::String.
4315 One solution to you problem might be L<autoupgrading|upgrading>.
4319 C<bsqrt()> works only good if the result is a big integer, e.g. the square
4320 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4323 If you want a better approximation of the square root, then use:
4325 $x = Math::BigFloat->new(12);
4326 Math::BigFloat->precision(0);
4327 Math::BigFloat->round_mode('even');
4328 print $x->copy->bsqrt(),"\n"; # 4
4330 Math::BigFloat->precision(2);
4331 print $x->bsqrt(),"\n"; # 3.46
4332 print $x->bsqrt(3),"\n"; # 3.464
4336 For negative numbers in base see also L<brsft|brsft>.
4342 This program is free software; you may redistribute it and/or modify it under
4343 the same terms as Perl itself.
4347 L<Math::BigFloat> and L<Math::Big> as well as L<Math::BigInt::BitVect>,
4348 L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4351 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4352 more documentation including a full version history, testcases, empty
4353 subclass files and benchmarks.
4357 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4358 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001.