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]*$/))
411 $self->{sign} = $1 || '+';
413 if ($wanted =~ /^[+-]/)
415 # remove sign without touching wanted
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 # e can only be positive
667 # MBF: my $s = $e->{sign}; $s = '' if $s eq '-'; my $sep = 'e'.$s;
668 return $m->bstr().$sign.$e->bstr();
673 # make a string from bigint object
674 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
675 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
677 if ($x->{sign} !~ /^[+-]$/)
679 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
682 my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
683 return $es.${$CALC->_str($x->{value})};
688 # Make a "normal" scalar from a BigInt object
689 my $x = shift; $x = $class->new($x) unless ref $x;
690 return $x->{sign} if $x->{sign} !~ /^[+-]$/;
691 my $num = $CALC->_num($x->{value});
692 return -$num if $x->{sign} eq '-';
696 ##############################################################################
697 # public stuff (usually prefixed with "b")
701 # return the sign of the number: +/-/-inf/+inf/NaN
702 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
707 sub _find_round_parameters
709 # After any operation or when calling round(), the result is rounded by
710 # regarding the A & P from arguments, local parameters, or globals.
712 # This procedure finds the round parameters, but it is for speed reasons
713 # duplicated in round. Otherwise, it is tested by the testsuite and used
716 my ($self,$a,$p,$r,@args) = @_;
717 # $a accuracy, if given by caller
718 # $p precision, if given by caller
719 # $r round_mode, if given by caller
720 # @args all 'other' arguments (0 for unary, 1 for binary ops)
722 # leave bigfloat parts alone
723 return ($self) if exists $self->{_f} && $self->{_f} & MB_NEVER_ROUND != 0;
725 my $c = ref($self); # find out class of argument(s)
728 # now pick $a or $p, but only if we have got "arguments"
731 foreach ($self,@args)
733 # take the defined one, or if both defined, the one that is smaller
734 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
739 # even if $a is defined, take $p, to signal error for both defined
740 foreach ($self,@args)
742 # take the defined one, or if both defined, the one that is bigger
744 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
747 # if still none defined, use globals (#2)
748 $a = ${"$c\::accuracy"} unless defined $a;
749 $p = ${"$c\::precision"} unless defined $p;
752 return ($self) unless defined $a || defined $p; # early out
754 # set A and set P is an fatal error
755 return ($self->bnan()) if defined $a && defined $p;
757 $r = ${"$c\::round_mode"} unless defined $r;
758 die "Unknown round mode '$r'" if $r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/;
760 return ($self,$a,$p,$r);
765 # Round $self according to given parameters, or given second argument's
766 # parameters or global defaults
768 # for speed reasons, _find_round_parameters is embeded here:
770 my ($self,$a,$p,$r,@args) = @_;
771 # $a accuracy, if given by caller
772 # $p precision, if given by caller
773 # $r round_mode, if given by caller
774 # @args all 'other' arguments (0 for unary, 1 for binary ops)
776 # leave bigfloat parts alone
777 return ($self) if exists $self->{_f} && $self->{_f} & MB_NEVER_ROUND != 0;
779 my $c = ref($self); # find out class of argument(s)
782 # now pick $a or $p, but only if we have got "arguments"
785 foreach ($self,@args)
787 # take the defined one, or if both defined, the one that is smaller
788 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
793 # even if $a is defined, take $p, to signal error for both defined
794 foreach ($self,@args)
796 # take the defined one, or if both defined, the one that is bigger
798 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
801 # if still none defined, use globals (#2)
802 $a = ${"$c\::accuracy"} unless defined $a;
803 $p = ${"$c\::precision"} unless defined $p;
806 return $self unless defined $a || defined $p; # early out
808 # set A and set P is an fatal error
809 return $self->bnan() if defined $a && defined $p;
811 $r = ${"$c\::round_mode"} unless defined $r;
812 die "Unknown round mode '$r'" if $r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/;
814 # now round, by calling either fround or ffround:
817 $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a;
819 else # both can't be undefined due to early out
821 $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p;
823 $self->bnorm(); # after round, normalize
828 # (numstr or BINT) return BINT
829 # Normalize number -- no-op here
830 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
836 # (BINT or num_str) return BINT
837 # make number absolute, or return absolute BINT from string
838 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
840 return $x if $x->modify('babs');
841 # post-normalized abs for internal use (does nothing for NaN)
842 $x->{sign} =~ s/^-/+/;
848 # (BINT or num_str) return BINT
849 # negate number or make a negated number from string
850 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
852 return $x if $x->modify('bneg');
854 # for +0 dont negate (to have always normalized)
855 $x->{sign} =~ tr/+-/-+/ if !$x->is_zero(); # does nothing for NaN
861 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
862 # (BINT or num_str, BINT or num_str) return cond_code
865 my ($self,$x,$y) = (ref($_[0]),@_);
867 # objectify is costly, so avoid it
868 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
870 ($self,$x,$y) = objectify(2,@_);
873 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
875 # handle +-inf and NaN
876 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
877 return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
878 return +1 if $x->{sign} eq '+inf';
879 return -1 if $x->{sign} eq '-inf';
880 return -1 if $y->{sign} eq '+inf';
883 # check sign for speed first
884 return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
885 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
887 # have same sign, so compare absolute values. Don't make tests for zero here
888 # because it's actually slower than testin in Calc (especially w/ Pari et al)
890 # post-normalized compare for internal use (honors signs)
891 if ($x->{sign} eq '+')
894 return $CALC->_acmp($x->{value},$y->{value});
898 $CALC->_acmp($y->{value},$x->{value}); # swaped (lib returns 0,1,-1)
903 # Compares 2 values, ignoring their signs.
904 # Returns one of undef, <0, =0, >0. (suitable for sort)
905 # (BINT, BINT) return cond_code
908 my ($self,$x,$y) = (ref($_[0]),@_);
909 # objectify is costly, so avoid it
910 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
912 ($self,$x,$y) = objectify(2,@_);
915 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
917 # handle +-inf and NaN
918 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
919 return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
920 return +1; # inf is always bigger
922 $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
927 # add second arg (BINT or string) to first (BINT) (modifies first)
928 # return result as BINT
931 my ($self,$x,$y,@r) = (ref($_[0]),@_);
932 # objectify is costly, so avoid it
933 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
935 ($self,$x,$y,@r) = objectify(2,@_);
938 return $x if $x->modify('badd');
939 return $upgrade->badd($x,$y,@r) if defined $upgrade &&
940 ((!$x->isa($self)) || (!$y->isa($self)));
942 $r[3] = $y; # no push!
943 # inf and NaN handling
944 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
947 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
949 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
951 # +inf++inf or -inf+-inf => same, rest is NaN
952 return $x if $x->{sign} eq $y->{sign};
955 # +-inf + something => +inf
956 # something +-inf => +-inf
957 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
961 my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
965 $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
970 my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
973 #print "swapped sub (a=$a)\n";
974 $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
979 # speedup, if equal, set result to 0
980 #print "equal sub, result = 0\n";
981 $x->{value} = $CALC->_zero();
986 #print "unswapped sub (a=$a)\n";
987 $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
991 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
997 # (BINT or num_str, BINT or num_str) return num_str
998 # subtract second arg from first, modify first
1001 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1002 # objectify is costly, so avoid it
1003 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1005 ($self,$x,$y,@r) = objectify(2,@_);
1008 return $x if $x->modify('bsub');
1010 # upgrade done by badd():
1011 # return $upgrade->badd($x,$y,@r) if defined $upgrade &&
1012 # ((!$x->isa($self)) || (!$y->isa($self)));
1016 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1020 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
1021 $x->badd($y,@r); # badd does not leave internal zeros
1022 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
1023 $x; # already rounded by badd() or no round necc.
1028 # increment arg by one
1029 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1030 return $x if $x->modify('binc');
1032 if ($x->{sign} eq '+')
1034 $x->{value} = $CALC->_inc($x->{value});
1035 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1038 elsif ($x->{sign} eq '-')
1040 $x->{value} = $CALC->_dec($x->{value});
1041 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1042 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1045 # inf, nan handling etc
1046 $x->badd($self->__one(),$a,$p,$r); # badd does round
1051 # decrement arg by one
1052 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1053 return $x if $x->modify('bdec');
1055 my $zero = $CALC->_is_zero($x->{value}) && $x->{sign} eq '+';
1057 if (($x->{sign} eq '-') || $zero)
1059 $x->{value} = $CALC->_inc($x->{value});
1060 $x->{sign} = '-' if $zero; # 0 => 1 => -1
1061 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1062 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1066 elsif ($x->{sign} eq '+')
1068 $x->{value} = $CALC->_dec($x->{value});
1069 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1072 # inf, nan handling etc
1073 $x->badd($self->__one('-'),$a,$p,$r); # badd does round
1078 # not implemented yet
1079 my ($self,$x,$base,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1081 return $upgrade->blog($x,$base,$a,$p,$r) if defined $upgrade;
1088 # (BINT or num_str, BINT or num_str) return BINT
1089 # does not modify arguments, but returns new object
1090 # Lowest Common Multiplicator
1092 my $y = shift; my ($x);
1099 $x = $class->new($y);
1101 while (@_) { $x = __lcm($x,shift); }
1107 # (BINT or num_str, BINT or num_str) return BINT
1108 # does not modify arguments, but returns new object
1109 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1112 $y = __PACKAGE__->new($y) if !ref($y);
1114 my $x = $y->copy(); # keep arguments
1115 if ($CALC->can('_gcd'))
1119 $y = shift; $y = $self->new($y) if !ref($y);
1120 next if $y->is_zero();
1121 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1122 $x->{value} = $CALC->_gcd($x->{value},$y->{value}); last if $x->is_one();
1129 $y = shift; $y = $self->new($y) if !ref($y);
1130 $x = __gcd($x,$y->copy()); last if $x->is_one(); # _gcd handles NaN
1138 # (num_str or BINT) return BINT
1139 # represent ~x as twos-complement number
1140 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1141 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1143 return $x if $x->modify('bnot');
1144 $x->bneg()->bdec(); # bdec already does round
1147 # is_foo test routines
1151 # return true if arg (BINT or num_str) is zero (array '+', '0')
1152 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1153 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1155 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1156 $CALC->_is_zero($x->{value});
1161 # return true if arg (BINT or num_str) is NaN
1162 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1164 return 1 if $x->{sign} eq $nan;
1170 # return true if arg (BINT or num_str) is +-inf
1171 my ($self,$x,$sign) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1173 $sign = '' if !defined $sign;
1174 return 1 if $sign eq $x->{sign}; # match ("+inf" eq "+inf")
1175 return 0 if $sign !~ /^([+-]|)$/;
1179 return 1 if ($x->{sign} =~ /^[+-]inf$/);
1182 $sign = quotemeta($sign.'inf');
1183 return 1 if ($x->{sign} =~ /^$sign$/);
1189 # return true if arg (BINT or num_str) is +1
1190 # or -1 if sign is given
1191 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1192 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1194 $sign = '' if !defined $sign; $sign = '+' if $sign ne '-';
1196 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1197 $CALC->_is_one($x->{value});
1202 # return true when arg (BINT or num_str) is odd, false for even
1203 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1204 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1206 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1207 $CALC->_is_odd($x->{value});
1212 # return true when arg (BINT or num_str) is even, false for odd
1213 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1214 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1216 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1217 $CALC->_is_even($x->{value});
1222 # return true when arg (BINT or num_str) is positive (>= 0)
1223 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1224 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1226 return 1 if $x->{sign} =~ /^\+/;
1232 # return true when arg (BINT or num_str) is negative (< 0)
1233 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1234 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1236 return 1 if ($x->{sign} =~ /^-/);
1242 # return true when arg (BINT or num_str) is an integer
1243 # always true for BigInt, but different for Floats
1244 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1245 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1247 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1250 ###############################################################################
1254 # multiply two numbers -- stolen from Knuth Vol 2 pg 233
1255 # (BINT or num_str, BINT or num_str) return BINT
1258 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1259 # objectify is costly, so avoid it
1260 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1262 ($self,$x,$y,@r) = objectify(2,@_);
1265 return $x if $x->modify('bmul');
1267 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1270 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1272 return $x->bnan() if $x->is_zero() || $y->is_zero();
1273 # result will always be +-inf:
1274 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1275 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1276 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1277 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1278 return $x->binf('-');
1281 return $upgrade->bmul($x,$y,@r)
1282 if defined $upgrade && $y->isa($upgrade);
1284 $r[3] = $y; # no push here
1286 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1288 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1289 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1291 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1297 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1298 my ($self,$x,$y) = @_;
1300 # NaN if x == NaN or y == NaN or x==y==0
1301 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1302 if (($x->is_nan() || $y->is_nan()) ||
1303 ($x->is_zero() && $y->is_zero()));
1305 # +-inf / +-inf == NaN, reminder also NaN
1306 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1308 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1310 # x / +-inf => 0, remainder x (works even if x == 0)
1311 if ($y->{sign} =~ /^[+-]inf$/)
1313 my $t = $x->copy(); # bzero clobbers up $x
1314 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1317 # 5 / 0 => +inf, -6 / 0 => -inf
1318 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1319 # exception: -8 / 0 has remainder -8, not 8
1320 # exception: -inf / 0 has remainder -inf, not inf
1323 # +-inf / 0 => special case for -inf
1324 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1325 if (!$x->is_zero() && !$x->is_inf())
1327 my $t = $x->copy(); # binf clobbers up $x
1329 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1333 # last case: +-inf / ordinary number
1335 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1337 return wantarray ? ($x,$self->bzero()) : $x;
1342 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1343 # (BINT,BINT) (quo,rem) or BINT (only rem)
1346 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1347 # objectify is costly, so avoid it
1348 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1350 ($self,$x,$y,@r) = objectify(2,@_);
1353 return $x if $x->modify('bdiv');
1355 return $self->_div_inf($x,$y)
1356 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1358 return $upgrade->bdiv($upgrade->new($x),$y,@r)
1359 if defined $upgrade && !$y->isa($self);
1361 $r[3] = $y; # no push!
1365 wantarray ? ($x->round(@r),$self->bzero(@r)):$x->round(@r) if $x->is_zero();
1367 # Is $x in the interval [0, $y) (aka $x <= $y) ?
1368 my $cmp = $CALC->_acmp($x->{value},$y->{value});
1369 if (($cmp < 0) and (($x->{sign} eq $y->{sign}) or !wantarray))
1371 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1372 if defined $upgrade;
1374 return $x->bzero()->round(@r) unless wantarray;
1375 my $t = $x->copy(); # make copy first, because $x->bzero() clobbers $x
1376 return ($x->bzero()->round(@r),$t);
1380 # shortcut, both are the same, so set to +/- 1
1381 $x->__one( ($x->{sign} ne $y->{sign} ? '-' : '+') );
1382 return $x unless wantarray;
1383 return ($x->round(@r),$self->bzero(@r));
1385 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1386 if defined $upgrade;
1388 # calc new sign and in case $y == +/- 1, return $x
1389 my $xsign = $x->{sign}; # keep
1390 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1391 # check for / +-1 (cant use $y->is_one due to '-'
1392 if ($CALC->_is_one($y->{value}))
1394 return wantarray ? ($x->round(@r),$self->bzero(@r)) : $x->round(@r);
1399 my $rem = $self->bzero();
1400 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1401 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1402 $rem->{_a} = $x->{_a};
1403 $rem->{_p} = $x->{_p};
1405 if (! $CALC->_is_zero($rem->{value}))
1407 $rem->{sign} = $y->{sign};
1408 $rem = $y-$rem if $xsign ne $y->{sign}; # one of them '-'
1412 $rem->{sign} = '+'; # dont leave -0
1414 return ($x,$rem->round(@r));
1417 $x->{value} = $CALC->_div($x->{value},$y->{value});
1418 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1420 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1424 ###############################################################################
1429 # modulus (or remainder)
1430 # (BINT or num_str, BINT or num_str) return BINT
1433 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1434 # objectify is costly, so avoid it
1435 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1437 ($self,$x,$y,@r) = objectify(2,@_);
1440 return $x if $x->modify('bmod');
1441 $r[3] = $y; # no push!
1442 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1444 my ($d,$r) = $self->_div_inf($x,$y);
1445 $x->{sign} = $r->{sign};
1446 $x->{value} = $r->{value};
1447 return $x->round(@r);
1450 if ($CALC->can('_mod'))
1452 # calc new sign and in case $y == +/- 1, return $x
1453 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1454 if (!$CALC->_is_zero($x->{value}))
1456 my $xsign = $x->{sign};
1457 $x->{sign} = $y->{sign};
1458 if ($xsign ne $y->{sign})
1460 my $t = $CALC->_copy($x->{value}); # copy $x
1461 $x->{value} = $CALC->_copy($y->{value}); # copy $y to $x
1462 $x->{value} = $CALC->_sub($y->{value},$t,1); # $y-$x
1467 $x->{sign} = '+'; # dont leave -0
1469 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1472 my ($t,$rem) = $self->bdiv($x->copy(),$y,@r); # slow way (also rounds)
1474 foreach (qw/value sign _a _p/)
1476 $x->{$_} = $rem->{$_};
1483 # modular inverse. given a number which is (hopefully) relatively
1484 # prime to the modulus, calculate its inverse using Euclid's
1485 # alogrithm. if the number is not relatively prime to the modulus
1486 # (i.e. their gcd is not one) then NaN is returned.
1489 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1490 # objectify is costly, so avoid it
1491 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1493 ($self,$x,$y,@r) = objectify(2,@_);
1496 return $x if $x->modify('bmodinv');
1499 if ($y->{sign} ne '+' # -, NaN, +inf, -inf
1500 || $x->is_zero() # or num == 0
1501 || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1504 # put least residue into $x if $x was negative, and thus make it positive
1505 $x->bmod($y) if $x->{sign} eq '-';
1507 if ($CALC->can('_modinv'))
1509 $x->{value} = $CALC->_modinv($x->{value},$y->{value});
1510 $x->bnan() if !defined $x->{value} ; # in case there was none
1514 my ($u, $u1) = ($self->bzero(), $self->bone());
1515 my ($a, $b) = ($y->copy(), $x->copy());
1517 # first step need always be done since $num (and thus $b) is never 0
1518 # Note that the loop is aligned so that the check occurs between #2 and #1
1519 # thus saving us one step #2 at the loop end. Typical loop count is 1. Even
1520 # a case with 28 loops still gains about 3% with this layout.
1522 ($a, $q, $b) = ($b, $a->bdiv($b)); # step #1
1523 # Euclid's Algorithm
1524 while (!$b->is_zero())
1526 ($u, $u1) = ($u1, $u->bsub($u1->copy()->bmul($q))); # step #2
1527 ($a, $q, $b) = ($b, $a->bdiv($b)); # step #1 again
1530 # if the gcd is not 1, then return NaN! It would be pointless to
1531 # have called bgcd to check this first, because we would then be performing
1532 # the same Euclidean Algorithm *twice*
1533 return $x->bnan() unless $a->is_one();
1536 $x->{value} = $u1->{value};
1537 $x->{sign} = $u1->{sign};
1543 # takes a very large number to a very large exponent in a given very
1544 # large modulus, quickly, thanks to binary exponentation. supports
1545 # negative exponents.
1546 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1548 return $num if $num->modify('bmodpow');
1550 # check modulus for valid values
1551 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1552 || $mod->is_zero());
1554 # check exponent for valid values
1555 if ($exp->{sign} =~ /\w/)
1557 # i.e., if it's NaN, +inf, or -inf...
1558 return $num->bnan();
1561 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
1563 # check num for valid values (also NaN if there was no inverse but $exp < 0)
1564 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1566 if ($CALC->can('_modpow'))
1568 # $mod is positive, sign on $exp is ignored, result also positive
1569 $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
1573 # in the trivial case,
1574 return $num->bzero(@r) if $mod->is_one();
1575 return $num->bone('+',@r) if $num->is_zero() or $num->is_one();
1577 # $num->bmod($mod); # if $x is large, make it smaller first
1578 my $acc = $num->copy(); # but this is not really faster...
1580 $num->bone(); # keep ref to $num
1582 my $expbin = $exp->as_bin(); $expbin =~ s/^[-]?0b//; # ignore sign and prefix
1583 my $len = length($expbin);
1586 if( substr($expbin,$len,1) eq '1')
1588 $num->bmul($acc)->bmod($mod);
1590 $acc->bmul($acc)->bmod($mod);
1596 ###############################################################################
1600 # (BINT or num_str, BINT or num_str) return BINT
1601 # compute factorial numbers
1602 # modifies first argument
1603 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1605 return $x if $x->modify('bfac');
1607 return $x->bnan() if $x->{sign} ne '+'; # inf, NnN, <0 etc => NaN
1608 return $x->bone('+',@r) if $x->is_zero() || $x->is_one(); # 0 or 1 => 1
1610 if ($CALC->can('_fac'))
1612 $x->{value} = $CALC->_fac($x->{value});
1613 return $x->round(@r);
1618 # seems we need not to temp. clear A/P of $x since the result is the same
1619 my $f = $self->new(2);
1620 while ($f->bacmp($n) < 0)
1622 $x->bmul($f); $f->binc();
1624 $x->bmul($f,@r); # last step and also round
1629 # (BINT or num_str, BINT or num_str) return BINT
1630 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1631 # modifies first argument
1634 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1635 # objectify is costly, so avoid it
1636 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1638 ($self,$x,$y,@r) = objectify(2,@_);
1641 return $x if $x->modify('bpow');
1643 return $upgrade->bpow($upgrade->new($x),$y,@r)
1644 if defined $upgrade && !$y->isa($self);
1646 $r[3] = $y; # no push!
1647 return $x if $x->{sign} =~ /^[+-]inf$/; # -inf/+inf ** x
1648 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1649 return $x->bone('+',@r) if $y->is_zero();
1650 return $x->round(@r) if $x->is_one() || $y->is_one();
1651 if ($x->{sign} eq '-' && $CALC->_is_one($x->{value}))
1653 # if $x == -1 and odd/even y => +1/-1
1654 return $y->is_odd() ? $x->round(@r) : $x->babs()->round(@r);
1655 # my Casio FX-5500L has a bug here: -1 ** 2 is -1, but -1 * -1 is 1;
1657 # 1 ** -y => 1 / (1 ** |y|)
1658 # so do test for negative $y after above's clause
1659 return $x->bnan() if $y->{sign} eq '-';
1660 return $x->round(@r) if $x->is_zero(); # 0**y => 0 (if not y <= 0)
1662 if ($CALC->can('_pow'))
1664 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1665 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1669 # based on the assumption that shifting in base 10 is fast, and that mul
1670 # works faster if numbers are small: we count trailing zeros (this step is
1671 # O(1)..O(N), but in case of O(N) we save much more time due to this),
1672 # stripping them out of the multiplication, and add $count * $y zeros
1673 # afterwards like this:
1674 # 300 ** 3 == 300*300*300 == 3*3*3 . '0' x 2 * 3 == 27 . '0' x 6
1675 # creates deep recursion since brsft/blsft use bpow sometimes.
1676 # my $zeros = $x->_trailing_zeros();
1679 # $x->brsft($zeros,10); # remove zeros
1680 # $x->bpow($y); # recursion (will not branch into here again)
1681 # $zeros = $y * $zeros; # real number of zeros to add
1682 # $x->blsft($zeros,10);
1683 # return $x->round(@r);
1686 my $pow2 = $self->__one();
1687 my $y_bin = $y->as_bin(); $y_bin =~ s/^0b//;
1688 my $len = length($y_bin);
1691 $pow2->bmul($x) if substr($y_bin,$len,1) eq '1'; # is odd?
1695 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1701 # (BINT or num_str, BINT or num_str) return BINT
1702 # compute x << y, base n, y >= 0
1705 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1706 # objectify is costly, so avoid it
1707 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1709 ($self,$x,$y,$n,@r) = objectify(2,@_);
1712 return $x if $x->modify('blsft');
1713 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1714 return $x->round(@r) if $y->is_zero();
1716 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1718 my $t; $t = $CALC->_lsft($x->{value},$y->{value},$n) if $CALC->can('_lsft');
1721 $x->{value} = $t; return $x->round(@r);
1724 return $x->bmul( $self->bpow($n, $y, @r), @r );
1729 # (BINT or num_str, BINT or num_str) return BINT
1730 # compute x >> y, base n, y >= 0
1733 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1734 # objectify is costly, so avoid it
1735 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1737 ($self,$x,$y,$n,@r) = objectify(2,@_);
1740 return $x if $x->modify('brsft');
1741 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1742 return $x->round(@r) if $y->is_zero();
1743 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
1745 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1747 # this only works for negative numbers when shifting in base 2
1748 if (($x->{sign} eq '-') && ($n == 2))
1750 return $x->round(@r) if $x->is_one('-'); # -1 => -1
1753 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1754 # but perhaps there is a better emulation for two's complement shift...
1755 # if $y != 1, we must simulate it by doing:
1756 # convert to bin, flip all bits, shift, and be done
1757 $x->binc(); # -3 => -2
1758 my $bin = $x->as_bin();
1759 $bin =~ s/^-0b//; # strip '-0b' prefix
1760 $bin =~ tr/10/01/; # flip bits
1762 if (CORE::length($bin) <= $y)
1764 $bin = '0'; # shifting to far right creates -1
1765 # 0, because later increment makes
1766 # that 1, attached '-' makes it '-1'
1767 # because -1 >> x == -1 !
1771 $bin =~ s/.{$y}$//; # cut off at the right side
1772 $bin = '1' . $bin; # extend left side by one dummy '1'
1773 $bin =~ tr/10/01/; # flip bits back
1775 my $res = $self->new('0b'.$bin); # add prefix and convert back
1776 $res->binc(); # remember to increment
1777 $x->{value} = $res->{value}; # take over value
1778 return $x->round(@r); # we are done now, magic, isn't?
1780 $x->bdec(); # n == 2, but $y == 1: this fixes it
1783 my $t; $t = $CALC->_rsft($x->{value},$y->{value},$n) if $CALC->can('_rsft');
1787 return $x->round(@r);
1790 $x->bdiv($self->bpow($n,$y, @r), @r);
1796 #(BINT or num_str, BINT or num_str) return BINT
1800 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1801 # objectify is costly, so avoid it
1802 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1804 ($self,$x,$y,@r) = objectify(2,@_);
1807 return $x if $x->modify('band');
1809 $r[3] = $y; # no push!
1810 local $Math::BigInt::upgrade = undef;
1812 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1813 return $x->bzero(@r) if $y->is_zero() || $x->is_zero();
1815 my $sign = 0; # sign of result
1816 $sign = 1 if ($x->{sign} eq '-') && ($y->{sign} eq '-');
1817 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1818 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1820 if ($CALC->can('_and') && $sx == 1 && $sy == 1)
1822 $x->{value} = $CALC->_and($x->{value},$y->{value});
1823 return $x->round(@r);
1826 my $m = $self->bone(); my ($xr,$yr);
1827 my $x10000 = $self->new (0x1000);
1828 my $y1 = copy(ref($x),$y); # make copy
1829 $y1->babs(); # and positive
1830 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1831 use integer; # need this for negative bools
1832 while (!$x1->is_zero() && !$y1->is_zero())
1834 ($x1, $xr) = bdiv($x1, $x10000);
1835 ($y1, $yr) = bdiv($y1, $x10000);
1836 # make both op's numbers!
1837 $x->badd( bmul( $class->new(
1838 abs($sx*int($xr->numify()) & $sy*int($yr->numify()))),
1842 $x->bneg() if $sign;
1848 #(BINT or num_str, BINT or num_str) return BINT
1852 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1853 # objectify is costly, so avoid it
1854 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1856 ($self,$x,$y,@r) = objectify(2,@_);
1859 return $x if $x->modify('bior');
1860 $r[3] = $y; # no push!
1862 local $Math::BigInt::upgrade = undef;
1864 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1865 return $x->round(@r) if $y->is_zero();
1867 my $sign = 0; # sign of result
1868 $sign = 1 if ($x->{sign} eq '-') || ($y->{sign} eq '-');
1869 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1870 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1872 # don't use lib for negative values
1873 if ($CALC->can('_or') && $sx == 1 && $sy == 1)
1875 $x->{value} = $CALC->_or($x->{value},$y->{value});
1876 return $x->round(@r);
1879 my $m = $self->bone(); my ($xr,$yr);
1880 my $x10000 = $self->new(0x10000);
1881 my $y1 = copy(ref($x),$y); # make copy
1882 $y1->babs(); # and positive
1883 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1884 use integer; # need this for negative bools
1885 while (!$x1->is_zero() || !$y1->is_zero())
1887 ($x1, $xr) = bdiv($x1,$x10000);
1888 ($y1, $yr) = bdiv($y1,$x10000);
1889 # make both op's numbers!
1890 $x->badd( bmul( $class->new(
1891 abs($sx*int($xr->numify()) | $sy*int($yr->numify()))),
1895 $x->bneg() if $sign;
1901 #(BINT or num_str, BINT or num_str) return BINT
1905 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1906 # objectify is costly, so avoid it
1907 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1909 ($self,$x,$y,@r) = objectify(2,@_);
1912 return $x if $x->modify('bxor');
1913 $r[3] = $y; # no push!
1915 local $Math::BigInt::upgrade = undef;
1917 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1918 return $x->round(@r) if $y->is_zero();
1920 my $sign = 0; # sign of result
1921 $sign = 1 if $x->{sign} ne $y->{sign};
1922 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1923 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1925 # don't use lib for negative values
1926 if ($CALC->can('_xor') && $sx == 1 && $sy == 1)
1928 $x->{value} = $CALC->_xor($x->{value},$y->{value});
1929 return $x->round(@r);
1932 my $m = $self->bone(); my ($xr,$yr);
1933 my $x10000 = $self->new(0x10000);
1934 my $y1 = copy(ref($x),$y); # make copy
1935 $y1->babs(); # and positive
1936 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1937 use integer; # need this for negative bools
1938 while (!$x1->is_zero() || !$y1->is_zero())
1940 ($x1, $xr) = bdiv($x1, $x10000);
1941 ($y1, $yr) = bdiv($y1, $x10000);
1942 # make both op's numbers!
1943 $x->badd( bmul( $class->new(
1944 abs($sx*int($xr->numify()) ^ $sy*int($yr->numify()))),
1948 $x->bneg() if $sign;
1954 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1956 my $e = $CALC->_len($x->{value});
1957 return wantarray ? ($e,0) : $e;
1962 # return the nth decimal digit, negative values count backward, 0 is right
1963 my ($self,$x,$n) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1965 $CALC->_digit($x->{value},$n||0);
1970 # return the amount of trailing zeros in $x
1972 $x = $class->new($x) unless ref $x;
1974 return 0 if $x->is_zero() || $x->is_odd() || $x->{sign} !~ /^[+-]$/;
1976 return $CALC->_zeros($x->{value}) if $CALC->can('_zeros');
1978 # if not: since we do not know underlying internal representation:
1979 my $es = "$x"; $es =~ /([0]*)$/;
1980 return 0 if !defined $1; # no zeros
1981 CORE::length("$1"); # as string, not as +0!
1986 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1988 return $x if $x->modify('bsqrt');
1990 return $x->bnan() if $x->{sign} ne '+'; # -x or inf or NaN => NaN
1991 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
1992 return $x->round(@r) if $x->is_one(); # 1 => 1
1994 return $upgrade->bsqrt($x,@r) if defined $upgrade;
1996 if ($CALC->can('_sqrt'))
1998 $x->{value} = $CALC->_sqrt($x->{value});
1999 return $x->round(@r);
2002 return $x->bone('+',@r) if $x < 4; # 2,3 => 1
2004 my $l = int($x->length()/2);
2006 $x->bone(); # keep ref($x), but modify it
2009 my $last = $self->bzero();
2010 my $two = $self->new(2);
2011 my $lastlast = $x+$two;
2012 while ($last != $x && $lastlast != $x)
2014 $lastlast = $last; $last = $x;
2018 $x-- if $x * $x > $y; # overshot?
2024 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
2025 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2027 if ($x->{sign} !~ /^[+-]$/)
2029 my $s = $x->{sign}; $s =~ s/^[+-]//;
2030 return $self->new($s); # -inf,+inf => inf
2032 my $e = $class->bzero();
2033 return $e->binc() if $x->is_zero();
2034 $e += $x->_trailing_zeros();
2040 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
2041 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2043 if ($x->{sign} !~ /^[+-]$/)
2045 return $self->new($x->{sign}); # keep + or - sign
2048 # that's inefficient
2049 my $zeros = $m->_trailing_zeros();
2050 $m->brsft($zeros,10) if $zeros != 0;
2056 # return a copy of both the exponent and the mantissa
2057 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2059 return ($x->mantissa(),$x->exponent());
2062 ##############################################################################
2063 # rounding functions
2067 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
2068 # $n == 0 || $n == 1 => round to integer
2069 my $x = shift; $x = $class->new($x) unless ref $x;
2070 my ($scale,$mode) = $x->_scale_p($x->precision(),$x->round_mode(),@_);
2071 return $x if !defined $scale; # no-op
2072 return $x if $x->modify('bfround');
2074 # no-op for BigInts if $n <= 0
2077 $x->{_a} = undef; # clear an eventual set A
2078 $x->{_p} = $scale; return $x;
2081 $x->bround( $x->length()-$scale, $mode);
2082 $x->{_a} = undef; # bround sets {_a}
2083 $x->{_p} = $scale; # so correct it
2087 sub _scan_for_nonzero
2093 my $len = $x->length();
2094 return 0 if $len == 1; # '5' is trailed by invisible zeros
2095 my $follow = $pad - 1;
2096 return 0 if $follow > $len || $follow < 1;
2098 # since we do not know underlying represention of $x, use decimal string
2099 #my $r = substr ($$xs,-$follow);
2100 my $r = substr ("$x",-$follow);
2101 return 1 if $r =~ /[^0]/;
2107 # to make life easier for switch between MBF and MBI (autoload fxxx()
2108 # like MBF does for bxxx()?)
2110 return $x->bround(@_);
2115 # accuracy: +$n preserve $n digits from left,
2116 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
2118 # and overwrite the rest with 0's, return normalized number
2119 # do not return $x->bnorm(), but $x
2121 my $x = shift; $x = $class->new($x) unless ref $x;
2122 my ($scale,$mode) = $x->_scale_a($x->accuracy(),$x->round_mode(),@_);
2123 return $x if !defined $scale; # no-op
2124 return $x if $x->modify('bround');
2126 if ($x->is_zero() || $scale == 0)
2128 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2131 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2133 # we have fewer digits than we want to scale to
2134 my $len = $x->length();
2135 # scale < 0, but > -len (not >=!)
2136 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2138 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2142 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2143 my ($pad,$digit_round,$digit_after);
2144 $pad = $len - $scale;
2145 $pad = abs($scale-1) if $scale < 0;
2147 # do not use digit(), it is costly for binary => decimal
2149 my $xs = $CALC->_str($x->{value});
2152 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2153 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2154 $digit_round = '0'; $digit_round = substr($$xs,$pl,1) if $pad <= $len;
2155 $pl++; $pl ++ if $pad >= $len;
2156 $digit_after = '0'; $digit_after = substr($$xs,$pl,1) if $pad > 0;
2158 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2159 # closer at the remaining digits of the original $x, remember decision
2160 my $round_up = 1; # default round up
2162 ($mode eq 'trunc') || # trunc by round down
2163 ($digit_after =~ /[01234]/) || # round down anyway,
2165 ($digit_after eq '5') && # not 5000...0000
2166 ($x->_scan_for_nonzero($pad,$xs) == 0) &&
2168 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2169 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2170 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2171 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2172 ($mode eq 'zero') # round down if zero, sign adjusted below
2174 my $put_back = 0; # not yet modified
2176 if (($pad > 0) && ($pad <= $len))
2178 substr($$xs,-$pad,$pad) = '0' x $pad;
2183 $x->bzero(); # round to '0'
2186 if ($round_up) # what gave test above?
2189 $pad = $len, $$xs = '0'x$pad if $scale < 0; # tlr: whack 0.51=>1.0
2191 # we modify directly the string variant instead of creating a number and
2192 # adding it, since that is faster (we already have the string)
2193 my $c = 0; $pad ++; # for $pad == $len case
2194 while ($pad <= $len)
2196 $c = substr($$xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2197 substr($$xs,-$pad,1) = $c; $pad++;
2198 last if $c != 0; # no overflow => early out
2200 $$xs = '1'.$$xs if $c == 0;
2203 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back in if needed
2205 $x->{_a} = $scale if $scale >= 0;
2208 $x->{_a} = $len+$scale;
2209 $x->{_a} = 0 if $scale < -$len;
2216 # return integer less or equal then number, since it is already integer,
2217 # always returns $self
2218 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2225 # return integer greater or equal then number, since it is already integer,
2226 # always returns $self
2227 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2232 ##############################################################################
2233 # private stuff (internal use only)
2237 # internal speedup, set argument to 1, or create a +/- 1
2239 my $x = $self->bone(); # $x->{value} = $CALC->_one();
2240 $x->{sign} = shift || '+';
2246 # Overload will swap params if first one is no object ref so that the first
2247 # one is always an object ref. In this case, third param is true.
2248 # This routine is to overcome the effect of scalar,$object creating an object
2249 # of the class of this package, instead of the second param $object. This
2250 # happens inside overload, when the overload section of this package is
2251 # inherited by sub classes.
2252 # For overload cases (and this is used only there), we need to preserve the
2253 # args, hence the copy().
2254 # You can override this method in a subclass, the overload section will call
2255 # $object->_swap() to make sure it arrives at the proper subclass, with some
2256 # exceptions like '+' and '-'. To make '+' and '-' work, you also need to
2257 # specify your own overload for them.
2259 # object, (object|scalar) => preserve first and make copy
2260 # scalar, object => swapped, re-swap and create new from first
2261 # (using class of second object, not $class!!)
2262 my $self = shift; # for override in subclass
2265 my $c = ref ($_[0]) || $class; # fallback $class should not happen
2266 return ( $c->new($_[1]), $_[0] );
2268 return ( $_[0]->copy(), $_[1] );
2273 # check for strings, if yes, return objects instead
2275 # the first argument is number of args objectify() should look at it will
2276 # return $count+1 elements, the first will be a classname. This is because
2277 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2278 # useless objects beeing created and thrown away. So we cannot simple loop
2279 # over @_. If the given count is 0, all arguments will be used.
2281 # If the second arg is a ref, use it as class.
2282 # If not, try to use it as classname, unless undef, then use $class
2283 # (aka Math::BigInt). The latter shouldn't happen,though.
2286 # $x->badd(1); => ref x, scalar y
2287 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2288 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2289 # Math::BigInt::badd(1,2); => scalar x, scalar y
2290 # In the last case we check number of arguments to turn it silently into
2291 # $class,1,2. (We can not take '1' as class ;o)
2292 # badd($class,1) is not supported (it should, eventually, try to add undef)
2293 # currently it tries 'Math::BigInt' + 1, which will not work.
2295 # some shortcut for the common cases
2297 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2299 my $count = abs(shift || 0);
2301 my (@a,$k,$d); # resulting array, temp, and downgrade
2304 # okay, got object as first
2309 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2311 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2315 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2316 if (defined ${"$a[0]::downgrade"})
2318 $d = ${"$a[0]::downgrade"};
2319 ${"$a[0]::downgrade"} = undef;
2322 my $up = ${"$a[0]::upgrade"};
2323 # print "Now in objectify, my class is today $a[0]\n";
2331 $k = $a[0]->new($k);
2333 elsif (!defined $up && ref($k) ne $a[0])
2335 # foreign object, try to convert to integer
2336 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2349 $k = $a[0]->new($k);
2351 elsif (!defined $up && ref($k) ne $a[0])
2353 # foreign object, try to convert to integer
2354 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2358 push @a,@_; # return other params, too
2360 die "$class objectify needs list context" unless wantarray;
2361 ${"$a[0]::downgrade"} = $d;
2370 my @a; my $l = scalar @_;
2371 for ( my $i = 0; $i < $l ; $i++ )
2373 if ($_[$i] eq ':constant')
2375 # this causes overlord er load to step in
2376 overload::constant integer => sub { $self->new(shift) };
2377 overload::constant binary => sub { $self->new(shift) };
2379 elsif ($_[$i] eq 'upgrade')
2381 # this causes upgrading
2382 $upgrade = $_[$i+1]; # or undef to disable
2385 elsif ($_[$i] =~ /^lib$/i)
2387 # this causes a different low lib to take care...
2388 $CALC = $_[$i+1] || '';
2396 # any non :constant stuff is handled by our parent, Exporter
2397 # even if @_ is empty, to give it a chance
2398 $self->SUPER::import(@a); # need it for subclasses
2399 $self->export_to_level(1,$self,@a); # need it for MBF
2401 # try to load core math lib
2402 my @c = split /\s*,\s*/,$CALC;
2403 push @c,'Calc'; # if all fail, try this
2404 $CALC = ''; # signal error
2405 foreach my $lib (@c)
2407 next if ($lib || '') eq '';
2408 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2412 # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
2413 # used in the same script, or eval inside import().
2414 my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
2415 my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
2417 $file = File::Spec->catfile (@parts, $file);
2418 eval { require "$file"; $lib->import( @c ); }
2422 eval "use $lib qw/@c/;";
2424 $CALC = $lib, last if $@ eq ''; # no error in loading lib?
2426 die "Couldn't load any math lib, not even the default" if $CALC eq '';
2431 # convert a (ref to) big hex string to BigInt, return undef for error
2434 my $x = Math::BigInt->bzero();
2437 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2438 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2440 return $x->bnan() if $$hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2442 my $sign = '+'; $sign = '-' if ($$hs =~ /^-/);
2444 $$hs =~ s/^[+-]//; # strip sign
2445 if ($CALC->can('_from_hex'))
2447 $x->{value} = $CALC->_from_hex($hs);
2451 # fallback to pure perl
2452 my $mul = Math::BigInt->bzero(); $mul++;
2453 my $x65536 = Math::BigInt->new(65536);
2454 my $len = CORE::length($$hs)-2;
2455 $len = int($len/4); # 4-digit parts, w/o '0x'
2456 my $val; my $i = -4;
2459 $val = substr($$hs,$i,4);
2460 $val =~ s/^[+-]?0x// if $len == 0; # for last part only because
2461 $val = hex($val); # hex does not like wrong chars
2463 $x += $mul * $val if $val != 0;
2464 $mul *= $x65536 if $len >= 0; # skip last mul
2467 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2473 # convert a (ref to) big binary string to BigInt, return undef for error
2476 my $x = Math::BigInt->bzero();
2478 $$bs =~ s/([01])_([01])/$1$2/g;
2479 $$bs =~ s/([01])_([01])/$1$2/g;
2480 return $x->bnan() if $$bs !~ /^[+-]?0b[01]+$/;
2482 my $sign = '+'; $sign = '-' if ($$bs =~ /^\-/);
2483 $$bs =~ s/^[+-]//; # strip sign
2484 if ($CALC->can('_from_bin'))
2486 $x->{value} = $CALC->_from_bin($bs);
2490 my $mul = Math::BigInt->bzero(); $mul++;
2491 my $x256 = Math::BigInt->new(256);
2492 my $len = CORE::length($$bs)-2;
2493 $len = int($len/8); # 8-digit parts, w/o '0b'
2494 my $val; my $i = -8;
2497 $val = substr($$bs,$i,8);
2498 $val =~ s/^[+-]?0b// if $len == 0; # for last part only
2499 #$val = oct('0b'.$val); # does not work on Perl prior to 5.6.0
2501 # $val = ('0' x (8-CORE::length($val))).$val if CORE::length($val) < 8;
2502 $val = ord(pack('B8',substr('00000000'.$val,-8,8)));
2504 $x += $mul * $val if $val != 0;
2505 $mul *= $x256 if $len >= 0; # skip last mul
2508 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2514 # (ref to num_str) return num_str
2515 # internal, take apart a string and return the pieces
2516 # strip leading/trailing whitespace, leading zeros, underscore and reject
2520 # strip white space at front, also extranous leading zeros
2521 $$x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2522 $$x =~ s/^\s+//; # but this will
2523 $$x =~ s/\s+$//g; # strip white space at end
2525 # shortcut, if nothing to split, return early
2526 if ($$x =~ /^[+-]?\d+$/)
2528 $$x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2529 return (\$sign, $x, \'', \'', \0);
2532 # invalid starting char?
2533 return if $$x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2535 return __from_hex($x) if $$x =~ /^[\-\+]?0x/; # hex string
2536 return __from_bin($x) if $$x =~ /^[\-\+]?0b/; # binary string
2538 # strip underscores between digits
2539 $$x =~ s/(\d)_(\d)/$1$2/g;
2540 $$x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3
2542 # some possible inputs:
2543 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2544 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2
2546 return if $$x =~ /[Ee].*[Ee]/; # more than one E => error
2548 my ($m,$e) = split /[Ee]/,$$x;
2549 $e = '0' if !defined $e || $e eq "";
2550 # sign,value for exponent,mantint,mantfrac
2551 my ($es,$ev,$mis,$miv,$mfv);
2553 if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2557 return if $m eq '.' || $m eq '';
2558 my ($mi,$mf,$last) = split /\./,$m;
2559 return if defined $last; # last defined => 1.2.3 or others
2560 $mi = '0' if !defined $mi;
2561 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2562 $mf = '0' if !defined $mf || $mf eq '';
2563 if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2565 $mis = $1||'+'; $miv = $2;
2566 return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
2568 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2571 return; # NaN, not a number
2576 # an object might be asked to return itself as bigint on certain overloaded
2577 # operations, this does exactly this, so that sub classes can simple inherit
2578 # it or override with their own integer conversion routine
2586 # return as hex string, with prefixed 0x
2587 my $x = shift; $x = $class->new($x) if !ref($x);
2589 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2590 return '0x0' if $x->is_zero();
2592 my $es = ''; my $s = '';
2593 $s = $x->{sign} if $x->{sign} eq '-';
2594 if ($CALC->can('_as_hex'))
2596 $es = ${$CALC->_as_hex($x->{value})};
2600 my $x1 = $x->copy()->babs(); my ($xr,$x10000,$h);
2603 $x10000 = Math::BigInt->new (0x10000); $h = 'h4';
2607 $x10000 = Math::BigInt->new (0x1000); $h = 'h3';
2609 while (!$x1->is_zero())
2611 ($x1, $xr) = bdiv($x1,$x10000);
2612 $es .= unpack($h,pack('v',$xr->numify()));
2615 $es =~ s/^[0]+//; # strip leading zeros
2623 # return as binary string, with prefixed 0b
2624 my $x = shift; $x = $class->new($x) if !ref($x);
2626 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2627 return '0b0' if $x->is_zero();
2629 my $es = ''; my $s = '';
2630 $s = $x->{sign} if $x->{sign} eq '-';
2631 if ($CALC->can('_as_bin'))
2633 $es = ${$CALC->_as_bin($x->{value})};
2637 my $x1 = $x->copy()->babs(); my ($xr,$x10000,$b);
2640 $x10000 = Math::BigInt->new (0x10000); $b = 'b16';
2644 $x10000 = Math::BigInt->new (0x1000); $b = 'b12';
2646 while (!$x1->is_zero())
2648 ($x1, $xr) = bdiv($x1,$x10000);
2649 $es .= unpack($b,pack('v',$xr->numify()));
2652 $es =~ s/^[0]+//; # strip leading zeros
2658 ##############################################################################
2659 # internal calculation routines (others are in Math::BigInt::Calc etc)
2663 # (BINT or num_str, BINT or num_str) return BINT
2664 # does modify first argument
2667 my $x = shift; my $ty = shift;
2668 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2669 return $x * $ty / bgcd($x,$ty);
2674 # (BINT or num_str, BINT or num_str) return BINT
2675 # does modify both arguments
2676 # GCD -- Euclids algorithm E, Knuth Vol 2 pg 296
2679 return $x->bnan() if $x->{sign} !~ /^[+-]$/ || $ty->{sign} !~ /^[+-]$/;
2681 while (!$ty->is_zero())
2683 ($x, $ty) = ($ty,bmod($x,$ty));
2688 ###############################################################################
2689 # this method return 0 if the object can be modified, or 1 for not
2690 # We use a fast use constant statement here, to avoid costly calls. Subclasses
2691 # may override it with special code (f.i. Math::BigInt::Constant does so)
2693 sub modify () { 0; }
2700 Math::BigInt - Arbitrary size integer math package
2707 $x = Math::BigInt->new($str); # defaults to 0
2708 $nan = Math::BigInt->bnan(); # create a NotANumber
2709 $zero = Math::BigInt->bzero(); # create a +0
2710 $inf = Math::BigInt->binf(); # create a +inf
2711 $inf = Math::BigInt->binf('-'); # create a -inf
2712 $one = Math::BigInt->bone(); # create a +1
2713 $one = Math::BigInt->bone('-'); # create a -1
2716 $x->is_zero(); # true if arg is +0
2717 $x->is_nan(); # true if arg is NaN
2718 $x->is_one(); # true if arg is +1
2719 $x->is_one('-'); # true if arg is -1
2720 $x->is_odd(); # true if odd, false for even
2721 $x->is_even(); # true if even, false for odd
2722 $x->is_positive(); # true if >= 0
2723 $x->is_negative(); # true if < 0
2724 $x->is_inf(sign); # true if +inf, or -inf (sign is default '+')
2725 $x->is_int(); # true if $x is an integer (not a float)
2727 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2728 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2729 $x->sign(); # return the sign, either +,- or NaN
2730 $x->digit($n); # return the nth digit, counting from right
2731 $x->digit(-$n); # return the nth digit, counting from left
2733 # The following all modify their first argument:
2736 $x->bzero(); # set $x to 0
2737 $x->bnan(); # set $x to NaN
2738 $x->bone(); # set $x to +1
2739 $x->bone('-'); # set $x to -1
2740 $x->binf(); # set $x to inf
2741 $x->binf('-'); # set $x to -inf
2743 $x->bneg(); # negation
2744 $x->babs(); # absolute value
2745 $x->bnorm(); # normalize (no-op)
2746 $x->bnot(); # two's complement (bit wise not)
2747 $x->binc(); # increment x by 1
2748 $x->bdec(); # decrement x by 1
2750 $x->badd($y); # addition (add $y to $x)
2751 $x->bsub($y); # subtraction (subtract $y from $x)
2752 $x->bmul($y); # multiplication (multiply $x by $y)
2753 $x->bdiv($y); # divide, set $x to quotient
2754 # return (quo,rem) or quo if scalar
2756 $x->bmod($y); # modulus (x % y)
2757 $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
2758 $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
2760 $x->bpow($y); # power of arguments (x ** y)
2761 $x->blsft($y); # left shift
2762 $x->brsft($y); # right shift
2763 $x->blsft($y,$n); # left shift, by base $n (like 10)
2764 $x->brsft($y,$n); # right shift, by base $n (like 10)
2766 $x->band($y); # bitwise and
2767 $x->bior($y); # bitwise inclusive or
2768 $x->bxor($y); # bitwise exclusive or
2769 $x->bnot(); # bitwise not (two's complement)
2771 $x->bsqrt(); # calculate square-root
2772 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2774 $x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r
2775 $x->bround($N); # accuracy: preserve $N digits
2776 $x->bfround($N); # round to $Nth digit, no-op for BigInts
2778 # The following do not modify their arguments in BigInt, but do in BigFloat:
2779 $x->bfloor(); # return integer less or equal than $x
2780 $x->bceil(); # return integer greater or equal than $x
2782 # The following do not modify their arguments:
2784 bgcd(@values); # greatest common divisor (no OO style)
2785 blcm(@values); # lowest common multiplicator (no OO style)
2787 $x->length(); # return number of digits in number
2788 ($x,$f) = $x->length(); # length of number and length of fraction part,
2789 # latter is always 0 digits long for BigInt's
2791 $x->exponent(); # return exponent as BigInt
2792 $x->mantissa(); # return (signed) mantissa as BigInt
2793 $x->parts(); # return (mantissa,exponent) as BigInt
2794 $x->copy(); # make a true copy of $x (unlike $y = $x;)
2795 $x->as_number(); # return as BigInt (in BigInt: same as copy())
2797 # conversation to string
2798 $x->bstr(); # normalized string
2799 $x->bsstr(); # normalized string in scientific notation
2800 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
2801 $x->as_bin(); # as signed binary string with prefixed 0b
2803 Math::BigInt->config(); # return hash containing configuration/version
2805 # precision and accuracy (see section about rounding for more)
2806 $x->precision(); # return P of $x (or global, if P of $x undef)
2807 $x->precision($n); # set P of $x to $n
2808 $x->accuracy(); # return A of $x (or global, if A of $x undef)
2809 $x->accuracy($n); # set A $x to $n
2811 Math::BigInt->precision(); # get/set global P for all BigInt objects
2812 Math::BigInt->accuracy(); # get/set global A for all BigInt objects
2816 All operators (inlcuding basic math operations) are overloaded if you
2817 declare your big integers as
2819 $i = new Math::BigInt '123_456_789_123_456_789';
2821 Operations with overloaded operators preserve the arguments which is
2822 exactly what you expect.
2826 =item Canonical notation
2828 Big integer values are strings of the form C</^[+-]\d+$/> with leading
2831 '-0' canonical value '-0', normalized '0'
2832 ' -123_123_123' canonical value '-123123123'
2833 '1_23_456_7890' canonical value '1234567890'
2837 Input values to these routines may be either Math::BigInt objects or
2838 strings of the form C</^[+-]?[\d]+\.?[\d]*E?[+-]?[\d]*$/>.
2840 You can include one underscore between any two digits. The input string may
2841 have leading and trailing whitespace, which will be ignored. In later
2842 versions, a more strict (no whitespace at all) or more lax (whitespace
2843 allowed everywhere) input checking will also be possible.
2845 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
2846 Non integer values result in NaN.
2848 Math::BigInt::new() defaults to 0, while Math::BigInt::new('') results
2851 bnorm() on a BigInt object is now effectively a no-op, since the numbers
2852 are always stored in normalized form. On a string, it creates a BigInt
2857 Output values are BigInt objects (normalized), except for bstr(), which
2858 returns a string in normalized form.
2859 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
2860 C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
2861 return either undef, <0, 0 or >0 and are suited for sort.
2867 Each of the methods below accepts three additional parameters. These arguments
2868 $A, $P and $R are accuracy, precision and round_mode. Please see more in the
2869 section about ACCURACY and ROUNDIND.
2875 print Dumper ( Math::BigInt->config() );
2877 Returns a hash containing the configuration, e.g. the version number, lib
2882 $x->accuracy(5); # local for $x
2883 $class->accuracy(5); # global for all members of $class
2885 Set or get the global or local accuracy, aka how many significant digits the
2886 results have. Please see the section about L<ACCURACY AND PRECISION> for
2889 Value must be greater than zero. Pass an undef value to disable it:
2891 $x->accuracy(undef);
2892 Math::BigInt->accuracy(undef);
2894 Returns the current accuracy. For C<$x->accuracy()> it will return either the
2895 local accuracy, or if not defined, the global. This means the return value
2896 represents the accuracy that will be in effect for $x:
2898 $y = Math::BigInt->new(1234567); # unrounded
2899 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
2900 $x = Math::BigInt->new(123456); # will be automatically rounded
2901 print "$x $y\n"; # '123500 1234567'
2902 print $x->accuracy(),"\n"; # will be 4
2903 print $y->accuracy(),"\n"; # also 4, since global is 4
2904 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
2905 print $x->accuracy(),"\n"; # still 4
2906 print $y->accuracy(),"\n"; # 5, since global is 5
2912 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
2913 2, but others work, too.
2915 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
2919 $x = Math::BigInt->new(10);
2920 $x->brsft(1); # same as $x >> 1: 5
2921 $x = Math::BigInt->new(1234);
2922 $x->brsft(2,10); # result 12
2924 There is one exception, and that is base 2 with negative $x:
2927 $x = Math::BigInt->new(-5);
2930 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
2935 $x = Math::BigInt->new($str,$A,$P,$R);
2937 Creates a new BigInt object from a string or another BigInt object. The
2938 input is accepted as decimal, hex (with leading '0x') or binary (with leading
2943 $x = Math::BigInt->bnan();
2945 Creates a new BigInt object representing NaN (Not A Number).
2946 If used on an object, it will set it to NaN:
2952 $x = Math::BigInt->bzero();
2954 Creates a new BigInt object representing zero.
2955 If used on an object, it will set it to zero:
2961 $x = Math::BigInt->binf($sign);
2963 Creates a new BigInt object representing infinity. The optional argument is
2964 either '-' or '+', indicating whether you want infinity or minus infinity.
2965 If used on an object, it will set it to infinity:
2972 $x = Math::BigInt->binf($sign);
2974 Creates a new BigInt object representing one. The optional argument is
2975 either '-' or '+', indicating whether you want one or minus one.
2976 If used on an object, it will set it to one:
2981 =head2 is_one()/is_zero()/is_nan()/is_inf()
2984 $x->is_zero(); # true if arg is +0
2985 $x->is_nan(); # true if arg is NaN
2986 $x->is_one(); # true if arg is +1
2987 $x->is_one('-'); # true if arg is -1
2988 $x->is_inf(); # true if +inf
2989 $x->is_inf('-'); # true if -inf (sign is default '+')
2991 These methods all test the BigInt for beeing one specific value and return
2992 true or false depending on the input. These are faster than doing something
2997 =head2 is_positive()/is_negative()
2999 $x->is_positive(); # true if >= 0
3000 $x->is_negative(); # true if < 0
3002 The methods return true if the argument is positive or negative, respectively.
3003 C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
3004 C<-inf> is negative. A C<zero> is positive.
3006 These methods are only testing the sign, and not the value.
3008 =head2 is_odd()/is_even()/is_int()
3010 $x->is_odd(); # true if odd, false for even
3011 $x->is_even(); # true if even, false for odd
3012 $x->is_int(); # true if $x is an integer
3014 The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
3015 C<-inf> are not integers and are neither odd nor even.
3021 Compares $x with $y and takes the sign into account.
3022 Returns -1, 0, 1 or undef.
3028 Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
3034 Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
3038 $x->digit($n); # return the nth digit, counting from right
3044 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
3045 and '-inf', respectively. Does nothing for NaN or zero.
3051 Set the number to it's absolute value, e.g. change the sign from '-' to '+'
3052 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
3057 $x->bnorm(); # normalize (no-op)
3061 $x->bnot(); # two's complement (bit wise not)
3065 $x->binc(); # increment x by 1
3069 $x->bdec(); # decrement x by 1
3073 $x->badd($y); # addition (add $y to $x)
3077 $x->bsub($y); # subtraction (subtract $y from $x)
3081 $x->bmul($y); # multiplication (multiply $x by $y)
3085 $x->bdiv($y); # divide, set $x to quotient
3086 # return (quo,rem) or quo if scalar
3090 $x->bmod($y); # modulus (x % y)
3094 $num->bmodinv($mod); # modular inverse
3096 Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
3097 returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
3098 C<bgcd($num, $mod)==1>.
3102 $num->bmodpow($exp,$mod); # modular exponentation ($num**$exp % $mod)
3104 Returns the value of C<$num> taken to the power C<$exp> in the modulus
3105 C<$mod> using binary exponentation. C<bmodpow> is far superior to
3110 because C<bmodpow> is much faster--it reduces internal variables into
3111 the modulus whenever possible, so it operates on smaller numbers.
3113 C<bmodpow> also supports negative exponents.
3115 bmodpow($num, -1, $mod)
3117 is exactly equivalent to
3123 $x->bpow($y); # power of arguments (x ** y)
3127 $x->blsft($y); # left shift
3128 $x->blsft($y,$n); # left shift, by base $n (like 10)
3132 $x->brsft($y); # right shift
3133 $x->brsft($y,$n); # right shift, by base $n (like 10)
3137 $x->band($y); # bitwise and
3141 $x->bior($y); # bitwise inclusive or
3145 $x->bxor($y); # bitwise exclusive or
3149 $x->bnot(); # bitwise not (two's complement)
3153 $x->bsqrt(); # calculate square-root
3157 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3161 $x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r
3165 $x->bround($N); # accuracy: preserve $N digits
3169 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3175 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3176 does change $x in BigFloat.
3182 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3183 does change $x in BigFloat.
3187 bgcd(@values); # greatest common divisor (no OO style)
3191 blcm(@values); # lowest common multiplicator (no OO style)
3196 ($xl,$fl) = $x->length();
3198 Returns the number of digits in the decimal representation of the number.
3199 In list context, returns the length of the integer and fraction part. For
3200 BigInt's, the length of the fraction part will always be 0.
3206 Return the exponent of $x as BigInt.
3212 Return the signed mantissa of $x as BigInt.
3216 $x->parts(); # return (mantissa,exponent) as BigInt
3220 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3224 $x->as_number(); # return as BigInt (in BigInt: same as copy())
3228 $x->bstr(); # normalized string
3232 $x->bsstr(); # normalized string in scientific notation
3236 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3240 $x->as_bin(); # as signed binary string with prefixed 0b
3242 =head1 ACCURACY and PRECISION
3244 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3245 accuracy and precision based rounding, both automatically after every
3246 operation as well as manually.
3248 This section describes the accuracy/precision handling in Math::Big* as it
3249 used to be and as it is now, complete with an explanation of all terms and
3252 Not yet implemented things (but with correct description) are marked with '!',
3253 things that need to be answered are marked with '?'.
3255 In the next paragraph follows a short description of terms used here (because
3256 these may differ from terms used by others people or documentation).
3258 During the rest of this document, the shortcuts A (for accuracy), P (for
3259 precision), F (fallback) and R (rounding mode) will be used.
3263 A fixed number of digits before (positive) or after (negative)
3264 the decimal point. For example, 123.45 has a precision of -2. 0 means an
3265 integer like 123 (or 120). A precision of 2 means two digits to the left
3266 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3267 numbers with zeros before the decimal point may have different precisions,
3268 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3269 was). It could also have p < 0, when the digits after the decimal point
3272 The string output (of floating point numbers) will be padded with zeros:
3274 Initial value P A Result String
3275 ------------------------------------------------------------
3276 1234.01 -3 1000 1000
3279 1234.001 1 1234 1234.0
3281 1234.01 2 1234.01 1234.01
3282 1234.01 5 1234.01 1234.01000
3284 For BigInts, no padding occurs.
3288 Number of significant digits. Leading zeros are not counted. A
3289 number may have an accuracy greater than the non-zero digits
3290 when there are zeros in it or trailing zeros. For example, 123.456 has
3291 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3293 The string output (of floating point numbers) will be padded with zeros:
3295 Initial value P A Result String
3296 ------------------------------------------------------------
3298 1234.01 6 1234.01 1234.01
3299 1234.1 8 1234.1 1234.1000
3301 For BigInts, no padding occurs.
3305 When both A and P are undefined, this is used as a fallback accuracy when
3308 =head2 Rounding mode R
3310 When rounding a number, different 'styles' or 'kinds'
3311 of rounding are possible. (Note that random rounding, as in
3312 Math::Round, is not implemented.)
3318 truncation invariably removes all digits following the
3319 rounding place, replacing them with zeros. Thus, 987.65 rounded
3320 to tens (P=1) becomes 980, and rounded to the fourth sigdig
3321 becomes 987.6 (A=4). 123.456 rounded to the second place after the
3322 decimal point (P=-2) becomes 123.46.
3324 All other implemented styles of rounding attempt to round to the
3325 "nearest digit." If the digit D immediately to the right of the
3326 rounding place (skipping the decimal point) is greater than 5, the
3327 number is incremented at the rounding place (possibly causing a
3328 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3329 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3330 truncated at the rounding place: e.g. when rounding to units, 0.4
3331 rounds to 0, and -19.4 rounds to -19.
3333 However the results of other styles of rounding differ if the
3334 digit immediately to the right of the rounding place (skipping the
3335 decimal point) is 5 and if there are no digits, or no digits other
3336 than 0, after that 5. In such cases:
3340 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3341 if it is not already. E.g., when rounding to the first sigdig, 0.45
3342 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3346 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3347 it is not already. E.g., when rounding to the first sigdig, 0.45
3348 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3352 round to plus infinity, i.e. always round up. E.g., when
3353 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3354 and 0.4501 also becomes 0.5.
3358 round to minus infinity, i.e. always round down. E.g., when
3359 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3360 but 0.4501 becomes 0.5.
3364 round to zero, i.e. positive numbers down, negative ones up.
3365 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3366 becomes -0.5, but 0.4501 becomes 0.5.
3370 The handling of A & P in MBI/MBF (the old core code shipped with Perl
3371 versions <= 5.7.2) is like this:
3377 * ffround($p) is able to round to $p number of digits after the decimal
3379 * otherwise P is unused
3381 =item Accuracy (significant digits)
3383 * fround($a) rounds to $a significant digits
3384 * only fdiv() and fsqrt() take A as (optional) paramater
3385 + other operations simply create the same number (fneg etc), or more (fmul)
3387 + rounding/truncating is only done when explicitly calling one of fround
3388 or ffround, and never for BigInt (not implemented)
3389 * fsqrt() simply hands its accuracy argument over to fdiv.
3390 * the documentation and the comment in the code indicate two different ways
3391 on how fdiv() determines the maximum number of digits it should calculate,
3392 and the actual code does yet another thing
3394 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3396 result has at most max(scale, length(dividend), length(divisor)) digits
3398 scale = max(scale, length(dividend)-1,length(divisor)-1);
3399 scale += length(divisior) - length(dividend);
3400 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3401 Actually, the 'difference' added to the scale is calculated from the
3402 number of "significant digits" in dividend and divisor, which is derived
3403 by looking at the length of the mantissa. Which is wrong, since it includes
3404 the + sign (oups) and actually gets 2 for '+100' and 4 for '+101'. Oups
3405 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3406 assumption that 124 has 3 significant digits, while 120/7 will get you
3407 '17', not '17.1' since 120 is thought to have 2 significant digits.
3408 The rounding after the division then uses the remainder and $y to determine
3409 wether it must round up or down.
3410 ? I have no idea which is the right way. That's why I used a slightly more
3411 ? simple scheme and tweaked the few failing testcases to match it.
3415 This is how it works now:
3419 =item Setting/Accessing
3421 * You can set the A global via Math::BigInt->accuracy() or
3422 Math::BigFloat->accuracy() or whatever class you are using.
3423 * You can also set P globally by using Math::SomeClass->precision() likewise.
3424 * Globals are classwide, and not inherited by subclasses.
3425 * to undefine A, use Math::SomeCLass->accuracy(undef);
3426 * to undefine P, use Math::SomeClass->precision(undef);
3427 * Setting Math::SomeClass->accuracy() clears automatically
3428 Math::SomeClass->precision(), and vice versa.
3429 * To be valid, A must be > 0, P can have any value.
3430 * If P is negative, this means round to the P'th place to the right of the
3431 decimal point; positive values mean to the left of the decimal point.
3432 P of 0 means round to integer.
3433 * to find out the current global A, take Math::SomeClass->accuracy()
3434 * to find out the current global P, take Math::SomeClass->precision()
3435 * use $x->accuracy() respective $x->precision() for the local setting of $x.
3436 * Please note that $x->accuracy() respecive $x->precision() fall back to the
3437 defined globals, when $x's A or P is not set.
3439 =item Creating numbers
3441 * When you create a number, you can give it's desired A or P via:
3442 $x = Math::BigInt->new($number,$A,$P);
3443 * Only one of A or P can be defined, otherwise the result is NaN
3444 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
3445 globals (if set) will be used. Thus changing the global defaults later on
3446 will not change the A or P of previously created numbers (i.e., A and P of
3447 $x will be what was in effect when $x was created)
3448 * If given undef for A and P, B<no> rounding will occur, and the globals will
3449 B<not> be used. This is used by subclasses to create numbers without
3450 suffering rounding in the parent. Thus a subclass is able to have it's own
3451 globals enforced upon creation of a number by using
3452 $x = Math::BigInt->new($number,undef,undef):
3454 use Math::Bigint::SomeSubclass;
3457 Math::BigInt->accuracy(2);
3458 Math::BigInt::SomeSubClass->accuracy(3);
3459 $x = Math::BigInt::SomeSubClass->new(1234);
3461 $x is now 1230, and not 1200. A subclass might choose to implement
3462 this otherwise, e.g. falling back to the parent's A and P.
3466 * If A or P are enabled/defined, they are used to round the result of each
3467 operation according to the rules below
3468 * Negative P is ignored in Math::BigInt, since BigInts never have digits
3469 after the decimal point
3470 * Math::BigFloat uses Math::BigInts internally, but setting A or P inside
3471 Math::BigInt as globals should not tamper with the parts of a BigFloat.
3472 Thus a flag is used to mark all Math::BigFloat numbers as 'never round'
3476 * It only makes sense that a number has only one of A or P at a time.
3477 Since you can set/get both A and P, there is a rule that will practically
3478 enforce only A or P to be in effect at a time, even if both are set.
3479 This is called precedence.
3480 * If two objects are involved in an operation, and one of them has A in
3481 effect, and the other P, this results in an error (NaN).
3482 * A takes precendence over P (Hint: A comes before P). If A is defined, it
3483 is used, otherwise P is used. If neither of them is defined, nothing is
3484 used, i.e. the result will have as many digits as it can (with an
3485 exception for fdiv/fsqrt) and will not be rounded.
3486 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
3487 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
3488 If either the dividend's or the divisor's mantissa has more digits than
3489 the value of F, the higher value will be used instead of F.
3490 This is to limit the digits (A) of the result (just consider what would
3491 happen with unlimited A and P in the case of 1/3 :-)
3492 * fdiv will calculate (at least) 4 more digits than required (determined by
3493 A, P or F), and, if F is not used, round the result
3494 (this will still fail in the case of a result like 0.12345000000001 with A
3495 or P of 5, but this can not be helped - or can it?)
3496 * Thus you can have the math done by on Math::Big* class in three modes:
3497 + never round (this is the default):
3498 This is done by setting A and P to undef. No math operation
3499 will round the result, with fdiv() and fsqrt() as exceptions to guard
3500 against overflows. You must explicitely call bround(), bfround() or
3501 round() (the latter with parameters).
3502 Note: Once you have rounded a number, the settings will 'stick' on it
3503 and 'infect' all other numbers engaged in math operations with it, since
3504 local settings have the highest precedence. So, to get SaferRound[tm],
3505 use a copy() before rounding like this:
3507 $x = Math::BigFloat->new(12.34);
3508 $y = Math::BigFloat->new(98.76);
3509 $z = $x * $y; # 1218.6984
3510 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
3511 $z = $x * $y; # still 1218.6984, without
3512 # copy would have been 1210!
3514 + round after each op:
3515 After each single operation (except for testing like is_zero()), the
3516 method round() is called and the result is rounded appropriately. By
3517 setting proper values for A and P, you can have all-the-same-A or
3518 all-the-same-P modes. For example, Math::Currency might set A to undef,
3519 and P to -2, globally.
3521 ?Maybe an extra option that forbids local A & P settings would be in order,
3522 ?so that intermediate rounding does not 'poison' further math?
3524 =item Overriding globals
3526 * you will be able to give A, P and R as an argument to all the calculation
3527 routines; the second parameter is A, the third one is P, and the fourth is
3528 R (shift right by one for binary operations like badd). P is used only if
3529 the first parameter (A) is undefined. These three parameters override the
3530 globals in the order detailed as follows, i.e. the first defined value
3532 (local: per object, global: global default, parameter: argument to sub)
3535 + local A (if defined on both of the operands: smaller one is taken)
3536 + local P (if defined on both of the operands: bigger one is taken)
3540 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
3541 arguments (A and P) instead of one
3543 =item Local settings
3545 * You can set A and P locally by using $x->accuracy() and $x->precision()
3546 and thus force different A and P for different objects/numbers.
3547 * Setting A or P this way immediately rounds $x to the new value.
3548 * $x->accuracy() clears $x->precision(), and vice versa.
3552 * the rounding routines will use the respective global or local settings.
3553 fround()/bround() is for accuracy rounding, while ffround()/bfround()
3555 * the two rounding functions take as the second parameter one of the
3556 following rounding modes (R):
3557 'even', 'odd', '+inf', '-inf', 'zero', 'trunc'
3558 * you can set and get the global R by using Math::SomeClass->round_mode()
3559 or by setting $Math::SomeClass::round_mode
3560 * after each operation, $result->round() is called, and the result may
3561 eventually be rounded (that is, if A or P were set either locally,
3562 globally or as parameter to the operation)
3563 * to manually round a number, call $x->round($A,$P,$round_mode);
3564 this will round the number by using the appropriate rounding function
3565 and then normalize it.
3566 * rounding modifies the local settings of the number:
3568 $x = Math::BigFloat->new(123.456);
3572 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
3573 will be 4 from now on.
3575 =item Default values
3584 * The defaults are set up so that the new code gives the same results as
3585 the old code (except in a few cases on fdiv):
3586 + Both A and P are undefined and thus will not be used for rounding
3587 after each operation.
3588 + round() is thus a no-op, unless given extra parameters A and P
3594 The actual numbers are stored as unsigned big integers (with seperate sign).
3595 You should neither care about nor depend on the internal representation; it
3596 might change without notice. Use only method calls like C<< $x->sign(); >>
3597 instead relying on the internal hash keys like in C<< $x->{sign}; >>.
3601 Math with the numbers is done (by default) by a module called
3602 Math::BigInt::Calc. This is equivalent to saying:
3604 use Math::BigInt lib => 'Calc';
3606 You can change this by using:
3608 use Math::BigInt lib => 'BitVect';
3610 The following would first try to find Math::BigInt::Foo, then
3611 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
3613 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
3615 Calc.pm uses as internal format an array of elements of some decimal base
3616 (usually 1e5 or 1e7) with the least significant digit first, while BitVect.pm
3617 uses a bit vector of base 2, most significant bit first. Other modules might
3618 use even different means of representing the numbers. See the respective
3619 module documentation for further details.
3623 The sign is either '+', '-', 'NaN', '+inf' or '-inf' and stored seperately.
3625 A sign of 'NaN' is used to represent the result when input arguments are not
3626 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
3627 minus infinity. You will get '+inf' when dividing a positive number by 0, and
3628 '-inf' when dividing any negative number by 0.
3630 =head2 mantissa(), exponent() and parts()
3632 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
3635 $m = $x->mantissa();
3636 $e = $x->exponent();
3637 $y = $m * ( 10 ** $e );
3638 print "ok\n" if $x == $y;
3640 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
3641 in one go. Both the returned mantissa and exponent have a sign.
3643 Currently, for BigInts C<$e> will be always 0, except for NaN, +inf and -inf,
3644 where it will be NaN; and for $x == 0, where it will be 1
3645 (to be compatible with Math::BigFloat's internal representation of a zero as
3648 C<$m> will always be a copy of the original number. The relation between $e
3649 and $m might change in the future, but will always be equivalent in a
3650 numerical sense, e.g. $m might get minimized.
3656 sub bint { Math::BigInt->new(shift); }
3658 $x = Math::BigInt->bstr("1234") # string "1234"
3659 $x = "$x"; # same as bstr()
3660 $x = Math::BigInt->bneg("1234"); # Bigint "-1234"
3661 $x = Math::BigInt->babs("-12345"); # Bigint "12345"
3662 $x = Math::BigInt->bnorm("-0 00"); # BigInt "0"
3663 $x = bint(1) + bint(2); # BigInt "3"
3664 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
3665 $x = bint(1); # BigInt "1"
3666 $x = $x + 5 / 2; # BigInt "3"
3667 $x = $x ** 3; # BigInt "27"
3668 $x *= 2; # BigInt "54"
3669 $x = Math::BigInt->new(0); # BigInt "0"
3671 $x = Math::BigInt->badd(4,5) # BigInt "9"
3672 print $x->bsstr(); # 9e+0
3674 Examples for rounding:
3679 $x = Math::BigFloat->new(123.4567);
3680 $y = Math::BigFloat->new(123.456789);
3681 Math::BigFloat->accuracy(4); # no more A than 4
3683 ok ($x->copy()->fround(),123.4); # even rounding
3684 print $x->copy()->fround(),"\n"; # 123.4
3685 Math::BigFloat->round_mode('odd'); # round to odd
3686 print $x->copy()->fround(),"\n"; # 123.5
3687 Math::BigFloat->accuracy(5); # no more A than 5
3688 Math::BigFloat->round_mode('odd'); # round to odd
3689 print $x->copy()->fround(),"\n"; # 123.46
3690 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
3691 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
3693 Math::BigFloat->accuracy(undef); # A not important now
3694 Math::BigFloat->precision(2); # P important
3695 print $x->copy()->bnorm(),"\n"; # 123.46
3696 print $x->copy()->fround(),"\n"; # 123.46
3698 Examples for converting:
3700 my $x = Math::BigInt->new('0b1'.'01' x 123);
3701 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
3703 =head1 Autocreating constants
3705 After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
3706 and binary constants in the given scope are converted to C<Math::BigInt>.
3707 This conversion happens at compile time.
3711 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
3713 prints the integer value of C<2**100>. Note that without conversion of
3714 constants the expression 2**100 will be calculated as perl scalar.
3716 Please note that strings and floating point constants are not affected,
3719 use Math::BigInt qw/:constant/;
3721 $x = 1234567890123456789012345678901234567890
3722 + 123456789123456789;
3723 $y = '1234567890123456789012345678901234567890'
3724 + '123456789123456789';
3726 do not work. You need an explicit Math::BigInt->new() around one of the
3727 operands. You should also quote large constants to protect loss of precision:
3731 $x = Math::BigInt->new('1234567889123456789123456789123456789');
3733 Without the quotes Perl would convert the large number to a floating point
3734 constant at compile time and then hand the result to BigInt, which results in
3735 an truncated result or a NaN.
3737 This also applies to integers that look like floating point constants:
3739 use Math::BigInt ':constant';
3741 print ref(123e2),"\n";
3742 print ref(123.2e2),"\n";
3744 will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
3745 to get this to work.
3749 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
3750 must be made in the second case. For long numbers, the copy can eat up to 20%
3751 of the work (in the case of addition/subtraction, less for
3752 multiplication/division). If $y is very small compared to $x, the form
3753 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
3754 more time then the actual addition.
3756 With a technique called copy-on-write, the cost of copying with overload could
3757 be minimized or even completely avoided. A test implementation of COW did show
3758 performance gains for overloaded math, but introduced a performance loss due
3759 to a constant overhead for all other operatons.
3761 The rewritten version of this module is slower on certain operations, like
3762 new(), bstr() and numify(). The reason are that it does now more work and
3763 handles more cases. The time spent in these operations is usually gained in
3764 the other operations so that programs on the average should get faster. If
3765 they don't, please contect the author.
3767 Some operations may be slower for small numbers, but are significantly faster
3768 for big numbers. Other operations are now constant (O(1), like bneg(), babs()
3769 etc), instead of O(N) and thus nearly always take much less time. These
3770 optimizations were done on purpose.
3772 If you find the Calc module to slow, try to install any of the replacement
3773 modules and see if they help you.
3775 =head2 Alternative math libraries
3777 You can use an alternative library to drive Math::BigInt via:
3779 use Math::BigInt lib => 'Module';
3781 See L<MATH LIBRARY> for more information.
3783 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
3787 =head1 Subclassing Math::BigInt
3789 The basic design of Math::BigInt allows simple subclasses with very little
3790 work, as long as a few simple rules are followed:
3796 The public API must remain consistent, i.e. if a sub-class is overloading
3797 addition, the sub-class must use the same name, in this case badd(). The
3798 reason for this is that Math::BigInt is optimized to call the object methods
3803 The private object hash keys like C<$x->{sign}> may not be changed, but
3804 additional keys can be added, like C<$x->{_custom}>.
3808 Accessor functions are available for all existing object hash keys and should
3809 be used instead of directly accessing the internal hash keys. The reason for
3810 this is that Math::BigInt itself has a pluggable interface which permits it
3811 to support different storage methods.
3815 More complex sub-classes may have to replicate more of the logic internal of
3816 Math::BigInt if they need to change more basic behaviors. A subclass that
3817 needs to merely change the output only needs to overload C<bstr()>.
3819 All other object methods and overloaded functions can be directly inherited
3820 from the parent class.
3822 At the very minimum, any subclass will need to provide it's own C<new()> and can
3823 store additional hash keys in the object. There are also some package globals
3824 that must be defined, e.g.:
3828 $precision = -2; # round to 2 decimal places
3829 $round_mode = 'even';
3832 Additionally, you might want to provide the following two globals to allow
3833 auto-upgrading and auto-downgrading to work correctly:
3838 This allows Math::BigInt to correctly retrieve package globals from the
3839 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
3840 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
3846 in your subclass to automatically inherit the overloading from the parent. If
3847 you like, you can change part of the overloading, look at Math::String for an
3852 When used like this:
3854 use Math::BigInt upgrade => 'Foo::Bar';
3856 certain operations will 'upgrade' their calculation and thus the result to
3857 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
3859 use Math::BigInt upgrade => 'Math::BigFloat';
3861 As a shortcut, you can use the module C<bignum>:
3865 Also good for oneliners:
3867 perl -Mbignum -le 'print 2 ** 255'
3869 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
3870 as well es preserve accuracy (as in sqrt(3)).
3872 Beware: This feature is not fully implemented yet.
3876 The following methods upgrade themselves unconditionally; that is if upgrade
3877 is in effect, they will always hand up their work:
3889 Beware: This list is not complete.
3891 All other methods upgrade themselves only when one (or all) of their
3892 arguments are of the class mentioned in $upgrade (This might change in later
3893 versions to a more sophisticated scheme):
3899 =item Input with trailing newlines
3901 Input with trailing newlines is handled wrong (e.g. lead to corrupted numbers)
3902 in some cases, for instance "123\n" or "123456789\n".
3904 =item Out of Memory!
3906 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
3907 C<eval()> in your code will crash with "Out of memory". This is probably an
3908 overload/exporter bug. You can workaround by not having C<eval()>
3909 and ':constant' at the same time or upgrade your Perl to a newer version.
3911 =item Fails to load Calc on Perl prior 5.6.0
3913 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
3914 will fall back to eval { require ... } when loading the math lib on Perls
3915 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
3916 filesystems using a different seperator.
3922 Some things might not work as you expect them. Below is documented what is
3923 known to be troublesome:
3927 =item stringify, bstr(), bsstr() and 'cmp'
3929 Both stringify and bstr() now drop the leading '+'. The old code would return
3930 '+3', the new returns '3'. This is to be consistent with Perl and to make
3931 cmp (especially with overloading) to work as you expect. It also solves
3932 problems with Test.pm, it's ok() uses 'eq' internally.
3934 Mark said, when asked about to drop the '+' altogether, or make only cmp work:
3936 I agree (with the first alternative), don't add the '+' on positive
3937 numbers. It's not as important anymore with the new internal
3938 form for numbers. It made doing things like abs and neg easier,
3939 but those have to be done differently now anyway.
3941 So, the following examples will now work all as expected:
3944 BEGIN { plan tests => 1 }
3947 my $x = new Math::BigInt 3*3;
3948 my $y = new Math::BigInt 3*3;
3951 print "$x eq 9" if $x eq $y;
3952 print "$x eq 9" if $x eq '9';
3953 print "$x eq 9" if $x eq 3*3;
3955 Additionally, the following still works:
3957 print "$x == 9" if $x == $y;
3958 print "$x == 9" if $x == 9;
3959 print "$x == 9" if $x == 3*3;
3961 There is now a C<bsstr()> method to get the string in scientific notation aka
3962 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
3963 for comparisation, but Perl will represent some numbers as 100 and others
3964 as 1e+308. If in doubt, convert both arguments to Math::BigInt before doing eq:
3967 BEGIN { plan tests => 3 }
3970 $x = Math::BigInt->new('1e56'); $y = 1e56;
3971 ok ($x,$y); # will fail
3972 ok ($x->bsstr(),$y); # okay
3973 $y = Math::BigInt->new($y);
3976 Alternatively, simple use <=> for comparisations, that will get it always
3977 right. There is not yet a way to get a number automatically represented as
3978 a string that matches exactly the way Perl represents it.
3982 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
3985 $x = Math::BigInt->new(123);
3986 $y = int($x); # BigInt 123
3987 $x = Math::BigFloat->new(123.45);
3988 $y = int($x); # BigInt 123
3990 In all Perl versions you can use C<as_number()> for the same effect:
3992 $x = Math::BigFloat->new(123.45);
3993 $y = $x->as_number(); # BigInt 123
3995 This also works for other subclasses, like Math::String.
3997 It is yet unlcear whether overloaded int() should return a scalar or a BigInt.
4001 The following will probably not do what you expect:
4003 $c = Math::BigInt->new(123);
4004 print $c->length(),"\n"; # prints 30
4006 It prints both the number of digits in the number and in the fraction part
4007 since print calls C<length()> in list context. Use something like:
4009 print scalar $c->length(),"\n"; # prints 3
4013 The following will probably not do what you expect:
4015 print $c->bdiv(10000),"\n";
4017 It prints both quotient and remainder since print calls C<bdiv()> in list
4018 context. Also, C<bdiv()> will modify $c, so be carefull. You probably want
4021 print $c / 10000,"\n";
4022 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
4026 The quotient is always the greatest integer less than or equal to the
4027 real-valued quotient of the two operands, and the remainder (when it is
4028 nonzero) always has the same sign as the second operand; so, for
4038 As a consequence, the behavior of the operator % agrees with the
4039 behavior of Perl's built-in % operator (as documented in the perlop
4040 manpage), and the equation
4042 $x == ($x / $y) * $y + ($x % $y)
4044 holds true for any $x and $y, which justifies calling the two return
4045 values of bdiv() the quotient and remainder. The only exception to this rule
4046 are when $y == 0 and $x is negative, then the remainder will also be
4047 negative. See below under "infinity handling" for the reasoning behing this.
4049 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
4050 not change BigInt's way to do things. This is because under 'use integer' Perl
4051 will do what the underlying C thinks is right and this is different for each
4052 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
4053 the author to implement it ;)
4055 =item infinity handling
4057 Here are some examples that explain the reasons why certain results occur while
4060 The following table shows the result of the division and the remainder, so that
4061 the equation above holds true. Some "ordinary" cases are strewn in to show more
4062 clearly the reasoning:
4064 A / B = C, R so that C * B + R = A
4065 =========================================================
4066 5 / 8 = 0, 5 0 * 8 + 5 = 5
4067 0 / 8 = 0, 0 0 * 8 + 0 = 0
4068 0 / inf = 0, 0 0 * inf + 0 = 0
4069 0 /-inf = 0, 0 0 * -inf + 0 = 0
4070 5 / inf = 0, 5 0 * inf + 5 = 5
4071 5 /-inf = 0, 5 0 * -inf + 5 = 5
4072 -5/ inf = 0, -5 0 * inf + -5 = -5
4073 -5/-inf = 0, -5 0 * -inf + -5 = -5
4074 inf/ 5 = inf, 0 inf * 5 + 0 = inf
4075 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
4076 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
4077 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
4078 5/ 5 = 1, 0 1 * 5 + 0 = 5
4079 -5/ -5 = 1, 0 1 * -5 + 0 = -5
4080 inf/ inf = 1, 0 1 * inf + 0 = inf
4081 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
4082 inf/-inf = -1, 0 -1 * -inf + 0 = inf
4083 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
4084 8/ 0 = inf, 8 inf * 0 + 8 = 8
4085 inf/ 0 = inf, inf inf * 0 + inf = inf
4088 These cases below violate the "remainder has the sign of the second of the two
4089 arguments", since they wouldn't match up otherwise.
4091 A / B = C, R so that C * B + R = A
4092 ========================================================
4093 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
4094 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
4096 =item Modifying and =
4100 $x = Math::BigFloat->new(5);
4103 It will not do what you think, e.g. making a copy of $x. Instead it just makes
4104 a second reference to the B<same> object and stores it in $y. Thus anything
4105 that modifies $x (except overloaded operators) will modify $y, and vice versa.
4106 Or in other words, C<=> is only safe if you modify your BigInts only via
4107 overloaded math. As soon as you use a method call it breaks:
4110 print "$x, $y\n"; # prints '10, 10'
4112 If you want a true copy of $x, use:
4116 You can also chain the calls like this, this will make first a copy and then
4119 $y = $x->copy()->bmul(2);
4121 See also the documentation for overload.pm regarding C<=>.
4125 C<bpow()> (and the rounding functions) now modifies the first argument and
4126 returns it, unlike the old code which left it alone and only returned the
4127 result. This is to be consistent with C<badd()> etc. The first three will
4128 modify $x, the last one won't:
4130 print bpow($x,$i),"\n"; # modify $x
4131 print $x->bpow($i),"\n"; # ditto
4132 print $x **= $i,"\n"; # the same
4133 print $x ** $i,"\n"; # leave $x alone
4135 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4137 =item Overloading -$x
4147 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4148 needs to preserve $x since it does not know that it later will get overwritten.
4149 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4151 With Copy-On-Write, this issue would be gone, but C-o-W is not implemented
4152 since it is slower for all other things.
4154 =item Mixing different object types
4156 In Perl you will get a floating point value if you do one of the following:
4162 With overloaded math, only the first two variants will result in a BigFloat:
4167 $mbf = Math::BigFloat->new(5);
4168 $mbi2 = Math::BigInteger->new(5);
4169 $mbi = Math::BigInteger->new(2);
4171 # what actually gets called:
4172 $float = $mbf + $mbi; # $mbf->badd()
4173 $float = $mbf / $mbi; # $mbf->bdiv()
4174 $integer = $mbi + $mbf; # $mbi->badd()
4175 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4176 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4178 This is because math with overloaded operators follows the first (dominating)
4179 operand, and the operation of that is called and returns thus the result. So,
4180 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4181 the result should be a Math::BigFloat or the second operant is one.
4183 To get a Math::BigFloat you either need to call the operation manually,
4184 make sure the operands are already of the proper type or casted to that type
4185 via Math::BigFloat->new():
4187 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4189 Beware of simple "casting" the entire expression, this would only convert
4190 the already computed result:
4192 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4194 Beware also of the order of more complicated expressions like:
4196 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4197 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4199 If in doubt, break the expression into simpler terms, or cast all operands
4200 to the desired resulting type.
4202 Scalar values are a bit different, since:
4207 will both result in the proper type due to the way the overloaded math works.
4209 This section also applies to other overloaded math packages, like Math::String.
4211 One solution to you problem might be L<autoupgrading|upgrading>.
4215 C<bsqrt()> works only good if the result is a big integer, e.g. the square
4216 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4219 If you want a better approximation of the square root, then use:
4221 $x = Math::BigFloat->new(12);
4222 Math::BigFloat->precision(0);
4223 Math::BigFloat->round_mode('even');
4224 print $x->copy->bsqrt(),"\n"; # 4
4226 Math::BigFloat->precision(2);
4227 print $x->bsqrt(),"\n"; # 3.46
4228 print $x->bsqrt(3),"\n"; # 3.464
4232 For negative numbers in base see also L<brsft|brsft>.
4238 This program is free software; you may redistribute it and/or modify it under
4239 the same terms as Perl itself.
4243 L<Math::BigFloat> and L<Math::Big> as well as L<Math::BigInt::BitVect>,
4244 L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4247 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4248 more documentation including a full version history, testcases, empty
4249 subclass files and benchmarks.
4253 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4254 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001.