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]) :
70 ref($_[0])->bcmp($_[0],$_[1])},
73 "$_[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
239 # $object->accuracy() or fallback to global
240 return $x->{_a} || ${"${class}::accuracy"};
242 return ${"${class}::accuracy"};
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
278 # $object->precision() or fallback to global
279 return $x->{_p} || ${"${class}::precision"};
281 return ${"${class}::precision"};
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();
589 if (defined $self->{_a} && defined $_[0] && $_[0] > $self->{_a});
591 if (defined $self->{_p} && defined $_[1] && $_[1] < $self->{_p});
598 # create a bigint '+1' (or -1 if given sign '-'),
599 # if given a BigInt, set it to +1 or -1, respecively
601 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
602 $self = $class if !defined $self;
606 my $c = $self; $self = {}; bless $self, $c;
608 $self->import() if $IMPORT == 0; # make require work
609 return if $self->modify('bone');
611 if ($self->can('_bone'))
613 # use subclass to initialize
618 # otherwise do our own thing
619 $self->{value} = $CALC->_one();
621 $self->{sign} = $sign;
625 if (defined $self->{_a} && defined $_[0] && $_[0] > $self->{_a});
627 if (defined $self->{_p} && defined $_[1] && $_[1] < $self->{_p});
632 ##############################################################################
633 # string conversation
637 # (ref to BFLOAT or num_str ) return num_str
638 # Convert number from internal format to scientific string format.
639 # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
640 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
641 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
643 if ($x->{sign} !~ /^[+-]$/)
645 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
648 my ($m,$e) = $x->parts();
649 # e can only be positive
651 # MBF: my $s = $e->{sign}; $s = '' if $s eq '-'; my $sep = 'e'.$s;
652 return $m->bstr().$sign.$e->bstr();
657 # make a string from bigint object
658 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
659 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
661 if ($x->{sign} !~ /^[+-]$/)
663 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
666 my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
667 return $es.${$CALC->_str($x->{value})};
672 # Make a "normal" scalar from a BigInt object
673 my $x = shift; $x = $class->new($x) unless ref $x;
674 return $x->{sign} if $x->{sign} !~ /^[+-]$/;
675 my $num = $CALC->_num($x->{value});
676 return -$num if $x->{sign} eq '-';
680 ##############################################################################
681 # public stuff (usually prefixed with "b")
685 # return the sign of the number: +/-/-inf/+inf/NaN
686 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
691 sub _find_round_parameters
693 # After any operation or when calling round(), the result is rounded by
694 # regarding the A & P from arguments, local parameters, or globals.
696 # This procedure finds the round parameters, but it is for speed reasons
697 # duplicated in round. Otherwise, it is tested by the testsuite and used
700 my ($self,$a,$p,$r,@args) = @_;
701 # $a accuracy, if given by caller
702 # $p precision, if given by caller
703 # $r round_mode, if given by caller
704 # @args all 'other' arguments (0 for unary, 1 for binary ops)
706 # leave bigfloat parts alone
707 return ($self) if exists $self->{_f} && $self->{_f} & MB_NEVER_ROUND != 0;
709 my $c = ref($self); # find out class of argument(s)
712 # now pick $a or $p, but only if we have got "arguments"
715 foreach ($self,@args)
717 # take the defined one, or if both defined, the one that is smaller
718 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
723 # even if $a is defined, take $p, to signal error for both defined
724 foreach ($self,@args)
726 # take the defined one, or if both defined, the one that is bigger
728 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
731 # if still none defined, use globals (#2)
732 $a = ${"$c\::accuracy"} unless defined $a;
733 $p = ${"$c\::precision"} unless defined $p;
736 return ($self) unless defined $a || defined $p; # early out
738 # set A and set P is an fatal error
739 return ($self->bnan()) if defined $a && defined $p;
741 $r = ${"$c\::round_mode"} unless defined $r;
742 die "Unknown round mode '$r'" if $r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/;
744 return ($self,$a,$p,$r);
749 # Round $self according to given parameters, or given second argument's
750 # parameters or global defaults
752 # for speed reasons, _find_round_parameters is embeded here:
754 my ($self,$a,$p,$r,@args) = @_;
755 # $a accuracy, if given by caller
756 # $p precision, if given by caller
757 # $r round_mode, if given by caller
758 # @args all 'other' arguments (0 for unary, 1 for binary ops)
760 # leave bigfloat parts alone
761 return ($self) if exists $self->{_f} && $self->{_f} & MB_NEVER_ROUND != 0;
763 my $c = ref($self); # find out class of argument(s)
766 # now pick $a or $p, but only if we have got "arguments"
769 foreach ($self,@args)
771 # take the defined one, or if both defined, the one that is smaller
772 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
777 # even if $a is defined, take $p, to signal error for both defined
778 foreach ($self,@args)
780 # take the defined one, or if both defined, the one that is bigger
782 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
785 # if still none defined, use globals (#2)
786 $a = ${"$c\::accuracy"} unless defined $a;
787 $p = ${"$c\::precision"} unless defined $p;
790 return $self unless defined $a || defined $p; # early out
792 # set A and set P is an fatal error
793 return $self->bnan() if defined $a && defined $p;
795 $r = ${"$c\::round_mode"} unless defined $r;
796 die "Unknown round mode '$r'" if $r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/;
798 # now round, by calling either fround or ffround:
801 $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a;
803 else # both can't be undefined due to early out
805 $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p;
807 $self->bnorm(); # after round, normalize
812 # (numstr or BINT) return BINT
813 # Normalize number -- no-op here
814 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
820 # (BINT or num_str) return BINT
821 # make number absolute, or return absolute BINT from string
822 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
824 return $x if $x->modify('babs');
825 # post-normalized abs for internal use (does nothing for NaN)
826 $x->{sign} =~ s/^-/+/;
832 # (BINT or num_str) return BINT
833 # negate number or make a negated number from string
834 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
836 return $x if $x->modify('bneg');
838 # for +0 dont negate (to have always normalized)
839 $x->{sign} =~ tr/+-/-+/ if !$x->is_zero(); # does nothing for NaN
845 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
846 # (BINT or num_str, BINT or num_str) return cond_code
847 my ($self,$x,$y) = objectify(2,@_);
849 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
851 # handle +-inf and NaN
852 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
853 return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
854 return +1 if $x->{sign} eq '+inf';
855 return -1 if $x->{sign} eq '-inf';
856 return -1 if $y->{sign} eq '+inf';
859 # check sign for speed first
860 return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
861 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
864 my $xz = $x->is_zero();
865 my $yz = $y->is_zero();
866 return 0 if $xz && $yz; # 0 <=> 0
867 return -1 if $xz && $y->{sign} eq '+'; # 0 <=> +y
868 return 1 if $yz && $x->{sign} eq '+'; # +x <=> 0
870 # post-normalized compare for internal use (honors signs)
871 if ($x->{sign} eq '+')
874 return $CALC->_acmp($x->{value},$y->{value});
878 $CALC->_acmp($y->{value},$x->{value}); # swaped (lib does only 0,1,-1)
883 # Compares 2 values, ignoring their signs.
884 # Returns one of undef, <0, =0, >0. (suitable for sort)
885 # (BINT, BINT) return cond_code
886 my ($self,$x,$y) = objectify(2,@_);
888 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
890 # handle +-inf and NaN
891 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
892 return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
893 return +1; # inf is always bigger
895 $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
900 # add second arg (BINT or string) to first (BINT) (modifies first)
901 # return result as BINT
902 my ($self,$x,$y,@r) = objectify(2,@_);
904 return $x if $x->modify('badd');
905 return $upgrade->badd($x,$y,@r) if defined $upgrade &&
906 ((!$x->isa($self)) || (!$y->isa($self)));
908 $r[3] = $y; # no push!
909 # inf and NaN handling
910 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
913 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
915 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
917 # +inf++inf or -inf+-inf => same, rest is NaN
918 return $x if $x->{sign} eq $y->{sign};
921 # +-inf + something => +inf
922 # something +-inf => +-inf
923 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
927 my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
931 $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
936 my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
939 #print "swapped sub (a=$a)\n";
940 $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
945 # speedup, if equal, set result to 0
946 #print "equal sub, result = 0\n";
947 $x->{value} = $CALC->_zero();
952 #print "unswapped sub (a=$a)\n";
953 $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
962 # (BINT or num_str, BINT or num_str) return num_str
963 # subtract second arg from first, modify first
964 my ($self,$x,$y,@r) = objectify(2,@_);
966 return $x if $x->modify('bsub');
968 # upgrade done by badd():
969 # return $upgrade->badd($x,$y,@r) if defined $upgrade &&
970 # ((!$x->isa($self)) || (!$y->isa($self)));
974 return $x->round(@r);
977 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
978 $x->badd($y,@r); # badd does not leave internal zeros
979 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
980 $x; # already rounded by badd() or no round necc.
985 # increment arg by one
986 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
987 return $x if $x->modify('binc');
989 if ($x->{sign} eq '+')
991 $x->{value} = $CALC->_inc($x->{value});
992 return $x->round($a,$p,$r);
994 elsif ($x->{sign} eq '-')
996 $x->{value} = $CALC->_dec($x->{value});
997 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
998 return $x->round($a,$p,$r);
1000 # inf, nan handling etc
1001 $x->badd($self->__one(),$a,$p,$r); # badd does round
1006 # decrement arg by one
1007 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1008 return $x if $x->modify('bdec');
1010 my $zero = $CALC->_is_zero($x->{value}) && $x->{sign} eq '+';
1012 if (($x->{sign} eq '-') || $zero)
1014 $x->{value} = $CALC->_inc($x->{value});
1015 $x->{sign} = '-' if $zero; # 0 => 1 => -1
1016 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1017 return $x->round($a,$p,$r);
1020 elsif ($x->{sign} eq '+')
1022 $x->{value} = $CALC->_dec($x->{value});
1023 return $x->round($a,$p,$r);
1025 # inf, nan handling etc
1026 $x->badd($self->__one('-'),$a,$p,$r); # badd does round
1031 # not implemented yet
1032 my ($self,$x,$base,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1034 return $upgrade->blog($x,$base,$a,$p,$r) if defined $upgrade;
1041 # (BINT or num_str, BINT or num_str) return BINT
1042 # does not modify arguments, but returns new object
1043 # Lowest Common Multiplicator
1045 my $y = shift; my ($x);
1052 $x = $class->new($y);
1054 while (@_) { $x = __lcm($x,shift); }
1060 # (BINT or num_str, BINT or num_str) return BINT
1061 # does not modify arguments, but returns new object
1062 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1065 $y = __PACKAGE__->new($y) if !ref($y);
1067 my $x = $y->copy(); # keep arguments
1068 if ($CALC->can('_gcd'))
1072 $y = shift; $y = $self->new($y) if !ref($y);
1073 next if $y->is_zero();
1074 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1075 $x->{value} = $CALC->_gcd($x->{value},$y->{value}); last if $x->is_one();
1082 $y = shift; $y = $self->new($y) if !ref($y);
1083 $x = __gcd($x,$y->copy()); last if $x->is_one(); # _gcd handles NaN
1091 # (num_str or BINT) return BINT
1092 # represent ~x as twos-complement number
1093 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1094 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1096 return $x if $x->modify('bnot');
1097 $x->bneg()->bdec(); # bdec already does round
1100 # is_foo test routines
1104 # return true if arg (BINT or num_str) is zero (array '+', '0')
1105 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1106 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1108 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1109 $CALC->_is_zero($x->{value});
1114 # return true if arg (BINT or num_str) is NaN
1115 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1117 return 1 if $x->{sign} eq $nan;
1123 # return true if arg (BINT or num_str) is +-inf
1124 my ($self,$x,$sign) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1126 $sign = '' if !defined $sign;
1127 return 1 if $sign eq $x->{sign}; # match ("+inf" eq "+inf")
1128 return 0 if $sign !~ /^([+-]|)$/;
1132 return 1 if ($x->{sign} =~ /^[+-]inf$/);
1135 $sign = quotemeta($sign.'inf');
1136 return 1 if ($x->{sign} =~ /^$sign$/);
1142 # return true if arg (BINT or num_str) is +1
1143 # or -1 if sign is given
1144 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1145 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1147 $sign = '' if !defined $sign; $sign = '+' if $sign ne '-';
1149 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1150 $CALC->_is_one($x->{value});
1155 # return true when arg (BINT or num_str) is odd, false for even
1156 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1157 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1159 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1160 $CALC->_is_odd($x->{value});
1165 # return true when arg (BINT or num_str) is even, false for odd
1166 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1167 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1169 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1170 $CALC->_is_even($x->{value});
1175 # return true when arg (BINT or num_str) is positive (>= 0)
1176 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1177 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1179 return 1 if $x->{sign} =~ /^\+/;
1185 # return true when arg (BINT or num_str) is negative (< 0)
1186 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1187 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1189 return 1 if ($x->{sign} =~ /^-/);
1195 # return true when arg (BINT or num_str) is an integer
1196 # always true for BigInt, but different for Floats
1197 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1198 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1200 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1203 ###############################################################################
1207 # multiply two numbers -- stolen from Knuth Vol 2 pg 233
1208 # (BINT or num_str, BINT or num_str) return BINT
1209 my ($self,$x,$y,@r) = objectify(2,@_);
1211 return $x if $x->modify('bmul');
1213 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1216 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1218 return $x->bnan() if $x->is_zero() || $y->is_zero();
1219 # result will always be +-inf:
1220 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1221 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1222 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1223 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1224 return $x->binf('-');
1227 return $upgrade->bmul($x,$y,@r)
1228 if defined $upgrade && $y->isa($upgrade);
1230 $r[3] = $y; # no push here
1232 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1234 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1235 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1241 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1242 my ($self,$x,$y) = @_;
1244 # NaN if x == NaN or y == NaN or x==y==0
1245 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1246 if (($x->is_nan() || $y->is_nan()) ||
1247 ($x->is_zero() && $y->is_zero()));
1249 # +-inf / +-inf == NaN, reminder also NaN
1250 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1252 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1254 # x / +-inf => 0, remainder x (works even if x == 0)
1255 if ($y->{sign} =~ /^[+-]inf$/)
1257 my $t = $x->copy(); # binf clobbers up $x
1258 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1261 # 5 / 0 => +inf, -6 / 0 => -inf
1262 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1263 # exception: -8 / 0 has remainder -8, not 8
1264 # exception: -inf / 0 has remainder -inf, not inf
1267 # +-inf / 0 => special case for -inf
1268 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1269 if (!$x->is_zero() && !$x->is_inf())
1271 my $t = $x->copy(); # binf clobbers up $x
1273 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1277 # last case: +-inf / ordinary number
1279 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1281 return wantarray ? ($x,$self->bzero()) : $x;
1286 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1287 # (BINT,BINT) (quo,rem) or BINT (only rem)
1288 my ($self,$x,$y,@r) = objectify(2,@_);
1290 return $x if $x->modify('bdiv');
1292 return $self->_div_inf($x,$y)
1293 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1295 #print "mbi bdiv $x $y\n";
1296 return $upgrade->bdiv($upgrade->new($x),$y,@r)
1297 if defined $upgrade && !$y->isa($self);
1299 $r[3] = $y; # no push!
1303 wantarray ? ($x->round(@r),$self->bzero(@r)):$x->round(@r) if $x->is_zero();
1305 # Is $x in the interval [0, $y) (aka $x <= $y) ?
1306 my $cmp = $CALC->_acmp($x->{value},$y->{value});
1307 if (($cmp < 0) and (($x->{sign} eq $y->{sign}) or !wantarray))
1309 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1310 if defined $upgrade;
1312 return $x->bzero()->round(@r) unless wantarray;
1313 my $t = $x->copy(); # make copy first, because $x->bzero() clobbers $x
1314 return ($x->bzero()->round(@r),$t);
1318 # shortcut, both are the same, so set to +/- 1
1319 $x->__one( ($x->{sign} ne $y->{sign} ? '-' : '+') );
1320 return $x unless wantarray;
1321 return ($x->round(@r),$self->bzero(@r));
1323 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1324 if defined $upgrade;
1326 # calc new sign and in case $y == +/- 1, return $x
1327 my $xsign = $x->{sign}; # keep
1328 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1329 # check for / +-1 (cant use $y->is_one due to '-'
1330 if ($CALC->_is_one($y->{value}))
1332 return wantarray ? ($x->round(@r),$self->bzero(@r)) : $x->round(@r);
1337 my $rem = $self->bzero();
1338 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1339 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1341 if (! $CALC->_is_zero($rem->{value}))
1343 $rem->{sign} = $y->{sign};
1344 $rem = $y-$rem if $xsign ne $y->{sign}; # one of them '-'
1348 $rem->{sign} = '+'; # dont leave -0
1354 $x->{value} = $CALC->_div($x->{value},$y->{value});
1355 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1359 ###############################################################################
1364 # modulus (or remainder)
1365 # (BINT or num_str, BINT or num_str) return BINT
1366 my ($self,$x,$y,@r) = objectify(2,@_);
1368 return $x if $x->modify('bmod');
1369 $r[3] = $y; # no push!
1370 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1372 my ($d,$r) = $self->_div_inf($x,$y);
1373 return $r->round(@r);
1376 if ($CALC->can('_mod'))
1378 # calc new sign and in case $y == +/- 1, return $x
1379 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1380 if (!$CALC->_is_zero($x->{value}))
1382 my $xsign = $x->{sign};
1383 $x->{sign} = $y->{sign};
1384 if ($xsign ne $y->{sign})
1386 my $t = [ @{$x->{value}} ]; # copy $x
1387 $x->{value} = [ @{$y->{value}} ]; # copy $y to $x
1388 $x->{value} = $CALC->_sub($y->{value},$t,1); # $y-$x
1393 $x->{sign} = '+'; # dont leave -0
1395 return $x->round(@r);
1397 my ($t,$rem) = $self->bdiv($x->copy(),$y,@r); # slow way (also rounds)
1399 foreach (qw/value sign _a _p/)
1401 $x->{$_} = $rem->{$_};
1408 # modular inverse. given a number which is (hopefully) relatively
1409 # prime to the modulus, calculate its inverse using Euclid's
1410 # alogrithm. if the number is not relatively prime to the modulus
1411 # (i.e. their gcd is not one) then NaN is returned.
1413 my ($self,$num,$mod,@r) = objectify(2,@_);
1415 return $num if $num->modify('bmodinv');
1418 if ($mod->{sign} ne '+' # -, NaN, +inf, -inf
1419 || $num->is_zero() # or num == 0
1420 || $num->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1422 return $num # i.e., NaN or some kind of infinity,
1423 if ($num->{sign} !~ /^[+-]$/);
1425 if ($CALC->can('_modinv'))
1427 $num->{value} = $CALC->_modinv($mod->{value});
1431 # the remaining case, nonpositive case, $num < 0, is addressed below.
1433 my ($u, $u1) = ($self->bzero(), $self->bone());
1434 my ($a, $b) = ($mod->copy(), $num->copy());
1436 # put least residue into $b if $num was negative
1437 $b->bmod($mod) if $b->{sign} eq '-';
1439 # Euclid's Algorithm
1440 while (!$b->is_zero())
1442 ($a, my $q, $b) = ($b, $a->copy()->bdiv($b));
1443 ($u, $u1) = ($u1, $u - $u1 * $q);
1446 # if the gcd is not 1, then return NaN! It would be pointless to
1447 # have called bgcd first, because we would then be performing the
1448 # same Euclidean Algorithm *twice*
1449 return $num->bnan() unless $a->is_one();
1452 $num->{value} = $u->{value};
1453 $num->{sign} = $u->{sign};
1459 # takes a very large number to a very large exponent in a given very
1460 # large modulus, quickly, thanks to binary exponentation. supports
1461 # negative exponents.
1462 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1464 return $num if $num->modify('bmodpow');
1466 # check modulus for valid values
1467 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1468 || $mod->is_zero());
1470 # check exponent for valid values
1471 if ($exp->{sign} =~ /\w/)
1473 # i.e., if it's NaN, +inf, or -inf...
1474 return $num->bnan();
1477 my $exp1 = $exp->copy();
1478 if ($exp->{sign} eq '-')
1481 $num->bmodinv ($mod);
1482 # return $num if $num->{sign} !~ /^[+-]/; # see next check
1485 # check num for valid values (also NaN if there was no inverse)
1486 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1488 if ($CALC->can('_modpow'))
1490 # $exp and $mod are positive, result is also positive
1491 $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
1495 # in the trivial case,
1496 return $num->bzero() if $mod->is_one();
1497 return $num->bone() if $num->is_zero() or $num->is_one();
1499 $num->bmod($mod); # if $x is large, make it smaller first
1500 my $acc = $num->copy(); $num->bone(); # keep ref to $num
1502 while( !$exp1->is_zero() )
1504 if( $exp1->is_odd() )
1506 $num->bmul($acc)->bmod($mod);
1508 $acc->bmul($acc)->bmod($mod);
1509 $exp1->brsft( 1, 2); # remove last (binary) digit
1514 ###############################################################################
1518 # (BINT or num_str, BINT or num_str) return BINT
1519 # compute factorial numbers
1520 # modifies first argument
1521 my ($self,$x,@r) = objectify(1,@_);
1523 return $x if $x->modify('bfac');
1525 return $x->bnan() if $x->{sign} ne '+'; # inf, NnN, <0 etc => NaN
1526 return $x->bone(@r) if $x->is_zero() || $x->is_one(); # 0 or 1 => 1
1528 if ($CALC->can('_fac'))
1530 $x->{value} = $CALC->_fac($x->{value});
1531 return $x->round(@r);
1536 my $f = $self->new(2);
1537 while ($f->bacmp($n) < 0)
1539 $x->bmul($f); $f->binc();
1541 $x->bmul($f); # last step
1542 $x->round(@r); # round
1547 # (BINT or num_str, BINT or num_str) return BINT
1548 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1549 # modifies first argument
1550 my ($self,$x,$y,@r) = objectify(2,@_);
1552 return $x if $x->modify('bpow');
1554 return $upgrade->bpow($upgrade->new($x),$y,@r)
1555 if defined $upgrade && !$y->isa($self);
1557 $r[3] = $y; # no push!
1558 return $x if $x->{sign} =~ /^[+-]inf$/; # -inf/+inf ** x
1559 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1560 return $x->bone(@r) if $y->is_zero();
1561 return $x->round(@r) if $x->is_one() || $y->is_one();
1562 if ($x->{sign} eq '-' && $CALC->_is_one($x->{value}))
1564 # if $x == -1 and odd/even y => +1/-1
1565 return $y->is_odd() ? $x->round(@r) : $x->babs()->round(@r);
1566 # my Casio FX-5500L has a bug here: -1 ** 2 is -1, but -1 * -1 is 1;
1568 # 1 ** -y => 1 / (1 ** |y|)
1569 # so do test for negative $y after above's clause
1570 return $x->bnan() if $y->{sign} eq '-';
1571 return $x->round(@r) if $x->is_zero(); # 0**y => 0 (if not y <= 0)
1573 if ($CALC->can('_pow'))
1575 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1576 return $x->round(@r);
1579 # based on the assumption that shifting in base 10 is fast, and that mul
1580 # works faster if numbers are small: we count trailing zeros (this step is
1581 # O(1)..O(N), but in case of O(N) we save much more time due to this),
1582 # stripping them out of the multiplication, and add $count * $y zeros
1583 # afterwards like this:
1584 # 300 ** 3 == 300*300*300 == 3*3*3 . '0' x 2 * 3 == 27 . '0' x 6
1585 # creates deep recursion?
1586 # my $zeros = $x->_trailing_zeros();
1589 # $x->brsft($zeros,10); # remove zeros
1590 # $x->bpow($y); # recursion (will not branch into here again)
1591 # $zeros = $y * $zeros; # real number of zeros to add
1592 # $x->blsft($zeros,10);
1593 # return $x->round($a,$p,$r);
1596 my $pow2 = $self->__one();
1597 my $y1 = $class->new($y);
1598 my $two = $self->new(2);
1599 while (!$y1->is_one())
1601 $pow2->bmul($x) if $y1->is_odd();
1605 $x->bmul($pow2) unless $pow2->is_one();
1611 # (BINT or num_str, BINT or num_str) return BINT
1612 # compute x << y, base n, y >= 0
1613 my ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_);
1615 return $x if $x->modify('blsft');
1616 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1617 return $x->round($a,$p,$r) if $y->is_zero();
1619 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1621 my $t; $t = $CALC->_lsft($x->{value},$y->{value},$n) if $CALC->can('_lsft');
1624 $x->{value} = $t; return $x->round($a,$p,$r);
1627 return $x->bmul( $self->bpow($n, $y, $a, $p, $r), $a, $p, $r );
1632 # (BINT or num_str, BINT or num_str) return BINT
1633 # compute x >> y, base n, y >= 0
1634 my ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_);
1636 return $x if $x->modify('brsft');
1637 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1638 return $x->round($a,$p,$r) if $y->is_zero();
1639 return $x->bzero($a,$p,$r) if $x->is_zero(); # 0 => 0
1641 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1643 # this only works for negative numbers when shifting in base 2
1644 if (($x->{sign} eq '-') && ($n == 2))
1646 return $x->round($a,$p,$r) if $x->is_one('-'); # -1 => -1
1649 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1650 # but perhaps there is a better emulation for two's complement shift...
1651 # if $y != 1, we must simulate it by doing:
1652 # convert to bin, flip all bits, shift, and be done
1653 $x->binc(); # -3 => -2
1654 my $bin = $x->as_bin();
1655 $bin =~ s/^-0b//; # strip '-0b' prefix
1656 $bin =~ tr/10/01/; # flip bits
1658 if (CORE::length($bin) <= $y)
1660 $bin = '0'; # shifting to far right creates -1
1661 # 0, because later increment makes
1662 # that 1, attached '-' makes it '-1'
1663 # because -1 >> x == -1 !
1667 $bin =~ s/.{$y}$//; # cut off at the right side
1668 $bin = '1' . $bin; # extend left side by one dummy '1'
1669 $bin =~ tr/10/01/; # flip bits back
1671 my $res = $self->new('0b'.$bin); # add prefix and convert back
1672 $res->binc(); # remember to increment
1673 $x->{value} = $res->{value}; # take over value
1674 return $x->round($a,$p,$r); # we are done now, magic, isn't?
1676 $x->bdec(); # n == 2, but $y == 1: this fixes it
1679 my $t; $t = $CALC->_rsft($x->{value},$y->{value},$n) if $CALC->can('_rsft');
1683 return $x->round($a,$p,$r);
1686 $x->bdiv($self->bpow($n,$y, $a,$p,$r), $a,$p,$r);
1692 #(BINT or num_str, BINT or num_str) return BINT
1694 my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
1696 return $x if $x->modify('band');
1698 local $Math::BigInt::upgrade = undef;
1700 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1701 return $x->bzero() if $y->is_zero() || $x->is_zero();
1703 my $sign = 0; # sign of result
1704 $sign = 1 if ($x->{sign} eq '-') && ($y->{sign} eq '-');
1705 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1706 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1708 if ($CALC->can('_and') && $sx == 1 && $sy == 1)
1710 $x->{value} = $CALC->_and($x->{value},$y->{value});
1711 return $x->round($a,$p,$r);
1714 my $m = $self->bone(); my ($xr,$yr);
1715 my $x10000 = $self->new (0x1000);
1716 my $y1 = copy(ref($x),$y); # make copy
1717 $y1->babs(); # and positive
1718 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1719 use integer; # need this for negative bools
1720 while (!$x1->is_zero() && !$y1->is_zero())
1722 ($x1, $xr) = bdiv($x1, $x10000);
1723 ($y1, $yr) = bdiv($y1, $x10000);
1724 # make both op's numbers!
1725 $x->badd( bmul( $class->new(
1726 abs($sx*int($xr->numify()) & $sy*int($yr->numify()))),
1730 $x->bneg() if $sign;
1731 return $x->round($a,$p,$r);
1736 #(BINT or num_str, BINT or num_str) return BINT
1738 my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
1740 return $x if $x->modify('bior');
1742 local $Math::BigInt::upgrade = undef;
1744 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1745 return $x if $y->is_zero();
1747 my $sign = 0; # sign of result
1748 $sign = 1 if ($x->{sign} eq '-') || ($y->{sign} eq '-');
1749 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1750 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1752 # don't use lib for negative values
1753 if ($CALC->can('_or') && $sx == 1 && $sy == 1)
1755 $x->{value} = $CALC->_or($x->{value},$y->{value});
1756 return $x->round($a,$p,$r);
1759 my $m = $self->bone(); my ($xr,$yr);
1760 my $x10000 = $self->new(0x10000);
1761 my $y1 = copy(ref($x),$y); # make copy
1762 $y1->babs(); # and positive
1763 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1764 use integer; # need this for negative bools
1765 while (!$x1->is_zero() || !$y1->is_zero())
1767 ($x1, $xr) = bdiv($x1,$x10000);
1768 ($y1, $yr) = bdiv($y1,$x10000);
1769 # make both op's numbers!
1770 $x->badd( bmul( $class->new(
1771 abs($sx*int($xr->numify()) | $sy*int($yr->numify()))),
1775 $x->bneg() if $sign;
1776 return $x->round($a,$p,$r);
1781 #(BINT or num_str, BINT or num_str) return BINT
1783 my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
1785 return $x if $x->modify('bxor');
1787 local $Math::BigInt::upgrade = undef;
1789 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1790 return $x if $y->is_zero();
1792 my $sign = 0; # sign of result
1793 $sign = 1 if $x->{sign} ne $y->{sign};
1794 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1795 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1797 # don't use lib for negative values
1798 if ($CALC->can('_xor') && $sx == 1 && $sy == 1)
1800 $x->{value} = $CALC->_xor($x->{value},$y->{value});
1801 return $x->round($a,$p,$r);
1804 my $m = $self->bone(); my ($xr,$yr);
1805 my $x10000 = $self->new(0x10000);
1806 my $y1 = copy(ref($x),$y); # make copy
1807 $y1->babs(); # and positive
1808 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1809 use integer; # need this for negative bools
1810 while (!$x1->is_zero() || !$y1->is_zero())
1812 ($x1, $xr) = bdiv($x1, $x10000);
1813 ($y1, $yr) = bdiv($y1, $x10000);
1814 # make both op's numbers!
1815 $x->badd( bmul( $class->new(
1816 abs($sx*int($xr->numify()) ^ $sy*int($yr->numify()))),
1820 $x->bneg() if $sign;
1821 return $x->round($a,$p,$r);
1826 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1828 my $e = $CALC->_len($x->{value});
1829 return wantarray ? ($e,0) : $e;
1834 # return the nth decimal digit, negative values count backward, 0 is right
1835 my ($self,$x,$n) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1836 $n = 0 if !defined $n;
1838 $CALC->_digit($x->{value},$n);
1843 # return the amount of trailing zeros in $x
1845 $x = $class->new($x) unless ref $x;
1847 return 0 if $x->is_zero() || $x->is_odd() || $x->{sign} !~ /^[+-]$/;
1849 return $CALC->_zeros($x->{value}) if $CALC->can('_zeros');
1851 # if not: since we do not know underlying internal representation:
1852 my $es = "$x"; $es =~ /([0]*)$/;
1853 return 0 if !defined $1; # no zeros
1854 return CORE::length("$1"); # as string, not as +0!
1859 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1861 return $x if $x->modify('bsqrt');
1863 return $x->bnan() if $x->{sign} ne '+'; # -x or inf or NaN => NaN
1864 return $x->bzero($a,$p) if $x->is_zero(); # 0 => 0
1865 return $x->round($a,$p,$r) if $x->is_one(); # 1 => 1
1867 return $upgrade->bsqrt($x,$a,$p,$r) if defined $upgrade;
1869 if ($CALC->can('_sqrt'))
1871 $x->{value} = $CALC->_sqrt($x->{value});
1872 return $x->round($a,$p,$r);
1875 return $x->bone($a,$p) if $x < 4; # 2,3 => 1
1877 my $l = int($x->length()/2);
1879 $x->bone(); # keep ref($x), but modify it
1882 my $last = $self->bzero();
1883 my $two = $self->new(2);
1884 my $lastlast = $x+$two;
1885 while ($last != $x && $lastlast != $x)
1887 $lastlast = $last; $last = $x;
1891 $x-- if $x * $x > $y; # overshot?
1892 $x->round($a,$p,$r);
1897 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
1898 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1900 if ($x->{sign} !~ /^[+-]$/)
1902 my $s = $x->{sign}; $s =~ s/^[+-]//;
1903 return $self->new($s); # -inf,+inf => inf
1905 my $e = $class->bzero();
1906 return $e->binc() if $x->is_zero();
1907 $e += $x->_trailing_zeros();
1913 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
1914 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1916 if ($x->{sign} !~ /^[+-]$/)
1918 return $self->new($x->{sign}); # keep + or - sign
1921 # that's inefficient
1922 my $zeros = $m->_trailing_zeros();
1923 $m->brsft($zeros,10) if $zeros != 0;
1924 # $m /= 10 ** $zeros if $zeros != 0;
1930 # return a copy of both the exponent and the mantissa
1931 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1933 return ($x->mantissa(),$x->exponent());
1936 ##############################################################################
1937 # rounding functions
1941 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
1942 # $n == 0 || $n == 1 => round to integer
1943 my $x = shift; $x = $class->new($x) unless ref $x;
1944 my ($scale,$mode) = $x->_scale_p($x->precision(),$x->round_mode(),@_);
1945 return $x if !defined $scale; # no-op
1946 return $x if $x->modify('bfround');
1948 # no-op for BigInts if $n <= 0
1951 $x->{_a} = undef; # clear an eventual set A
1952 $x->{_p} = $scale; return $x;
1955 $x->bround( $x->length()-$scale, $mode);
1956 $x->{_a} = undef; # bround sets {_a}
1957 $x->{_p} = $scale; # so correct it
1961 sub _scan_for_nonzero
1967 my $len = $x->length();
1968 return 0 if $len == 1; # '5' is trailed by invisible zeros
1969 my $follow = $pad - 1;
1970 return 0 if $follow > $len || $follow < 1;
1972 # since we do not know underlying represention of $x, use decimal string
1973 #my $r = substr ($$xs,-$follow);
1974 my $r = substr ("$x",-$follow);
1975 return 1 if $r =~ /[^0]/; return 0;
1980 # to make life easier for switch between MBF and MBI (autoload fxxx()
1981 # like MBF does for bxxx()?)
1983 return $x->bround(@_);
1988 # accuracy: +$n preserve $n digits from left,
1989 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
1991 # and overwrite the rest with 0's, return normalized number
1992 # do not return $x->bnorm(), but $x
1994 my $x = shift; $x = $class->new($x) unless ref $x;
1995 my ($scale,$mode) = $x->_scale_a($x->accuracy(),$x->round_mode(),@_);
1996 return $x if !defined $scale; # no-op
1997 return $x if $x->modify('bround');
1999 if ($x->is_zero() || $scale == 0)
2001 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2004 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2006 # we have fewer digits than we want to scale to
2007 my $len = $x->length();
2008 # scale < 0, but > -len (not >=!)
2009 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2011 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2015 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2016 my ($pad,$digit_round,$digit_after);
2017 $pad = $len - $scale;
2018 $pad = abs($scale-1) if $scale < 0;
2020 # do not use digit(), it is costly for binary => decimal
2022 my $xs = $CALC->_str($x->{value});
2025 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2026 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2027 $digit_round = '0'; $digit_round = substr($$xs,$pl,1) if $pad <= $len;
2028 $pl++; $pl ++ if $pad >= $len;
2029 $digit_after = '0'; $digit_after = substr($$xs,$pl,1) if $pad > 0;
2031 # print "$pad $pl $$xs dr $digit_round da $digit_after\n";
2033 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2034 # closer at the remaining digits of the original $x, remember decision
2035 my $round_up = 1; # default round up
2037 ($mode eq 'trunc') || # trunc by round down
2038 ($digit_after =~ /[01234]/) || # round down anyway,
2040 ($digit_after eq '5') && # not 5000...0000
2041 ($x->_scan_for_nonzero($pad,$xs) == 0) &&
2043 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2044 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2045 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2046 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2047 ($mode eq 'zero') # round down if zero, sign adjusted below
2049 my $put_back = 0; # not yet modified
2051 # old code, depend on internal representation
2052 # split mantissa at $pad and then pad with zeros
2053 #my $s5 = int($pad / 5);
2057 # $x->{value}->[$i++] = 0; # replace with 5 x 0
2059 #$x->{value}->[$s5] = '00000'.$x->{value}->[$s5]; # pad with 0
2060 #my $rem = $pad % 5; # so much left over
2063 # #print "remainder $rem\n";
2064 ## #print "elem $x->{value}->[$s5]\n";
2065 # substr($x->{value}->[$s5],-$rem,$rem) = '0' x $rem; # stamp w/ '0'
2067 #$x->{value}->[$s5] = int ($x->{value}->[$s5]); # str '05' => int '5'
2068 #print ${$CALC->_str($pad->{value})}," $len\n";
2070 if (($pad > 0) && ($pad <= $len))
2072 substr($$xs,-$pad,$pad) = '0' x $pad;
2077 $x->bzero(); # round to '0'
2080 if ($round_up) # what gave test above?
2083 $pad = $len, $$xs = '0'x$pad if $scale < 0; # tlr: whack 0.51=>1.0
2085 # we modify directly the string variant instead of creating a number and
2087 my $c = 0; $pad ++; # for $pad == $len case
2088 while ($pad <= $len)
2090 $c = substr($$xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2091 substr($$xs,-$pad,1) = $c; $pad++;
2092 last if $c != 0; # no overflow => early out
2094 $$xs = '1'.$$xs if $c == 0;
2096 # $x->badd( Math::BigInt->new($x->{sign}.'1'. '0' x $pad) );
2098 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back in
2100 $x->{_a} = $scale if $scale >= 0;
2103 $x->{_a} = $len+$scale;
2104 $x->{_a} = 0 if $scale < -$len;
2111 # return integer less or equal then number, since it is already integer,
2112 # always returns $self
2113 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2115 # not needed: return $x if $x->modify('bfloor');
2116 return $x->round($a,$p,$r);
2121 # return integer greater or equal then number, since it is already integer,
2122 # always returns $self
2123 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2125 # not needed: return $x if $x->modify('bceil');
2126 return $x->round($a,$p,$r);
2129 ##############################################################################
2130 # private stuff (internal use only)
2134 # internal speedup, set argument to 1, or create a +/- 1
2136 my $x = $self->bone(); # $x->{value} = $CALC->_one();
2137 $x->{sign} = shift || '+';
2143 # Overload will swap params if first one is no object ref so that the first
2144 # one is always an object ref. In this case, third param is true.
2145 # This routine is to overcome the effect of scalar,$object creating an object
2146 # of the class of this package, instead of the second param $object. This
2147 # happens inside overload, when the overload section of this package is
2148 # inherited by sub classes.
2149 # For overload cases (and this is used only there), we need to preserve the
2150 # args, hence the copy().
2151 # You can override this method in a subclass, the overload section will call
2152 # $object->_swap() to make sure it arrives at the proper subclass, with some
2153 # exceptions like '+' and '-'. To make '+' and '-' work, you also need to
2154 # specify your own overload for them.
2156 # object, (object|scalar) => preserve first and make copy
2157 # scalar, object => swapped, re-swap and create new from first
2158 # (using class of second object, not $class!!)
2159 my $self = shift; # for override in subclass
2162 my $c = ref ($_[0]) || $class; # fallback $class should not happen
2163 return ( $c->new($_[1]), $_[0] );
2165 return ( $_[0]->copy(), $_[1] );
2170 # check for strings, if yes, return objects instead
2172 # the first argument is number of args objectify() should look at it will
2173 # return $count+1 elements, the first will be a classname. This is because
2174 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2175 # useless objects beeing created and thrown away. So we cannot simple loop
2176 # over @_. If the given count is 0, all arguments will be used.
2178 # If the second arg is a ref, use it as class.
2179 # If not, try to use it as classname, unless undef, then use $class
2180 # (aka Math::BigInt). The latter shouldn't happen,though.
2183 # $x->badd(1); => ref x, scalar y
2184 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2185 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2186 # Math::BigInt::badd(1,2); => scalar x, scalar y
2187 # In the last case we check number of arguments to turn it silently into
2188 # $class,1,2. (We can not take '1' as class ;o)
2189 # badd($class,1) is not supported (it should, eventually, try to add undef)
2190 # currently it tries 'Math::BigInt' + 1, which will not work.
2192 # some shortcut for the common cases
2194 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2196 my $count = abs(shift || 0);
2198 my (@a,$k,$d); # resulting array, temp, and downgrade
2201 # okay, got object as first
2206 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2208 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2212 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2213 if (defined ${"$a[0]::downgrade"})
2215 $d = ${"$a[0]::downgrade"};
2216 ${"$a[0]::downgrade"} = undef;
2219 my $up = ${"$a[0]::upgrade"};
2220 # print "Now in objectify, my class is today $a[0]\n";
2228 $k = $a[0]->new($k);
2230 elsif (!defined $up && ref($k) ne $a[0])
2232 # foreign object, try to convert to integer
2233 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2246 $k = $a[0]->new($k);
2248 elsif (!defined $up && ref($k) ne $a[0])
2250 # foreign object, try to convert to integer
2251 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2255 push @a,@_; # return other params, too
2257 die "$class objectify needs list context" unless wantarray;
2258 ${"$a[0]::downgrade"} = $d;
2267 my @a; my $l = scalar @_;
2268 for ( my $i = 0; $i < $l ; $i++ )
2270 if ($_[$i] eq ':constant')
2272 # this causes overlord er load to step in
2273 overload::constant integer => sub { $self->new(shift) };
2274 overload::constant binary => sub { $self->new(shift) };
2276 elsif ($_[$i] eq 'upgrade')
2278 # this causes upgrading
2279 $upgrade = $_[$i+1]; # or undef to disable
2282 elsif ($_[$i] =~ /^lib$/i)
2284 # this causes a different low lib to take care...
2285 $CALC = $_[$i+1] || '';
2293 # any non :constant stuff is handled by our parent, Exporter
2294 # even if @_ is empty, to give it a chance
2295 $self->SUPER::import(@a); # need it for subclasses
2296 $self->export_to_level(1,$self,@a); # need it for MBF
2298 # try to load core math lib
2299 my @c = split /\s*,\s*/,$CALC;
2300 push @c,'Calc'; # if all fail, try this
2301 $CALC = ''; # signal error
2302 foreach my $lib (@c)
2304 next if ($lib || '') eq '';
2305 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2309 # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
2310 # used in the same script, or eval inside import().
2311 my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
2312 my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
2314 $file = File::Spec->catfile (@parts, $file);
2315 eval { require "$file"; $lib->import( @c ); }
2319 eval "use $lib qw/@c/;";
2321 $CALC = $lib, last if $@ eq ''; # no error in loading lib?
2323 die "Couldn't load any math lib, not even the default" if $CALC eq '';
2328 # convert a (ref to) big hex string to BigInt, return undef for error
2331 my $x = Math::BigInt->bzero();
2334 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2335 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2337 return $x->bnan() if $$hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2339 my $sign = '+'; $sign = '-' if ($$hs =~ /^-/);
2341 $$hs =~ s/^[+-]//; # strip sign
2342 if ($CALC->can('_from_hex'))
2344 $x->{value} = $CALC->_from_hex($hs);
2348 # fallback to pure perl
2349 my $mul = Math::BigInt->bzero(); $mul++;
2350 my $x65536 = Math::BigInt->new(65536);
2351 my $len = CORE::length($$hs)-2;
2352 $len = int($len/4); # 4-digit parts, w/o '0x'
2353 my $val; my $i = -4;
2356 $val = substr($$hs,$i,4);
2357 $val =~ s/^[+-]?0x// if $len == 0; # for last part only because
2358 $val = hex($val); # hex does not like wrong chars
2360 $x += $mul * $val if $val != 0;
2361 $mul *= $x65536 if $len >= 0; # skip last mul
2364 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2370 # convert a (ref to) big binary string to BigInt, return undef for error
2373 my $x = Math::BigInt->bzero();
2375 $$bs =~ s/([01])_([01])/$1$2/g;
2376 $$bs =~ s/([01])_([01])/$1$2/g;
2377 return $x->bnan() if $$bs !~ /^[+-]?0b[01]+$/;
2379 my $sign = '+'; $sign = '-' if ($$bs =~ /^\-/);
2380 $$bs =~ s/^[+-]//; # strip sign
2381 if ($CALC->can('_from_bin'))
2383 $x->{value} = $CALC->_from_bin($bs);
2387 my $mul = Math::BigInt->bzero(); $mul++;
2388 my $x256 = Math::BigInt->new(256);
2389 my $len = CORE::length($$bs)-2;
2390 $len = int($len/8); # 8-digit parts, w/o '0b'
2391 my $val; my $i = -8;
2394 $val = substr($$bs,$i,8);
2395 $val =~ s/^[+-]?0b// if $len == 0; # for last part only
2396 #$val = oct('0b'.$val); # does not work on Perl prior to 5.6.0
2398 # $val = ('0' x (8-CORE::length($val))).$val if CORE::length($val) < 8;
2399 $val = ord(pack('B8',substr('00000000'.$val,-8,8)));
2401 $x += $mul * $val if $val != 0;
2402 $mul *= $x256 if $len >= 0; # skip last mul
2405 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2411 # (ref to num_str) return num_str
2412 # internal, take apart a string and return the pieces
2413 # strip leading/trailing whitespace, leading zeros, underscore and reject
2417 # strip white space at front, also extranous leading zeros
2418 $$x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2419 $$x =~ s/^\s+//; # but this will
2420 $$x =~ s/\s+$//g; # strip white space at end
2422 # shortcut, if nothing to split, return early
2423 if ($$x =~ /^[+-]?\d+$/)
2425 $$x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2426 return (\$sign, $x, \'', \'', \0);
2429 # invalid starting char?
2430 return if $$x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2432 return __from_hex($x) if $$x =~ /^[\-\+]?0x/; # hex string
2433 return __from_bin($x) if $$x =~ /^[\-\+]?0b/; # binary string
2435 # strip underscores between digits
2436 $$x =~ s/(\d)_(\d)/$1$2/g;
2437 $$x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3
2439 # some possible inputs:
2440 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2441 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2
2443 return if $$x =~ /[Ee].*[Ee]/; # more than one E => error
2445 my ($m,$e) = split /[Ee]/,$$x;
2446 $e = '0' if !defined $e || $e eq "";
2447 # sign,value for exponent,mantint,mantfrac
2448 my ($es,$ev,$mis,$miv,$mfv);
2450 if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2454 return if $m eq '.' || $m eq '';
2455 my ($mi,$mf,$last) = split /\./,$m;
2456 return if defined $last; # last defined => 1.2.3 or others
2457 $mi = '0' if !defined $mi;
2458 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2459 $mf = '0' if !defined $mf || $mf eq '';
2460 if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2462 $mis = $1||'+'; $miv = $2;
2463 return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
2465 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2468 return; # NaN, not a number
2473 # an object might be asked to return itself as bigint on certain overloaded
2474 # operations, this does exactly this, so that sub classes can simple inherit
2475 # it or override with their own integer conversion routine
2483 # return as hex string, with prefixed 0x
2484 my $x = shift; $x = $class->new($x) if !ref($x);
2486 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2487 return '0x0' if $x->is_zero();
2489 my $es = ''; my $s = '';
2490 $s = $x->{sign} if $x->{sign} eq '-';
2491 if ($CALC->can('_as_hex'))
2493 $es = ${$CALC->_as_hex($x->{value})};
2497 my $x1 = $x->copy()->babs(); my $xr;
2498 my $x10000 = Math::BigInt->new (0x10000);
2499 while (!$x1->is_zero())
2501 ($x1, $xr) = bdiv($x1,$x10000);
2502 $es .= unpack('h4',pack('v',$xr->numify()));
2505 $es =~ s/^[0]+//; # strip leading zeros
2513 # return as binary string, with prefixed 0b
2514 my $x = shift; $x = $class->new($x) if !ref($x);
2516 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2517 return '0b0' if $x->is_zero();
2519 my $es = ''; my $s = '';
2520 $s = $x->{sign} if $x->{sign} eq '-';
2521 if ($CALC->can('_as_bin'))
2523 $es = ${$CALC->_as_bin($x->{value})};
2527 my $x1 = $x->copy()->babs(); my $xr;
2528 my $x10000 = Math::BigInt->new (0x10000);
2529 while (!$x1->is_zero())
2531 ($x1, $xr) = bdiv($x1,$x10000);
2532 $es .= unpack('b16',pack('v',$xr->numify()));
2535 $es =~ s/^[0]+//; # strip leading zeros
2541 ##############################################################################
2542 # internal calculation routines (others are in Math::BigInt::Calc etc)
2546 # (BINT or num_str, BINT or num_str) return BINT
2547 # does modify first argument
2550 my $x = shift; my $ty = shift;
2551 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2552 return $x * $ty / bgcd($x,$ty);
2557 # (BINT or num_str, BINT or num_str) return BINT
2558 # does modify both arguments
2559 # GCD -- Euclids algorithm E, Knuth Vol 2 pg 296
2562 return $x->bnan() if $x->{sign} !~ /^[+-]$/ || $ty->{sign} !~ /^[+-]$/;
2564 while (!$ty->is_zero())
2566 ($x, $ty) = ($ty,bmod($x,$ty));
2571 ###############################################################################
2572 # this method return 0 if the object can be modified, or 1 for not
2573 # We use a fast use constant statement here, to avoid costly calls. Subclasses
2574 # may override it with special code (f.i. Math::BigInt::Constant does so)
2576 sub modify () { 0; }
2583 Math::BigInt - Arbitrary size integer math package
2590 $x = Math::BigInt->new($str); # defaults to 0
2591 $nan = Math::BigInt->bnan(); # create a NotANumber
2592 $zero = Math::BigInt->bzero(); # create a +0
2593 $inf = Math::BigInt->binf(); # create a +inf
2594 $inf = Math::BigInt->binf('-'); # create a -inf
2595 $one = Math::BigInt->bone(); # create a +1
2596 $one = Math::BigInt->bone('-'); # create a -1
2599 $x->is_zero(); # true if arg is +0
2600 $x->is_nan(); # true if arg is NaN
2601 $x->is_one(); # true if arg is +1
2602 $x->is_one('-'); # true if arg is -1
2603 $x->is_odd(); # true if odd, false for even
2604 $x->is_even(); # true if even, false for odd
2605 $x->is_positive(); # true if >= 0
2606 $x->is_negative(); # true if < 0
2607 $x->is_inf(sign); # true if +inf, or -inf (sign is default '+')
2608 $x->is_int(); # true if $x is an integer (not a float)
2610 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2611 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2612 $x->sign(); # return the sign, either +,- or NaN
2613 $x->digit($n); # return the nth digit, counting from right
2614 $x->digit(-$n); # return the nth digit, counting from left
2616 # The following all modify their first argument:
2619 $x->bzero(); # set $x to 0
2620 $x->bnan(); # set $x to NaN
2621 $x->bone(); # set $x to +1
2622 $x->bone('-'); # set $x to -1
2623 $x->binf(); # set $x to inf
2624 $x->binf('-'); # set $x to -inf
2626 $x->bneg(); # negation
2627 $x->babs(); # absolute value
2628 $x->bnorm(); # normalize (no-op)
2629 $x->bnot(); # two's complement (bit wise not)
2630 $x->binc(); # increment x by 1
2631 $x->bdec(); # decrement x by 1
2633 $x->badd($y); # addition (add $y to $x)
2634 $x->bsub($y); # subtraction (subtract $y from $x)
2635 $x->bmul($y); # multiplication (multiply $x by $y)
2636 $x->bdiv($y); # divide, set $x to quotient
2637 # return (quo,rem) or quo if scalar
2639 $x->bmod($y); # modulus (x % y)
2640 $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
2641 $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
2643 $x->bpow($y); # power of arguments (x ** y)
2644 $x->blsft($y); # left shift
2645 $x->brsft($y); # right shift
2646 $x->blsft($y,$n); # left shift, by base $n (like 10)
2647 $x->brsft($y,$n); # right shift, by base $n (like 10)
2649 $x->band($y); # bitwise and
2650 $x->bior($y); # bitwise inclusive or
2651 $x->bxor($y); # bitwise exclusive or
2652 $x->bnot(); # bitwise not (two's complement)
2654 $x->bsqrt(); # calculate square-root
2655 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2657 $x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r
2658 $x->bround($N); # accuracy: preserve $N digits
2659 $x->bfround($N); # round to $Nth digit, no-op for BigInts
2661 # The following do not modify their arguments in BigInt, but do in BigFloat:
2662 $x->bfloor(); # return integer less or equal than $x
2663 $x->bceil(); # return integer greater or equal than $x
2665 # The following do not modify their arguments:
2667 bgcd(@values); # greatest common divisor (no OO style)
2668 blcm(@values); # lowest common multiplicator (no OO style)
2670 $x->length(); # return number of digits in number
2671 ($x,$f) = $x->length(); # length of number and length of fraction part,
2672 # latter is always 0 digits long for BigInt's
2674 $x->exponent(); # return exponent as BigInt
2675 $x->mantissa(); # return (signed) mantissa as BigInt
2676 $x->parts(); # return (mantissa,exponent) as BigInt
2677 $x->copy(); # make a true copy of $x (unlike $y = $x;)
2678 $x->as_number(); # return as BigInt (in BigInt: same as copy())
2680 # conversation to string
2681 $x->bstr(); # normalized string
2682 $x->bsstr(); # normalized string in scientific notation
2683 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
2684 $x->as_bin(); # as signed binary string with prefixed 0b
2686 Math::BigInt->config(); # return hash containing configuration/version
2690 All operators (inlcuding basic math operations) are overloaded if you
2691 declare your big integers as
2693 $i = new Math::BigInt '123_456_789_123_456_789';
2695 Operations with overloaded operators preserve the arguments which is
2696 exactly what you expect.
2700 =item Canonical notation
2702 Big integer values are strings of the form C</^[+-]\d+$/> with leading
2705 '-0' canonical value '-0', normalized '0'
2706 ' -123_123_123' canonical value '-123123123'
2707 '1_23_456_7890' canonical value '1234567890'
2711 Input values to these routines may be either Math::BigInt objects or
2712 strings of the form C</^\s*[+-]?[\d]+\.?[\d]*E?[+-]?[\d]*$/>.
2714 You can include one underscore between any two digits.
2716 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
2717 Non integer values result in NaN.
2719 Math::BigInt::new() defaults to 0, while Math::BigInt::new('') results
2722 bnorm() on a BigInt object is now effectively a no-op, since the numbers
2723 are always stored in normalized form. On a string, it creates a BigInt
2728 Output values are BigInt objects (normalized), except for bstr(), which
2729 returns a string in normalized form.
2730 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
2731 C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
2732 return either undef, <0, 0 or >0 and are suited for sort.
2738 Each of the methods below accepts three additional parameters. These arguments
2739 $A, $P and $R are accuracy, precision and round_mode. Please see more in the
2740 section about ACCURACY and ROUNDIND.
2746 print Dumper ( Math::BigInt->config() );
2748 Returns a hash containing the configuration, e.g. the version number, lib
2753 $x->accuracy(5); # local for $x
2754 $class->accuracy(5); # global for all members of $class
2756 Set or get the global or local accuracy, aka how many significant digits the
2757 results have. Please see the section about L<ACCURACY AND PRECISION> for
2760 Value must be greater than zero. Pass an undef value to disable it:
2762 $x->accuracy(undef);
2763 Math::BigInt->accuracy(undef);
2765 Returns the current accuracy. For C<$x->accuracy()> it will return either the
2766 local accuracy, or if not defined, the global. This means the return value
2767 represents the accuracy that will be in effect for $x:
2769 $y = Math::BigInt->new(1234567); # unrounded
2770 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
2771 $x = Math::BigInt->new(123456); # will be automatically rounded
2772 print "$x $y\n"; # '123500 1234567'
2773 print $x->accuracy(),"\n"; # will be 4
2774 print $y->accuracy(),"\n"; # also 4, since global is 4
2775 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
2776 print $x->accuracy(),"\n"; # still 4
2777 print $y->accuracy(),"\n"; # 5, since global is 5
2783 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
2784 2, but others work, too.
2786 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
2790 $x = Math::BigInt->new(10);
2791 $x->brsft(1); # same as $x >> 1: 5
2792 $x = Math::BigInt->new(1234);
2793 $x->brsft(2,10); # result 12
2795 There is one exception, and that is base 2 with negative $x:
2798 $x = Math::BigInt->new(-5);
2801 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
2806 $x = Math::BigInt->new($str,$A,$P,$R);
2808 Creates a new BigInt object from a string or another BigInt object. The
2809 input is accepted as decimal, hex (with leading '0x') or binary (with leading
2814 $x = Math::BigInt->bnan();
2816 Creates a new BigInt object representing NaN (Not A Number).
2817 If used on an object, it will set it to NaN:
2823 $x = Math::BigInt->bzero();
2825 Creates a new BigInt object representing zero.
2826 If used on an object, it will set it to zero:
2832 $x = Math::BigInt->binf($sign);
2834 Creates a new BigInt object representing infinity. The optional argument is
2835 either '-' or '+', indicating whether you want infinity or minus infinity.
2836 If used on an object, it will set it to infinity:
2843 $x = Math::BigInt->binf($sign);
2845 Creates a new BigInt object representing one. The optional argument is
2846 either '-' or '+', indicating whether you want one or minus one.
2847 If used on an object, it will set it to one:
2852 =head2 is_one()/is_zero()/is_nan()/is_inf()
2855 $x->is_zero(); # true if arg is +0
2856 $x->is_nan(); # true if arg is NaN
2857 $x->is_one(); # true if arg is +1
2858 $x->is_one('-'); # true if arg is -1
2859 $x->is_inf(); # true if +inf
2860 $x->is_inf('-'); # true if -inf (sign is default '+')
2862 These methods all test the BigInt for beeing one specific value and return
2863 true or false depending on the input. These are faster than doing something
2868 =head2 is_positive()/is_negative()
2870 $x->is_positive(); # true if >= 0
2871 $x->is_negative(); # true if < 0
2873 The methods return true if the argument is positive or negative, respectively.
2874 C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
2875 C<-inf> is negative. A C<zero> is positive.
2877 These methods are only testing the sign, and not the value.
2879 =head2 is_odd()/is_even()/is_int()
2881 $x->is_odd(); # true if odd, false for even
2882 $x->is_even(); # true if even, false for odd
2883 $x->is_int(); # true if $x is an integer
2885 The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
2886 C<-inf> are not integers and are neither odd nor even.
2892 Compares $x with $y and takes the sign into account.
2893 Returns -1, 0, 1 or undef.
2899 Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
2905 Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
2909 $x->digit($n); # return the nth digit, counting from right
2915 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
2916 and '-inf', respectively. Does nothing for NaN or zero.
2922 Set the number to it's absolute value, e.g. change the sign from '-' to '+'
2923 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
2928 $x->bnorm(); # normalize (no-op)
2932 $x->bnot(); # two's complement (bit wise not)
2936 $x->binc(); # increment x by 1
2940 $x->bdec(); # decrement x by 1
2944 $x->badd($y); # addition (add $y to $x)
2948 $x->bsub($y); # subtraction (subtract $y from $x)
2952 $x->bmul($y); # multiplication (multiply $x by $y)
2956 $x->bdiv($y); # divide, set $x to quotient
2957 # return (quo,rem) or quo if scalar
2961 $x->bmod($y); # modulus (x % y)
2965 bmodinv($num,$mod); # modular inverse (no OO style)
2967 Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
2968 returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
2969 C<bgcd($num, $mod)==1>.
2973 bmodpow($num,$exp,$mod); # modular exponentation ($num**$exp % $mod)
2975 Returns the value of C<$num> taken to the power C<$exp> in the modulus
2976 C<$mod> using binary exponentation. C<bmodpow> is far superior to
2981 because C<bmodpow> is much faster--it reduces internal variables into
2982 the modulus whenever possible, so it operates on smaller numbers.
2984 C<bmodpow> also supports negative exponents.
2986 bmodpow($num, -1, $mod)
2988 is exactly equivalent to
2994 $x->bpow($y); # power of arguments (x ** y)
2998 $x->blsft($y); # left shift
2999 $x->blsft($y,$n); # left shift, by base $n (like 10)
3003 $x->brsft($y); # right shift
3004 $x->brsft($y,$n); # right shift, by base $n (like 10)
3008 $x->band($y); # bitwise and
3012 $x->bior($y); # bitwise inclusive or
3016 $x->bxor($y); # bitwise exclusive or
3020 $x->bnot(); # bitwise not (two's complement)
3024 $x->bsqrt(); # calculate square-root
3028 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3032 $x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r
3036 $x->bround($N); # accuracy: preserve $N digits
3040 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3046 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3047 does change $x in BigFloat.
3053 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3054 does change $x in BigFloat.
3058 bgcd(@values); # greatest common divisor (no OO style)
3062 blcm(@values); # lowest common multiplicator (no OO style)
3067 ($xl,$fl) = $x->length();
3069 Returns the number of digits in the decimal representation of the number.
3070 In list context, returns the length of the integer and fraction part. For
3071 BigInt's, the length of the fraction part will always be 0.
3077 Return the exponent of $x as BigInt.
3083 Return the signed mantissa of $x as BigInt.
3087 $x->parts(); # return (mantissa,exponent) as BigInt
3091 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3095 $x->as_number(); # return as BigInt (in BigInt: same as copy())
3099 $x->bstr(); # normalized string
3103 $x->bsstr(); # normalized string in scientific notation
3107 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3111 $x->as_bin(); # as signed binary string with prefixed 0b
3113 =head1 ACCURACY and PRECISION
3115 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3116 accuracy and precision based rounding, both automatically after every
3117 operation as well as manually.
3119 This section describes the accuracy/precision handling in Math::Big* as it
3120 used to be and as it is now, complete with an explanation of all terms and
3123 Not yet implemented things (but with correct description) are marked with '!',
3124 things that need to be answered are marked with '?'.
3126 In the next paragraph follows a short description of terms used here (because
3127 these may differ from terms used by others people or documentation).
3129 During the rest of this document, the shortcuts A (for accuracy), P (for
3130 precision), F (fallback) and R (rounding mode) will be used.
3134 A fixed number of digits before (positive) or after (negative)
3135 the decimal point. For example, 123.45 has a precision of -2. 0 means an
3136 integer like 123 (or 120). A precision of 2 means two digits to the left
3137 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3138 numbers with zeros before the decimal point may have different precisions,
3139 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3140 was). It could also have p < 0, when the digits after the decimal point
3143 The string output (of floating point numbers) will be padded with zeros:
3145 Initial value P A Result String
3146 ------------------------------------------------------------
3147 1234.01 -3 1000 1000
3150 1234.001 1 1234 1234.0
3152 1234.01 2 1234.01 1234.01
3153 1234.01 5 1234.01 1234.01000
3155 For BigInts, no padding occurs.
3159 Number of significant digits. Leading zeros are not counted. A
3160 number may have an accuracy greater than the non-zero digits
3161 when there are zeros in it or trailing zeros. For example, 123.456 has
3162 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3164 The string output (of floating point numbers) will be padded with zeros:
3166 Initial value P A Result String
3167 ------------------------------------------------------------
3169 1234.01 6 1234.01 1234.01
3170 1234.1 8 1234.1 1234.1000
3172 For BigInts, no padding occurs.
3176 When both A and P are undefined, this is used as a fallback accuracy when
3179 =head2 Rounding mode R
3181 When rounding a number, different 'styles' or 'kinds'
3182 of rounding are possible. (Note that random rounding, as in
3183 Math::Round, is not implemented.)
3189 truncation invariably removes all digits following the
3190 rounding place, replacing them with zeros. Thus, 987.65 rounded
3191 to tens (P=1) becomes 980, and rounded to the fourth sigdig
3192 becomes 987.6 (A=4). 123.456 rounded to the second place after the
3193 decimal point (P=-2) becomes 123.46.
3195 All other implemented styles of rounding attempt to round to the
3196 "nearest digit." If the digit D immediately to the right of the
3197 rounding place (skipping the decimal point) is greater than 5, the
3198 number is incremented at the rounding place (possibly causing a
3199 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3200 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3201 truncated at the rounding place: e.g. when rounding to units, 0.4
3202 rounds to 0, and -19.4 rounds to -19.
3204 However the results of other styles of rounding differ if the
3205 digit immediately to the right of the rounding place (skipping the
3206 decimal point) is 5 and if there are no digits, or no digits other
3207 than 0, after that 5. In such cases:
3211 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3212 if it is not already. E.g., when rounding to the first sigdig, 0.45
3213 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3217 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3218 it is not already. E.g., when rounding to the first sigdig, 0.45
3219 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3223 round to plus infinity, i.e. always round up. E.g., when
3224 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3225 and 0.4501 also becomes 0.5.
3229 round to minus infinity, i.e. always round down. E.g., when
3230 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3231 but 0.4501 becomes 0.5.
3235 round to zero, i.e. positive numbers down, negative ones up.
3236 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3237 becomes -0.5, but 0.4501 becomes 0.5.
3241 The handling of A & P in MBI/MBF (the old core code shipped with Perl
3242 versions <= 5.7.2) is like this:
3248 * ffround($p) is able to round to $p number of digits after the decimal
3250 * otherwise P is unused
3252 =item Accuracy (significant digits)
3254 * fround($a) rounds to $a significant digits
3255 * only fdiv() and fsqrt() take A as (optional) paramater
3256 + other operations simply create the same number (fneg etc), or more (fmul)
3258 + rounding/truncating is only done when explicitly calling one of fround
3259 or ffround, and never for BigInt (not implemented)
3260 * fsqrt() simply hands its accuracy argument over to fdiv.
3261 * the documentation and the comment in the code indicate two different ways
3262 on how fdiv() determines the maximum number of digits it should calculate,
3263 and the actual code does yet another thing
3265 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3267 result has at most max(scale, length(dividend), length(divisor)) digits
3269 scale = max(scale, length(dividend)-1,length(divisor)-1);
3270 scale += length(divisior) - length(dividend);
3271 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3272 Actually, the 'difference' added to the scale is calculated from the
3273 number of "significant digits" in dividend and divisor, which is derived
3274 by looking at the length of the mantissa. Which is wrong, since it includes
3275 the + sign (oups) and actually gets 2 for '+100' and 4 for '+101'. Oups
3276 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3277 assumption that 124 has 3 significant digits, while 120/7 will get you
3278 '17', not '17.1' since 120 is thought to have 2 significant digits.
3279 The rounding after the division then uses the remainder and $y to determine
3280 wether it must round up or down.
3281 ? I have no idea which is the right way. That's why I used a slightly more
3282 ? simple scheme and tweaked the few failing testcases to match it.
3286 This is how it works now:
3290 =item Setting/Accessing
3292 * You can set the A global via Math::BigInt->accuracy() or
3293 Math::BigFloat->accuracy() or whatever class you are using.
3294 * You can also set P globally by using Math::SomeClass->precision() likewise.
3295 * Globals are classwide, and not inherited by subclasses.
3296 * to undefine A, use Math::SomeCLass->accuracy(undef);
3297 * to undefine P, use Math::SomeClass->precision(undef);
3298 * Setting Math::SomeClass->accuracy() clears automatically
3299 Math::SomeClass->precision(), and vice versa.
3300 * To be valid, A must be > 0, P can have any value.
3301 * If P is negative, this means round to the P'th place to the right of the
3302 decimal point; positive values mean to the left of the decimal point.
3303 P of 0 means round to integer.
3304 * to find out the current global A, take Math::SomeClass->accuracy()
3305 * to find out the current global P, take Math::SomeClass->precision()
3306 * use $x->accuracy() respective $x->precision() for the local setting of $x.
3307 * Please note that $x->accuracy() respecive $x->precision() fall back to the
3308 defined globals, when $x's A or P is not set.
3310 =item Creating numbers
3312 * When you create a number, you can give it's desired A or P via:
3313 $x = Math::BigInt->new($number,$A,$P);
3314 * Only one of A or P can be defined, otherwise the result is NaN
3315 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
3316 globals (if set) will be used. Thus changing the global defaults later on
3317 will not change the A or P of previously created numbers (i.e., A and P of
3318 $x will be what was in effect when $x was created)
3319 * If given undef for A and P, B<no> rounding will occur, and the globals will
3320 B<not> be used. This is used by subclasses to create numbers without
3321 suffering rounding in the parent. Thus a subclass is able to have it's own
3322 globals enforced upon creation of a number by using
3323 $x = Math::BigInt->new($number,undef,undef):
3325 use Math::Bigint::SomeSubclass;
3328 Math::BigInt->accuracy(2);
3329 Math::BigInt::SomeSubClass->accuracy(3);
3330 $x = Math::BigInt::SomeSubClass->new(1234);
3332 $x is now 1230, and not 1200. A subclass might choose to implement
3333 this otherwise, e.g. falling back to the parent's A and P.
3337 * If A or P are enabled/defined, they are used to round the result of each
3338 operation according to the rules below
3339 * Negative P is ignored in Math::BigInt, since BigInts never have digits
3340 after the decimal point
3341 * Math::BigFloat uses Math::BigInts internally, but setting A or P inside
3342 Math::BigInt as globals should not tamper with the parts of a BigFloat.
3343 Thus a flag is used to mark all Math::BigFloat numbers as 'never round'
3347 * It only makes sense that a number has only one of A or P at a time.
3348 Since you can set/get both A and P, there is a rule that will practically
3349 enforce only A or P to be in effect at a time, even if both are set.
3350 This is called precedence.
3351 * If two objects are involved in an operation, and one of them has A in
3352 effect, and the other P, this results in an error (NaN).
3353 * A takes precendence over P (Hint: A comes before P). If A is defined, it
3354 is used, otherwise P is used. If neither of them is defined, nothing is
3355 used, i.e. the result will have as many digits as it can (with an
3356 exception for fdiv/fsqrt) and will not be rounded.
3357 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
3358 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
3359 If either the dividend's or the divisor's mantissa has more digits than
3360 the value of F, the higher value will be used instead of F.
3361 This is to limit the digits (A) of the result (just consider what would
3362 happen with unlimited A and P in the case of 1/3 :-)
3363 * fdiv will calculate (at least) 4 more digits than required (determined by
3364 A, P or F), and, if F is not used, round the result
3365 (this will still fail in the case of a result like 0.12345000000001 with A
3366 or P of 5, but this can not be helped - or can it?)
3367 * Thus you can have the math done by on Math::Big* class in three modes:
3368 + never round (this is the default):
3369 This is done by setting A and P to undef. No math operation
3370 will round the result, with fdiv() and fsqrt() as exceptions to guard
3371 against overflows. You must explicitely call bround(), bfround() or
3372 round() (the latter with parameters).
3373 Note: Once you have rounded a number, the settings will 'stick' on it
3374 and 'infect' all other numbers engaged in math operations with it, since
3375 local settings have the highest precedence. So, to get SaferRound[tm],
3376 use a copy() before rounding like this:
3378 $x = Math::BigFloat->new(12.34);
3379 $y = Math::BigFloat->new(98.76);
3380 $z = $x * $y; # 1218.6984
3381 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
3382 $z = $x * $y; # still 1218.6984, without
3383 # copy would have been 1210!
3385 + round after each op:
3386 After each single operation (except for testing like is_zero()), the
3387 method round() is called and the result is rounded appropriately. By
3388 setting proper values for A and P, you can have all-the-same-A or
3389 all-the-same-P modes. For example, Math::Currency might set A to undef,
3390 and P to -2, globally.
3392 ?Maybe an extra option that forbids local A & P settings would be in order,
3393 ?so that intermediate rounding does not 'poison' further math?
3395 =item Overriding globals
3397 * you will be able to give A, P and R as an argument to all the calculation
3398 routines; the second parameter is A, the third one is P, and the fourth is
3399 R (shift right by one for binary operations like badd). P is used only if
3400 the first parameter (A) is undefined. These three parameters override the
3401 globals in the order detailed as follows, i.e. the first defined value
3403 (local: per object, global: global default, parameter: argument to sub)
3406 + local A (if defined on both of the operands: smaller one is taken)
3407 + local P (if defined on both of the operands: bigger one is taken)
3411 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
3412 arguments (A and P) instead of one
3414 =item Local settings
3416 * You can set A and P locally by using $x->accuracy() and $x->precision()
3417 and thus force different A and P for different objects/numbers.
3418 * Setting A or P this way immediately rounds $x to the new value.
3419 * $x->accuracy() clears $x->precision(), and vice versa.
3423 * the rounding routines will use the respective global or local settings.
3424 fround()/bround() is for accuracy rounding, while ffround()/bfround()
3426 * the two rounding functions take as the second parameter one of the
3427 following rounding modes (R):
3428 'even', 'odd', '+inf', '-inf', 'zero', 'trunc'
3429 * you can set and get the global R by using Math::SomeClass->round_mode()
3430 or by setting $Math::SomeClass::round_mode
3431 * after each operation, $result->round() is called, and the result may
3432 eventually be rounded (that is, if A or P were set either locally,
3433 globally or as parameter to the operation)
3434 * to manually round a number, call $x->round($A,$P,$round_mode);
3435 this will round the number by using the appropriate rounding function
3436 and then normalize it.
3437 * rounding modifies the local settings of the number:
3439 $x = Math::BigFloat->new(123.456);
3443 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
3444 will be 4 from now on.
3446 =item Default values
3455 * The defaults are set up so that the new code gives the same results as
3456 the old code (except in a few cases on fdiv):
3457 + Both A and P are undefined and thus will not be used for rounding
3458 after each operation.
3459 + round() is thus a no-op, unless given extra parameters A and P
3465 The actual numbers are stored as unsigned big integers (with seperate sign).
3466 You should neither care about nor depend on the internal representation; it
3467 might change without notice. Use only method calls like C<< $x->sign(); >>
3468 instead relying on the internal hash keys like in C<< $x->{sign}; >>.
3472 Math with the numbers is done (by default) by a module called
3473 Math::BigInt::Calc. This is equivalent to saying:
3475 use Math::BigInt lib => 'Calc';
3477 You can change this by using:
3479 use Math::BigInt lib => 'BitVect';
3481 The following would first try to find Math::BigInt::Foo, then
3482 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
3484 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
3486 Calc.pm uses as internal format an array of elements of some decimal base
3487 (usually 1e5 or 1e7) with the least significant digit first, while BitVect.pm
3488 uses a bit vector of base 2, most significant bit first. Other modules might
3489 use even different means of representing the numbers. See the respective
3490 module documentation for further details.
3494 The sign is either '+', '-', 'NaN', '+inf' or '-inf' and stored seperately.
3496 A sign of 'NaN' is used to represent the result when input arguments are not
3497 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
3498 minus infinity. You will get '+inf' when dividing a positive number by 0, and
3499 '-inf' when dividing any negative number by 0.
3501 =head2 mantissa(), exponent() and parts()
3503 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
3506 $m = $x->mantissa();
3507 $e = $x->exponent();
3508 $y = $m * ( 10 ** $e );
3509 print "ok\n" if $x == $y;
3511 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
3512 in one go. Both the returned mantissa and exponent have a sign.
3514 Currently, for BigInts C<$e> will be always 0, except for NaN, +inf and -inf,
3515 where it will be NaN; and for $x == 0, where it will be 1
3516 (to be compatible with Math::BigFloat's internal representation of a zero as
3519 C<$m> will always be a copy of the original number. The relation between $e
3520 and $m might change in the future, but will always be equivalent in a
3521 numerical sense, e.g. $m might get minimized.
3527 sub bint { Math::BigInt->new(shift); }
3529 $x = Math::BigInt->bstr("1234") # string "1234"
3530 $x = "$x"; # same as bstr()
3531 $x = Math::BigInt->bneg("1234"); # Bigint "-1234"
3532 $x = Math::BigInt->babs("-12345"); # Bigint "12345"
3533 $x = Math::BigInt->bnorm("-0 00"); # BigInt "0"
3534 $x = bint(1) + bint(2); # BigInt "3"
3535 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
3536 $x = bint(1); # BigInt "1"
3537 $x = $x + 5 / 2; # BigInt "3"
3538 $x = $x ** 3; # BigInt "27"
3539 $x *= 2; # BigInt "54"
3540 $x = Math::BigInt->new(0); # BigInt "0"
3542 $x = Math::BigInt->badd(4,5) # BigInt "9"
3543 print $x->bsstr(); # 9e+0
3545 Examples for rounding:
3550 $x = Math::BigFloat->new(123.4567);
3551 $y = Math::BigFloat->new(123.456789);
3552 Math::BigFloat->accuracy(4); # no more A than 4
3554 ok ($x->copy()->fround(),123.4); # even rounding
3555 print $x->copy()->fround(),"\n"; # 123.4
3556 Math::BigFloat->round_mode('odd'); # round to odd
3557 print $x->copy()->fround(),"\n"; # 123.5
3558 Math::BigFloat->accuracy(5); # no more A than 5
3559 Math::BigFloat->round_mode('odd'); # round to odd
3560 print $x->copy()->fround(),"\n"; # 123.46
3561 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
3562 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
3564 Math::BigFloat->accuracy(undef); # A not important now
3565 Math::BigFloat->precision(2); # P important
3566 print $x->copy()->bnorm(),"\n"; # 123.46
3567 print $x->copy()->fround(),"\n"; # 123.46
3569 Examples for converting:
3571 my $x = Math::BigInt->new('0b1'.'01' x 123);
3572 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
3574 =head1 Autocreating constants
3576 After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
3577 and binary constants in the given scope are converted to C<Math::BigInt>.
3578 This conversion happens at compile time.
3582 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
3584 prints the integer value of C<2**100>. Note that without conversion of
3585 constants the expression 2**100 will be calculated as perl scalar.
3587 Please note that strings and floating point constants are not affected,
3590 use Math::BigInt qw/:constant/;
3592 $x = 1234567890123456789012345678901234567890
3593 + 123456789123456789;
3594 $y = '1234567890123456789012345678901234567890'
3595 + '123456789123456789';
3597 do not work. You need an explicit Math::BigInt->new() around one of the
3598 operands. You should also quote large constants to protect loss of precision:
3602 $x = Math::BigInt->new('1234567889123456789123456789123456789');
3604 Without the quotes Perl would convert the large number to a floating point
3605 constant at compile time and then hand the result to BigInt, which results in
3606 an truncated result or a NaN.
3608 This also applies to integers that look like floating point constants:
3610 use Math::BigInt ':constant';
3612 print ref(123e2),"\n";
3613 print ref(123.2e2),"\n";
3615 will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
3616 to get this to work.
3620 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
3621 must be made in the second case. For long numbers, the copy can eat up to 20%
3622 of the work (in the case of addition/subtraction, less for
3623 multiplication/division). If $y is very small compared to $x, the form
3624 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
3625 more time then the actual addition.
3627 With a technique called copy-on-write, the cost of copying with overload could
3628 be minimized or even completely avoided. A test implementation of COW did show
3629 performance gains for overloaded math, but introduced a performance loss due
3630 to a constant overhead for all other operatons.
3632 The rewritten version of this module is slower on certain operations, like
3633 new(), bstr() and numify(). The reason are that it does now more work and
3634 handles more cases. The time spent in these operations is usually gained in
3635 the other operations so that programs on the average should get faster. If
3636 they don't, please contect the author.
3638 Some operations may be slower for small numbers, but are significantly faster
3639 for big numbers. Other operations are now constant (O(1), like bneg(), babs()
3640 etc), instead of O(N) and thus nearly always take much less time. These
3641 optimizations were done on purpose.
3643 If you find the Calc module to slow, try to install any of the replacement
3644 modules and see if they help you.
3646 =head2 Alternative math libraries
3648 You can use an alternative library to drive Math::BigInt via:
3650 use Math::BigInt lib => 'Module';
3652 See L<MATH LIBRARY> for more information.
3654 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
3658 =head1 Subclassing Math::BigInt
3660 The basic design of Math::BigInt allows simple subclasses with very little
3661 work, as long as a few simple rules are followed:
3667 The public API must remain consistent, i.e. if a sub-class is overloading
3668 addition, the sub-class must use the same name, in this case badd(). The
3669 reason for this is that Math::BigInt is optimized to call the object methods
3674 The private object hash keys like C<$x->{sign}> may not be changed, but
3675 additional keys can be added, like C<$x->{_custom}>.
3679 Accessor functions are available for all existing object hash keys and should
3680 be used instead of directly accessing the internal hash keys. The reason for
3681 this is that Math::BigInt itself has a pluggable interface which permits it
3682 to support different storage methods.
3686 More complex sub-classes may have to replicate more of the logic internal of
3687 Math::BigInt if they need to change more basic behaviors. A subclass that
3688 needs to merely change the output only needs to overload C<bstr()>.
3690 All other object methods and overloaded functions can be directly inherited
3691 from the parent class.
3693 At the very minimum, any subclass will need to provide it's own C<new()> and can
3694 store additional hash keys in the object. There are also some package globals
3695 that must be defined, e.g.:
3699 $precision = -2; # round to 2 decimal places
3700 $round_mode = 'even';
3703 Additionally, you might want to provide the following two globals to allow
3704 auto-upgrading and auto-downgrading to work correctly:
3709 This allows Math::BigInt to correctly retrieve package globals from the
3710 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
3711 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
3717 in your subclass to automatically inherit the overloading from the parent. If
3718 you like, you can change part of the overloading, look at Math::String for an
3723 When used like this:
3725 use Math::BigInt upgrade => 'Foo::Bar';
3727 certain operations will 'upgrade' their calculation and thus the result to
3728 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
3730 use Math::BigInt upgrade => 'Math::BigFloat';
3732 As a shortcut, you can use the module C<bignum>:
3736 Also good for oneliners:
3738 perl -Mbignum -le 'print 2 ** 255'
3740 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
3741 as well es preserve accuracy (as in sqrt(3)).
3743 Beware: This feature is not fully implemented yet.
3747 The following methods upgrade themselves unconditionally; that is if upgrade
3748 is in effect, they will always hand up their work:
3760 Beware: This list is not complete.
3762 All other methods upgrade themselves only when one (or all) of their
3763 arguments are of the class mentioned in $upgrade (This might change in later
3764 versions to a more sophisticated scheme):
3770 =item Out of Memory!
3772 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
3773 C<eval()> in your code will crash with "Out of memory". This is probably an
3774 overload/exporter bug. You can workaround by not having C<eval()>
3775 and ':constant' at the same time or upgrade your Perl to a newer version.
3777 =item Fails to load Calc on Perl prior 5.6.0
3779 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
3780 will fall back to eval { require ... } when loading the math lib on Perls
3781 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
3782 filesystems using a different seperator.
3788 Some things might not work as you expect them. Below is documented what is
3789 known to be troublesome:
3793 =item stringify, bstr(), bsstr() and 'cmp'
3795 Both stringify and bstr() now drop the leading '+'. The old code would return
3796 '+3', the new returns '3'. This is to be consistent with Perl and to make
3797 cmp (especially with overloading) to work as you expect. It also solves
3798 problems with Test.pm, it's ok() uses 'eq' internally.
3800 Mark said, when asked about to drop the '+' altogether, or make only cmp work:
3802 I agree (with the first alternative), don't add the '+' on positive
3803 numbers. It's not as important anymore with the new internal
3804 form for numbers. It made doing things like abs and neg easier,
3805 but those have to be done differently now anyway.
3807 So, the following examples will now work all as expected:
3810 BEGIN { plan tests => 1 }
3813 my $x = new Math::BigInt 3*3;
3814 my $y = new Math::BigInt 3*3;
3817 print "$x eq 9" if $x eq $y;
3818 print "$x eq 9" if $x eq '9';
3819 print "$x eq 9" if $x eq 3*3;
3821 Additionally, the following still works:
3823 print "$x == 9" if $x == $y;
3824 print "$x == 9" if $x == 9;
3825 print "$x == 9" if $x == 3*3;
3827 There is now a C<bsstr()> method to get the string in scientific notation aka
3828 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
3829 for comparisation, but Perl will represent some numbers as 100 and others
3830 as 1e+308. If in doubt, convert both arguments to Math::BigInt before doing eq:
3833 BEGIN { plan tests => 3 }
3836 $x = Math::BigInt->new('1e56'); $y = 1e56;
3837 ok ($x,$y); # will fail
3838 ok ($x->bsstr(),$y); # okay
3839 $y = Math::BigInt->new($y);
3842 Alternatively, simple use <=> for comparisations, that will get it always
3843 right. There is not yet a way to get a number automatically represented as
3844 a string that matches exactly the way Perl represents it.
3848 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
3851 $x = Math::BigInt->new(123);
3852 $y = int($x); # BigInt 123
3853 $x = Math::BigFloat->new(123.45);
3854 $y = int($x); # BigInt 123
3856 In all Perl versions you can use C<as_number()> for the same effect:
3858 $x = Math::BigFloat->new(123.45);
3859 $y = $x->as_number(); # BigInt 123
3861 This also works for other subclasses, like Math::String.
3863 It is yet unlcear whether overloaded int() should return a scalar or a BigInt.
3867 The following will probably not do what you expect:
3869 $c = Math::BigInt->new(123);
3870 print $c->length(),"\n"; # prints 30
3872 It prints both the number of digits in the number and in the fraction part
3873 since print calls C<length()> in list context. Use something like:
3875 print scalar $c->length(),"\n"; # prints 3
3879 The following will probably not do what you expect:
3881 print $c->bdiv(10000),"\n";
3883 It prints both quotient and remainder since print calls C<bdiv()> in list
3884 context. Also, C<bdiv()> will modify $c, so be carefull. You probably want
3887 print $c / 10000,"\n";
3888 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
3892 The quotient is always the greatest integer less than or equal to the
3893 real-valued quotient of the two operands, and the remainder (when it is
3894 nonzero) always has the same sign as the second operand; so, for
3904 As a consequence, the behavior of the operator % agrees with the
3905 behavior of Perl's built-in % operator (as documented in the perlop
3906 manpage), and the equation
3908 $x == ($x / $y) * $y + ($x % $y)
3910 holds true for any $x and $y, which justifies calling the two return
3911 values of bdiv() the quotient and remainder. The only exception to this rule
3912 are when $y == 0 and $x is negative, then the remainder will also be
3913 negative. See below under "infinity handling" for the reasoning behing this.
3915 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
3916 not change BigInt's way to do things. This is because under 'use integer' Perl
3917 will do what the underlying C thinks is right and this is different for each
3918 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
3919 the author to implement it ;)
3921 =item infinity handling
3923 Here are some examples that explain the reasons why certain results occur while
3926 The following table shows the result of the division and the remainder, so that
3927 the equation above holds true. Some "ordinary" cases are strewn in to show more
3928 clearly the reasoning:
3930 A / B = C, R so that C * B + R = A
3931 =========================================================
3932 5 / 8 = 0, 5 0 * 8 + 5 = 5
3933 0 / 8 = 0, 0 0 * 8 + 0 = 0
3934 0 / inf = 0, 0 0 * inf + 0 = 0
3935 0 /-inf = 0, 0 0 * -inf + 0 = 0
3936 5 / inf = 0, 5 0 * inf + 5 = 5
3937 5 /-inf = 0, 5 0 * -inf + 5 = 5
3938 -5/ inf = 0, -5 0 * inf + -5 = -5
3939 -5/-inf = 0, -5 0 * -inf + -5 = -5
3940 inf/ 5 = inf, 0 inf * 5 + 0 = inf
3941 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
3942 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
3943 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
3944 5/ 5 = 1, 0 1 * 5 + 0 = 5
3945 -5/ -5 = 1, 0 1 * -5 + 0 = -5
3946 inf/ inf = 1, 0 1 * inf + 0 = inf
3947 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
3948 inf/-inf = -1, 0 -1 * -inf + 0 = inf
3949 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
3950 8/ 0 = inf, 8 inf * 0 + 8 = 8
3951 inf/ 0 = inf, inf inf * 0 + inf = inf
3954 These cases below violate the "remainder has the sign of the second of the two
3955 arguments", since they wouldn't match up otherwise.
3957 A / B = C, R so that C * B + R = A
3958 ========================================================
3959 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
3960 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
3962 =item Modifying and =
3966 $x = Math::BigFloat->new(5);
3969 It will not do what you think, e.g. making a copy of $x. Instead it just makes
3970 a second reference to the B<same> object and stores it in $y. Thus anything
3971 that modifies $x (except overloaded operators) will modify $y, and vice versa.
3972 Or in other words, C<=> is only safe if you modify your BigInts only via
3973 overloaded math. As soon as you use a method call it breaks:
3976 print "$x, $y\n"; # prints '10, 10'
3978 If you want a true copy of $x, use:
3982 You can also chain the calls like this, this will make first a copy and then
3985 $y = $x->copy()->bmul(2);
3987 See also the documentation for overload.pm regarding C<=>.
3991 C<bpow()> (and the rounding functions) now modifies the first argument and
3992 returns it, unlike the old code which left it alone and only returned the
3993 result. This is to be consistent with C<badd()> etc. The first three will
3994 modify $x, the last one won't:
3996 print bpow($x,$i),"\n"; # modify $x
3997 print $x->bpow($i),"\n"; # ditto
3998 print $x **= $i,"\n"; # the same
3999 print $x ** $i,"\n"; # leave $x alone
4001 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4003 =item Overloading -$x
4013 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4014 needs to preserve $x since it does not know that it later will get overwritten.
4015 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4017 With Copy-On-Write, this issue would be gone, but C-o-W is not implemented
4018 since it is slower for all other things.
4020 =item Mixing different object types
4022 In Perl you will get a floating point value if you do one of the following:
4028 With overloaded math, only the first two variants will result in a BigFloat:
4033 $mbf = Math::BigFloat->new(5);
4034 $mbi2 = Math::BigInteger->new(5);
4035 $mbi = Math::BigInteger->new(2);
4037 # what actually gets called:
4038 $float = $mbf + $mbi; # $mbf->badd()
4039 $float = $mbf / $mbi; # $mbf->bdiv()
4040 $integer = $mbi + $mbf; # $mbi->badd()
4041 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4042 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4044 This is because math with overloaded operators follows the first (dominating)
4045 operand, and the operation of that is called and returns thus the result. So,
4046 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4047 the result should be a Math::BigFloat or the second operant is one.
4049 To get a Math::BigFloat you either need to call the operation manually,
4050 make sure the operands are already of the proper type or casted to that type
4051 via Math::BigFloat->new():
4053 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4055 Beware of simple "casting" the entire expression, this would only convert
4056 the already computed result:
4058 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4060 Beware also of the order of more complicated expressions like:
4062 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4063 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4065 If in doubt, break the expression into simpler terms, or cast all operands
4066 to the desired resulting type.
4068 Scalar values are a bit different, since:
4073 will both result in the proper type due to the way the overloaded math works.
4075 This section also applies to other overloaded math packages, like Math::String.
4077 One solution to you problem might be L<autoupgrading|upgrading>.
4081 C<bsqrt()> works only good if the result is a big integer, e.g. the square
4082 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4085 If you want a better approximation of the square root, then use:
4087 $x = Math::BigFloat->new(12);
4088 Math::BigFloat->precision(0);
4089 Math::BigFloat->round_mode('even');
4090 print $x->copy->bsqrt(),"\n"; # 4
4092 Math::BigFloat->precision(2);
4093 print $x->bsqrt(),"\n"; # 3.46
4094 print $x->bsqrt(3),"\n"; # 3.464
4098 For negative numbers in base see also L<brsft|brsft>.
4104 This program is free software; you may redistribute it and/or modify it under
4105 the same terms as Perl itself.
4109 L<Math::BigFloat> and L<Math::Big> as well as L<Math::BigInt::BitVect>,
4110 L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4113 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4114 more documentation including a full version history, testcases, empty
4115 subclass files and benchmarks.
4119 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4120 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001.