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 $x = $y-$x if $xsign ne $y->{sign}; # one of them '-'
1388 $x->{sign} = '+'; # dont leave -0
1390 return $x->round(@r);
1392 my ($t,$rem) = $self->bdiv($x->copy(),$y,@r); # slow way (also rounds)
1394 foreach (qw/value sign _a _p/)
1396 $x->{$_} = $rem->{$_};
1401 sub bmodinv_not_yet_implemented
1403 # modular inverse. given a number which is (hopefully) relatively
1404 # prime to the modulus, calculate its inverse using Euclid's
1405 # alogrithm. if the number is not relatively prime to the modulus
1406 # (i.e. their gcd is not one) then NaN is returned.
1408 my ($self,$num,$mod,@r) = objectify(2,@_);
1410 return $num if $num->modify('bmodinv');
1413 if ($mod->{sign} ne '+' # -, NaN, +inf, -inf
1414 || $num->is_zero() # or num == 0
1415 || $num->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1417 # return $num # i.e., NaN or some kind of infinity,
1418 # if ($num->{sign} =~ /\w/);
1420 # the remaining case, nonpositive case, $num < 0, is addressed below.
1422 my ($u, $u1) = ($self->bzero(), $self->bone());
1423 my ($a, $b) = ($mod->copy(), $num->copy());
1425 # put least residue into $b if $num was negative
1426 $b %= $mod if $b->{sign} eq '-';
1428 # Euclid's Algorithm
1429 while( ! $b->is_zero()) {
1430 ($a, my $q, $b) = ($b, $self->bdiv( $a->copy(), $b));
1431 ($u, $u1) = ($u1, $u - $u1 * $q);
1434 # if the gcd is not 1, then return NaN! It would be pointless to
1435 # have called bgcd first, because we would then be performing the
1436 # same Euclidean Algorithm *twice*
1437 return $self->bnan() unless $a->is_one();
1443 sub bmodpow_not_yet_implemented
1445 # takes a very large number to a very large exponent in a given very
1446 # large modulus, quickly, thanks to binary exponentation. supports
1447 # negative exponents.
1448 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1450 return $num if $num->modify('bmodpow');
1452 # check modulus for valid values
1453 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1454 || $mod->is_zero());
1456 # check exponent for valid values
1457 if ($exp->{sign} =~ /\w/)
1459 # i.e., if it's NaN, +inf, or -inf...
1460 return $num->bnan();
1462 elsif ($exp->{sign} eq '-')
1465 $num->bmodinv ($mod);
1466 return $num if $num->{sign} !~ /^[+-]/; # i.e. if there was no inverse
1469 # check num for valid values
1470 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1472 # in the trivial case,
1473 return $num->bzero() if $mod->is_one();
1474 return $num->bone() if $num->is_zero() or $num->is_one();
1476 my $acc = $num->copy(); $num->bone(); # keep ref to $num
1478 print "$num $acc $exp\n";
1479 while( !$exp->is_zero() ) {
1480 if( $exp->is_odd() ) {
1481 $num->bmul($acc)->bmod($mod);
1483 $acc->bmul($acc)->bmod($mod);
1484 $exp->brsft( 1, 2); # remove last (binary) digit
1485 print "$num $acc $exp\n";
1490 ###############################################################################
1494 # (BINT or num_str, BINT or num_str) return BINT
1495 # compute factorial numbers
1496 # modifies first argument
1497 my ($self,$x,@r) = objectify(1,@_);
1499 return $x if $x->modify('bfac');
1501 return $x->bnan() if $x->{sign} ne '+'; # inf, NnN, <0 etc => NaN
1502 return $x->bone(@r) if $x->is_zero() || $x->is_one(); # 0 or 1 => 1
1504 if ($CALC->can('_fac'))
1506 $x->{value} = $CALC->_fac($x->{value});
1507 return $x->round(@r);
1512 my $f = $self->new(2);
1513 while ($f->bacmp($n) < 0)
1515 $x->bmul($f); $f->binc();
1517 $x->bmul($f); # last step
1518 $x->round(@r); # round
1523 # (BINT or num_str, BINT or num_str) return BINT
1524 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1525 # modifies first argument
1526 my ($self,$x,$y,@r) = objectify(2,@_);
1528 return $x if $x->modify('bpow');
1530 return $upgrade->bpow($upgrade->new($x),$y,@r)
1531 if defined $upgrade && !$y->isa($self);
1533 $r[3] = $y; # no push!
1534 return $x if $x->{sign} =~ /^[+-]inf$/; # -inf/+inf ** x
1535 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1536 return $x->bone(@r) if $y->is_zero();
1537 return $x->round(@r) if $x->is_one() || $y->is_one();
1538 if ($x->{sign} eq '-' && $CALC->_is_one($x->{value}))
1540 # if $x == -1 and odd/even y => +1/-1
1541 return $y->is_odd() ? $x->round(@r) : $x->babs()->round(@r);
1542 # my Casio FX-5500L has a bug here: -1 ** 2 is -1, but -1 * -1 is 1;
1544 # 1 ** -y => 1 / (1 ** |y|)
1545 # so do test for negative $y after above's clause
1546 return $x->bnan() if $y->{sign} eq '-';
1547 return $x->round(@r) if $x->is_zero(); # 0**y => 0 (if not y <= 0)
1549 if ($CALC->can('_pow'))
1551 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1552 return $x->round(@r);
1555 # based on the assumption that shifting in base 10 is fast, and that mul
1556 # works faster if numbers are small: we count trailing zeros (this step is
1557 # O(1)..O(N), but in case of O(N) we save much more time due to this),
1558 # stripping them out of the multiplication, and add $count * $y zeros
1559 # afterwards like this:
1560 # 300 ** 3 == 300*300*300 == 3*3*3 . '0' x 2 * 3 == 27 . '0' x 6
1561 # creates deep recursion?
1562 # my $zeros = $x->_trailing_zeros();
1565 # $x->brsft($zeros,10); # remove zeros
1566 # $x->bpow($y); # recursion (will not branch into here again)
1567 # $zeros = $y * $zeros; # real number of zeros to add
1568 # $x->blsft($zeros,10);
1569 # return $x->round($a,$p,$r);
1572 my $pow2 = $self->__one();
1573 my $y1 = $class->new($y);
1574 my $two = $self->new(2);
1575 while (!$y1->is_one())
1577 $pow2->bmul($x) if $y1->is_odd();
1581 $x->bmul($pow2) unless $pow2->is_one();
1587 # (BINT or num_str, BINT or num_str) return BINT
1588 # compute x << y, base n, y >= 0
1589 my ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_);
1591 return $x if $x->modify('blsft');
1592 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1593 return $x->round($a,$p,$r) if $y->is_zero();
1595 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1597 my $t; $t = $CALC->_lsft($x->{value},$y->{value},$n) if $CALC->can('_lsft');
1600 $x->{value} = $t; return $x->round($a,$p,$r);
1603 return $x->bmul( $self->bpow($n, $y, $a, $p, $r), $a, $p, $r );
1608 # (BINT or num_str, BINT or num_str) return BINT
1609 # compute x >> y, base n, y >= 0
1610 my ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_);
1612 return $x if $x->modify('brsft');
1613 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1614 return $x->round($a,$p,$r) if $y->is_zero();
1615 return $x->bzero($a,$p,$r) if $x->is_zero(); # 0 => 0
1617 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1619 # this only works for negative numbers when shifting in base 2
1620 if (($x->{sign} eq '-') && ($n == 2))
1622 return $x->round($a,$p,$r) if $x->is_one('-'); # -1 => -1
1625 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1626 # but perhaps there is a better emulation for two's complement shift...
1627 # if $y != 1, we must simulate it by doing:
1628 # convert to bin, flip all bits, shift, and be done
1629 $x->binc(); # -3 => -2
1630 my $bin = $x->as_bin();
1631 $bin =~ s/^-0b//; # strip '-0b' prefix
1632 $bin =~ tr/10/01/; # flip bits
1634 if (CORE::length($bin) <= $y)
1636 $bin = '0'; # shifting to far right creates -1
1637 # 0, because later increment makes
1638 # that 1, attached '-' makes it '-1'
1639 # because -1 >> x == -1 !
1643 $bin =~ s/.{$y}$//; # cut off at the right side
1644 $bin = '1' . $bin; # extend left side by one dummy '1'
1645 $bin =~ tr/10/01/; # flip bits back
1647 my $res = $self->new('0b'.$bin); # add prefix and convert back
1648 $res->binc(); # remember to increment
1649 $x->{value} = $res->{value}; # take over value
1650 return $x->round($a,$p,$r); # we are done now, magic, isn't?
1652 $x->bdec(); # n == 2, but $y == 1: this fixes it
1655 my $t; $t = $CALC->_rsft($x->{value},$y->{value},$n) if $CALC->can('_rsft');
1659 return $x->round($a,$p,$r);
1662 $x->bdiv($self->bpow($n,$y, $a,$p,$r), $a,$p,$r);
1668 #(BINT or num_str, BINT or num_str) return BINT
1670 my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
1672 return $x if $x->modify('band');
1674 local $Math::BigInt::upgrade = undef;
1676 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1677 return $x->bzero() if $y->is_zero() || $x->is_zero();
1679 my $sign = 0; # sign of result
1680 $sign = 1 if ($x->{sign} eq '-') && ($y->{sign} eq '-');
1681 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1682 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1684 if ($CALC->can('_and') && $sx == 1 && $sy == 1)
1686 $x->{value} = $CALC->_and($x->{value},$y->{value});
1687 return $x->round($a,$p,$r);
1690 my $m = $self->bone(); my ($xr,$yr);
1691 my $x10000 = $self->new (0x1000);
1692 my $y1 = copy(ref($x),$y); # make copy
1693 $y1->babs(); # and positive
1694 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1695 use integer; # need this for negative bools
1696 while (!$x1->is_zero() && !$y1->is_zero())
1698 ($x1, $xr) = bdiv($x1, $x10000);
1699 ($y1, $yr) = bdiv($y1, $x10000);
1700 # make both op's numbers!
1701 $x->badd( bmul( $class->new(
1702 abs($sx*int($xr->numify()) & $sy*int($yr->numify()))),
1706 $x->bneg() if $sign;
1707 return $x->round($a,$p,$r);
1712 #(BINT or num_str, BINT or num_str) return BINT
1714 my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
1716 return $x if $x->modify('bior');
1718 local $Math::BigInt::upgrade = undef;
1720 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1721 return $x if $y->is_zero();
1723 my $sign = 0; # sign of result
1724 $sign = 1 if ($x->{sign} eq '-') || ($y->{sign} eq '-');
1725 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1726 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1728 # don't use lib for negative values
1729 if ($CALC->can('_or') && $sx == 1 && $sy == 1)
1731 $x->{value} = $CALC->_or($x->{value},$y->{value});
1732 return $x->round($a,$p,$r);
1735 my $m = $self->bone(); my ($xr,$yr);
1736 my $x10000 = $self->new(0x10000);
1737 my $y1 = copy(ref($x),$y); # make copy
1738 $y1->babs(); # and positive
1739 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1740 use integer; # need this for negative bools
1741 while (!$x1->is_zero() || !$y1->is_zero())
1743 ($x1, $xr) = bdiv($x1,$x10000);
1744 ($y1, $yr) = bdiv($y1,$x10000);
1745 # make both op's numbers!
1746 $x->badd( bmul( $class->new(
1747 abs($sx*int($xr->numify()) | $sy*int($yr->numify()))),
1751 $x->bneg() if $sign;
1752 return $x->round($a,$p,$r);
1757 #(BINT or num_str, BINT or num_str) return BINT
1759 my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
1761 return $x if $x->modify('bxor');
1763 local $Math::BigInt::upgrade = undef;
1765 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1766 return $x if $y->is_zero();
1768 my $sign = 0; # sign of result
1769 $sign = 1 if $x->{sign} ne $y->{sign};
1770 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1771 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1773 # don't use lib for negative values
1774 if ($CALC->can('_xor') && $sx == 1 && $sy == 1)
1776 $x->{value} = $CALC->_xor($x->{value},$y->{value});
1777 return $x->round($a,$p,$r);
1780 my $m = $self->bone(); my ($xr,$yr);
1781 my $x10000 = $self->new(0x10000);
1782 my $y1 = copy(ref($x),$y); # make copy
1783 $y1->babs(); # and positive
1784 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1785 use integer; # need this for negative bools
1786 while (!$x1->is_zero() || !$y1->is_zero())
1788 ($x1, $xr) = bdiv($x1, $x10000);
1789 ($y1, $yr) = bdiv($y1, $x10000);
1790 # make both op's numbers!
1791 $x->badd( bmul( $class->new(
1792 abs($sx*int($xr->numify()) ^ $sy*int($yr->numify()))),
1796 $x->bneg() if $sign;
1797 return $x->round($a,$p,$r);
1802 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1804 my $e = $CALC->_len($x->{value});
1805 return wantarray ? ($e,0) : $e;
1810 # return the nth decimal digit, negative values count backward, 0 is right
1811 my ($self,$x,$n) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1812 $n = 0 if !defined $n;
1814 $CALC->_digit($x->{value},$n);
1819 # return the amount of trailing zeros in $x
1821 $x = $class->new($x) unless ref $x;
1823 return 0 if $x->is_zero() || $x->is_odd() || $x->{sign} !~ /^[+-]$/;
1825 return $CALC->_zeros($x->{value}) if $CALC->can('_zeros');
1827 # if not: since we do not know underlying internal representation:
1828 my $es = "$x"; $es =~ /([0]*)$/;
1829 return 0 if !defined $1; # no zeros
1830 return CORE::length("$1"); # as string, not as +0!
1835 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1837 return $x if $x->modify('bsqrt');
1839 return $x->bnan() if $x->{sign} ne '+'; # -x or inf or NaN => NaN
1840 return $x->bzero($a,$p) if $x->is_zero(); # 0 => 0
1841 return $x->round($a,$p,$r) if $x->is_one(); # 1 => 1
1843 return $upgrade->bsqrt($x,$a,$p,$r) if defined $upgrade;
1845 if ($CALC->can('_sqrt'))
1847 $x->{value} = $CALC->_sqrt($x->{value});
1848 return $x->round($a,$p,$r);
1851 return $x->bone($a,$p) if $x < 4; # 2,3 => 1
1853 my $l = int($x->length()/2);
1855 $x->bone(); # keep ref($x), but modify it
1858 my $last = $self->bzero();
1859 my $two = $self->new(2);
1860 my $lastlast = $x+$two;
1861 while ($last != $x && $lastlast != $x)
1863 $lastlast = $last; $last = $x;
1867 $x-- if $x * $x > $y; # overshot?
1868 $x->round($a,$p,$r);
1873 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
1874 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1876 if ($x->{sign} !~ /^[+-]$/)
1878 my $s = $x->{sign}; $s =~ s/^[+-]//;
1879 return $self->new($s); # -inf,+inf => inf
1881 my $e = $class->bzero();
1882 return $e->binc() if $x->is_zero();
1883 $e += $x->_trailing_zeros();
1889 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
1890 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1892 if ($x->{sign} !~ /^[+-]$/)
1894 return $self->new($x->{sign}); # keep + or - sign
1897 # that's inefficient
1898 my $zeros = $m->_trailing_zeros();
1899 $m->brsft($zeros,10) if $zeros != 0;
1900 # $m /= 10 ** $zeros if $zeros != 0;
1906 # return a copy of both the exponent and the mantissa
1907 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1909 return ($x->mantissa(),$x->exponent());
1912 ##############################################################################
1913 # rounding functions
1917 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
1918 # $n == 0 || $n == 1 => round to integer
1919 my $x = shift; $x = $class->new($x) unless ref $x;
1920 my ($scale,$mode) = $x->_scale_p($x->precision(),$x->round_mode(),@_);
1921 return $x if !defined $scale; # no-op
1922 return $x if $x->modify('bfround');
1924 # no-op for BigInts if $n <= 0
1927 $x->{_a} = undef; # clear an eventual set A
1928 $x->{_p} = $scale; return $x;
1931 $x->bround( $x->length()-$scale, $mode);
1932 $x->{_a} = undef; # bround sets {_a}
1933 $x->{_p} = $scale; # so correct it
1937 sub _scan_for_nonzero
1943 my $len = $x->length();
1944 return 0 if $len == 1; # '5' is trailed by invisible zeros
1945 my $follow = $pad - 1;
1946 return 0 if $follow > $len || $follow < 1;
1948 # since we do not know underlying represention of $x, use decimal string
1949 #my $r = substr ($$xs,-$follow);
1950 my $r = substr ("$x",-$follow);
1951 return 1 if $r =~ /[^0]/; return 0;
1956 # to make life easier for switch between MBF and MBI (autoload fxxx()
1957 # like MBF does for bxxx()?)
1959 return $x->bround(@_);
1964 # accuracy: +$n preserve $n digits from left,
1965 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
1967 # and overwrite the rest with 0's, return normalized number
1968 # do not return $x->bnorm(), but $x
1970 my $x = shift; $x = $class->new($x) unless ref $x;
1971 my ($scale,$mode) = $x->_scale_a($x->accuracy(),$x->round_mode(),@_);
1972 return $x if !defined $scale; # no-op
1973 return $x if $x->modify('bround');
1975 if ($x->is_zero() || $scale == 0)
1977 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
1980 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
1982 # we have fewer digits than we want to scale to
1983 my $len = $x->length();
1984 # scale < 0, but > -len (not >=!)
1985 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
1987 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
1991 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
1992 my ($pad,$digit_round,$digit_after);
1993 $pad = $len - $scale;
1994 $pad = abs($scale-1) if $scale < 0;
1996 # do not use digit(), it is costly for binary => decimal
1998 my $xs = $CALC->_str($x->{value});
2001 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2002 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2003 $digit_round = '0'; $digit_round = substr($$xs,$pl,1) if $pad <= $len;
2004 $pl++; $pl ++ if $pad >= $len;
2005 $digit_after = '0'; $digit_after = substr($$xs,$pl,1) if $pad > 0;
2007 # print "$pad $pl $$xs dr $digit_round da $digit_after\n";
2009 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2010 # closer at the remaining digits of the original $x, remember decision
2011 my $round_up = 1; # default round up
2013 ($mode eq 'trunc') || # trunc by round down
2014 ($digit_after =~ /[01234]/) || # round down anyway,
2016 ($digit_after eq '5') && # not 5000...0000
2017 ($x->_scan_for_nonzero($pad,$xs) == 0) &&
2019 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2020 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2021 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2022 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2023 ($mode eq 'zero') # round down if zero, sign adjusted below
2025 my $put_back = 0; # not yet modified
2027 # old code, depend on internal representation
2028 # split mantissa at $pad and then pad with zeros
2029 #my $s5 = int($pad / 5);
2033 # $x->{value}->[$i++] = 0; # replace with 5 x 0
2035 #$x->{value}->[$s5] = '00000'.$x->{value}->[$s5]; # pad with 0
2036 #my $rem = $pad % 5; # so much left over
2039 # #print "remainder $rem\n";
2040 ## #print "elem $x->{value}->[$s5]\n";
2041 # substr($x->{value}->[$s5],-$rem,$rem) = '0' x $rem; # stamp w/ '0'
2043 #$x->{value}->[$s5] = int ($x->{value}->[$s5]); # str '05' => int '5'
2044 #print ${$CALC->_str($pad->{value})}," $len\n";
2046 if (($pad > 0) && ($pad <= $len))
2048 substr($$xs,-$pad,$pad) = '0' x $pad;
2053 $x->bzero(); # round to '0'
2056 if ($round_up) # what gave test above?
2059 $pad = $len, $$xs = '0'x$pad if $scale < 0; # tlr: whack 0.51=>1.0
2061 # we modify directly the string variant instead of creating a number and
2063 my $c = 0; $pad ++; # for $pad == $len case
2064 while ($pad <= $len)
2066 $c = substr($$xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2067 substr($$xs,-$pad,1) = $c; $pad++;
2068 last if $c != 0; # no overflow => early out
2070 $$xs = '1'.$$xs if $c == 0;
2072 # $x->badd( Math::BigInt->new($x->{sign}.'1'. '0' x $pad) );
2074 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back in
2076 $x->{_a} = $scale if $scale >= 0;
2079 $x->{_a} = $len+$scale;
2080 $x->{_a} = 0 if $scale < -$len;
2087 # return integer less or equal then number, since it is already integer,
2088 # always returns $self
2089 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2091 # not needed: return $x if $x->modify('bfloor');
2092 return $x->round($a,$p,$r);
2097 # return integer greater or equal then number, since it is already integer,
2098 # always returns $self
2099 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2101 # not needed: return $x if $x->modify('bceil');
2102 return $x->round($a,$p,$r);
2105 ##############################################################################
2106 # private stuff (internal use only)
2110 # internal speedup, set argument to 1, or create a +/- 1
2112 my $x = $self->bone(); # $x->{value} = $CALC->_one();
2113 $x->{sign} = shift || '+';
2119 # Overload will swap params if first one is no object ref so that the first
2120 # one is always an object ref. In this case, third param is true.
2121 # This routine is to overcome the effect of scalar,$object creating an object
2122 # of the class of this package, instead of the second param $object. This
2123 # happens inside overload, when the overload section of this package is
2124 # inherited by sub classes.
2125 # For overload cases (and this is used only there), we need to preserve the
2126 # args, hence the copy().
2127 # You can override this method in a subclass, the overload section will call
2128 # $object->_swap() to make sure it arrives at the proper subclass, with some
2129 # exceptions like '+' and '-'. To make '+' and '-' work, you also need to
2130 # specify your own overload for them.
2132 # object, (object|scalar) => preserve first and make copy
2133 # scalar, object => swapped, re-swap and create new from first
2134 # (using class of second object, not $class!!)
2135 my $self = shift; # for override in subclass
2138 my $c = ref ($_[0]) || $class; # fallback $class should not happen
2139 return ( $c->new($_[1]), $_[0] );
2141 return ( $_[0]->copy(), $_[1] );
2146 # check for strings, if yes, return objects instead
2148 # the first argument is number of args objectify() should look at it will
2149 # return $count+1 elements, the first will be a classname. This is because
2150 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2151 # useless objects beeing created and thrown away. So we cannot simple loop
2152 # over @_. If the given count is 0, all arguments will be used.
2154 # If the second arg is a ref, use it as class.
2155 # If not, try to use it as classname, unless undef, then use $class
2156 # (aka Math::BigInt). The latter shouldn't happen,though.
2159 # $x->badd(1); => ref x, scalar y
2160 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2161 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2162 # Math::BigInt::badd(1,2); => scalar x, scalar y
2163 # In the last case we check number of arguments to turn it silently into
2164 # $class,1,2. (We can not take '1' as class ;o)
2165 # badd($class,1) is not supported (it should, eventually, try to add undef)
2166 # currently it tries 'Math::BigInt' + 1, which will not work.
2168 # some shortcut for the common cases
2170 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2172 my $count = abs(shift || 0);
2174 my (@a,$k,$d); # resulting array, temp, and downgrade
2177 # okay, got object as first
2182 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2184 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2188 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2189 if (defined ${"$a[0]::downgrade"})
2191 $d = ${"$a[0]::downgrade"};
2192 ${"$a[0]::downgrade"} = undef;
2195 my $up = ${"$a[0]::upgrade"};
2196 # print "Now in objectify, my class is today $a[0]\n";
2204 $k = $a[0]->new($k);
2206 elsif (!defined $up && ref($k) ne $a[0])
2208 # foreign object, try to convert to integer
2209 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2222 $k = $a[0]->new($k);
2224 elsif (!defined $up && ref($k) ne $a[0])
2226 # foreign object, try to convert to integer
2227 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2231 push @a,@_; # return other params, too
2233 die "$class objectify needs list context" unless wantarray;
2234 ${"$a[0]::downgrade"} = $d;
2243 my @a; my $l = scalar @_;
2244 for ( my $i = 0; $i < $l ; $i++ )
2246 if ($_[$i] eq ':constant')
2248 # this causes overlord er load to step in
2249 overload::constant integer => sub { $self->new(shift) };
2250 overload::constant binary => sub { $self->new(shift) };
2252 elsif ($_[$i] eq 'upgrade')
2254 # this causes upgrading
2255 $upgrade = $_[$i+1]; # or undef to disable
2258 elsif ($_[$i] =~ /^lib$/i)
2260 # this causes a different low lib to take care...
2261 $CALC = $_[$i+1] || '';
2269 # any non :constant stuff is handled by our parent, Exporter
2270 # even if @_ is empty, to give it a chance
2271 $self->SUPER::import(@a); # need it for subclasses
2272 $self->export_to_level(1,$self,@a); # need it for MBF
2274 # try to load core math lib
2275 my @c = split /\s*,\s*/,$CALC;
2276 push @c,'Calc'; # if all fail, try this
2277 $CALC = ''; # signal error
2278 foreach my $lib (@c)
2280 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2284 # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
2285 # used in the same script, or eval inside import().
2286 (my $mod = $lib . '.pm') =~ s!::!/!g;
2287 # require does not automatically :: => /, so portability problems arise
2288 eval { require $mod; $lib->import( @c ); }
2292 eval "use $lib qw/@c/;";
2294 $CALC = $lib, last if $@ eq ''; # no error in loading lib?
2296 die "Couldn't load any math lib, not even the default" if $CALC eq '';
2301 # convert a (ref to) big hex string to BigInt, return undef for error
2304 my $x = Math::BigInt->bzero();
2307 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2308 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2310 return $x->bnan() if $$hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2312 my $sign = '+'; $sign = '-' if ($$hs =~ /^-/);
2314 $$hs =~ s/^[+-]//; # strip sign
2315 if ($CALC->can('_from_hex'))
2317 $x->{value} = $CALC->_from_hex($hs);
2321 # fallback to pure perl
2322 my $mul = Math::BigInt->bzero(); $mul++;
2323 my $x65536 = Math::BigInt->new(65536);
2324 my $len = CORE::length($$hs)-2;
2325 $len = int($len/4); # 4-digit parts, w/o '0x'
2326 my $val; my $i = -4;
2329 $val = substr($$hs,$i,4);
2330 $val =~ s/^[+-]?0x// if $len == 0; # for last part only because
2331 $val = hex($val); # hex does not like wrong chars
2333 $x += $mul * $val if $val != 0;
2334 $mul *= $x65536 if $len >= 0; # skip last mul
2337 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2343 # convert a (ref to) big binary string to BigInt, return undef for error
2346 my $x = Math::BigInt->bzero();
2348 $$bs =~ s/([01])_([01])/$1$2/g;
2349 $$bs =~ s/([01])_([01])/$1$2/g;
2350 return $x->bnan() if $$bs !~ /^[+-]?0b[01]+$/;
2352 my $sign = '+'; $sign = '-' if ($$bs =~ /^\-/);
2353 $$bs =~ s/^[+-]//; # strip sign
2354 if ($CALC->can('_from_bin'))
2356 $x->{value} = $CALC->_from_bin($bs);
2360 my $mul = Math::BigInt->bzero(); $mul++;
2361 my $x256 = Math::BigInt->new(256);
2362 my $len = CORE::length($$bs)-2;
2363 $len = int($len/8); # 8-digit parts, w/o '0b'
2364 my $val; my $i = -8;
2367 $val = substr($$bs,$i,8);
2368 $val =~ s/^[+-]?0b// if $len == 0; # for last part only
2369 #$val = oct('0b'.$val); # does not work on Perl prior to 5.6.0
2371 # $val = ('0' x (8-CORE::length($val))).$val if CORE::length($val) < 8;
2372 $val = ord(pack('B8',substr('00000000'.$val,-8,8)));
2374 $x += $mul * $val if $val != 0;
2375 $mul *= $x256 if $len >= 0; # skip last mul
2378 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2384 # (ref to num_str) return num_str
2385 # internal, take apart a string and return the pieces
2386 # strip leading/trailing whitespace, leading zeros, underscore and reject
2390 # strip white space at front, also extranous leading zeros
2391 $$x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2392 $$x =~ s/^\s+//; # but this will
2393 $$x =~ s/\s+$//g; # strip white space at end
2395 # shortcut, if nothing to split, return early
2396 if ($$x =~ /^[+-]?\d+$/)
2398 $$x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2399 return (\$sign, $x, \'', \'', \0);
2402 # invalid starting char?
2403 return if $$x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2405 return __from_hex($x) if $$x =~ /^[\-\+]?0x/; # hex string
2406 return __from_bin($x) if $$x =~ /^[\-\+]?0b/; # binary string
2408 # strip underscores between digits
2409 $$x =~ s/(\d)_(\d)/$1$2/g;
2410 $$x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3
2412 # some possible inputs:
2413 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2414 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2
2416 return if $$x =~ /[Ee].*[Ee]/; # more than one E => error
2418 my ($m,$e) = split /[Ee]/,$$x;
2419 $e = '0' if !defined $e || $e eq "";
2420 # sign,value for exponent,mantint,mantfrac
2421 my ($es,$ev,$mis,$miv,$mfv);
2423 if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2427 return if $m eq '.' || $m eq '';
2428 my ($mi,$mf) = split /\./,$m;
2429 $mi = '0' if !defined $mi;
2430 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2431 $mf = '0' if !defined $mf || $mf eq '';
2432 if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2434 $mis = $1||'+'; $miv = $2;
2435 return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
2437 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2440 return; # NaN, not a number
2445 # an object might be asked to return itself as bigint on certain overloaded
2446 # operations, this does exactly this, so that sub classes can simple inherit
2447 # it or override with their own integer conversion routine
2455 # return as hex string, with prefixed 0x
2456 my $x = shift; $x = $class->new($x) if !ref($x);
2458 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2459 return '0x0' if $x->is_zero();
2461 my $es = ''; my $s = '';
2462 $s = $x->{sign} if $x->{sign} eq '-';
2463 if ($CALC->can('_as_hex'))
2465 $es = ${$CALC->_as_hex($x->{value})};
2469 my $x1 = $x->copy()->babs(); my $xr;
2470 my $x10000 = Math::BigInt->new (0x10000);
2471 while (!$x1->is_zero())
2473 ($x1, $xr) = bdiv($x1,$x10000);
2474 $es .= unpack('h4',pack('v',$xr->numify()));
2477 $es =~ s/^[0]+//; # strip leading zeros
2485 # return as binary string, with prefixed 0b
2486 my $x = shift; $x = $class->new($x) if !ref($x);
2488 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2489 return '0b0' if $x->is_zero();
2491 my $es = ''; my $s = '';
2492 $s = $x->{sign} if $x->{sign} eq '-';
2493 if ($CALC->can('_as_bin'))
2495 $es = ${$CALC->_as_bin($x->{value})};
2499 my $x1 = $x->copy()->babs(); my $xr;
2500 my $x10000 = Math::BigInt->new (0x10000);
2501 while (!$x1->is_zero())
2503 ($x1, $xr) = bdiv($x1,$x10000);
2504 $es .= unpack('b16',pack('v',$xr->numify()));
2507 $es =~ s/^[0]+//; # strip leading zeros
2513 ##############################################################################
2514 # internal calculation routines (others are in Math::BigInt::Calc etc)
2518 # (BINT or num_str, BINT or num_str) return BINT
2519 # does modify first argument
2522 my $x = shift; my $ty = shift;
2523 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2524 return $x * $ty / bgcd($x,$ty);
2529 # (BINT or num_str, BINT or num_str) return BINT
2530 # does modify both arguments
2531 # GCD -- Euclids algorithm E, Knuth Vol 2 pg 296
2534 return $x->bnan() if $x->{sign} !~ /^[+-]$/ || $ty->{sign} !~ /^[+-]$/;
2536 while (!$ty->is_zero())
2538 ($x, $ty) = ($ty,bmod($x,$ty));
2543 ###############################################################################
2544 # this method return 0 if the object can be modified, or 1 for not
2545 # We use a fast use constant statement here, to avoid costly calls. Subclasses
2546 # may override it with special code (f.i. Math::BigInt::Constant does so)
2548 sub modify () { 0; }
2555 Math::BigInt - Arbitrary size integer math package
2562 $x = Math::BigInt->new($str); # defaults to 0
2563 $nan = Math::BigInt->bnan(); # create a NotANumber
2564 $zero = Math::BigInt->bzero(); # create a +0
2565 $inf = Math::BigInt->binf(); # create a +inf
2566 $inf = Math::BigInt->binf('-'); # create a -inf
2567 $one = Math::BigInt->bone(); # create a +1
2568 $one = Math::BigInt->bone('-'); # create a -1
2571 $x->is_zero(); # true if arg is +0
2572 $x->is_nan(); # true if arg is NaN
2573 $x->is_one(); # true if arg is +1
2574 $x->is_one('-'); # true if arg is -1
2575 $x->is_odd(); # true if odd, false for even
2576 $x->is_even(); # true if even, false for odd
2577 $x->is_positive(); # true if >= 0
2578 $x->is_negative(); # true if < 0
2579 $x->is_inf(sign); # true if +inf, or -inf (sign is default '+')
2580 $x->is_int(); # true if $x is an integer (not a float)
2582 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2583 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2584 $x->sign(); # return the sign, either +,- or NaN
2585 $x->digit($n); # return the nth digit, counting from right
2586 $x->digit(-$n); # return the nth digit, counting from left
2588 # The following all modify their first argument:
2591 $x->bzero(); # set $x to 0
2592 $x->bnan(); # set $x to NaN
2593 $x->bone(); # set $x to +1
2594 $x->bone('-'); # set $x to -1
2595 $x->binf(); # set $x to inf
2596 $x->binf('-'); # set $x to -inf
2598 $x->bneg(); # negation
2599 $x->babs(); # absolute value
2600 $x->bnorm(); # normalize (no-op)
2601 $x->bnot(); # two's complement (bit wise not)
2602 $x->binc(); # increment x by 1
2603 $x->bdec(); # decrement x by 1
2605 $x->badd($y); # addition (add $y to $x)
2606 $x->bsub($y); # subtraction (subtract $y from $x)
2607 $x->bmul($y); # multiplication (multiply $x by $y)
2608 $x->bdiv($y); # divide, set $x to quotient
2609 # return (quo,rem) or quo if scalar
2611 $x->bmod($y); # modulus (x % y)
2613 $x->bpow($y); # power of arguments (x ** y)
2614 $x->blsft($y); # left shift
2615 $x->brsft($y); # right shift
2616 $x->blsft($y,$n); # left shift, by base $n (like 10)
2617 $x->brsft($y,$n); # right shift, by base $n (like 10)
2619 $x->band($y); # bitwise and
2620 $x->bior($y); # bitwise inclusive or
2621 $x->bxor($y); # bitwise exclusive or
2622 $x->bnot(); # bitwise not (two's complement)
2624 $x->bsqrt(); # calculate square-root
2625 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2627 $x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r
2628 $x->bround($N); # accuracy: preserve $N digits
2629 $x->bfround($N); # round to $Nth digit, no-op for BigInts
2631 # The following do not modify their arguments in BigInt, but do in BigFloat:
2632 $x->bfloor(); # return integer less or equal than $x
2633 $x->bceil(); # return integer greater or equal than $x
2635 # The following do not modify their arguments:
2637 bgcd(@values); # greatest common divisor (no OO style)
2638 blcm(@values); # lowest common multiplicator (no OO style)
2640 $x->length(); # return number of digits in number
2641 ($x,$f) = $x->length(); # length of number and length of fraction part,
2642 # latter is always 0 digits long for BigInt's
2644 $x->exponent(); # return exponent as BigInt
2645 $x->mantissa(); # return (signed) mantissa as BigInt
2646 $x->parts(); # return (mantissa,exponent) as BigInt
2647 $x->copy(); # make a true copy of $x (unlike $y = $x;)
2648 $x->as_number(); # return as BigInt (in BigInt: same as copy())
2650 # conversation to string
2651 $x->bstr(); # normalized string
2652 $x->bsstr(); # normalized string in scientific notation
2653 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
2654 $x->as_bin(); # as signed binary string with prefixed 0b
2656 Math::BigInt->config(); # return hash containing configuration/version
2660 All operators (inlcuding basic math operations) are overloaded if you
2661 declare your big integers as
2663 $i = new Math::BigInt '123_456_789_123_456_789';
2665 Operations with overloaded operators preserve the arguments which is
2666 exactly what you expect.
2670 =item Canonical notation
2672 Big integer values are strings of the form C</^[+-]\d+$/> with leading
2675 '-0' canonical value '-0', normalized '0'
2676 ' -123_123_123' canonical value '-123123123'
2677 '1_23_456_7890' canonical value '1234567890'
2681 Input values to these routines may be either Math::BigInt objects or
2682 strings of the form C</^\s*[+-]?[\d]+\.?[\d]*E?[+-]?[\d]*$/>.
2684 You can include one underscore between any two digits.
2686 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
2687 Non integer values result in NaN.
2689 Math::BigInt::new() defaults to 0, while Math::BigInt::new('') results
2692 bnorm() on a BigInt object is now effectively a no-op, since the numbers
2693 are always stored in normalized form. On a string, it creates a BigInt
2698 Output values are BigInt objects (normalized), except for bstr(), which
2699 returns a string in normalized form.
2700 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
2701 C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
2702 return either undef, <0, 0 or >0 and are suited for sort.
2708 Each of the methods below accepts three additional parameters. These arguments
2709 $A, $P and $R are accuracy, precision and round_mode. Please see more in the
2710 section about ACCURACY and ROUNDIND.
2716 print Dumper ( Math::BigInt->config() );
2718 Returns a hash containing the configuration, e.g. the version number, lib
2723 $x->accuracy(5); # local for $x
2724 $class->accuracy(5); # global for all members of $class
2726 Set or get the global or local accuracy, aka how many significant digits the
2727 results have. Please see the section about L<ACCURACY AND PRECISION> for
2730 Value must be greater than zero. Pass an undef value to disable it:
2732 $x->accuracy(undef);
2733 Math::BigInt->accuracy(undef);
2735 Returns the current accuracy. For C<$x->accuracy()> it will return either the
2736 local accuracy, or if not defined, the global. This means the return value
2737 represents the accuracy that will be in effect for $x:
2739 $y = Math::BigInt->new(1234567); # unrounded
2740 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
2741 $x = Math::BigInt->new(123456); # will be automatically rounded
2742 print "$x $y\n"; # '123500 1234567'
2743 print $x->accuracy(),"\n"; # will be 4
2744 print $y->accuracy(),"\n"; # also 4, since global is 4
2745 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
2746 print $x->accuracy(),"\n"; # still 4
2747 print $y->accuracy(),"\n"; # 5, since global is 5
2753 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
2754 2, but others work, too.
2756 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
2760 $x = Math::BigInt->new(10);
2761 $x->brsft(1); # same as $x >> 1: 5
2762 $x = Math::BigInt->new(1234);
2763 $x->brsft(2,10); # result 12
2765 There is one exception, and that is base 2 with negative $x:
2768 $x = Math::BigInt->new(-5);
2771 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
2776 $x = Math::BigInt->new($str,$A,$P,$R);
2778 Creates a new BigInt object from a string or another BigInt object. The
2779 input is accepted as decimal, hex (with leading '0x') or binary (with leading
2784 $x = Math::BigInt->bnan();
2786 Creates a new BigInt object representing NaN (Not A Number).
2787 If used on an object, it will set it to NaN:
2793 $x = Math::BigInt->bzero();
2795 Creates a new BigInt object representing zero.
2796 If used on an object, it will set it to zero:
2802 $x = Math::BigInt->binf($sign);
2804 Creates a new BigInt object representing infinity. The optional argument is
2805 either '-' or '+', indicating whether you want infinity or minus infinity.
2806 If used on an object, it will set it to infinity:
2813 $x = Math::BigInt->binf($sign);
2815 Creates a new BigInt object representing one. The optional argument is
2816 either '-' or '+', indicating whether you want one or minus one.
2817 If used on an object, it will set it to one:
2822 =head2 is_one()/is_zero()/is_nan()/is_inf()
2825 $x->is_zero(); # true if arg is +0
2826 $x->is_nan(); # true if arg is NaN
2827 $x->is_one(); # true if arg is +1
2828 $x->is_one('-'); # true if arg is -1
2829 $x->is_inf(); # true if +inf
2830 $x->is_inf('-'); # true if -inf (sign is default '+')
2832 These methods all test the BigInt for beeing one specific value and return
2833 true or false depending on the input. These are faster than doing something
2838 =head2 is_positive()/is_negative()
2840 $x->is_positive(); # true if >= 0
2841 $x->is_negative(); # true if < 0
2843 The methods return true if the argument is positive or negative, respectively.
2844 C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
2845 C<-inf> is negative. A C<zero> is positive.
2847 These methods are only testing the sign, and not the value.
2849 =head2 is_odd()/is_even()/is_int()
2851 $x->is_odd(); # true if odd, false for even
2852 $x->is_even(); # true if even, false for odd
2853 $x->is_int(); # true if $x is an integer
2855 The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
2856 C<-inf> are not integers and are neither odd nor even.
2862 Compares $x with $y and takes the sign into account.
2863 Returns -1, 0, 1 or undef.
2869 Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
2875 Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
2879 $x->digit($n); # return the nth digit, counting from right
2885 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
2886 and '-inf', respectively. Does nothing for NaN or zero.
2892 Set the number to it's absolute value, e.g. change the sign from '-' to '+'
2893 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
2898 $x->bnorm(); # normalize (no-op)
2902 $x->bnot(); # two's complement (bit wise not)
2906 $x->binc(); # increment x by 1
2910 $x->bdec(); # decrement x by 1
2914 $x->badd($y); # addition (add $y to $x)
2918 $x->bsub($y); # subtraction (subtract $y from $x)
2922 $x->bmul($y); # multiplication (multiply $x by $y)
2926 $x->bdiv($y); # divide, set $x to quotient
2927 # return (quo,rem) or quo if scalar
2931 $x->bmod($y); # modulus (x % y)
2935 Not yet implemented.
2937 bmodinv($num,$mod); # modular inverse (no OO style)
2939 Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
2940 returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
2941 C<bgcd($num, $mod)==1>.
2945 Not yet implemented.
2947 bmodpow($num,$exp,$mod); # modular exponentation ($num**$exp % $mod)
2949 Returns the value of C<$num> taken to the power C<$exp> in the modulus
2950 C<$mod> using binary exponentation. C<bmodpow> is far superior to
2955 because C<bmodpow> is much faster--it reduces internal variables into
2956 the modulus whenever possible, so it operates on smaller numbers.
2958 C<bmodpow> also supports negative exponents.
2960 bmodpow($num, -1, $mod)
2962 is exactly equivalent to
2968 $x->bpow($y); # power of arguments (x ** y)
2972 $x->blsft($y); # left shift
2973 $x->blsft($y,$n); # left shift, by base $n (like 10)
2977 $x->brsft($y); # right shift
2978 $x->brsft($y,$n); # right shift, by base $n (like 10)
2982 $x->band($y); # bitwise and
2986 $x->bior($y); # bitwise inclusive or
2990 $x->bxor($y); # bitwise exclusive or
2994 $x->bnot(); # bitwise not (two's complement)
2998 $x->bsqrt(); # calculate square-root
3002 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3006 $x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r
3010 $x->bround($N); # accuracy: preserve $N digits
3014 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3020 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3021 does change $x in BigFloat.
3027 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3028 does change $x in BigFloat.
3032 bgcd(@values); # greatest common divisor (no OO style)
3036 blcm(@values); # lowest common multiplicator (no OO style)
3041 ($xl,$fl) = $x->length();
3043 Returns the number of digits in the decimal representation of the number.
3044 In list context, returns the length of the integer and fraction part. For
3045 BigInt's, the length of the fraction part will always be 0.
3051 Return the exponent of $x as BigInt.
3057 Return the signed mantissa of $x as BigInt.
3061 $x->parts(); # return (mantissa,exponent) as BigInt
3065 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3069 $x->as_number(); # return as BigInt (in BigInt: same as copy())
3073 $x->bstr(); # normalized string
3077 $x->bsstr(); # normalized string in scientific notation
3081 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3085 $x->as_bin(); # as signed binary string with prefixed 0b
3087 =head1 ACCURACY and PRECISION
3089 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3090 accuracy and precision based rounding, both automatically after every
3091 operation as well as manually.
3093 This section describes the accuracy/precision handling in Math::Big* as it
3094 used to be and as it is now, complete with an explanation of all terms and
3097 Not yet implemented things (but with correct description) are marked with '!',
3098 things that need to be answered are marked with '?'.
3100 In the next paragraph follows a short description of terms used here (because
3101 these may differ from terms used by others people or documentation).
3103 During the rest of this document, the shortcuts A (for accuracy), P (for
3104 precision), F (fallback) and R (rounding mode) will be used.
3108 A fixed number of digits before (positive) or after (negative)
3109 the decimal point. For example, 123.45 has a precision of -2. 0 means an
3110 integer like 123 (or 120). A precision of 2 means two digits to the left
3111 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3112 numbers with zeros before the decimal point may have different precisions,
3113 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3114 was). It could also have p < 0, when the digits after the decimal point
3117 The string output (of floating point numbers) will be padded with zeros:
3119 Initial value P A Result String
3120 ------------------------------------------------------------
3121 1234.01 -3 1000 1000
3124 1234.001 1 1234 1234.0
3126 1234.01 2 1234.01 1234.01
3127 1234.01 5 1234.01 1234.01000
3129 For BigInts, no padding occurs.
3133 Number of significant digits. Leading zeros are not counted. A
3134 number may have an accuracy greater than the non-zero digits
3135 when there are zeros in it or trailing zeros. For example, 123.456 has
3136 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3138 The string output (of floating point numbers) will be padded with zeros:
3140 Initial value P A Result String
3141 ------------------------------------------------------------
3143 1234.01 6 1234.01 1234.01
3144 1234.1 8 1234.1 1234.1000
3146 For BigInts, no padding occurs.
3150 When both A and P are undefined, this is used as a fallback accuracy when
3153 =head2 Rounding mode R
3155 When rounding a number, different 'styles' or 'kinds'
3156 of rounding are possible. (Note that random rounding, as in
3157 Math::Round, is not implemented.)
3163 truncation invariably removes all digits following the
3164 rounding place, replacing them with zeros. Thus, 987.65 rounded
3165 to tens (P=1) becomes 980, and rounded to the fourth sigdig
3166 becomes 987.6 (A=4). 123.456 rounded to the second place after the
3167 decimal point (P=-2) becomes 123.46.
3169 All other implemented styles of rounding attempt to round to the
3170 "nearest digit." If the digit D immediately to the right of the
3171 rounding place (skipping the decimal point) is greater than 5, the
3172 number is incremented at the rounding place (possibly causing a
3173 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3174 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3175 truncated at the rounding place: e.g. when rounding to units, 0.4
3176 rounds to 0, and -19.4 rounds to -19.
3178 However the results of other styles of rounding differ if the
3179 digit immediately to the right of the rounding place (skipping the
3180 decimal point) is 5 and if there are no digits, or no digits other
3181 than 0, after that 5. In such cases:
3185 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3186 if it is not already. E.g., when rounding to the first sigdig, 0.45
3187 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3191 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3192 it is not already. E.g., when rounding to the first sigdig, 0.45
3193 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3197 round to plus infinity, i.e. always round up. E.g., when
3198 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3199 and 0.4501 also becomes 0.5.
3203 round to minus infinity, i.e. always round down. E.g., when
3204 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3205 but 0.4501 becomes 0.5.
3209 round to zero, i.e. positive numbers down, negative ones up.
3210 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3211 becomes -0.5, but 0.4501 becomes 0.5.
3215 The handling of A & P in MBI/MBF (the old core code shipped with Perl
3216 versions <= 5.7.2) is like this:
3222 * ffround($p) is able to round to $p number of digits after the decimal
3224 * otherwise P is unused
3226 =item Accuracy (significant digits)
3228 * fround($a) rounds to $a significant digits
3229 * only fdiv() and fsqrt() take A as (optional) paramater
3230 + other operations simply create the same number (fneg etc), or more (fmul)
3232 + rounding/truncating is only done when explicitly calling one of fround
3233 or ffround, and never for BigInt (not implemented)
3234 * fsqrt() simply hands its accuracy argument over to fdiv.
3235 * the documentation and the comment in the code indicate two different ways
3236 on how fdiv() determines the maximum number of digits it should calculate,
3237 and the actual code does yet another thing
3239 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3241 result has at most max(scale, length(dividend), length(divisor)) digits
3243 scale = max(scale, length(dividend)-1,length(divisor)-1);
3244 scale += length(divisior) - length(dividend);
3245 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3246 Actually, the 'difference' added to the scale is calculated from the
3247 number of "significant digits" in dividend and divisor, which is derived
3248 by looking at the length of the mantissa. Which is wrong, since it includes
3249 the + sign (oups) and actually gets 2 for '+100' and 4 for '+101'. Oups
3250 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3251 assumption that 124 has 3 significant digits, while 120/7 will get you
3252 '17', not '17.1' since 120 is thought to have 2 significant digits.
3253 The rounding after the division then uses the remainder and $y to determine
3254 wether it must round up or down.
3255 ? I have no idea which is the right way. That's why I used a slightly more
3256 ? simple scheme and tweaked the few failing testcases to match it.
3260 This is how it works now:
3264 =item Setting/Accessing
3266 * You can set the A global via Math::BigInt->accuracy() or
3267 Math::BigFloat->accuracy() or whatever class you are using.
3268 * You can also set P globally by using Math::SomeClass->precision() likewise.
3269 * Globals are classwide, and not inherited by subclasses.
3270 * to undefine A, use Math::SomeCLass->accuracy(undef);
3271 * to undefine P, use Math::SomeClass->precision(undef);
3272 * Setting Math::SomeClass->accuracy() clears automatically
3273 Math::SomeClass->precision(), and vice versa.
3274 * To be valid, A must be > 0, P can have any value.
3275 * If P is negative, this means round to the P'th place to the right of the
3276 decimal point; positive values mean to the left of the decimal point.
3277 P of 0 means round to integer.
3278 * to find out the current global A, take Math::SomeClass->accuracy()
3279 * to find out the current global P, take Math::SomeClass->precision()
3280 * use $x->accuracy() respective $x->precision() for the local setting of $x.
3281 * Please note that $x->accuracy() respecive $x->precision() fall back to the
3282 defined globals, when $x's A or P is not set.
3284 =item Creating numbers
3286 * When you create a number, you can give it's desired A or P via:
3287 $x = Math::BigInt->new($number,$A,$P);
3288 * Only one of A or P can be defined, otherwise the result is NaN
3289 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
3290 globals (if set) will be used. Thus changing the global defaults later on
3291 will not change the A or P of previously created numbers (i.e., A and P of
3292 $x will be what was in effect when $x was created)
3293 * If given undef for A and P, B<no> rounding will occur, and the globals will
3294 B<not> be used. This is used by subclasses to create numbers without
3295 suffering rounding in the parent. Thus a subclass is able to have it's own
3296 globals enforced upon creation of a number by using
3297 $x = Math::BigInt->new($number,undef,undef):
3299 use Math::Bigint::SomeSubclass;
3302 Math::BigInt->accuracy(2);
3303 Math::BigInt::SomeSubClass->accuracy(3);
3304 $x = Math::BigInt::SomeSubClass->new(1234);
3306 $x is now 1230, and not 1200. A subclass might choose to implement
3307 this otherwise, e.g. falling back to the parent's A and P.
3311 * If A or P are enabled/defined, they are used to round the result of each
3312 operation according to the rules below
3313 * Negative P is ignored in Math::BigInt, since BigInts never have digits
3314 after the decimal point
3315 * Math::BigFloat uses Math::BigInts internally, but setting A or P inside
3316 Math::BigInt as globals should not tamper with the parts of a BigFloat.
3317 Thus a flag is used to mark all Math::BigFloat numbers as 'never round'
3321 * It only makes sense that a number has only one of A or P at a time.
3322 Since you can set/get both A and P, there is a rule that will practically
3323 enforce only A or P to be in effect at a time, even if both are set.
3324 This is called precedence.
3325 * If two objects are involved in an operation, and one of them has A in
3326 effect, and the other P, this results in an error (NaN).
3327 * A takes precendence over P (Hint: A comes before P). If A is defined, it
3328 is used, otherwise P is used. If neither of them is defined, nothing is
3329 used, i.e. the result will have as many digits as it can (with an
3330 exception for fdiv/fsqrt) and will not be rounded.
3331 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
3332 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
3333 If either the dividend's or the divisor's mantissa has more digits than
3334 the value of F, the higher value will be used instead of F.
3335 This is to limit the digits (A) of the result (just consider what would
3336 happen with unlimited A and P in the case of 1/3 :-)
3337 * fdiv will calculate (at least) 4 more digits than required (determined by
3338 A, P or F), and, if F is not used, round the result
3339 (this will still fail in the case of a result like 0.12345000000001 with A
3340 or P of 5, but this can not be helped - or can it?)
3341 * Thus you can have the math done by on Math::Big* class in three modes:
3342 + never round (this is the default):
3343 This is done by setting A and P to undef. No math operation
3344 will round the result, with fdiv() and fsqrt() as exceptions to guard
3345 against overflows. You must explicitely call bround(), bfround() or
3346 round() (the latter with parameters).
3347 Note: Once you have rounded a number, the settings will 'stick' on it
3348 and 'infect' all other numbers engaged in math operations with it, since
3349 local settings have the highest precedence. So, to get SaferRound[tm],
3350 use a copy() before rounding like this:
3352 $x = Math::BigFloat->new(12.34);
3353 $y = Math::BigFloat->new(98.76);
3354 $z = $x * $y; # 1218.6984
3355 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
3356 $z = $x * $y; # still 1218.6984, without
3357 # copy would have been 1210!
3359 + round after each op:
3360 After each single operation (except for testing like is_zero()), the
3361 method round() is called and the result is rounded appropriately. By
3362 setting proper values for A and P, you can have all-the-same-A or
3363 all-the-same-P modes. For example, Math::Currency might set A to undef,
3364 and P to -2, globally.
3366 ?Maybe an extra option that forbids local A & P settings would be in order,
3367 ?so that intermediate rounding does not 'poison' further math?
3369 =item Overriding globals
3371 * you will be able to give A, P and R as an argument to all the calculation
3372 routines; the second parameter is A, the third one is P, and the fourth is
3373 R (shift right by one for binary operations like badd). P is used only if
3374 the first parameter (A) is undefined. These three parameters override the
3375 globals in the order detailed as follows, i.e. the first defined value
3377 (local: per object, global: global default, parameter: argument to sub)
3380 + local A (if defined on both of the operands: smaller one is taken)
3381 + local P (if defined on both of the operands: bigger one is taken)
3385 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
3386 arguments (A and P) instead of one
3388 =item Local settings
3390 * You can set A and P locally by using $x->accuracy() and $x->precision()
3391 and thus force different A and P for different objects/numbers.
3392 * Setting A or P this way immediately rounds $x to the new value.
3393 * $x->accuracy() clears $x->precision(), and vice versa.
3397 * the rounding routines will use the respective global or local settings.
3398 fround()/bround() is for accuracy rounding, while ffround()/bfround()
3400 * the two rounding functions take as the second parameter one of the
3401 following rounding modes (R):
3402 'even', 'odd', '+inf', '-inf', 'zero', 'trunc'
3403 * you can set and get the global R by using Math::SomeClass->round_mode()
3404 or by setting $Math::SomeClass::round_mode
3405 * after each operation, $result->round() is called, and the result may
3406 eventually be rounded (that is, if A or P were set either locally,
3407 globally or as parameter to the operation)
3408 * to manually round a number, call $x->round($A,$P,$round_mode);
3409 this will round the number by using the appropriate rounding function
3410 and then normalize it.
3411 * rounding modifies the local settings of the number:
3413 $x = Math::BigFloat->new(123.456);
3417 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
3418 will be 4 from now on.
3420 =item Default values
3429 * The defaults are set up so that the new code gives the same results as
3430 the old code (except in a few cases on fdiv):
3431 + Both A and P are undefined and thus will not be used for rounding
3432 after each operation.
3433 + round() is thus a no-op, unless given extra parameters A and P
3439 The actual numbers are stored as unsigned big integers (with seperate sign).
3440 You should neither care about nor depend on the internal representation; it
3441 might change without notice. Use only method calls like C<< $x->sign(); >>
3442 instead relying on the internal hash keys like in C<< $x->{sign}; >>.
3446 Math with the numbers is done (by default) by a module called
3447 Math::BigInt::Calc. This is equivalent to saying:
3449 use Math::BigInt lib => 'Calc';
3451 You can change this by using:
3453 use Math::BigInt lib => 'BitVect';
3455 The following would first try to find Math::BigInt::Foo, then
3456 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
3458 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
3460 Calc.pm uses as internal format an array of elements of some decimal base
3461 (usually 1e5 or 1e7) with the least significant digit first, while BitVect.pm
3462 uses a bit vector of base 2, most significant bit first. Other modules might
3463 use even different means of representing the numbers. See the respective
3464 module documentation for further details.
3468 The sign is either '+', '-', 'NaN', '+inf' or '-inf' and stored seperately.
3470 A sign of 'NaN' is used to represent the result when input arguments are not
3471 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
3472 minus infinity. You will get '+inf' when dividing a positive number by 0, and
3473 '-inf' when dividing any negative number by 0.
3475 =head2 mantissa(), exponent() and parts()
3477 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
3480 $m = $x->mantissa();
3481 $e = $x->exponent();
3482 $y = $m * ( 10 ** $e );
3483 print "ok\n" if $x == $y;
3485 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
3486 in one go. Both the returned mantissa and exponent have a sign.
3488 Currently, for BigInts C<$e> will be always 0, except for NaN, +inf and -inf,
3489 where it will be NaN; and for $x == 0, where it will be 1
3490 (to be compatible with Math::BigFloat's internal representation of a zero as
3493 C<$m> will always be a copy of the original number. The relation between $e
3494 and $m might change in the future, but will always be equivalent in a
3495 numerical sense, e.g. $m might get minimized.
3501 sub bint { Math::BigInt->new(shift); }
3503 $x = Math::BigInt->bstr("1234") # string "1234"
3504 $x = "$x"; # same as bstr()
3505 $x = Math::BigInt->bneg("1234"); # Bigint "-1234"
3506 $x = Math::BigInt->babs("-12345"); # Bigint "12345"
3507 $x = Math::BigInt->bnorm("-0 00"); # BigInt "0"
3508 $x = bint(1) + bint(2); # BigInt "3"
3509 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
3510 $x = bint(1); # BigInt "1"
3511 $x = $x + 5 / 2; # BigInt "3"
3512 $x = $x ** 3; # BigInt "27"
3513 $x *= 2; # BigInt "54"
3514 $x = Math::BigInt->new(0); # BigInt "0"
3516 $x = Math::BigInt->badd(4,5) # BigInt "9"
3517 print $x->bsstr(); # 9e+0
3519 Examples for rounding:
3524 $x = Math::BigFloat->new(123.4567);
3525 $y = Math::BigFloat->new(123.456789);
3526 Math::BigFloat->accuracy(4); # no more A than 4
3528 ok ($x->copy()->fround(),123.4); # even rounding
3529 print $x->copy()->fround(),"\n"; # 123.4
3530 Math::BigFloat->round_mode('odd'); # round to odd
3531 print $x->copy()->fround(),"\n"; # 123.5
3532 Math::BigFloat->accuracy(5); # no more A than 5
3533 Math::BigFloat->round_mode('odd'); # round to odd
3534 print $x->copy()->fround(),"\n"; # 123.46
3535 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
3536 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
3538 Math::BigFloat->accuracy(undef); # A not important now
3539 Math::BigFloat->precision(2); # P important
3540 print $x->copy()->bnorm(),"\n"; # 123.46
3541 print $x->copy()->fround(),"\n"; # 123.46
3543 Examples for converting:
3545 my $x = Math::BigInt->new('0b1'.'01' x 123);
3546 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
3548 =head1 Autocreating constants
3550 After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
3551 and binary constants in the given scope are converted to C<Math::BigInt>.
3552 This conversion happens at compile time.
3556 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
3558 prints the integer value of C<2**100>. Note that without conversion of
3559 constants the expression 2**100 will be calculated as perl scalar.
3561 Please note that strings and floating point constants are not affected,
3564 use Math::BigInt qw/:constant/;
3566 $x = 1234567890123456789012345678901234567890
3567 + 123456789123456789;
3568 $y = '1234567890123456789012345678901234567890'
3569 + '123456789123456789';
3571 do not work. You need an explicit Math::BigInt->new() around one of the
3572 operands. You should also quote large constants to protect loss of precision:
3576 $x = Math::BigInt->new('1234567889123456789123456789123456789');
3578 Without the quotes Perl would convert the large number to a floating point
3579 constant at compile time and then hand the result to BigInt, which results in
3580 an truncated result or a NaN.
3582 This also applies to integers that look like floating point constants:
3584 use Math::BigInt ':constant';
3586 print ref(123e2),"\n";
3587 print ref(123.2e2),"\n";
3589 will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
3590 to get this to work.
3594 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
3595 must be made in the second case. For long numbers, the copy can eat up to 20%
3596 of the work (in the case of addition/subtraction, less for
3597 multiplication/division). If $y is very small compared to $x, the form
3598 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
3599 more time then the actual addition.
3601 With a technique called copy-on-write, the cost of copying with overload could
3602 be minimized or even completely avoided. A test implementation of COW did show
3603 performance gains for overloaded math, but introduced a performance loss due
3604 to a constant overhead for all other operatons.
3606 The rewritten version of this module is slower on certain operations, like
3607 new(), bstr() and numify(). The reason are that it does now more work and
3608 handles more cases. The time spent in these operations is usually gained in
3609 the other operations so that programs on the average should get faster. If
3610 they don't, please contect the author.
3612 Some operations may be slower for small numbers, but are significantly faster
3613 for big numbers. Other operations are now constant (O(1), like bneg(), babs()
3614 etc), instead of O(N) and thus nearly always take much less time. These
3615 optimizations were done on purpose.
3617 If you find the Calc module to slow, try to install any of the replacement
3618 modules and see if they help you.
3620 =head2 Alternative math libraries
3622 You can use an alternative library to drive Math::BigInt via:
3624 use Math::BigInt lib => 'Module';
3626 See L<MATH LIBRARY> for more information.
3628 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
3632 =head1 Subclassing Math::BigInt
3634 The basic design of Math::BigInt allows simple subclasses with very little
3635 work, as long as a few simple rules are followed:
3641 The public API must remain consistent, i.e. if a sub-class is overloading
3642 addition, the sub-class must use the same name, in this case badd(). The
3643 reason for this is that Math::BigInt is optimized to call the object methods
3648 The private object hash keys like C<$x->{sign}> may not be changed, but
3649 additional keys can be added, like C<$x->{_custom}>.
3653 Accessor functions are available for all existing object hash keys and should
3654 be used instead of directly accessing the internal hash keys. The reason for
3655 this is that Math::BigInt itself has a pluggable interface which permits it
3656 to support different storage methods.
3660 More complex sub-classes may have to replicate more of the logic internal of
3661 Math::BigInt if they need to change more basic behaviors. A subclass that
3662 needs to merely change the output only needs to overload C<bstr()>.
3664 All other object methods and overloaded functions can be directly inherited
3665 from the parent class.
3667 At the very minimum, any subclass will need to provide it's own C<new()> and can
3668 store additional hash keys in the object. There are also some package globals
3669 that must be defined, e.g.:
3673 $precision = -2; # round to 2 decimal places
3674 $round_mode = 'even';
3677 Additionally, you might want to provide the following two globals to allow
3678 auto-upgrading and auto-downgrading to work correctly:
3683 This allows Math::BigInt to correctly retrieve package globals from the
3684 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
3685 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
3691 in your subclass to automatically inherit the overloading from the parent. If
3692 you like, you can change part of the overloading, look at Math::String for an
3697 When used like this:
3699 use Math::BigInt upgrade => 'Foo::Bar';
3701 certain operations will 'upgrade' their calculation and thus the result to
3702 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
3704 use Math::BigInt upgrade => 'Math::BigFloat';
3706 As a shortcut, you can use the module C<bignum>:
3710 Also good for oneliners:
3712 perl -Mbignum -le 'print 2 ** 255'
3714 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
3715 as well es preserve accuracy (as in sqrt(3)).
3717 Beware: This feature is not fully implemented yet.
3721 The following methods upgrade themselves unconditionally; that is if upgrade
3722 is in effect, they will always hand up their work:
3734 Beware: This list is not complete.
3736 All other methods upgrade themselves only when one (or all) of their
3737 arguments are of the class mentioned in $upgrade (This might change in later
3738 versions to a more sophisticated scheme):
3744 =item Out of Memory!
3746 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
3747 C<eval()> in your code will crash with "Out of memory". This is probably an
3748 overload/exporter bug. You can workaround by not having C<eval()>
3749 and ':constant' at the same time or upgrade your Perl to a newer version.
3751 =item Fails to load Calc on Perl prior 5.6.0
3753 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
3754 will fall back to eval { require ... } when loading the math lib on Perls
3755 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
3756 filesystems using a different seperator.
3762 Some things might not work as you expect them. Below is documented what is
3763 known to be troublesome:
3767 =item stringify, bstr(), bsstr() and 'cmp'
3769 Both stringify and bstr() now drop the leading '+'. The old code would return
3770 '+3', the new returns '3'. This is to be consistent with Perl and to make
3771 cmp (especially with overloading) to work as you expect. It also solves
3772 problems with Test.pm, it's ok() uses 'eq' internally.
3774 Mark said, when asked about to drop the '+' altogether, or make only cmp work:
3776 I agree (with the first alternative), don't add the '+' on positive
3777 numbers. It's not as important anymore with the new internal
3778 form for numbers. It made doing things like abs and neg easier,
3779 but those have to be done differently now anyway.
3781 So, the following examples will now work all as expected:
3784 BEGIN { plan tests => 1 }
3787 my $x = new Math::BigInt 3*3;
3788 my $y = new Math::BigInt 3*3;
3791 print "$x eq 9" if $x eq $y;
3792 print "$x eq 9" if $x eq '9';
3793 print "$x eq 9" if $x eq 3*3;
3795 Additionally, the following still works:
3797 print "$x == 9" if $x == $y;
3798 print "$x == 9" if $x == 9;
3799 print "$x == 9" if $x == 3*3;
3801 There is now a C<bsstr()> method to get the string in scientific notation aka
3802 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
3803 for comparisation, but Perl will represent some numbers as 100 and others
3804 as 1e+308. If in doubt, convert both arguments to Math::BigInt before doing eq:
3807 BEGIN { plan tests => 3 }
3810 $x = Math::BigInt->new('1e56'); $y = 1e56;
3811 ok ($x,$y); # will fail
3812 ok ($x->bsstr(),$y); # okay
3813 $y = Math::BigInt->new($y);
3816 Alternatively, simple use <=> for comparisations, that will get it always
3817 right. There is not yet a way to get a number automatically represented as
3818 a string that matches exactly the way Perl represents it.
3822 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
3825 $x = Math::BigInt->new(123);
3826 $y = int($x); # BigInt 123
3827 $x = Math::BigFloat->new(123.45);
3828 $y = int($x); # BigInt 123
3830 In all Perl versions you can use C<as_number()> for the same effect:
3832 $x = Math::BigFloat->new(123.45);
3833 $y = $x->as_number(); # BigInt 123
3835 This also works for other subclasses, like Math::String.
3837 It is yet unlcear whether overloaded int() should return a scalar or a BigInt.
3841 The following will probably not do what you expect:
3843 $c = Math::BigInt->new(123);
3844 print $c->length(),"\n"; # prints 30
3846 It prints both the number of digits in the number and in the fraction part
3847 since print calls C<length()> in list context. Use something like:
3849 print scalar $c->length(),"\n"; # prints 3
3853 The following will probably not do what you expect:
3855 print $c->bdiv(10000),"\n";
3857 It prints both quotient and remainder since print calls C<bdiv()> in list
3858 context. Also, C<bdiv()> will modify $c, so be carefull. You probably want
3861 print $c / 10000,"\n";
3862 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
3866 The quotient is always the greatest integer less than or equal to the
3867 real-valued quotient of the two operands, and the remainder (when it is
3868 nonzero) always has the same sign as the second operand; so, for
3878 As a consequence, the behavior of the operator % agrees with the
3879 behavior of Perl's built-in % operator (as documented in the perlop
3880 manpage), and the equation
3882 $x == ($x / $y) * $y + ($x % $y)
3884 holds true for any $x and $y, which justifies calling the two return
3885 values of bdiv() the quotient and remainder. The only exception to this rule
3886 are when $y == 0 and $x is negative, then the remainder will also be
3887 negative. See below under "infinity handling" for the reasoning behing this.
3889 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
3890 not change BigInt's way to do things. This is because under 'use integer' Perl
3891 will do what the underlying C thinks is right and this is different for each
3892 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
3893 the author to implement it ;)
3895 =item infinity handling
3897 Here are some examples that explain the reasons why certain results occur while
3900 The following table shows the result of the division and the remainder, so that
3901 the equation above holds true. Some "ordinary" cases are strewn in to show more
3902 clearly the reasoning:
3904 A / B = C, R so that C * B + R = A
3905 =========================================================
3906 5 / 8 = 0, 5 0 * 8 + 5 = 5
3907 0 / 8 = 0, 0 0 * 8 + 0 = 0
3908 0 / inf = 0, 0 0 * inf + 0 = 0
3909 0 /-inf = 0, 0 0 * -inf + 0 = 0
3910 5 / inf = 0, 5 0 * inf + 5 = 5
3911 5 /-inf = 0, 5 0 * -inf + 5 = 5
3912 -5/ inf = 0, -5 0 * inf + -5 = -5
3913 -5/-inf = 0, -5 0 * -inf + -5 = -5
3914 inf/ 5 = inf, 0 inf * 5 + 0 = inf
3915 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
3916 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
3917 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
3918 5/ 5 = 1, 0 1 * 5 + 0 = 5
3919 -5/ -5 = 1, 0 1 * -5 + 0 = -5
3920 inf/ inf = 1, 0 1 * inf + 0 = inf
3921 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
3922 inf/-inf = -1, 0 -1 * -inf + 0 = inf
3923 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
3924 8/ 0 = inf, 8 inf * 0 + 8 = 8
3925 inf/ 0 = inf, inf inf * 0 + inf = inf
3928 These cases below violate the "remainder has the sign of the second of the two
3929 arguments", since they wouldn't match up otherwise.
3931 A / B = C, R so that C * B + R = A
3932 ========================================================
3933 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
3934 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
3936 =item Modifying and =
3940 $x = Math::BigFloat->new(5);
3943 It will not do what you think, e.g. making a copy of $x. Instead it just makes
3944 a second reference to the B<same> object and stores it in $y. Thus anything
3945 that modifies $x (except overloaded operators) will modify $y, and vice versa.
3946 Or in other words, C<=> is only safe if you modify your BigInts only via
3947 overloaded math. As soon as you use a method call it breaks:
3950 print "$x, $y\n"; # prints '10, 10'
3952 If you want a true copy of $x, use:
3956 You can also chain the calls like this, this will make first a copy and then
3959 $y = $x->copy()->bmul(2);
3961 See also the documentation for overload.pm regarding C<=>.
3965 C<bpow()> (and the rounding functions) now modifies the first argument and
3966 returns it, unlike the old code which left it alone and only returned the
3967 result. This is to be consistent with C<badd()> etc. The first three will
3968 modify $x, the last one won't:
3970 print bpow($x,$i),"\n"; # modify $x
3971 print $x->bpow($i),"\n"; # ditto
3972 print $x **= $i,"\n"; # the same
3973 print $x ** $i,"\n"; # leave $x alone
3975 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
3977 =item Overloading -$x
3987 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
3988 needs to preserve $x since it does not know that it later will get overwritten.
3989 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
3991 With Copy-On-Write, this issue would be gone, but C-o-W is not implemented
3992 since it is slower for all other things.
3994 =item Mixing different object types
3996 In Perl you will get a floating point value if you do one of the following:
4002 With overloaded math, only the first two variants will result in a BigFloat:
4007 $mbf = Math::BigFloat->new(5);
4008 $mbi2 = Math::BigInteger->new(5);
4009 $mbi = Math::BigInteger->new(2);
4011 # what actually gets called:
4012 $float = $mbf + $mbi; # $mbf->badd()
4013 $float = $mbf / $mbi; # $mbf->bdiv()
4014 $integer = $mbi + $mbf; # $mbi->badd()
4015 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4016 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4018 This is because math with overloaded operators follows the first (dominating)
4019 operand, and the operation of that is called and returns thus the result. So,
4020 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4021 the result should be a Math::BigFloat or the second operant is one.
4023 To get a Math::BigFloat you either need to call the operation manually,
4024 make sure the operands are already of the proper type or casted to that type
4025 via Math::BigFloat->new():
4027 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4029 Beware of simple "casting" the entire expression, this would only convert
4030 the already computed result:
4032 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4034 Beware also of the order of more complicated expressions like:
4036 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4037 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4039 If in doubt, break the expression into simpler terms, or cast all operands
4040 to the desired resulting type.
4042 Scalar values are a bit different, since:
4047 will both result in the proper type due to the way the overloaded math works.
4049 This section also applies to other overloaded math packages, like Math::String.
4051 One solution to you problem might be L<autoupgrading|upgrading>.
4055 C<bsqrt()> works only good if the result is a big integer, e.g. the square
4056 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4059 If you want a better approximation of the square root, then use:
4061 $x = Math::BigFloat->new(12);
4062 Math::BigFloat->precision(0);
4063 Math::BigFloat->round_mode('even');
4064 print $x->copy->bsqrt(),"\n"; # 4
4066 Math::BigFloat->precision(2);
4067 print $x->bsqrt(),"\n"; # 3.46
4068 print $x->bsqrt(3),"\n"; # 3.464
4072 For negative numbers in base see also L<brsft|brsft>.
4078 This program is free software; you may redistribute it and/or modify it under
4079 the same terms as Perl itself.
4083 L<Math::BigFloat> and L<Math::Big> as well as L<Math::BigInt::BitVect>,
4084 L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4087 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4088 more documentation including a full version history, testcases, empty
4089 subclass files and benchmarks.
4093 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4094 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001.