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->round_mode() work
166 my $class = ref($self) || $self || __PACKAGE__;
170 return ${"${class}::upgrade"} = $u;
172 return ${"${class}::upgrade"};
178 # make Class->round_mode() work
180 my $class = ref($self) || $self || __PACKAGE__;
183 die ('div_scale must be greater than zero') if $_[0] < 0;
184 ${"${class}::div_scale"} = shift;
186 return ${"${class}::div_scale"};
191 # $x->accuracy($a); ref($x) $a
192 # $x->accuracy(); ref($x)
193 # Class->accuracy(); class
194 # Class->accuracy($a); class $a
197 my $class = ref($x) || $x || __PACKAGE__;
200 # need to set new value?
204 die ('accuracy must not be zero') if defined $a && $a == 0;
207 # $object->accuracy() or fallback to global
208 $x->bround($a) if defined $a;
209 $x->{_a} = $a; # set/overwrite, even if not rounded
210 $x->{_p} = undef; # clear P
215 ${"${class}::accuracy"} = $a;
216 ${"${class}::precision"} = undef; # clear P
218 return $a; # shortcut
223 # $object->accuracy() or fallback to global
224 return $x->{_a} || ${"${class}::accuracy"};
226 return ${"${class}::accuracy"};
231 # $x->precision($p); ref($x) $p
232 # $x->precision(); ref($x)
233 # Class->precision(); class
234 # Class->precision($p); class $p
237 my $class = ref($x) || $x || __PACKAGE__;
240 # need to set new value?
246 # $object->precision() or fallback to global
247 $x->bfround($p) if defined $p;
248 $x->{_p} = $p; # set/overwrite, even if not rounded
249 $x->{_a} = undef; # clear A
254 ${"${class}::precision"} = $p;
255 ${"${class}::accuracy"} = undef; # clear A
257 return $p; # shortcut
262 # $object->precision() or fallback to global
263 return $x->{_p} || ${"${class}::precision"};
265 return ${"${class}::precision"};
270 # return (later set?) configuration data as hash ref
271 my $class = shift || 'Math::BigInt';
277 lib_version => ${"${lib}::VERSION"},
281 qw/upgrade downgrade precisison accuracy round_mode VERSION div_scale/)
283 $cfg->{lc($_)} = ${"${class}::$_"};
290 # select accuracy parameter based on precedence,
291 # used by bround() and bfround(), may return undef for scale (means no op)
292 my ($x,$s,$m,$scale,$mode) = @_;
293 $scale = $x->{_a} if !defined $scale;
294 $scale = $s if (!defined $scale);
295 $mode = $m if !defined $mode;
296 return ($scale,$mode);
301 # select precision parameter based on precedence,
302 # used by bround() and bfround(), may return undef for scale (means no op)
303 my ($x,$s,$m,$scale,$mode) = @_;
304 $scale = $x->{_p} if !defined $scale;
305 $scale = $s if (!defined $scale);
306 $mode = $m if !defined $mode;
307 return ($scale,$mode);
310 ##############################################################################
318 # if two arguments, the first one is the class to "swallow" subclasses
326 return unless ref($x); # only for objects
328 my $self = {}; bless $self,$c;
330 foreach my $k (keys %$x)
334 $self->{value} = $CALC->_copy($x->{value}); next;
336 if (!($r = ref($x->{$k})))
338 $self->{$k} = $x->{$k}; next;
342 $self->{$k} = \${$x->{$k}};
344 elsif ($r eq 'ARRAY')
346 $self->{$k} = [ @{$x->{$k}} ];
350 # only one level deep!
351 foreach my $h (keys %{$x->{$k}})
353 $self->{$k}->{$h} = $x->{$k}->{$h};
359 if ($xk->can('copy'))
361 $self->{$k} = $xk->copy();
365 $self->{$k} = $xk->new($xk);
374 # create a new BigInt object from a string or another BigInt object.
375 # see hash keys documented at top
377 # the argument could be an object, so avoid ||, && etc on it, this would
378 # cause costly overloaded code to be called. The only allowed ops are
381 my ($class,$wanted,$a,$p,$r) = @_;
383 # avoid numify-calls by not using || on $wanted!
384 return $class->bzero($a,$p) if !defined $wanted; # default to 0
385 return $class->copy($wanted,$a,$p,$r) if ref($wanted);
387 $class->import() if $IMPORT == 0; # make require work
389 my $self = {}; bless $self, $class;
390 # handle '+inf', '-inf' first
391 if ($wanted =~ /^[+-]?inf$/)
393 $self->{value} = $CALC->_zero();
394 $self->{sign} = $wanted; $self->{sign} = '+inf' if $self->{sign} eq 'inf';
397 # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign
398 my ($mis,$miv,$mfv,$es,$ev) = _split(\$wanted);
401 die "$wanted is not a number initialized to $class" if !$NaNOK;
403 $self->{value} = $CALC->_zero();
404 $self->{sign} = $nan;
409 # _from_hex or _from_bin
410 $self->{value} = $mis->{value};
411 $self->{sign} = $mis->{sign};
412 return $self; # throw away $mis
414 # make integer from mantissa by adjusting exp, then convert to bigint
415 $self->{sign} = $$mis; # store sign
416 $self->{value} = $CALC->_zero(); # for all the NaN cases
417 my $e = int("$$es$$ev"); # exponent (avoid recursion)
420 my $diff = $e - CORE::length($$mfv);
421 if ($diff < 0) # Not integer
424 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
425 $self->{sign} = $nan;
429 # adjust fraction and add it to value
430 # print "diff > 0 $$miv\n";
431 $$miv = $$miv . ($$mfv . '0' x $diff);
436 if ($$mfv ne '') # e <= 0
438 # fraction and negative/zero E => NOI
439 #print "NOI 2 \$\$mfv '$$mfv'\n";
440 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
441 $self->{sign} = $nan;
445 # xE-y, and empty mfv
448 if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
451 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
452 $self->{sign} = $nan;
456 $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
457 $self->{value} = $CALC->_new($miv) if $self->{sign} =~ /^[+-]$/;
458 # if any of the globals is set, use them to round and store them inside $self
459 # do not round for new($x,undef,undef) since that is used by MBF to signal
461 $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p;
462 # print "mbi new $self\n";
468 # create a bigint 'NaN', if given a BigInt, set it to 'NaN'
470 $self = $class if !defined $self;
473 my $c = $self; $self = {}; bless $self, $c;
475 $self->import() if $IMPORT == 0; # make require work
476 return if $self->modify('bnan');
478 if ($self->can('_bnan'))
480 # use subclass to initialize
485 # otherwise do our own thing
486 $self->{value} = $CALC->_zero();
488 $self->{value} = $CALC->_zero();
489 $self->{sign} = $nan;
490 delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly
496 # create a bigint '+-inf', if given a BigInt, set it to '+-inf'
497 # the sign is either '+', or if given, used from there
499 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
500 $self = $class if !defined $self;
503 my $c = $self; $self = {}; bless $self, $c;
505 $self->import() if $IMPORT == 0; # make require work
506 return if $self->modify('binf');
508 if ($self->can('_binf'))
510 # use subclass to initialize
515 # otherwise do our own thing
516 $self->{value} = $CALC->_zero();
518 $self->{sign} = $sign.'inf';
519 ($self->{_a},$self->{_p}) = @_; # take over requested rounding
525 # create a bigint '+0', if given a BigInt, set it to 0
527 $self = $class if !defined $self;
531 my $c = $self; $self = {}; bless $self, $c;
533 $self->import() if $IMPORT == 0; # make require work
534 return if $self->modify('bzero');
536 if ($self->can('_bzero'))
538 # use subclass to initialize
543 # otherwise do our own thing
544 $self->{value} = $CALC->_zero();
550 if (defined $self->{_a} && defined $_[0] && $_[0] > $self->{_a});
552 if (defined $self->{_p} && defined $_[1] && $_[1] < $self->{_p});
559 # create a bigint '+1' (or -1 if given sign '-'),
560 # if given a BigInt, set it to +1 or -1, respecively
562 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
563 $self = $class if !defined $self;
567 my $c = $self; $self = {}; bless $self, $c;
569 $self->import() if $IMPORT == 0; # make require work
570 return if $self->modify('bone');
572 if ($self->can('_bone'))
574 # use subclass to initialize
579 # otherwise do our own thing
580 $self->{value} = $CALC->_one();
582 $self->{sign} = $sign;
586 if (defined $self->{_a} && defined $_[0] && $_[0] > $self->{_a});
588 if (defined $self->{_p} && defined $_[1] && $_[1] < $self->{_p});
593 ##############################################################################
594 # string conversation
598 # (ref to BFLOAT or num_str ) return num_str
599 # Convert number from internal format to scientific string format.
600 # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
601 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
602 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
604 if ($x->{sign} !~ /^[+-]$/)
606 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
609 my ($m,$e) = $x->parts();
610 # e can only be positive
612 # MBF: my $s = $e->{sign}; $s = '' if $s eq '-'; my $sep = 'e'.$s;
613 return $m->bstr().$sign.$e->bstr();
618 # make a string from bigint object
619 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
620 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
622 if ($x->{sign} !~ /^[+-]$/)
624 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
627 my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
628 return $es.${$CALC->_str($x->{value})};
633 # Make a "normal" scalar from a BigInt object
634 my $x = shift; $x = $class->new($x) unless ref $x;
635 return $x->{sign} if $x->{sign} !~ /^[+-]$/;
636 my $num = $CALC->_num($x->{value});
637 return -$num if $x->{sign} eq '-';
641 ##############################################################################
642 # public stuff (usually prefixed with "b")
646 # return the sign of the number: +/-/NaN
647 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
652 sub _find_round_parameters
654 # After any operation or when calling round(), the result is rounded by
655 # regarding the A & P from arguments, local parameters, or globals.
657 # This procedure finds the round parameters, but it is for speed reasons
658 # duplicated in round. Otherwise, it is tested by the testsuite and used
661 my ($self,$a,$p,$r,@args) = @_;
662 # $a accuracy, if given by caller
663 # $p precision, if given by caller
664 # $r round_mode, if given by caller
665 # @args all 'other' arguments (0 for unary, 1 for binary ops)
667 # leave bigfloat parts alone
668 return ($self) if exists $self->{_f} && $self->{_f} & MB_NEVER_ROUND != 0;
670 my $c = ref($self); # find out class of argument(s)
673 # now pick $a or $p, but only if we have got "arguments"
676 foreach ($self,@args)
678 # take the defined one, or if both defined, the one that is smaller
679 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
684 # even if $a is defined, take $p, to signal error for both defined
685 foreach ($self,@args)
687 # take the defined one, or if both defined, the one that is bigger
689 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
692 # if still none defined, use globals (#2)
693 $a = ${"$c\::accuracy"} unless defined $a;
694 $p = ${"$c\::precision"} unless defined $p;
697 return ($self) unless defined $a || defined $p; # early out
699 # set A and set P is an fatal error
700 return ($self->bnan()) if defined $a && defined $p;
702 $r = ${"$c\::round_mode"} unless defined $r;
703 die "Unknown round mode '$r'" if $r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/;
705 return ($self,$a,$p,$r);
710 # Round $self according to given parameters, or given second argument's
711 # parameters or global defaults
713 # for speed reasons, _find_round_parameters is embeded here:
715 my ($self,$a,$p,$r,@args) = @_;
716 # $a accuracy, if given by caller
717 # $p precision, if given by caller
718 # $r round_mode, if given by caller
719 # @args all 'other' arguments (0 for unary, 1 for binary ops)
721 # leave bigfloat parts alone
722 return ($self) if exists $self->{_f} && $self->{_f} & MB_NEVER_ROUND != 0;
724 my $c = ref($self); # find out class of argument(s)
727 # now pick $a or $p, but only if we have got "arguments"
730 foreach ($self,@args)
732 # take the defined one, or if both defined, the one that is smaller
733 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
738 # even if $a is defined, take $p, to signal error for both defined
739 foreach ($self,@args)
741 # take the defined one, or if both defined, the one that is bigger
743 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
746 # if still none defined, use globals (#2)
747 $a = ${"$c\::accuracy"} unless defined $a;
748 $p = ${"$c\::precision"} unless defined $p;
751 return $self unless defined $a || defined $p; # early out
753 # set A and set P is an fatal error
754 return $self->bnan() if defined $a && defined $p;
756 $r = ${"$c\::round_mode"} unless defined $r;
757 die "Unknown round mode '$r'" if $r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/;
759 # now round, by calling either fround or ffround:
762 $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a;
764 else # both can't be undefined due to early out
766 $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p;
768 $self->bnorm(); # after round, normalize
773 # (numstr or BINT) return BINT
774 # Normalize number -- no-op here
775 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
781 # (BINT or num_str) return BINT
782 # make number absolute, or return absolute BINT from string
783 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
785 return $x if $x->modify('babs');
786 # post-normalized abs for internal use (does nothing for NaN)
787 $x->{sign} =~ s/^-/+/;
793 # (BINT or num_str) return BINT
794 # negate number or make a negated number from string
795 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
797 return $x if $x->modify('bneg');
799 # for +0 dont negate (to have always normalized)
800 $x->{sign} =~ tr/+-/-+/ if !$x->is_zero(); # does nothing for NaN
806 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
807 # (BINT or num_str, BINT or num_str) return cond_code
808 my ($self,$x,$y) = objectify(2,@_);
810 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
812 # handle +-inf and NaN
813 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
814 return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
815 return +1 if $x->{sign} eq '+inf';
816 return -1 if $x->{sign} eq '-inf';
817 return -1 if $y->{sign} eq '+inf';
820 # check sign for speed first
821 return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
822 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
825 my $xz = $x->is_zero();
826 my $yz = $y->is_zero();
827 return 0 if $xz && $yz; # 0 <=> 0
828 return -1 if $xz && $y->{sign} eq '+'; # 0 <=> +y
829 return 1 if $yz && $x->{sign} eq '+'; # +x <=> 0
831 # post-normalized compare for internal use (honors signs)
832 if ($x->{sign} eq '+')
834 return 1 if $y->{sign} eq '-'; # 0 check handled above
835 return $CALC->_acmp($x->{value},$y->{value});
839 return -1 if $y->{sign} eq '+';
840 $CALC->_acmp($y->{value},$x->{value}); # swaped (lib does only 0,1,-1)
845 # Compares 2 values, ignoring their signs.
846 # Returns one of undef, <0, =0, >0. (suitable for sort)
847 # (BINT, BINT) return cond_code
848 my ($self,$x,$y) = objectify(2,@_);
850 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
852 # handle +-inf and NaN
853 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
854 return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
855 return +1; # inf is always bigger
857 $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
862 # add second arg (BINT or string) to first (BINT) (modifies first)
863 # return result as BINT
864 my ($self,$x,$y,@r) = objectify(2,@_);
866 return $x if $x->modify('badd');
867 # print "mbi badd ",join(' ',caller()),"\n";
868 # print "upgrade => ",$upgrade||'undef',
869 # " \$x (",ref($x),") \$y (",ref($y),")\n";
870 # return $upgrade->badd($x,$y,@r) if defined $upgrade &&
871 # ((ref($x) eq $upgrade) || (ref($y) eq $upgrade));
872 # print "still badd\n";
874 $r[3] = $y; # no push!
875 # inf and NaN handling
876 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
879 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
881 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
883 # +inf++inf or -inf+-inf => same, rest is NaN
884 return $x if $x->{sign} eq $y->{sign};
887 # +-inf + something => +inf
888 # something +-inf => +-inf
889 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
893 my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
897 $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
902 my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
905 #print "swapped sub (a=$a)\n";
906 $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
911 # speedup, if equal, set result to 0
912 #print "equal sub, result = 0\n";
913 $x->{value} = $CALC->_zero();
918 #print "unswapped sub (a=$a)\n";
919 $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
928 # (BINT or num_str, BINT or num_str) return num_str
929 # subtract second arg from first, modify first
930 my ($self,$x,$y,@r) = objectify(2,@_);
932 return $x if $x->modify('bsub');
933 # return $upgrade->badd($x,$y,@r) if defined $upgrade &&
934 # ((ref($x) eq $upgrade) || (ref($y) eq $upgrade));
938 return $x->round(@r);
941 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
942 $x->badd($y,@r); # badd does not leave internal zeros
943 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
944 $x; # already rounded by badd() or no round necc.
949 # increment arg by one
950 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
951 return $x if $x->modify('binc');
953 if ($x->{sign} eq '+')
955 $x->{value} = $CALC->_inc($x->{value});
956 return $x->round($a,$p,$r);
958 elsif ($x->{sign} eq '-')
960 $x->{value} = $CALC->_dec($x->{value});
961 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
962 return $x->round($a,$p,$r);
964 # inf, nan handling etc
965 $x->badd($self->__one(),$a,$p,$r); # badd does round
970 # decrement arg by one
971 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
972 return $x if $x->modify('bdec');
974 my $zero = $CALC->_is_zero($x->{value}) && $x->{sign} eq '+';
976 if (($x->{sign} eq '-') || $zero)
978 $x->{value} = $CALC->_inc($x->{value});
979 $x->{sign} = '-' if $zero; # 0 => 1 => -1
980 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
981 return $x->round($a,$p,$r);
984 elsif ($x->{sign} eq '+')
986 $x->{value} = $CALC->_dec($x->{value});
987 return $x->round($a,$p,$r);
989 # inf, nan handling etc
990 $x->badd($self->__one('-'),$a,$p,$r); # badd does round
995 # not implemented yet
996 my ($self,$x,$base,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
998 return $upgrade->blog($x,$base,$a,$p,$r) if defined $upgrade;
1005 # (BINT or num_str, BINT or num_str) return BINT
1006 # does not modify arguments, but returns new object
1007 # Lowest Common Multiplicator
1009 my $y = shift; my ($x);
1016 $x = $class->new($y);
1018 while (@_) { $x = __lcm($x,shift); }
1024 # (BINT or num_str, BINT or num_str) return BINT
1025 # does not modify arguments, but returns new object
1026 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1029 $y = __PACKAGE__->new($y) if !ref($y);
1031 my $x = $y->copy(); # keep arguments
1032 if ($CALC->can('_gcd'))
1036 $y = shift; $y = $self->new($y) if !ref($y);
1037 next if $y->is_zero();
1038 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1039 $x->{value} = $CALC->_gcd($x->{value},$y->{value}); last if $x->is_one();
1046 $y = shift; $y = $self->new($y) if !ref($y);
1047 $x = __gcd($x,$y->copy()); last if $x->is_one(); # _gcd handles NaN
1055 # (num_str or BINT) return BINT
1056 # represent ~x as twos-complement number
1057 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1058 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1060 return $x if $x->modify('bnot');
1061 $x->bneg()->bdec(); # bdec already does round
1064 # is_foo test routines
1068 # return true if arg (BINT or num_str) is zero (array '+', '0')
1069 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1070 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1072 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1073 $CALC->_is_zero($x->{value});
1078 # return true if arg (BINT or num_str) is NaN
1079 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1081 return 1 if $x->{sign} eq $nan;
1087 # return true if arg (BINT or num_str) is +-inf
1088 my ($self,$x,$sign) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1090 $sign = '' if !defined $sign;
1091 return 0 if $sign !~ /^([+-]|)$/;
1095 return 1 if ($x->{sign} =~ /^[+-]inf$/);
1098 $sign = quotemeta($sign.'inf');
1099 return 1 if ($x->{sign} =~ /^$sign$/);
1105 # return true if arg (BINT or num_str) is +1
1106 # or -1 if sign is given
1107 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1108 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1110 $sign = '' if !defined $sign; $sign = '+' if $sign ne '-';
1112 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1113 $CALC->_is_one($x->{value});
1118 # return true when arg (BINT or num_str) is odd, false for even
1119 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1120 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1122 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1123 $CALC->_is_odd($x->{value});
1128 # return true when arg (BINT or num_str) is even, false for odd
1129 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1130 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1132 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1133 $CALC->_is_even($x->{value});
1138 # return true when arg (BINT or num_str) is positive (>= 0)
1139 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1140 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1142 return 1 if $x->{sign} =~ /^\+/;
1148 # return true when arg (BINT or num_str) is negative (< 0)
1149 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1150 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1152 return 1 if ($x->{sign} =~ /^-/);
1158 # return true when arg (BINT or num_str) is an integer
1159 # always true for BigInt, but different for Floats
1160 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1161 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1163 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1166 ###############################################################################
1170 # multiply two numbers -- stolen from Knuth Vol 2 pg 233
1171 # (BINT or num_str, BINT or num_str) return BINT
1172 my ($self,$x,$y,@r) = objectify(2,@_);
1174 return $x if $x->modify('bmul');
1176 $r[3] = $y; # no push here
1178 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1181 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1183 return $x->bnan() if $x->is_zero() || $y->is_zero();
1184 # result will always be +-inf:
1185 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1186 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1187 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1188 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1189 return $x->binf('-');
1192 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1194 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1195 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1201 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1202 my ($self,$x,$y) = @_;
1204 # NaN if x == NaN or y == NaN or x==y==0
1205 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1206 if (($x->is_nan() || $y->is_nan()) ||
1207 ($x->is_zero() && $y->is_zero()));
1209 # +-inf / +-inf == NaN, reminder also NaN
1210 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1212 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1214 # x / +-inf => 0, remainder x (works even if x == 0)
1215 if ($y->{sign} =~ /^[+-]inf$/)
1217 my $t = $x->copy(); # binf clobbers up $x
1218 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1221 # 5 / 0 => +inf, -6 / 0 => -inf
1222 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1223 # exception: -8 / 0 has remainder -8, not 8
1224 # exception: -inf / 0 has remainder -inf, not inf
1227 # +-inf / 0 => special case for -inf
1228 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1229 if (!$x->is_zero() && !$x->is_inf())
1231 my $t = $x->copy(); # binf clobbers up $x
1233 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1237 # last case: +-inf / ordinary number
1239 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1241 return wantarray ? ($x,$self->bzero()) : $x;
1246 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1247 # (BINT,BINT) (quo,rem) or BINT (only rem)
1248 my ($self,$x,$y,@r) = objectify(2,@_);
1250 return $x if $x->modify('bdiv');
1252 return $self->_div_inf($x,$y)
1253 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1255 $r[3] = $y; # no push!
1259 wantarray ? ($x->round(@r),$self->bzero(@r)):$x->round(@r) if $x->is_zero();
1261 # Is $x in the interval [0, $y) (aka $x <= $y) ?
1262 my $cmp = $CALC->_acmp($x->{value},$y->{value});
1263 if (($cmp < 0) and (($x->{sign} eq $y->{sign}) or !wantarray))
1265 return $upgrade->bdiv($x,$y,@r) if defined $upgrade;
1267 return $x->bzero()->round(@r) unless wantarray;
1268 my $t = $x->copy(); # make copy first, because $x->bzero() clobbers $x
1269 return ($x->bzero()->round(@r),$t);
1273 # shortcut, both are the same, so set to +/- 1
1274 $x->__one( ($x->{sign} ne $y->{sign} ? '-' : '+') );
1275 return $x unless wantarray;
1276 return ($x->round(@r),$self->bzero(@r));
1279 # calc new sign and in case $y == +/- 1, return $x
1280 my $xsign = $x->{sign}; # keep
1281 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1282 # check for / +-1 (cant use $y->is_one due to '-'
1283 if ($CALC->_is_one($y->{value}))
1285 return wantarray ? ($x->round(@r),$self->bzero(@r)) : $x->round(@r);
1290 my $rem = $self->bzero();
1291 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1292 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1294 if (! $CALC->_is_zero($rem->{value}))
1296 $rem->{sign} = $y->{sign};
1297 $rem = $y-$rem if $xsign ne $y->{sign}; # one of them '-'
1301 $rem->{sign} = '+'; # dont leave -0
1307 $x->{value} = $CALC->_div($x->{value},$y->{value});
1308 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1314 # modulus (or remainder)
1315 # (BINT or num_str, BINT or num_str) return BINT
1316 my ($self,$x,$y,@r) = objectify(2,@_);
1318 return $x if $x->modify('bmod');
1319 $r[3] = $y; # no push!
1320 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1322 my ($d,$r) = $self->_div_inf($x,$y);
1323 return $r->round(@r);
1326 if ($CALC->can('_mod'))
1328 # calc new sign and in case $y == +/- 1, return $x
1329 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1330 if (!$CALC->_is_zero($x->{value}))
1332 my $xsign = $x->{sign};
1333 $x->{sign} = $y->{sign};
1334 $x = $y-$x if $xsign ne $y->{sign}; # one of them '-'
1338 $x->{sign} = '+'; # dont leave -0
1340 return $x->round(@r);
1342 my ($t,$rem) = $self->bdiv($x->copy(),$y,@r); # slow way (also rounds)
1344 foreach (qw/value sign _a _p/)
1346 $x->{$_} = $rem->{$_};
1353 # (BINT or num_str, BINT or num_str) return BINT
1354 # compute factorial numbers
1355 # modifies first argument
1356 my ($self,$x,@r) = objectify(1,@_);
1358 return $x if $x->modify('bfac');
1360 return $x->bnan() if $x->{sign} ne '+'; # inf, NnN, <0 etc => NaN
1361 return $x->bone(@r) if $x->is_zero() || $x->is_one(); # 0 or 1 => 1
1363 if ($CALC->can('_fac'))
1365 $x->{value} = $CALC->_fac($x->{value});
1366 return $x->round(@r);
1371 my $f = $self->new(2);
1372 while ($f->bacmp($n) < 0)
1374 $x->bmul($f); $f->binc();
1376 $x->bmul($f); # last step
1377 $x->round(@r); # round
1382 # (BINT or num_str, BINT or num_str) return BINT
1383 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1384 # modifies first argument
1385 my ($self,$x,$y,@r) = objectify(2,@_);
1387 return $x if $x->modify('bpow');
1389 $r[3] = $y; # no push!
1390 return $x if $x->{sign} =~ /^[+-]inf$/; # -inf/+inf ** x
1391 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1392 return $x->bone(@r) if $y->is_zero();
1393 return $x->round(@r) if $x->is_one() || $y->is_one();
1394 if ($x->{sign} eq '-' && $CALC->_is_one($x->{value}))
1396 # if $x == -1 and odd/even y => +1/-1
1397 return $y->is_odd() ? $x->round(@r) : $x->babs()->round(@r);
1398 # my Casio FX-5500L has a bug here: -1 ** 2 is -1, but -1 * -1 is 1;
1400 # 1 ** -y => 1 / (1 ** |y|)
1401 # so do test for negative $y after above's clause
1402 return $x->bnan() if $y->{sign} eq '-';
1403 return $x->round(@r) if $x->is_zero(); # 0**y => 0 (if not y <= 0)
1405 if ($CALC->can('_pow'))
1407 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1408 return $x->round(@r);
1411 # based on the assumption that shifting in base 10 is fast, and that mul
1412 # works faster if numbers are small: we count trailing zeros (this step is
1413 # O(1)..O(N), but in case of O(N) we save much more time due to this),
1414 # stripping them out of the multiplication, and add $count * $y zeros
1415 # afterwards like this:
1416 # 300 ** 3 == 300*300*300 == 3*3*3 . '0' x 2 * 3 == 27 . '0' x 6
1417 # creates deep recursion?
1418 # my $zeros = $x->_trailing_zeros();
1421 # $x->brsft($zeros,10); # remove zeros
1422 # $x->bpow($y); # recursion (will not branch into here again)
1423 # $zeros = $y * $zeros; # real number of zeros to add
1424 # $x->blsft($zeros,10);
1425 # return $x->round($a,$p,$r);
1428 my $pow2 = $self->__one();
1429 my $y1 = $class->new($y);
1430 my $two = $self->new(2);
1431 while (!$y1->is_one())
1433 $pow2->bmul($x) if $y1->is_odd();
1437 $x->bmul($pow2) unless $pow2->is_one();
1438 return $x->round(@r);
1443 # (BINT or num_str, BINT or num_str) return BINT
1444 # compute x << y, base n, y >= 0
1445 my ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_);
1447 return $x if $x->modify('blsft');
1448 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1449 return $x->round($a,$p,$r) if $y->is_zero();
1451 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1453 my $t; $t = $CALC->_lsft($x->{value},$y->{value},$n) if $CALC->can('_lsft');
1456 $x->{value} = $t; return $x->round($a,$p,$r);
1459 return $x->bmul( $self->bpow($n, $y, $a, $p, $r), $a, $p, $r );
1464 # (BINT or num_str, BINT or num_str) return BINT
1465 # compute x >> y, base n, y >= 0
1466 my ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_);
1468 return $x if $x->modify('brsft');
1469 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1470 return $x->round($a,$p,$r) if $y->is_zero();
1471 return $x->bzero($a,$p,$r) if $x->is_zero(); # 0 => 0
1473 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1475 # this only works for negative numbers when shifting in base 2
1476 if (($x->{sign} eq '-') && ($n == 2))
1478 return $x->round($a,$p,$r) if $x->is_one('-'); # -1 => -1
1481 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1482 # but perhaps there is a better emulation for two's complement shift...
1483 # if $y != 1, we must simulate it by doing:
1484 # convert to bin, flip all bits, shift, and be done
1485 $x->binc(); # -3 => -2
1486 my $bin = $x->as_bin();
1487 $bin =~ s/^-0b//; # strip '-0b' prefix
1488 $bin =~ tr/10/01/; # flip bits
1490 if (length($bin) <= $y)
1492 $bin = '0'; # shifting to far right creates -1
1493 # 0, because later increment makes
1494 # that 1, attached '-' makes it '-1'
1495 # because -1 >> x == -1 !
1499 $bin =~ s/.{$y}$//; # cut off at the right side
1500 $bin = '1' . $bin; # extend left side by one dummy '1'
1501 $bin =~ tr/10/01/; # flip bits back
1503 my $res = $self->new('0b'.$bin); # add prefix and convert back
1504 $res->binc(); # remember to increment
1505 $x->{value} = $res->{value}; # take over value
1506 return $x->round($a,$p,$r); # we are done now, magic, isn't?
1508 $x->bdec(); # n == 2, but $y == 1: this fixes it
1511 my $t; $t = $CALC->_rsft($x->{value},$y->{value},$n) if $CALC->can('_rsft');
1515 return $x->round($a,$p,$r);
1518 $x->bdiv($self->bpow($n,$y, $a,$p,$r), $a,$p,$r);
1524 #(BINT or num_str, BINT or num_str) return BINT
1526 my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
1528 return $x if $x->modify('band');
1530 local $Math::BigInt::upgrade = undef;
1532 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1533 return $x->bzero() if $y->is_zero() || $x->is_zero();
1535 my $sign = 0; # sign of result
1536 $sign = 1 if ($x->{sign} eq '-') && ($y->{sign} eq '-');
1537 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1538 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1540 if ($CALC->can('_and') && $sx == 1 && $sy == 1)
1542 $x->{value} = $CALC->_and($x->{value},$y->{value});
1543 return $x->round($a,$p,$r);
1546 my $m = $self->bone(); my ($xr,$yr);
1547 my $x10000 = $self->new (0x1000);
1548 my $y1 = copy(ref($x),$y); # make copy
1549 $y1->babs(); # and positive
1550 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1551 use integer; # need this for negative bools
1552 while (!$x1->is_zero() && !$y1->is_zero())
1554 ($x1, $xr) = bdiv($x1, $x10000);
1555 ($y1, $yr) = bdiv($y1, $x10000);
1556 # make both op's numbers!
1557 $x->badd( bmul( $class->new(
1558 abs($sx*int($xr->numify()) & $sy*int($yr->numify()))),
1562 $x->bneg() if $sign;
1563 return $x->round($a,$p,$r);
1568 #(BINT or num_str, BINT or num_str) return BINT
1570 my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
1572 return $x if $x->modify('bior');
1574 local $Math::BigInt::upgrade = undef;
1576 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1577 return $x if $y->is_zero();
1579 my $sign = 0; # sign of result
1580 $sign = 1 if ($x->{sign} eq '-') || ($y->{sign} eq '-');
1581 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1582 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1584 # don't use lib for negative values
1585 if ($CALC->can('_or') && $sx == 1 && $sy == 1)
1587 $x->{value} = $CALC->_or($x->{value},$y->{value});
1588 return $x->round($a,$p,$r);
1591 my $m = $self->bone(); my ($xr,$yr);
1592 my $x10000 = $self->new(0x10000);
1593 my $y1 = copy(ref($x),$y); # make copy
1594 $y1->babs(); # and positive
1595 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1596 use integer; # need this for negative bools
1597 while (!$x1->is_zero() || !$y1->is_zero())
1599 ($x1, $xr) = bdiv($x1,$x10000);
1600 ($y1, $yr) = bdiv($y1,$x10000);
1601 # make both op's numbers!
1602 $x->badd( bmul( $class->new(
1603 abs($sx*int($xr->numify()) | $sy*int($yr->numify()))),
1607 $x->bneg() if $sign;
1608 return $x->round($a,$p,$r);
1613 #(BINT or num_str, BINT or num_str) return BINT
1615 my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
1617 return $x if $x->modify('bxor');
1619 local $Math::BigInt::upgrade = undef;
1621 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1622 return $x if $y->is_zero();
1624 my $sign = 0; # sign of result
1625 $sign = 1 if $x->{sign} ne $y->{sign};
1626 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1627 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1629 # don't use lib for negative values
1630 if ($CALC->can('_xor') && $sx == 1 && $sy == 1)
1632 $x->{value} = $CALC->_xor($x->{value},$y->{value});
1633 return $x->round($a,$p,$r);
1636 my $m = $self->bone(); my ($xr,$yr);
1637 my $x10000 = $self->new(0x10000);
1638 my $y1 = copy(ref($x),$y); # make copy
1639 $y1->babs(); # and positive
1640 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1641 use integer; # need this for negative bools
1642 while (!$x1->is_zero() || !$y1->is_zero())
1644 ($x1, $xr) = bdiv($x1, $x10000);
1645 ($y1, $yr) = bdiv($y1, $x10000);
1646 # make both op's numbers!
1647 $x->badd( bmul( $class->new(
1648 abs($sx*int($xr->numify()) ^ $sy*int($yr->numify()))),
1652 $x->bneg() if $sign;
1653 return $x->round($a,$p,$r);
1658 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1660 my $e = $CALC->_len($x->{value});
1661 return wantarray ? ($e,0) : $e;
1666 # return the nth decimal digit, negative values count backward, 0 is right
1670 return $CALC->_digit($x->{value},$n);
1675 # return the amount of trailing zeros in $x
1677 $x = $class->new($x) unless ref $x;
1679 return 0 if $x->is_zero() || $x->is_odd() || $x->{sign} !~ /^[+-]$/;
1681 return $CALC->_zeros($x->{value}) if $CALC->can('_zeros');
1683 # if not: since we do not know underlying internal representation:
1684 my $es = "$x"; $es =~ /([0]*)$/;
1685 return 0 if !defined $1; # no zeros
1686 return CORE::length("$1"); # as string, not as +0!
1691 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1693 return $x if $x->modify('bsqrt');
1695 return $x->bnan() if $x->{sign} ne '+'; # -x or inf or NaN => NaN
1696 return $x->bzero($a,$p) if $x->is_zero(); # 0 => 0
1697 return $x->round($a,$p,$r) if $x->is_one(); # 1 => 1
1699 return $upgrade->bsqrt($x,$a,$p,$r) if defined $upgrade;
1701 if ($CALC->can('_sqrt'))
1703 $x->{value} = $CALC->_sqrt($x->{value});
1704 return $x->round($a,$p,$r);
1707 return $x->bone($a,$p) if $x < 4; # 2,3 => 1
1709 my $l = int($x->length()/2);
1711 $x->bone(); # keep ref($x), but modify it
1714 my $last = $self->bzero();
1715 my $two = $self->new(2);
1716 my $lastlast = $x+$two;
1717 while ($last != $x && $lastlast != $x)
1719 $lastlast = $last; $last = $x;
1723 $x-- if $x * $x > $y; # overshot?
1724 $x->round($a,$p,$r);
1729 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
1730 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1732 if ($x->{sign} !~ /^[+-]$/)
1734 my $s = $x->{sign}; $s =~ s/^[+-]//;
1735 return $self->new($s); # -inf,+inf => inf
1737 my $e = $class->bzero();
1738 return $e->binc() if $x->is_zero();
1739 $e += $x->_trailing_zeros();
1745 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
1746 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1748 if ($x->{sign} !~ /^[+-]$/)
1750 my $s = $x->{sign}; $s =~ s/^[+]//;
1751 return $self->new($s); # +inf => inf
1754 # that's inefficient
1755 my $zeros = $m->_trailing_zeros();
1756 $m /= 10 ** $zeros if $zeros != 0;
1762 # return a copy of both the exponent and the mantissa
1763 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1765 return ($x->mantissa(),$x->exponent());
1768 ##############################################################################
1769 # rounding functions
1773 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
1774 # $n == 0 || $n == 1 => round to integer
1775 my $x = shift; $x = $class->new($x) unless ref $x;
1776 my ($scale,$mode) = $x->_scale_p($x->precision(),$x->round_mode(),@_);
1777 return $x if !defined $scale; # no-op
1778 return $x if $x->modify('bfround');
1780 # no-op for BigInts if $n <= 0
1783 $x->{_a} = undef; # clear an eventual set A
1784 $x->{_p} = $scale; return $x;
1787 $x->bround( $x->length()-$scale, $mode);
1788 $x->{_a} = undef; # bround sets {_a}
1789 $x->{_p} = $scale; # so correct it
1793 sub _scan_for_nonzero
1799 my $len = $x->length();
1800 return 0 if $len == 1; # '5' is trailed by invisible zeros
1801 my $follow = $pad - 1;
1802 return 0 if $follow > $len || $follow < 1;
1804 # since we do not know underlying represention of $x, use decimal string
1805 #my $r = substr ($$xs,-$follow);
1806 my $r = substr ("$x",-$follow);
1807 return 1 if $r =~ /[^0]/; return 0;
1812 # to make life easier for switch between MBF and MBI (autoload fxxx()
1813 # like MBF does for bxxx()?)
1815 return $x->bround(@_);
1820 # accuracy: +$n preserve $n digits from left,
1821 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
1823 # and overwrite the rest with 0's, return normalized number
1824 # do not return $x->bnorm(), but $x
1826 my $x = shift; $x = $class->new($x) unless ref $x;
1827 my ($scale,$mode) = $x->_scale_a($x->accuracy(),$x->round_mode(),@_);
1828 return $x if !defined $scale; # no-op
1829 return $x if $x->modify('bround');
1831 if ($x->is_zero() || $scale == 0)
1833 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
1836 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
1838 # we have fewer digits than we want to scale to
1839 my $len = $x->length();
1840 # scale < 0, but > -len (not >=!)
1841 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
1843 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
1847 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
1848 my ($pad,$digit_round,$digit_after);
1849 $pad = $len - $scale;
1850 $pad = abs($scale-1) if $scale < 0;
1852 # do not use digit(), it is costly for binary => decimal
1854 my $xs = $CALC->_str($x->{value});
1857 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
1858 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
1859 $digit_round = '0'; $digit_round = substr($$xs,$pl,1) if $pad <= $len;
1860 $pl++; $pl ++ if $pad >= $len;
1861 $digit_after = '0'; $digit_after = substr($$xs,$pl,1) if $pad > 0;
1863 # print "$pad $pl $$xs dr $digit_round da $digit_after\n";
1865 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
1866 # closer at the remaining digits of the original $x, remember decision
1867 my $round_up = 1; # default round up
1869 ($mode eq 'trunc') || # trunc by round down
1870 ($digit_after =~ /[01234]/) || # round down anyway,
1872 ($digit_after eq '5') && # not 5000...0000
1873 ($x->_scan_for_nonzero($pad,$xs) == 0) &&
1875 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
1876 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
1877 ($mode eq '+inf') && ($x->{sign} eq '-') ||
1878 ($mode eq '-inf') && ($x->{sign} eq '+') ||
1879 ($mode eq 'zero') # round down if zero, sign adjusted below
1881 my $put_back = 0; # not yet modified
1883 # old code, depend on internal representation
1884 # split mantissa at $pad and then pad with zeros
1885 #my $s5 = int($pad / 5);
1889 # $x->{value}->[$i++] = 0; # replace with 5 x 0
1891 #$x->{value}->[$s5] = '00000'.$x->{value}->[$s5]; # pad with 0
1892 #my $rem = $pad % 5; # so much left over
1895 # #print "remainder $rem\n";
1896 ## #print "elem $x->{value}->[$s5]\n";
1897 # substr($x->{value}->[$s5],-$rem,$rem) = '0' x $rem; # stamp w/ '0'
1899 #$x->{value}->[$s5] = int ($x->{value}->[$s5]); # str '05' => int '5'
1900 #print ${$CALC->_str($pad->{value})}," $len\n";
1902 if (($pad > 0) && ($pad <= $len))
1904 substr($$xs,-$pad,$pad) = '0' x $pad;
1909 $x->bzero(); # round to '0'
1912 if ($round_up) # what gave test above?
1915 $pad = $len, $$xs = '0'x$pad if $scale < 0; # tlr: whack 0.51=>1.0
1917 # we modify directly the string variant instead of creating a number and
1919 my $c = 0; $pad ++; # for $pad == $len case
1920 while ($pad <= $len)
1922 $c = substr($$xs,-$pad,1) + 1; $c = '0' if $c eq '10';
1923 substr($$xs,-$pad,1) = $c; $pad++;
1924 last if $c != 0; # no overflow => early out
1926 $$xs = '1'.$$xs if $c == 0;
1928 # $x->badd( Math::BigInt->new($x->{sign}.'1'. '0' x $pad) );
1930 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back in
1932 $x->{_a} = $scale if $scale >= 0;
1935 $x->{_a} = $len+$scale;
1936 $x->{_a} = 0 if $scale < -$len;
1943 # return integer less or equal then number, since it is already integer,
1944 # always returns $self
1945 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1947 # not needed: return $x if $x->modify('bfloor');
1948 return $x->round($a,$p,$r);
1953 # return integer greater or equal then number, since it is already integer,
1954 # always returns $self
1955 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1957 # not needed: return $x if $x->modify('bceil');
1958 return $x->round($a,$p,$r);
1961 ##############################################################################
1962 # private stuff (internal use only)
1966 # internal speedup, set argument to 1, or create a +/- 1
1968 my $x = $self->bone(); # $x->{value} = $CALC->_one();
1969 $x->{sign} = shift || '+';
1975 # Overload will swap params if first one is no object ref so that the first
1976 # one is always an object ref. In this case, third param is true.
1977 # This routine is to overcome the effect of scalar,$object creating an object
1978 # of the class of this package, instead of the second param $object. This
1979 # happens inside overload, when the overload section of this package is
1980 # inherited by sub classes.
1981 # For overload cases (and this is used only there), we need to preserve the
1982 # args, hence the copy().
1983 # You can override this method in a subclass, the overload section will call
1984 # $object->_swap() to make sure it arrives at the proper subclass, with some
1985 # exceptions like '+' and '-'. To make '+' and '-' work, you also need to
1986 # specify your own overload for them.
1988 # object, (object|scalar) => preserve first and make copy
1989 # scalar, object => swapped, re-swap and create new from first
1990 # (using class of second object, not $class!!)
1991 my $self = shift; # for override in subclass
1994 my $c = ref ($_[0]) || $class; # fallback $class should not happen
1995 return ( $c->new($_[1]), $_[0] );
1997 return ( $_[0]->copy(), $_[1] );
2002 # check for strings, if yes, return objects instead
2004 # the first argument is number of args objectify() should look at it will
2005 # return $count+1 elements, the first will be a classname. This is because
2006 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2007 # useless objects beeing created and thrown away. So we cannot simple loop
2008 # over @_. If the given count is 0, all arguments will be used.
2010 # If the second arg is a ref, use it as class.
2011 # If not, try to use it as classname, unless undef, then use $class
2012 # (aka Math::BigInt). The latter shouldn't happen,though.
2015 # $x->badd(1); => ref x, scalar y
2016 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2017 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2018 # Math::BigInt::badd(1,2); => scalar x, scalar y
2019 # In the last case we check number of arguments to turn it silently into
2020 # $class,1,2. (We can not take '1' as class ;o)
2021 # badd($class,1) is not supported (it should, eventually, try to add undef)
2022 # currently it tries 'Math::BigInt' + 1, which will not work.
2024 # some shortcut for the common cases
2027 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2028 # $x->binary_op($y);
2029 #return (ref($_[1]),$_[1],$_[2]) if (@_ == 3) && ($_[0]||0 == 2)
2030 # && ref($_[1]) && ref($_[2]);
2032 my $count = abs(shift || 0);
2034 my @a; # resulting array
2037 # okay, got object as first
2042 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2044 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2046 # print "Now in objectify, my class is today $a[0]\n";
2055 $k = $a[0]->new($k);
2057 elsif (ref($k) ne $a[0])
2059 # foreign object, try to convert to integer
2060 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2073 $k = $a[0]->new($k);
2075 elsif (ref($k) ne $a[0])
2077 # foreign object, try to convert to integer
2078 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2082 push @a,@_; # return other params, too
2084 die "$class objectify needs list context" unless wantarray;
2093 my @a = @_; my $l = scalar @_; my $j = 0;
2094 for ( my $i = 0; $i < $l ; $i++,$j++ )
2096 if ($_[$i] eq ':constant')
2098 # this causes overlord er load to step in
2099 overload::constant integer => sub { $self->new(shift) };
2100 splice @a, $j, 1; $j --;
2102 elsif ($_[$i] eq 'upgrade')
2104 # this causes upgrading
2105 $upgrade = $_[$i+1]; # or undef to disable
2106 my $s = 2; $s = 1 if @a-$j < 2; # avoid "can not modify non-existant..."
2107 splice @a, $j, $s; $j -= $s;
2109 elsif ($_[$i] =~ /^lib$/i)
2111 # this causes a different low lib to take care...
2112 $CALC = $_[$i+1] || '';
2113 my $s = 2; $s = 1 if @a-$j < 2; # avoid "can not modify non-existant..."
2114 splice @a, $j, $s; $j -= $s;
2117 # any non :constant stuff is handled by our parent, Exporter
2118 # even if @_ is empty, to give it a chance
2119 $self->SUPER::import(@a); # need it for subclasses
2120 $self->export_to_level(1,$self,@a); # need it for MBF
2122 # try to load core math lib
2123 my @c = split /\s*,\s*/,$CALC;
2124 push @c,'Calc'; # if all fail, try this
2125 $CALC = ''; # signal error
2126 foreach my $lib (@c)
2128 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2132 # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
2133 # used in the same script, or eval inside import().
2134 (my $mod = $lib . '.pm') =~ s!::!/!g;
2135 # require does not automatically :: => /, so portability problems arise
2136 eval { require $mod; $lib->import( @c ); }
2140 eval "use $lib qw/@c/;";
2142 $CALC = $lib, last if $@ eq ''; # no error in loading lib?
2144 die "Couldn't load any math lib, not even the default" if $CALC eq '';
2149 # convert a (ref to) big hex string to BigInt, return undef for error
2152 my $x = Math::BigInt->bzero();
2155 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2156 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2158 return $x->bnan() if $$hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2160 my $sign = '+'; $sign = '-' if ($$hs =~ /^-/);
2162 $$hs =~ s/^[+-]//; # strip sign
2163 if ($CALC->can('_from_hex'))
2165 $x->{value} = $CALC->_from_hex($hs);
2169 # fallback to pure perl
2170 my $mul = Math::BigInt->bzero(); $mul++;
2171 my $x65536 = Math::BigInt->new(65536);
2172 my $len = CORE::length($$hs)-2;
2173 $len = int($len/4); # 4-digit parts, w/o '0x'
2174 my $val; my $i = -4;
2177 $val = substr($$hs,$i,4);
2178 $val =~ s/^[+-]?0x// if $len == 0; # for last part only because
2179 $val = hex($val); # hex does not like wrong chars
2181 $x += $mul * $val if $val != 0;
2182 $mul *= $x65536 if $len >= 0; # skip last mul
2185 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2191 # convert a (ref to) big binary string to BigInt, return undef for error
2194 my $x = Math::BigInt->bzero();
2196 $$bs =~ s/([01])_([01])/$1$2/g;
2197 $$bs =~ s/([01])_([01])/$1$2/g;
2198 return $x->bnan() if $$bs !~ /^[+-]?0b[01]+$/;
2200 my $sign = '+'; $sign = '-' if ($$bs =~ /^\-/);
2201 $$bs =~ s/^[+-]//; # strip sign
2202 if ($CALC->can('_from_bin'))
2204 $x->{value} = $CALC->_from_bin($bs);
2208 my $mul = Math::BigInt->bzero(); $mul++;
2209 my $x256 = Math::BigInt->new(256);
2210 my $len = CORE::length($$bs)-2;
2211 $len = int($len/8); # 8-digit parts, w/o '0b'
2212 my $val; my $i = -8;
2215 $val = substr($$bs,$i,8);
2216 $val =~ s/^[+-]?0b// if $len == 0; # for last part only
2217 #$val = oct('0b'.$val); # does not work on Perl prior to 5.6.0
2219 # $val = ('0' x (8-CORE::length($val))).$val if CORE::length($val) < 8;
2220 $val = ord(pack('B8',substr('00000000'.$val,-8,8)));
2222 $x += $mul * $val if $val != 0;
2223 $mul *= $x256 if $len >= 0; # skip last mul
2226 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2232 # (ref to num_str) return num_str
2233 # internal, take apart a string and return the pieces
2234 # strip leading/trailing whitespace, leading zeros, underscore and reject
2238 # strip white space at front, also extranous leading zeros
2239 $$x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2240 $$x =~ s/^\s+//; # but this will
2241 $$x =~ s/\s+$//g; # strip white space at end
2243 # shortcut, if nothing to split, return early
2244 if ($$x =~ /^[+-]?\d+$/)
2246 $$x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2247 return (\$sign, $x, \'', \'', \0);
2250 # invalid starting char?
2251 return if $$x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2253 return __from_hex($x) if $$x =~ /^[\-\+]?0x/; # hex string
2254 return __from_bin($x) if $$x =~ /^[\-\+]?0b/; # binary string
2256 # strip underscores between digits
2257 $$x =~ s/(\d)_(\d)/$1$2/g;
2258 $$x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3
2260 # some possible inputs:
2261 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2262 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2
2264 return if $$x =~ /[Ee].*[Ee]/; # more than one E => error
2266 my ($m,$e) = split /[Ee]/,$$x;
2267 $e = '0' if !defined $e || $e eq "";
2268 # sign,value for exponent,mantint,mantfrac
2269 my ($es,$ev,$mis,$miv,$mfv);
2271 if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2275 return if $m eq '.' || $m eq '';
2276 my ($mi,$mf) = split /\./,$m;
2277 $mi = '0' if !defined $mi;
2278 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2279 $mf = '0' if !defined $mf || $mf eq '';
2280 if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2282 $mis = $1||'+'; $miv = $2;
2283 return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
2285 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2288 return; # NaN, not a number
2293 # an object might be asked to return itself as bigint on certain overloaded
2294 # operations, this does exactly this, so that sub classes can simple inherit
2295 # it or override with their own integer conversion routine
2303 # return as hex string, with prefixed 0x
2304 my $x = shift; $x = $class->new($x) if !ref($x);
2306 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2307 return '0x0' if $x->is_zero();
2309 my $es = ''; my $s = '';
2310 $s = $x->{sign} if $x->{sign} eq '-';
2311 if ($CALC->can('_as_hex'))
2313 $es = ${$CALC->_as_hex($x->{value})};
2317 my $x1 = $x->copy()->babs(); my $xr;
2318 my $x10000 = Math::BigInt->new (0x10000);
2319 while (!$x1->is_zero())
2321 ($x1, $xr) = bdiv($x1,$x10000);
2322 $es .= unpack('h4',pack('v',$xr->numify()));
2325 $es =~ s/^[0]+//; # strip leading zeros
2333 # return as binary string, with prefixed 0b
2334 my $x = shift; $x = $class->new($x) if !ref($x);
2336 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2337 return '0b0' if $x->is_zero();
2339 my $es = ''; my $s = '';
2340 $s = $x->{sign} if $x->{sign} eq '-';
2341 if ($CALC->can('_as_bin'))
2343 $es = ${$CALC->_as_bin($x->{value})};
2347 my $x1 = $x->copy()->babs(); my $xr;
2348 my $x10000 = Math::BigInt->new (0x10000);
2349 while (!$x1->is_zero())
2351 ($x1, $xr) = bdiv($x1,$x10000);
2352 $es .= unpack('b16',pack('v',$xr->numify()));
2355 $es =~ s/^[0]+//; # strip leading zeros
2361 ##############################################################################
2362 # internal calculation routines (others are in Math::BigInt::Calc etc)
2366 # (BINT or num_str, BINT or num_str) return BINT
2367 # does modify first argument
2370 my $x = shift; my $ty = shift;
2371 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2372 return $x * $ty / bgcd($x,$ty);
2377 # (BINT or num_str, BINT or num_str) return BINT
2378 # does modify both arguments
2379 # GCD -- Euclids algorithm E, Knuth Vol 2 pg 296
2382 return $x->bnan() if $x->{sign} !~ /^[+-]$/ || $ty->{sign} !~ /^[+-]$/;
2384 while (!$ty->is_zero())
2386 ($x, $ty) = ($ty,bmod($x,$ty));
2391 ###############################################################################
2392 # this method return 0 if the object can be modified, or 1 for not
2393 # We use a fast use constant statement here, to avoid costly calls. Subclasses
2394 # may override it with special code (f.i. Math::BigInt::Constant does so)
2396 sub modify () { 0; }
2403 Math::BigInt - Arbitrary size integer math package
2410 $x = Math::BigInt->new($str); # defaults to 0
2411 $nan = Math::BigInt->bnan(); # create a NotANumber
2412 $zero = Math::BigInt->bzero(); # create a +0
2413 $inf = Math::BigInt->binf(); # create a +inf
2414 $inf = Math::BigInt->binf('-'); # create a -inf
2415 $one = Math::BigInt->bone(); # create a +1
2416 $one = Math::BigInt->bone('-'); # create a -1
2419 $x->is_zero(); # true if arg is +0
2420 $x->is_nan(); # true if arg is NaN
2421 $x->is_one(); # true if arg is +1
2422 $x->is_one('-'); # true if arg is -1
2423 $x->is_odd(); # true if odd, false for even
2424 $x->is_even(); # true if even, false for odd
2425 $x->is_positive(); # true if >= 0
2426 $x->is_negative(); # true if < 0
2427 $x->is_inf(sign); # true if +inf, or -inf (sign is default '+')
2428 $x->is_int(); # true if $x is an integer (not a float)
2430 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2431 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2432 $x->sign(); # return the sign, either +,- or NaN
2433 $x->digit($n); # return the nth digit, counting from right
2434 $x->digit(-$n); # return the nth digit, counting from left
2436 # The following all modify their first argument:
2439 $x->bzero(); # set $x to 0
2440 $x->bnan(); # set $x to NaN
2441 $x->bone(); # set $x to +1
2442 $x->bone('-'); # set $x to -1
2443 $x->binf(); # set $x to inf
2444 $x->binf('-'); # set $x to -inf
2446 $x->bneg(); # negation
2447 $x->babs(); # absolute value
2448 $x->bnorm(); # normalize (no-op)
2449 $x->bnot(); # two's complement (bit wise not)
2450 $x->binc(); # increment x by 1
2451 $x->bdec(); # decrement x by 1
2453 $x->badd($y); # addition (add $y to $x)
2454 $x->bsub($y); # subtraction (subtract $y from $x)
2455 $x->bmul($y); # multiplication (multiply $x by $y)
2456 $x->bdiv($y); # divide, set $x to quotient
2457 # return (quo,rem) or quo if scalar
2459 $x->bmod($y); # modulus (x % y)
2460 $x->bpow($y); # power of arguments (x ** y)
2461 $x->blsft($y); # left shift
2462 $x->brsft($y); # right shift
2463 $x->blsft($y,$n); # left shift, by base $n (like 10)
2464 $x->brsft($y,$n); # right shift, by base $n (like 10)
2466 $x->band($y); # bitwise and
2467 $x->bior($y); # bitwise inclusive or
2468 $x->bxor($y); # bitwise exclusive or
2469 $x->bnot(); # bitwise not (two's complement)
2471 $x->bsqrt(); # calculate square-root
2472 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2474 $x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r
2475 $x->bround($N); # accuracy: preserve $N digits
2476 $x->bfround($N); # round to $Nth digit, no-op for BigInts
2478 # The following do not modify their arguments in BigInt, but do in BigFloat:
2479 $x->bfloor(); # return integer less or equal than $x
2480 $x->bceil(); # return integer greater or equal than $x
2482 # The following do not modify their arguments:
2484 bgcd(@values); # greatest common divisor (no OO style)
2485 blcm(@values); # lowest common multiplicator (no OO style)
2487 $x->length(); # return number of digits in number
2488 ($x,$f) = $x->length(); # length of number and length of fraction part,
2489 # latter is always 0 digits long for BigInt's
2491 $x->exponent(); # return exponent as BigInt
2492 $x->mantissa(); # return (signed) mantissa as BigInt
2493 $x->parts(); # return (mantissa,exponent) as BigInt
2494 $x->copy(); # make a true copy of $x (unlike $y = $x;)
2495 $x->as_number(); # return as BigInt (in BigInt: same as copy())
2497 # conversation to string
2498 $x->bstr(); # normalized string
2499 $x->bsstr(); # normalized string in scientific notation
2500 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
2501 $x->as_bin(); # as signed binary string with prefixed 0b
2505 All operators (inlcuding basic math operations) are overloaded if you
2506 declare your big integers as
2508 $i = new Math::BigInt '123_456_789_123_456_789';
2510 Operations with overloaded operators preserve the arguments which is
2511 exactly what you expect.
2515 =item Canonical notation
2517 Big integer values are strings of the form C</^[+-]\d+$/> with leading
2520 '-0' canonical value '-0', normalized '0'
2521 ' -123_123_123' canonical value '-123123123'
2522 '1_23_456_7890' canonical value '1234567890'
2526 Input values to these routines may be either Math::BigInt objects or
2527 strings of the form C</^\s*[+-]?[\d]+\.?[\d]*E?[+-]?[\d]*$/>.
2529 You can include one underscore between any two digits.
2531 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
2532 Non integer values result in NaN.
2534 Math::BigInt::new() defaults to 0, while Math::BigInt::new('') results
2537 bnorm() on a BigInt object is now effectively a no-op, since the numbers
2538 are always stored in normalized form. On a string, it creates a BigInt
2543 Output values are BigInt objects (normalized), except for bstr(), which
2544 returns a string in normalized form.
2545 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
2546 C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
2547 return either undef, <0, 0 or >0 and are suited for sort.
2553 Each of the methods below accepts three additional parameters. These arguments
2554 $A, $P and $R are accuracy, precision and round_mode. Please see more in the
2555 section about ACCURACY and ROUNDIND.
2559 $x->accuracy(5); # local for $x
2560 $class->accuracy(5); # global for all members of $class
2562 Set or get the global or local accuracy, aka how many significant digits the
2563 results have. Please see the section about L<ACCURACY AND PRECISION> for
2566 Value must be greater than zero. Pass an undef value to disable it:
2568 $x->accuracy(undef);
2569 Math::BigInt->accuracy(undef);
2571 Returns the current accuracy. For C<$x->accuracy()> it will return either the
2572 local accuracy, or if not defined, the global. This means the return value
2573 represents the accuracy that will be in effect for $x:
2575 $y = Math::BigInt->new(1234567); # unrounded
2576 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
2577 $x = Math::BigInt->new(123456); # will be automatically rounded
2578 print "$x $y\n"; # '123500 1234567'
2579 print $x->accuracy(),"\n"; # will be 4
2580 print $y->accuracy(),"\n"; # also 4, since global is 4
2581 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
2582 print $x->accuracy(),"\n"; # still 4
2583 print $y->accuracy(),"\n"; # 5, since global is 5
2589 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
2590 2, but others work, too.
2592 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
2596 $x = Math::BigInt->new(10);
2597 $x->brsft(1); # same as $x >> 1: 5
2598 $x = Math::BigInt->new(1234);
2599 $x->brsft(2,10); # result 12
2601 There is one exception, and that is base 2 with negative $x:
2604 $x = Math::BigInt->new(-5);
2607 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
2612 $x = Math::BigInt->new($str,$A,$P,$R);
2614 Creates a new BigInt object from a string or another BigInt object. The
2615 input is accepted as decimal, hex (with leading '0x') or binary (with leading
2620 $x = Math::BigInt->bnan();
2622 Creates a new BigInt object representing NaN (Not A Number).
2623 If used on an object, it will set it to NaN:
2629 $x = Math::BigInt->bzero();
2631 Creates a new BigInt object representing zero.
2632 If used on an object, it will set it to zero:
2638 $x = Math::BigInt->binf($sign);
2640 Creates a new BigInt object representing infinity. The optional argument is
2641 either '-' or '+', indicating whether you want infinity or minus infinity.
2642 If used on an object, it will set it to infinity:
2649 $x = Math::BigInt->binf($sign);
2651 Creates a new BigInt object representing one. The optional argument is
2652 either '-' or '+', indicating whether you want one or minus one.
2653 If used on an object, it will set it to one:
2658 =head2 is_one()/is_zero()/is_nan()/is_positive()/is_negative()/is_inf()/is_odd()/is_even()/is_int()
2660 $x->is_zero(); # true if arg is +0
2661 $x->is_nan(); # true if arg is NaN
2662 $x->is_one(); # true if arg is +1
2663 $x->is_one('-'); # true if arg is -1
2664 $x->is_odd(); # true if odd, false for even
2665 $x->is_even(); # true if even, false for odd
2666 $x->is_positive(); # true if >= 0
2667 $x->is_negative(); # true if < 0
2668 $x->is_inf(); # true if +inf
2669 $x->is_inf('-'); # true if -inf (sign is default '+')
2670 $x->is_int(); # true if $x is an integer
2672 These methods all test the BigInt for one condition and return true or false
2673 depending on the input.
2677 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2681 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2685 $x->sign(); # return the sign, either +,- or NaN
2689 $x->digit($n); # return the nth digit, counting from right
2695 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
2696 and '-inf', respectively. Does nothing for NaN or zero.
2702 Set the number to it's absolute value, e.g. change the sign from '-' to '+'
2703 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
2708 $x->bnorm(); # normalize (no-op)
2712 $x->bnot(); # two's complement (bit wise not)
2716 $x->binc(); # increment x by 1
2720 $x->bdec(); # decrement x by 1
2724 $x->badd($y); # addition (add $y to $x)
2728 $x->bsub($y); # subtraction (subtract $y from $x)
2732 $x->bmul($y); # multiplication (multiply $x by $y)
2736 $x->bdiv($y); # divide, set $x to quotient
2737 # return (quo,rem) or quo if scalar
2741 $x->bmod($y); # modulus (x % y)
2745 $x->bpow($y); # power of arguments (x ** y)
2749 $x->blsft($y); # left shift
2750 $x->blsft($y,$n); # left shift, by base $n (like 10)
2754 $x->brsft($y); # right shift
2755 $x->brsft($y,$n); # right shift, by base $n (like 10)
2759 $x->band($y); # bitwise and
2763 $x->bior($y); # bitwise inclusive or
2767 $x->bxor($y); # bitwise exclusive or
2771 $x->bnot(); # bitwise not (two's complement)
2775 $x->bsqrt(); # calculate square-root
2779 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2783 $x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r
2787 $x->bround($N); # accuracy: preserve $N digits
2791 $x->bfround($N); # round to $Nth digit, no-op for BigInts
2797 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
2798 does change $x in BigFloat.
2804 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
2805 does change $x in BigFloat.
2809 bgcd(@values); # greatest common divisor (no OO style)
2813 blcm(@values); # lowest common multiplicator (no OO style)
2818 ($xl,$fl) = $x->length();
2820 Returns the number of digits in the decimal representation of the number.
2821 In list context, returns the length of the integer and fraction part. For
2822 BigInt's, the length of the fraction part will always be 0.
2828 Return the exponent of $x as BigInt.
2834 Return the signed mantissa of $x as BigInt.
2838 $x->parts(); # return (mantissa,exponent) as BigInt
2842 $x->copy(); # make a true copy of $x (unlike $y = $x;)
2846 $x->as_number(); # return as BigInt (in BigInt: same as copy())
2850 $x->bstr(); # normalized string
2854 $x->bsstr(); # normalized string in scientific notation
2858 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
2862 $x->as_bin(); # as signed binary string with prefixed 0b
2864 =head1 ACCURACY and PRECISION
2866 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
2867 accuracy and precision based rounding, both automatically after every
2868 operation as well as manually.
2870 This section describes the accuracy/precision handling in Math::Big* as it
2871 used to be and as it is now, complete with an explanation of all terms and
2874 Not yet implemented things (but with correct description) are marked with '!',
2875 things that need to be answered are marked with '?'.
2877 In the next paragraph follows a short description of terms used here (because
2878 these may differ from terms used by others people or documentation).
2880 During the rest of this document, the shortcuts A (for accuracy), P (for
2881 precision), F (fallback) and R (rounding mode) will be used.
2885 A fixed number of digits before (positive) or after (negative)
2886 the decimal point. For example, 123.45 has a precision of -2. 0 means an
2887 integer like 123 (or 120). A precision of 2 means two digits to the left
2888 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
2889 numbers with zeros before the decimal point may have different precisions,
2890 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
2891 was). It could also have p < 0, when the digits after the decimal point
2894 The string output (of floating point numbers) will be padded with zeros:
2896 Initial value P A Result String
2897 ------------------------------------------------------------
2898 1234.01 -3 1000 1000
2901 1234.001 1 1234 1234.0
2903 1234.01 2 1234.01 1234.01
2904 1234.01 5 1234.01 1234.01000
2906 For BigInts, no padding occurs.
2910 Number of significant digits. Leading zeros are not counted. A
2911 number may have an accuracy greater than the non-zero digits
2912 when there are zeros in it or trailing zeros. For example, 123.456 has
2913 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
2915 The string output (of floating point numbers) will be padded with zeros:
2917 Initial value P A Result String
2918 ------------------------------------------------------------
2920 1234.01 6 1234.01 1234.01
2921 1234.1 8 1234.1 1234.1000
2923 For BigInts, no padding occurs.
2927 When both A and P are undefined, this is used as a fallback accuracy when
2930 =head2 Rounding mode R
2932 When rounding a number, different 'styles' or 'kinds'
2933 of rounding are possible. (Note that random rounding, as in
2934 Math::Round, is not implemented.)
2940 truncation invariably removes all digits following the
2941 rounding place, replacing them with zeros. Thus, 987.65 rounded
2942 to tens (P=1) becomes 980, and rounded to the fourth sigdig
2943 becomes 987.6 (A=4). 123.456 rounded to the second place after the
2944 decimal point (P=-2) becomes 123.46.
2946 All other implemented styles of rounding attempt to round to the
2947 "nearest digit." If the digit D immediately to the right of the
2948 rounding place (skipping the decimal point) is greater than 5, the
2949 number is incremented at the rounding place (possibly causing a
2950 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
2951 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
2952 truncated at the rounding place: e.g. when rounding to units, 0.4
2953 rounds to 0, and -19.4 rounds to -19.
2955 However the results of other styles of rounding differ if the
2956 digit immediately to the right of the rounding place (skipping the
2957 decimal point) is 5 and if there are no digits, or no digits other
2958 than 0, after that 5. In such cases:
2962 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
2963 if it is not already. E.g., when rounding to the first sigdig, 0.45
2964 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
2968 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
2969 it is not already. E.g., when rounding to the first sigdig, 0.45
2970 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
2974 round to plus infinity, i.e. always round up. E.g., when
2975 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
2976 and 0.4501 also becomes 0.5.
2980 round to minus infinity, i.e. always round down. E.g., when
2981 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
2982 but 0.4501 becomes 0.5.
2986 round to zero, i.e. positive numbers down, negative ones up.
2987 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
2988 becomes -0.5, but 0.4501 becomes 0.5.
2992 The handling of A & P in MBI/MBF (the old core code shipped with Perl
2993 versions <= 5.7.2) is like this:
2999 * ffround($p) is able to round to $p number of digits after the decimal
3001 * otherwise P is unused
3003 =item Accuracy (significant digits)
3005 * fround($a) rounds to $a significant digits
3006 * only fdiv() and fsqrt() take A as (optional) paramater
3007 + other operations simply create the same number (fneg etc), or more (fmul)
3009 + rounding/truncating is only done when explicitly calling one of fround
3010 or ffround, and never for BigInt (not implemented)
3011 * fsqrt() simply hands its accuracy argument over to fdiv.
3012 * the documentation and the comment in the code indicate two different ways
3013 on how fdiv() determines the maximum number of digits it should calculate,
3014 and the actual code does yet another thing
3016 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3018 result has at most max(scale, length(dividend), length(divisor)) digits
3020 scale = max(scale, length(dividend)-1,length(divisor)-1);
3021 scale += length(divisior) - length(dividend);
3022 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3023 Actually, the 'difference' added to the scale is calculated from the
3024 number of "significant digits" in dividend and divisor, which is derived
3025 by looking at the length of the mantissa. Which is wrong, since it includes
3026 the + sign (oups) and actually gets 2 for '+100' and 4 for '+101'. Oups
3027 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3028 assumption that 124 has 3 significant digits, while 120/7 will get you
3029 '17', not '17.1' since 120 is thought to have 2 significant digits.
3030 The rounding after the division then uses the remainder and $y to determine
3031 wether it must round up or down.
3032 ? I have no idea which is the right way. That's why I used a slightly more
3033 ? simple scheme and tweaked the few failing testcases to match it.
3037 This is how it works now:
3041 =item Setting/Accessing
3043 * You can set the A global via Math::BigInt->accuracy() or
3044 Math::BigFloat->accuracy() or whatever class you are using.
3045 * You can also set P globally by using Math::SomeClass->precision() likewise.
3046 * Globals are classwide, and not inherited by subclasses.
3047 * to undefine A, use Math::SomeCLass->accuracy(undef);
3048 * to undefine P, use Math::SomeClass->precision(undef);
3049 * Setting Math::SomeClass->accuracy() clears automatically
3050 Math::SomeClass->precision(), and vice versa.
3051 * To be valid, A must be > 0, P can have any value.
3052 * If P is negative, this means round to the P'th place to the right of the
3053 decimal point; positive values mean to the left of the decimal point.
3054 P of 0 means round to integer.
3055 * to find out the current global A, take Math::SomeClass->accuracy()
3056 * to find out the current global P, take Math::SomeClass->precision()
3057 * use $x->accuracy() respective $x->precision() for the local setting of $x.
3058 * Please note that $x->accuracy() respecive $x->precision() fall back to the
3059 defined globals, when $x's A or P is not set.
3061 =item Creating numbers
3063 * When you create a number, you can give it's desired A or P via:
3064 $x = Math::BigInt->new($number,$A,$P);
3065 * Only one of A or P can be defined, otherwise the result is NaN
3066 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
3067 globals (if set) will be used. Thus changing the global defaults later on
3068 will not change the A or P of previously created numbers (i.e., A and P of
3069 $x will be what was in effect when $x was created)
3070 * If given undef for A and P, B<no> rounding will occur, and the globals will
3071 B<not> be used. This is used by subclasses to create numbers without
3072 suffering rounding in the parent. Thus a subclass is able to have it's own
3073 globals enforced upon creation of a number by using
3074 $x = Math::BigInt->new($number,undef,undef):
3076 use Math::Bigint::SomeSubclass;
3079 Math::BigInt->accuracy(2);
3080 Math::BigInt::SomeSubClass->accuracy(3);
3081 $x = Math::BigInt::SomeSubClass->new(1234);
3083 $x is now 1230, and not 1200. A subclass might choose to implement
3084 this otherwise, e.g. falling back to the parent's A and P.
3088 * If A or P are enabled/defined, they are used to round the result of each
3089 operation according to the rules below
3090 * Negative P is ignored in Math::BigInt, since BigInts never have digits
3091 after the decimal point
3092 * Math::BigFloat uses Math::BigInts internally, but setting A or P inside
3093 Math::BigInt as globals should not tamper with the parts of a BigFloat.
3094 Thus a flag is used to mark all Math::BigFloat numbers as 'never round'
3098 * It only makes sense that a number has only one of A or P at a time.
3099 Since you can set/get both A and P, there is a rule that will practically
3100 enforce only A or P to be in effect at a time, even if both are set.
3101 This is called precedence.
3102 * If two objects are involved in an operation, and one of them has A in
3103 effect, and the other P, this results in an error (NaN).
3104 * A takes precendence over P (Hint: A comes before P). If A is defined, it
3105 is used, otherwise P is used. If neither of them is defined, nothing is
3106 used, i.e. the result will have as many digits as it can (with an
3107 exception for fdiv/fsqrt) and will not be rounded.
3108 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
3109 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
3110 If either the dividend's or the divisor's mantissa has more digits than
3111 the value of F, the higher value will be used instead of F.
3112 This is to limit the digits (A) of the result (just consider what would
3113 happen with unlimited A and P in the case of 1/3 :-)
3114 * fdiv will calculate (at least) 4 more digits than required (determined by
3115 A, P or F), and, if F is not used, round the result
3116 (this will still fail in the case of a result like 0.12345000000001 with A
3117 or P of 5, but this can not be helped - or can it?)
3118 * Thus you can have the math done by on Math::Big* class in three modes:
3119 + never round (this is the default):
3120 This is done by setting A and P to undef. No math operation
3121 will round the result, with fdiv() and fsqrt() as exceptions to guard
3122 against overflows. You must explicitely call bround(), bfround() or
3123 round() (the latter with parameters).
3124 Note: Once you have rounded a number, the settings will 'stick' on it
3125 and 'infect' all other numbers engaged in math operations with it, since
3126 local settings have the highest precedence. So, to get SaferRound[tm],
3127 use a copy() before rounding like this:
3129 $x = Math::BigFloat->new(12.34);
3130 $y = Math::BigFloat->new(98.76);
3131 $z = $x * $y; # 1218.6984
3132 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
3133 $z = $x * $y; # still 1218.6984, without
3134 # copy would have been 1210!
3136 + round after each op:
3137 After each single operation (except for testing like is_zero()), the
3138 method round() is called and the result is rounded appropriately. By
3139 setting proper values for A and P, you can have all-the-same-A or
3140 all-the-same-P modes. For example, Math::Currency might set A to undef,
3141 and P to -2, globally.
3143 ?Maybe an extra option that forbids local A & P settings would be in order,
3144 ?so that intermediate rounding does not 'poison' further math?
3146 =item Overriding globals
3148 * you will be able to give A, P and R as an argument to all the calculation
3149 routines; the second parameter is A, the third one is P, and the fourth is
3150 R (shift right by one for binary operations like badd). P is used only if
3151 the first parameter (A) is undefined. These three parameters override the
3152 globals in the order detailed as follows, i.e. the first defined value
3154 (local: per object, global: global default, parameter: argument to sub)
3157 + local A (if defined on both of the operands: smaller one is taken)
3158 + local P (if defined on both of the operands: bigger one is taken)
3162 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
3163 arguments (A and P) instead of one
3165 =item Local settings
3167 * You can set A and P locally by using $x->accuracy() and $x->precision()
3168 and thus force different A and P for different objects/numbers.
3169 * Setting A or P this way immediately rounds $x to the new value.
3170 * $x->accuracy() clears $x->precision(), and vice versa.
3174 * the rounding routines will use the respective global or local settings.
3175 fround()/bround() is for accuracy rounding, while ffround()/bfround()
3177 * the two rounding functions take as the second parameter one of the
3178 following rounding modes (R):
3179 'even', 'odd', '+inf', '-inf', 'zero', 'trunc'
3180 * you can set and get the global R by using Math::SomeClass->round_mode()
3181 or by setting $Math::SomeClass::round_mode
3182 * after each operation, $result->round() is called, and the result may
3183 eventually be rounded (that is, if A or P were set either locally,
3184 globally or as parameter to the operation)
3185 * to manually round a number, call $x->round($A,$P,$round_mode);
3186 this will round the number by using the appropriate rounding function
3187 and then normalize it.
3188 * rounding modifies the local settings of the number:
3190 $x = Math::BigFloat->new(123.456);
3194 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
3195 will be 4 from now on.
3197 =item Default values
3206 * The defaults are set up so that the new code gives the same results as
3207 the old code (except in a few cases on fdiv):
3208 + Both A and P are undefined and thus will not be used for rounding
3209 after each operation.
3210 + round() is thus a no-op, unless given extra parameters A and P
3216 The actual numbers are stored as unsigned big integers (with seperate sign).
3217 You should neither care about nor depend on the internal representation; it
3218 might change without notice. Use only method calls like C<< $x->sign(); >>
3219 instead relying on the internal hash keys like in C<< $x->{sign}; >>.
3223 Math with the numbers is done (by default) by a module called
3224 Math::BigInt::Calc. This is equivalent to saying:
3226 use Math::BigInt lib => 'Calc';
3228 You can change this by using:
3230 use Math::BigInt lib => 'BitVect';
3232 The following would first try to find Math::BigInt::Foo, then
3233 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
3235 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
3237 Calc.pm uses as internal format an array of elements of some decimal base
3238 (usually 1e5 or 1e7) with the least significant digit first, while BitVect.pm
3239 uses a bit vector of base 2, most significant bit first. Other modules might
3240 use even different means of representing the numbers. See the respective
3241 module documentation for further details.
3245 The sign is either '+', '-', 'NaN', '+inf' or '-inf' and stored seperately.
3247 A sign of 'NaN' is used to represent the result when input arguments are not
3248 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
3249 minus infinity. You will get '+inf' when dividing a positive number by 0, and
3250 '-inf' when dividing any negative number by 0.
3252 =head2 mantissa(), exponent() and parts()
3254 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
3257 $m = $x->mantissa();
3258 $e = $x->exponent();
3259 $y = $m * ( 10 ** $e );
3260 print "ok\n" if $x == $y;
3262 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
3263 in one go. Both the returned mantissa and exponent have a sign.
3265 Currently, for BigInts C<$e> will be always 0, except for NaN, +inf and -inf,
3266 where it will be NaN; and for $x == 0, where it will be 1
3267 (to be compatible with Math::BigFloat's internal representation of a zero as
3270 C<$m> will always be a copy of the original number. The relation between $e
3271 and $m might change in the future, but will always be equivalent in a
3272 numerical sense, e.g. $m might get minimized.
3278 sub bint { Math::BigInt->new(shift); }
3280 $x = Math::BigInt->bstr("1234") # string "1234"
3281 $x = "$x"; # same as bstr()
3282 $x = Math::BigInt->bneg("1234"); # Bigint "-1234"
3283 $x = Math::BigInt->babs("-12345"); # Bigint "12345"
3284 $x = Math::BigInt->bnorm("-0 00"); # BigInt "0"
3285 $x = bint(1) + bint(2); # BigInt "3"
3286 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
3287 $x = bint(1); # BigInt "1"
3288 $x = $x + 5 / 2; # BigInt "3"
3289 $x = $x ** 3; # BigInt "27"
3290 $x *= 2; # BigInt "54"
3291 $x = Math::BigInt->new(0); # BigInt "0"
3293 $x = Math::BigInt->badd(4,5) # BigInt "9"
3294 print $x->bsstr(); # 9e+0
3296 Examples for rounding:
3301 $x = Math::BigFloat->new(123.4567);
3302 $y = Math::BigFloat->new(123.456789);
3303 Math::BigFloat->accuracy(4); # no more A than 4
3305 ok ($x->copy()->fround(),123.4); # even rounding
3306 print $x->copy()->fround(),"\n"; # 123.4
3307 Math::BigFloat->round_mode('odd'); # round to odd
3308 print $x->copy()->fround(),"\n"; # 123.5
3309 Math::BigFloat->accuracy(5); # no more A than 5
3310 Math::BigFloat->round_mode('odd'); # round to odd
3311 print $x->copy()->fround(),"\n"; # 123.46
3312 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
3313 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
3315 Math::BigFloat->accuracy(undef); # A not important now
3316 Math::BigFloat->precision(2); # P important
3317 print $x->copy()->bnorm(),"\n"; # 123.46
3318 print $x->copy()->fround(),"\n"; # 123.46
3320 Examples for converting:
3322 my $x = Math::BigInt->new('0b1'.'01' x 123);
3323 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
3325 =head1 Autocreating constants
3327 After C<use Math::BigInt ':constant'> all the B<integer> decimal constants
3328 in the given scope are converted to C<Math::BigInt>. This conversion
3329 happens at compile time.
3333 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
3335 prints the integer value of C<2**100>. Note that without conversion of
3336 constants the expression 2**100 will be calculated as perl scalar.
3338 Please note that strings and floating point constants are not affected,
3341 use Math::BigInt qw/:constant/;
3343 $x = 1234567890123456789012345678901234567890
3344 + 123456789123456789;
3345 $y = '1234567890123456789012345678901234567890'
3346 + '123456789123456789';
3348 do not work. You need an explicit Math::BigInt->new() around one of the
3349 operands. You should also quote large constants to protect loss of precision:
3353 $x = Math::BigInt->new('1234567889123456789123456789123456789');
3355 Without the quotes Perl would convert the large number to a floating point
3356 constant at compile time and then hand the result to BigInt, which results in
3357 an truncated result or a NaN.
3361 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
3362 must be made in the second case. For long numbers, the copy can eat up to 20%
3363 of the work (in the case of addition/subtraction, less for
3364 multiplication/division). If $y is very small compared to $x, the form
3365 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
3366 more time then the actual addition.
3368 With a technique called copy-on-write, the cost of copying with overload could
3369 be minimized or even completely avoided. A test implementation of COW did show
3370 performance gains for overloaded math, but introduced a performance loss due
3371 to a constant overhead for all other operatons.
3373 The rewritten version of this module is slower on certain operations, like
3374 new(), bstr() and numify(). The reason are that it does now more work and
3375 handles more cases. The time spent in these operations is usually gained in
3376 the other operations so that programs on the average should get faster. If
3377 they don't, please contect the author.
3379 Some operations may be slower for small numbers, but are significantly faster
3380 for big numbers. Other operations are now constant (O(1), like bneg(), babs()
3381 etc), instead of O(N) and thus nearly always take much less time. These
3382 optimizations were done on purpose.
3384 If you find the Calc module to slow, try to install any of the replacement
3385 modules and see if they help you.
3387 =head2 Alternative math libraries
3389 You can use an alternative library to drive Math::BigInt via:
3391 use Math::BigInt lib => 'Module';
3393 See L<MATH LIBRARY> for more information.
3395 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
3399 =head1 Subclassing Math::BigInt
3401 The basic design of Math::BigInt allows simple subclasses with very little
3402 work, as long as a few simple rules are followed:
3408 The public API must remain consistent, i.e. if a sub-class is overloading
3409 addition, the sub-class must use the same name, in this case badd(). The
3410 reason for this is that Math::BigInt is optimized to call the object methods
3415 The private object hash keys like C<$x->{sign}> may not be changed, but
3416 additional keys can be added, like C<$x->{_custom}>.
3420 Accessor functions are available for all existing object hash keys and should
3421 be used instead of directly accessing the internal hash keys. The reason for
3422 this is that Math::BigInt itself has a pluggable interface which permits it
3423 to support different storage methods.
3427 More complex sub-classes may have to replicate more of the logic internal of
3428 Math::BigInt if they need to change more basic behaviors. A subclass that
3429 needs to merely change the output only needs to overload C<bstr()>.
3431 All other object methods and overloaded functions can be directly inherited
3432 from the parent class.
3434 At the very minimum, any subclass will need to provide it's own C<new()> and can
3435 store additional hash keys in the object. There are also some package globals
3436 that must be defined, e.g.:
3440 $precision = -2; # round to 2 decimal places
3441 $round_mode = 'even';
3444 Additionally, you might want to provide the following two globals to allow
3445 auto-upgrading and auto-downgrading to work correctly:
3450 This allows Math::BigInt to correctly retrieve package globals from the
3451 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
3452 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
3458 in your subclass to automatically inherit the overloading from the parent. If
3459 you like, you can change part of the overloading, look at Math::String for an
3464 When used like this:
3466 use Math::BigInt upgrade => 'Foo::Bar';
3468 certain operations will 'upgrade' their calculation and thus the result to
3469 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
3471 use Math::BigInt upgrade => 'Math::BigFloat';
3473 As a shortcut, you can use the module C<bignum>:
3477 Also good for oneliners:
3479 perl -Mbignum -le 'print 2 ** 255'
3481 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
3482 as well es preserve accuracy (as in sqrt(3)).
3484 Beware: This feature is not fully implemented yet.
3488 The following methods upgrade themselves unconditionally; that is if upgrade
3489 is in effect, they will always hand up their work:
3501 Beware: This list is not complete.
3503 All other methods upgrade themselves only when one (or all) of their
3504 arguments are of the class mentioned in $upgrade (This might change in later
3505 versions to a more sophisticated scheme):
3511 =item Out of Memory!
3513 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
3514 C<eval()> in your code will crash with "Out of memory". This is probably an
3515 overload/exporter bug. You can workaround by not having C<eval()>
3516 and ':constant' at the same time or upgrade your Perl to a newer version.
3518 =item Fails to load Calc on Perl prior 5.6.0
3520 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
3521 will fall back to eval { require ... } when loading the math lib on Perls
3522 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
3523 filesystems using a different seperator.
3529 Some things might not work as you expect them. Below is documented what is
3530 known to be troublesome:
3534 =item stringify, bstr(), bsstr() and 'cmp'
3536 Both stringify and bstr() now drop the leading '+'. The old code would return
3537 '+3', the new returns '3'. This is to be consistent with Perl and to make
3538 cmp (especially with overloading) to work as you expect. It also solves
3539 problems with Test.pm, it's ok() uses 'eq' internally.
3541 Mark said, when asked about to drop the '+' altogether, or make only cmp work:
3543 I agree (with the first alternative), don't add the '+' on positive
3544 numbers. It's not as important anymore with the new internal
3545 form for numbers. It made doing things like abs and neg easier,
3546 but those have to be done differently now anyway.
3548 So, the following examples will now work all as expected:
3551 BEGIN { plan tests => 1 }
3554 my $x = new Math::BigInt 3*3;
3555 my $y = new Math::BigInt 3*3;
3558 print "$x eq 9" if $x eq $y;
3559 print "$x eq 9" if $x eq '9';
3560 print "$x eq 9" if $x eq 3*3;
3562 Additionally, the following still works:
3564 print "$x == 9" if $x == $y;
3565 print "$x == 9" if $x == 9;
3566 print "$x == 9" if $x == 3*3;
3568 There is now a C<bsstr()> method to get the string in scientific notation aka
3569 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
3570 for comparisation, but Perl will represent some numbers as 100 and others
3571 as 1e+308. If in doubt, convert both arguments to Math::BigInt before doing eq:
3574 BEGIN { plan tests => 3 }
3577 $x = Math::BigInt->new('1e56'); $y = 1e56;
3578 ok ($x,$y); # will fail
3579 ok ($x->bsstr(),$y); # okay
3580 $y = Math::BigInt->new($y);
3583 Alternatively, simple use <=> for comparisations, that will get it always
3584 right. There is not yet a way to get a number automatically represented as
3585 a string that matches exactly the way Perl represents it.
3589 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
3592 $x = Math::BigInt->new(123);
3593 $y = int($x); # BigInt 123
3594 $x = Math::BigFloat->new(123.45);
3595 $y = int($x); # BigInt 123
3597 In all Perl versions you can use C<as_number()> for the same effect:
3599 $x = Math::BigFloat->new(123.45);
3600 $y = $x->as_number(); # BigInt 123
3602 This also works for other subclasses, like Math::String.
3604 It is yet unlcear whether overloaded int() should return a scalar or a BigInt.
3608 The following will probably not do what you expect:
3610 $c = Math::BigInt->new(123);
3611 print $c->length(),"\n"; # prints 30
3613 It prints both the number of digits in the number and in the fraction part
3614 since print calls C<length()> in list context. Use something like:
3616 print scalar $c->length(),"\n"; # prints 3
3620 The following will probably not do what you expect:
3622 print $c->bdiv(10000),"\n";
3624 It prints both quotient and remainder since print calls C<bdiv()> in list
3625 context. Also, C<bdiv()> will modify $c, so be carefull. You probably want
3628 print $c / 10000,"\n";
3629 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
3633 The quotient is always the greatest integer less than or equal to the
3634 real-valued quotient of the two operands, and the remainder (when it is
3635 nonzero) always has the same sign as the second operand; so, for
3645 As a consequence, the behavior of the operator % agrees with the
3646 behavior of Perl's built-in % operator (as documented in the perlop
3647 manpage), and the equation
3649 $x == ($x / $y) * $y + ($x % $y)
3651 holds true for any $x and $y, which justifies calling the two return
3652 values of bdiv() the quotient and remainder. The only exception to this rule
3653 are when $y == 0 and $x is negative, then the remainder will also be
3654 negative. See below under "infinity handling" for the reasoning behing this.
3656 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
3657 not change BigInt's way to do things. This is because under 'use integer' Perl
3658 will do what the underlying C thinks is right and this is different for each
3659 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
3660 the author to implement it ;)
3662 =item infinity handling
3664 Here are some examples that explain the reasons why certain results occur while
3667 The following table shows the result of the division and the remainder, so that
3668 the equation above holds true. Some "ordinary" cases are strewn in to show more
3669 clearly the reasoning:
3671 A / B = C, R so that C * B + R = A
3672 =========================================================
3673 5 / 8 = 0, 5 0 * 8 + 5 = 5
3674 0 / 8 = 0, 0 0 * 8 + 0 = 0
3675 0 / inf = 0, 0 0 * inf + 0 = 0
3676 0 /-inf = 0, 0 0 * -inf + 0 = 0
3677 5 / inf = 0, 5 0 * inf + 5 = 5
3678 5 /-inf = 0, 5 0 * -inf + 5 = 5
3679 -5/ inf = 0, -5 0 * inf + -5 = -5
3680 -5/-inf = 0, -5 0 * -inf + -5 = -5
3681 inf/ 5 = inf, 0 inf * 5 + 0 = inf
3682 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
3683 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
3684 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
3685 5/ 5 = 1, 0 1 * 5 + 0 = 5
3686 -5/ -5 = 1, 0 1 * -5 + 0 = -5
3687 inf/ inf = 1, 0 1 * inf + 0 = inf
3688 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
3689 inf/-inf = -1, 0 -1 * -inf + 0 = inf
3690 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
3691 8/ 0 = inf, 8 inf * 0 + 8 = 8
3692 inf/ 0 = inf, inf inf * 0 + inf = inf
3695 These cases below violate the "remainder has the sign of the second of the two
3696 arguments", since they wouldn't match up otherwise.
3698 A / B = C, R so that C * B + R = A
3699 ========================================================
3700 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
3701 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
3703 =item Modifying and =
3707 $x = Math::BigFloat->new(5);
3710 It will not do what you think, e.g. making a copy of $x. Instead it just makes
3711 a second reference to the B<same> object and stores it in $y. Thus anything
3712 that modifies $x (except overloaded operators) will modify $y, and vice versa.
3713 Or in other words, C<=> is only safe if you modify your BigInts only via
3714 overloaded math. As soon as you use a method call it breaks:
3717 print "$x, $y\n"; # prints '10, 10'
3719 If you want a true copy of $x, use:
3723 You can also chain the calls like this, this will make first a copy and then
3726 $y = $x->copy()->bmul(2);
3728 See also the documentation for overload.pm regarding C<=>.
3732 C<bpow()> (and the rounding functions) now modifies the first argument and
3733 returns it, unlike the old code which left it alone and only returned the
3734 result. This is to be consistent with C<badd()> etc. The first three will
3735 modify $x, the last one won't:
3737 print bpow($x,$i),"\n"; # modify $x
3738 print $x->bpow($i),"\n"; # ditto
3739 print $x **= $i,"\n"; # the same
3740 print $x ** $i,"\n"; # leave $x alone
3742 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
3744 =item Overloading -$x
3754 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
3755 needs to preserve $x since it does not know that it later will get overwritten.
3756 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
3758 With Copy-On-Write, this issue would be gone, but C-o-W is not implemented
3759 since it is slower for all other things.
3761 =item Mixing different object types
3763 In Perl you will get a floating point value if you do one of the following:
3769 With overloaded math, only the first two variants will result in a BigFloat:
3774 $mbf = Math::BigFloat->new(5);
3775 $mbi2 = Math::BigInteger->new(5);
3776 $mbi = Math::BigInteger->new(2);
3778 # what actually gets called:
3779 $float = $mbf + $mbi; # $mbf->badd()
3780 $float = $mbf / $mbi; # $mbf->bdiv()
3781 $integer = $mbi + $mbf; # $mbi->badd()
3782 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
3783 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
3785 This is because math with overloaded operators follows the first (dominating)
3786 operand, and the operation of that is called and returns thus the result. So,
3787 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
3788 the result should be a Math::BigFloat or the second operant is one.
3790 To get a Math::BigFloat you either need to call the operation manually,
3791 make sure the operands are already of the proper type or casted to that type
3792 via Math::BigFloat->new():
3794 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
3796 Beware of simple "casting" the entire expression, this would only convert
3797 the already computed result:
3799 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
3801 Beware also of the order of more complicated expressions like:
3803 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
3804 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
3806 If in doubt, break the expression into simpler terms, or cast all operands
3807 to the desired resulting type.
3809 Scalar values are a bit different, since:
3814 will both result in the proper type due to the way the overloaded math works.
3816 This section also applies to other overloaded math packages, like Math::String.
3818 One solution to you problem might be L<autoupgrading|upgrading>.
3822 C<bsqrt()> works only good if the result is a big integer, e.g. the square
3823 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
3826 If you want a better approximation of the square root, then use:
3828 $x = Math::BigFloat->new(12);
3829 Math::BigFloat->precision(0);
3830 Math::BigFloat->round_mode('even');
3831 print $x->copy->bsqrt(),"\n"; # 4
3833 Math::BigFloat->precision(2);
3834 print $x->bsqrt(),"\n"; # 3.46
3835 print $x->bsqrt(3),"\n"; # 3.464
3839 For negative numbers in base see also L<brsft|brsft>.
3845 This program is free software; you may redistribute it and/or modify it under
3846 the same terms as Perl itself.
3850 L<Math::BigFloat> and L<Math::Big> as well as L<Math::BigInt::BitVect>,
3851 L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
3854 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
3855 more documentation including a full version history, testcases, empty
3856 subclass files and benchmarks.
3860 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
3861 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001.