4 # "Mike had an infinite amount to do and a negative amount of time in which
5 # to do it." - Before and After
8 # The following hash values are used:
9 # value: unsigned int with actual value (as a Math::BigInt::Calc or similiar)
10 # sign : +,-,NaN,+inf,-inf
13 # _f : flags, used by MBF to flag parts of a float as untouchable
15 # Remember not to take shortcuts ala $xs = $x->{value}; $CALC->foo($xs); since
16 # underlying lib might change the reference!
18 my $class = "Math::BigInt";
23 @ISA = qw( Exporter );
24 @EXPORT_OK = qw( objectify _swap bgcd blcm);
25 use vars qw/$round_mode $accuracy $precision $div_scale $rnd_mode/;
26 use vars qw/$upgrade $downgrade/;
29 # Inside overload, the first arg is always an object. If the original code had
30 # it reversed (like $x = 2 * $y), then the third paramater indicates this
31 # swapping. To make it work, we use a helper routine which not only reswaps the
32 # params, but also makes a new object in this case. See _swap() for details,
33 # especially the cases of operators with different classes.
35 # For overloaded ops with only one argument we simple use $_[0]->copy() to
36 # preserve the argument.
38 # Thus inheritance of overload operators becomes possible and transparent for
39 # our subclasses without the need to repeat the entire overload section there.
42 '=' => sub { $_[0]->copy(); },
44 # '+' and '-' do not use _swap, since it is a triffle slower. If you want to
45 # override _swap (if ever), then override overload of '+' and '-', too!
46 # for sub it is a bit tricky to keep b: b-a => -a+b
47 '-' => sub { my $c = $_[0]->copy; $_[2] ?
48 $c->bneg()->badd($_[1]) :
50 '+' => sub { $_[0]->copy()->badd($_[1]); },
52 # some shortcuts for speed (assumes that reversed order of arguments is routed
53 # to normal '+' and we thus can always modify first arg. If this is changed,
54 # this breaks and must be adjusted.)
55 '+=' => sub { $_[0]->badd($_[1]); },
56 '-=' => sub { $_[0]->bsub($_[1]); },
57 '*=' => sub { $_[0]->bmul($_[1]); },
58 '/=' => sub { scalar $_[0]->bdiv($_[1]); },
59 '%=' => sub { $_[0]->bmod($_[1]); },
60 '^=' => sub { $_[0]->bxor($_[1]); },
61 '&=' => sub { $_[0]->band($_[1]); },
62 '|=' => sub { $_[0]->bior($_[1]); },
63 '**=' => sub { $_[0]->bpow($_[1]); },
65 # not supported by Perl yet
66 '..' => \&_pointpoint,
68 '<=>' => sub { $_[2] ?
69 ref($_[0])->bcmp($_[1],$_[0]) :
70 ref($_[0])->bcmp($_[0],$_[1])},
73 "$_[1]" cmp $_[0]->bstr() :
74 $_[0]->bstr() cmp "$_[1]" },
76 'log' => sub { $_[0]->copy()->blog(); },
77 'int' => sub { $_[0]->copy(); },
78 'neg' => sub { $_[0]->copy()->bneg(); },
79 'abs' => sub { $_[0]->copy()->babs(); },
80 'sqrt' => sub { $_[0]->copy()->bsqrt(); },
81 '~' => sub { $_[0]->copy()->bnot(); },
83 '*' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bmul($a[1]); },
84 '/' => sub { my @a = ref($_[0])->_swap(@_);scalar $a[0]->bdiv($a[1]);},
85 '%' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bmod($a[1]); },
86 '**' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bpow($a[1]); },
87 '<<' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->blsft($a[1]); },
88 '>>' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->brsft($a[1]); },
90 '&' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->band($a[1]); },
91 '|' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bior($a[1]); },
92 '^' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bxor($a[1]); },
94 # can modify arg of ++ and --, so avoid a new-copy for speed, but don't
95 # use $_[0]->__one(), it modifies $_[0] to be 1!
96 '++' => sub { $_[0]->binc() },
97 '--' => sub { $_[0]->bdec() },
99 # if overloaded, O(1) instead of O(N) and twice as fast for small numbers
101 # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/
102 # v5.6.1 dumps on that: return !$_[0]->is_zero() || undef; :-(
103 my $t = !$_[0]->is_zero();
108 # the original qw() does not work with the TIESCALAR below, why?
109 # Order of arguments unsignificant
110 '""' => sub { $_[0]->bstr(); },
111 '0+' => sub { $_[0]->numify(); }
114 ##############################################################################
115 # global constants, flags and accessory
117 use constant MB_NEVER_ROUND => 0x0001;
119 my $NaNOK=1; # are NaNs ok?
120 my $nan = 'NaN'; # constants for easier life
122 my $CALC = 'Math::BigInt::Calc'; # module to do low level math
123 my $IMPORT = 0; # did import() yet?
125 $round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
130 $upgrade = undef; # default is no upgrade
131 $downgrade = undef; # default is no downgrade
133 ##############################################################################
134 # the old code had $rnd_mode, so we need to support it, too
137 sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; }
138 sub FETCH { return $round_mode; }
139 sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
141 BEGIN { tie $rnd_mode, 'Math::BigInt'; }
143 ##############################################################################
148 # make Class->round_mode() work
150 my $class = ref($self) || $self || __PACKAGE__;
154 die "Unknown round mode $m"
155 if $m !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/;
156 return ${"${class}::round_mode"} = $m;
158 return ${"${class}::round_mode"};
164 # make Class->upgrade() work
166 my $class = ref($self) || $self || __PACKAGE__;
167 # need to set new value?
171 return ${"${class}::upgrade"} = $u;
173 return ${"${class}::upgrade"};
179 # make Class->downgrade() work
181 my $class = ref($self) || $self || __PACKAGE__;
182 # need to set new value?
186 return ${"${class}::downgrade"} = $u;
188 return ${"${class}::downgrade"};
194 # make Class->round_mode() work
196 my $class = ref($self) || $self || __PACKAGE__;
199 die ('div_scale must be greater than zero') if $_[0] < 0;
200 ${"${class}::div_scale"} = shift;
202 return ${"${class}::div_scale"};
207 # $x->accuracy($a); ref($x) $a
208 # $x->accuracy(); ref($x)
209 # Class->accuracy(); class
210 # Class->accuracy($a); class $a
213 my $class = ref($x) || $x || __PACKAGE__;
216 # need to set new value?
220 die ('accuracy must not be zero') if defined $a && $a == 0;
223 # $object->accuracy() or fallback to global
224 $x->bround($a) if defined $a;
225 $x->{_a} = $a; # set/overwrite, even if not rounded
226 $x->{_p} = undef; # clear P
231 ${"${class}::accuracy"} = $a;
232 ${"${class}::precision"} = undef; # clear P
234 return $a; # shortcut
239 # $object->accuracy() or fallback to global
240 return $x->{_a} || ${"${class}::accuracy"};
242 return ${"${class}::accuracy"};
247 # $x->precision($p); ref($x) $p
248 # $x->precision(); ref($x)
249 # Class->precision(); class
250 # Class->precision($p); class $p
253 my $class = ref($x) || $x || __PACKAGE__;
256 # need to set new value?
262 # $object->precision() or fallback to global
263 $x->bfround($p) if defined $p;
264 $x->{_p} = $p; # set/overwrite, even if not rounded
265 $x->{_a} = undef; # clear A
270 ${"${class}::precision"} = $p;
271 ${"${class}::accuracy"} = undef; # clear A
273 return $p; # shortcut
278 # $object->precision() or fallback to global
279 return $x->{_p} || ${"${class}::precision"};
281 return ${"${class}::precision"};
286 # return (later set?) configuration data as hash ref
287 my $class = shift || 'Math::BigInt';
293 lib_version => ${"${lib}::VERSION"},
297 qw/upgrade downgrade precision accuracy round_mode VERSION div_scale/)
299 $cfg->{lc($_)} = ${"${class}::$_"};
306 # select accuracy parameter based on precedence,
307 # used by bround() and bfround(), may return undef for scale (means no op)
308 my ($x,$s,$m,$scale,$mode) = @_;
309 $scale = $x->{_a} if !defined $scale;
310 $scale = $s if (!defined $scale);
311 $mode = $m if !defined $mode;
312 return ($scale,$mode);
317 # select precision parameter based on precedence,
318 # used by bround() and bfround(), may return undef for scale (means no op)
319 my ($x,$s,$m,$scale,$mode) = @_;
320 $scale = $x->{_p} if !defined $scale;
321 $scale = $s if (!defined $scale);
322 $mode = $m if !defined $mode;
323 return ($scale,$mode);
326 ##############################################################################
334 # if two arguments, the first one is the class to "swallow" subclasses
342 return unless ref($x); # only for objects
344 my $self = {}; bless $self,$c;
346 foreach my $k (keys %$x)
350 $self->{value} = $CALC->_copy($x->{value}); next;
352 if (!($r = ref($x->{$k})))
354 $self->{$k} = $x->{$k}; next;
358 $self->{$k} = \${$x->{$k}};
360 elsif ($r eq 'ARRAY')
362 $self->{$k} = [ @{$x->{$k}} ];
366 # only one level deep!
367 foreach my $h (keys %{$x->{$k}})
369 $self->{$k}->{$h} = $x->{$k}->{$h};
375 if ($xk->can('copy'))
377 $self->{$k} = $xk->copy();
381 $self->{$k} = $xk->new($xk);
390 # create a new BigInt object from a string or another BigInt object.
391 # see hash keys documented at top
393 # the argument could be an object, so avoid ||, && etc on it, this would
394 # cause costly overloaded code to be called. The only allowed ops are
397 my ($class,$wanted,$a,$p,$r) = @_;
399 # avoid numify-calls by not using || on $wanted!
400 return $class->bzero($a,$p) if !defined $wanted; # default to 0
401 return $class->copy($wanted,$a,$p,$r)
402 if ref($wanted) && $wanted->isa($class); # MBI or subclass
404 $class->import() if $IMPORT == 0; # make require work
406 my $self = bless {}, $class;
408 # shortcut for "normal" numbers
409 if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*$/))
411 $self->{sign} = $1 || '+';
413 if ($wanted =~ /^[+-]/)
415 # remove sign without touching wanted
416 my $t = $wanted; $t =~ s/^[+-]//; $ref = \$t;
418 $self->{value} = $CALC->_new($ref);
420 if ( (defined $a) || (defined $p)
421 || (defined ${"${class}::precision"})
422 || (defined ${"${class}::accuracy"})
425 $self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p);
430 # handle '+inf', '-inf' first
431 if ($wanted =~ /^[+-]?inf$/)
433 $self->{value} = $CALC->_zero();
434 $self->{sign} = $wanted; $self->{sign} = '+inf' if $self->{sign} eq 'inf';
437 # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign
438 my ($mis,$miv,$mfv,$es,$ev) = _split(\$wanted);
441 die "$wanted is not a number initialized to $class" if !$NaNOK;
443 $self->{value} = $CALC->_zero();
444 $self->{sign} = $nan;
449 # _from_hex or _from_bin
450 $self->{value} = $mis->{value};
451 $self->{sign} = $mis->{sign};
452 return $self; # throw away $mis
454 # make integer from mantissa by adjusting exp, then convert to bigint
455 $self->{sign} = $$mis; # store sign
456 $self->{value} = $CALC->_zero(); # for all the NaN cases
457 my $e = int("$$es$$ev"); # exponent (avoid recursion)
460 my $diff = $e - CORE::length($$mfv);
461 if ($diff < 0) # Not integer
464 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
465 $self->{sign} = $nan;
469 # adjust fraction and add it to value
470 # print "diff > 0 $$miv\n";
471 $$miv = $$miv . ($$mfv . '0' x $diff);
476 if ($$mfv ne '') # e <= 0
478 # fraction and negative/zero E => NOI
479 #print "NOI 2 \$\$mfv '$$mfv'\n";
480 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
481 $self->{sign} = $nan;
485 # xE-y, and empty mfv
488 if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
491 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
492 $self->{sign} = $nan;
496 $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
497 $self->{value} = $CALC->_new($miv) if $self->{sign} =~ /^[+-]$/;
498 # if any of the globals is set, use them to round and store them inside $self
499 # do not round for new($x,undef,undef) since that is used by MBF to signal
501 $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p;
507 # create a bigint 'NaN', if given a BigInt, set it to 'NaN'
509 $self = $class if !defined $self;
512 my $c = $self; $self = {}; bless $self, $c;
514 $self->import() if $IMPORT == 0; # make require work
515 return if $self->modify('bnan');
517 if ($self->can('_bnan'))
519 # use subclass to initialize
524 # otherwise do our own thing
525 $self->{value} = $CALC->_zero();
527 $self->{value} = $CALC->_zero();
528 $self->{sign} = $nan;
529 delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly
535 # create a bigint '+-inf', if given a BigInt, set it to '+-inf'
536 # the sign is either '+', or if given, used from there
538 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
539 $self = $class if !defined $self;
542 my $c = $self; $self = {}; bless $self, $c;
544 $self->import() if $IMPORT == 0; # make require work
545 return if $self->modify('binf');
547 if ($self->can('_binf'))
549 # use subclass to initialize
554 # otherwise do our own thing
555 $self->{value} = $CALC->_zero();
557 $self->{sign} = $sign.'inf';
558 ($self->{_a},$self->{_p}) = @_; # take over requested rounding
564 # create a bigint '+0', if given a BigInt, set it to 0
566 $self = $class if !defined $self;
570 my $c = $self; $self = {}; bless $self, $c;
572 $self->import() if $IMPORT == 0; # make require work
573 return if $self->modify('bzero');
575 if ($self->can('_bzero'))
577 # use subclass to initialize
582 # otherwise do our own thing
583 $self->{value} = $CALC->_zero();
589 if (defined $self->{_a} && defined $_[0] && $_[0] > $self->{_a});
591 if (defined $self->{_p} && defined $_[1] && $_[1] < $self->{_p});
598 # create a bigint '+1' (or -1 if given sign '-'),
599 # if given a BigInt, set it to +1 or -1, respecively
601 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
602 $self = $class if !defined $self;
606 my $c = $self; $self = {}; bless $self, $c;
608 $self->import() if $IMPORT == 0; # make require work
609 return if $self->modify('bone');
611 if ($self->can('_bone'))
613 # use subclass to initialize
618 # otherwise do our own thing
619 $self->{value} = $CALC->_one();
621 $self->{sign} = $sign;
625 if (defined $self->{_a} && defined $_[0] && $_[0] > $self->{_a});
627 if (defined $self->{_p} && defined $_[1] && $_[1] < $self->{_p});
632 ##############################################################################
633 # string conversation
637 # (ref to BFLOAT or num_str ) return num_str
638 # Convert number from internal format to scientific string format.
639 # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
640 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
641 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
643 if ($x->{sign} !~ /^[+-]$/)
645 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
648 my ($m,$e) = $x->parts();
649 # e can only be positive
651 # MBF: my $s = $e->{sign}; $s = '' if $s eq '-'; my $sep = 'e'.$s;
652 return $m->bstr().$sign.$e->bstr();
657 # make a string from bigint object
658 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
659 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
661 if ($x->{sign} !~ /^[+-]$/)
663 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
666 my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
667 return $es.${$CALC->_str($x->{value})};
672 # Make a "normal" scalar from a BigInt object
673 my $x = shift; $x = $class->new($x) unless ref $x;
674 return $x->{sign} if $x->{sign} !~ /^[+-]$/;
675 my $num = $CALC->_num($x->{value});
676 return -$num if $x->{sign} eq '-';
680 ##############################################################################
681 # public stuff (usually prefixed with "b")
685 # return the sign of the number: +/-/-inf/+inf/NaN
686 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
691 sub _find_round_parameters
693 # After any operation or when calling round(), the result is rounded by
694 # regarding the A & P from arguments, local parameters, or globals.
696 # This procedure finds the round parameters, but it is for speed reasons
697 # duplicated in round. Otherwise, it is tested by the testsuite and used
700 my ($self,$a,$p,$r,@args) = @_;
701 # $a accuracy, if given by caller
702 # $p precision, if given by caller
703 # $r round_mode, if given by caller
704 # @args all 'other' arguments (0 for unary, 1 for binary ops)
706 # leave bigfloat parts alone
707 return ($self) if exists $self->{_f} && $self->{_f} & MB_NEVER_ROUND != 0;
709 my $c = ref($self); # find out class of argument(s)
712 # now pick $a or $p, but only if we have got "arguments"
715 foreach ($self,@args)
717 # take the defined one, or if both defined, the one that is smaller
718 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
723 # even if $a is defined, take $p, to signal error for both defined
724 foreach ($self,@args)
726 # take the defined one, or if both defined, the one that is bigger
728 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
731 # if still none defined, use globals (#2)
732 $a = ${"$c\::accuracy"} unless defined $a;
733 $p = ${"$c\::precision"} unless defined $p;
736 return ($self) unless defined $a || defined $p; # early out
738 # set A and set P is an fatal error
739 return ($self->bnan()) if defined $a && defined $p;
741 $r = ${"$c\::round_mode"} unless defined $r;
742 die "Unknown round mode '$r'" if $r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/;
744 return ($self,$a,$p,$r);
749 # Round $self according to given parameters, or given second argument's
750 # parameters or global defaults
752 # for speed reasons, _find_round_parameters is embeded here:
754 my ($self,$a,$p,$r,@args) = @_;
755 # $a accuracy, if given by caller
756 # $p precision, if given by caller
757 # $r round_mode, if given by caller
758 # @args all 'other' arguments (0 for unary, 1 for binary ops)
760 # leave bigfloat parts alone
761 return ($self) if exists $self->{_f} && $self->{_f} & MB_NEVER_ROUND != 0;
763 my $c = ref($self); # find out class of argument(s)
766 # now pick $a or $p, but only if we have got "arguments"
769 foreach ($self,@args)
771 # take the defined one, or if both defined, the one that is smaller
772 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
777 # even if $a is defined, take $p, to signal error for both defined
778 foreach ($self,@args)
780 # take the defined one, or if both defined, the one that is bigger
782 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
785 # if still none defined, use globals (#2)
786 $a = ${"$c\::accuracy"} unless defined $a;
787 $p = ${"$c\::precision"} unless defined $p;
790 return $self unless defined $a || defined $p; # early out
792 # set A and set P is an fatal error
793 return $self->bnan() if defined $a && defined $p;
795 $r = ${"$c\::round_mode"} unless defined $r;
796 die "Unknown round mode '$r'" if $r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/;
798 # now round, by calling either fround or ffround:
801 $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a;
803 else # both can't be undefined due to early out
805 $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p;
807 $self->bnorm(); # after round, normalize
812 # (numstr or BINT) return BINT
813 # Normalize number -- no-op here
814 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
820 # (BINT or num_str) return BINT
821 # make number absolute, or return absolute BINT from string
822 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
824 return $x if $x->modify('babs');
825 # post-normalized abs for internal use (does nothing for NaN)
826 $x->{sign} =~ s/^-/+/;
832 # (BINT or num_str) return BINT
833 # negate number or make a negated number from string
834 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
836 return $x if $x->modify('bneg');
838 # for +0 dont negate (to have always normalized)
839 $x->{sign} =~ tr/+-/-+/ if !$x->is_zero(); # does nothing for NaN
845 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
846 # (BINT or num_str, BINT or num_str) return cond_code
847 my ($self,$x,$y) = objectify(2,@_);
849 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
851 # handle +-inf and NaN
852 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
853 return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
854 return +1 if $x->{sign} eq '+inf';
855 return -1 if $x->{sign} eq '-inf';
856 return -1 if $y->{sign} eq '+inf';
859 # check sign for speed first
860 return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
861 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
864 my $xz = $x->is_zero();
865 my $yz = $y->is_zero();
866 return 0 if $xz && $yz; # 0 <=> 0
867 return -1 if $xz && $y->{sign} eq '+'; # 0 <=> +y
868 return 1 if $yz && $x->{sign} eq '+'; # +x <=> 0
870 # post-normalized compare for internal use (honors signs)
871 if ($x->{sign} eq '+')
873 return 1 if $y->{sign} eq '-'; # 0 check handled above
874 return $CALC->_acmp($x->{value},$y->{value});
878 return -1 if $y->{sign} eq '+';
879 $CALC->_acmp($y->{value},$x->{value}); # swaped (lib does only 0,1,-1)
884 # Compares 2 values, ignoring their signs.
885 # Returns one of undef, <0, =0, >0. (suitable for sort)
886 # (BINT, BINT) return cond_code
887 my ($self,$x,$y) = objectify(2,@_);
889 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
891 # handle +-inf and NaN
892 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
893 return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
894 return +1; # inf is always bigger
896 $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
901 # add second arg (BINT or string) to first (BINT) (modifies first)
902 # return result as BINT
903 my ($self,$x,$y,@r) = objectify(2,@_);
905 return $x if $x->modify('badd');
906 # print "mbi badd ",join(' ',caller()),"\n";
907 # print "upgrade => ",$upgrade||'undef',
908 # " \$x (",ref($x),") \$y (",ref($y),")\n";
909 # return $upgrade->badd($x,$y,@r) if defined $upgrade &&
910 # ((ref($x) eq $upgrade) || (ref($y) eq $upgrade));
911 # print "still badd\n";
913 $r[3] = $y; # no push!
914 # inf and NaN handling
915 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
918 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
920 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
922 # +inf++inf or -inf+-inf => same, rest is NaN
923 return $x if $x->{sign} eq $y->{sign};
926 # +-inf + something => +inf
927 # something +-inf => +-inf
928 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
932 my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
936 $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
941 my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
944 #print "swapped sub (a=$a)\n";
945 $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
950 # speedup, if equal, set result to 0
951 #print "equal sub, result = 0\n";
952 $x->{value} = $CALC->_zero();
957 #print "unswapped sub (a=$a)\n";
958 $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
967 # (BINT or num_str, BINT or num_str) return num_str
968 # subtract second arg from first, modify first
969 my ($self,$x,$y,@r) = objectify(2,@_);
971 return $x if $x->modify('bsub');
972 # return $upgrade->badd($x,$y,@r) if defined $upgrade &&
973 # ((ref($x) eq $upgrade) || (ref($y) eq $upgrade));
977 return $x->round(@r);
980 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
981 $x->badd($y,@r); # badd does not leave internal zeros
982 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
983 $x; # already rounded by badd() or no round necc.
988 # increment arg by one
989 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
990 return $x if $x->modify('binc');
992 if ($x->{sign} eq '+')
994 $x->{value} = $CALC->_inc($x->{value});
995 return $x->round($a,$p,$r);
997 elsif ($x->{sign} eq '-')
999 $x->{value} = $CALC->_dec($x->{value});
1000 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1001 return $x->round($a,$p,$r);
1003 # inf, nan handling etc
1004 $x->badd($self->__one(),$a,$p,$r); # badd does round
1009 # decrement arg by one
1010 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1011 return $x if $x->modify('bdec');
1013 my $zero = $CALC->_is_zero($x->{value}) && $x->{sign} eq '+';
1015 if (($x->{sign} eq '-') || $zero)
1017 $x->{value} = $CALC->_inc($x->{value});
1018 $x->{sign} = '-' if $zero; # 0 => 1 => -1
1019 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1020 return $x->round($a,$p,$r);
1023 elsif ($x->{sign} eq '+')
1025 $x->{value} = $CALC->_dec($x->{value});
1026 return $x->round($a,$p,$r);
1028 # inf, nan handling etc
1029 $x->badd($self->__one('-'),$a,$p,$r); # badd does round
1034 # not implemented yet
1035 my ($self,$x,$base,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1037 return $upgrade->blog($x,$base,$a,$p,$r) if defined $upgrade;
1044 # (BINT or num_str, BINT or num_str) return BINT
1045 # does not modify arguments, but returns new object
1046 # Lowest Common Multiplicator
1048 my $y = shift; my ($x);
1055 $x = $class->new($y);
1057 while (@_) { $x = __lcm($x,shift); }
1063 # (BINT or num_str, BINT or num_str) return BINT
1064 # does not modify arguments, but returns new object
1065 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1068 $y = __PACKAGE__->new($y) if !ref($y);
1070 my $x = $y->copy(); # keep arguments
1071 if ($CALC->can('_gcd'))
1075 $y = shift; $y = $self->new($y) if !ref($y);
1076 next if $y->is_zero();
1077 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1078 $x->{value} = $CALC->_gcd($x->{value},$y->{value}); last if $x->is_one();
1085 $y = shift; $y = $self->new($y) if !ref($y);
1086 $x = __gcd($x,$y->copy()); last if $x->is_one(); # _gcd handles NaN
1094 # (num_str or BINT) return BINT
1095 # represent ~x as twos-complement number
1096 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1097 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1099 return $x if $x->modify('bnot');
1100 $x->bneg()->bdec(); # bdec already does round
1103 # is_foo test routines
1107 # return true if arg (BINT or num_str) is zero (array '+', '0')
1108 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1109 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1111 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1112 $CALC->_is_zero($x->{value});
1117 # return true if arg (BINT or num_str) is NaN
1118 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1120 return 1 if $x->{sign} eq $nan;
1126 # return true if arg (BINT or num_str) is +-inf
1127 my ($self,$x,$sign) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1129 $sign = '' if !defined $sign;
1130 return 1 if $sign eq $x->{sign}; # match ("+inf" eq "+inf")
1131 return 0 if $sign !~ /^([+-]|)$/;
1135 return 1 if ($x->{sign} =~ /^[+-]inf$/);
1138 $sign = quotemeta($sign.'inf');
1139 return 1 if ($x->{sign} =~ /^$sign$/);
1145 # return true if arg (BINT or num_str) is +1
1146 # or -1 if sign is given
1147 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1148 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1150 $sign = '' if !defined $sign; $sign = '+' if $sign ne '-';
1152 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1153 $CALC->_is_one($x->{value});
1158 # return true when arg (BINT or num_str) is odd, false for even
1159 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1160 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1162 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1163 $CALC->_is_odd($x->{value});
1168 # return true when arg (BINT or num_str) is even, false for odd
1169 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1170 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1172 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1173 $CALC->_is_even($x->{value});
1178 # return true when arg (BINT or num_str) is positive (>= 0)
1179 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1180 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1182 return 1 if $x->{sign} =~ /^\+/;
1188 # return true when arg (BINT or num_str) is negative (< 0)
1189 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1190 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1192 return 1 if ($x->{sign} =~ /^-/);
1198 # return true when arg (BINT or num_str) is an integer
1199 # always true for BigInt, but different for Floats
1200 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1201 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1203 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1206 ###############################################################################
1210 # multiply two numbers -- stolen from Knuth Vol 2 pg 233
1211 # (BINT or num_str, BINT or num_str) return BINT
1212 my ($self,$x,$y,@r) = objectify(2,@_);
1214 return $x if $x->modify('bmul');
1216 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1219 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1221 return $x->bnan() if $x->is_zero() || $y->is_zero();
1222 # result will always be +-inf:
1223 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1224 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1225 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1226 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1227 return $x->binf('-');
1230 return $upgrade->bmul($x,$y,@r)
1231 if defined $upgrade && $y->isa($upgrade);
1233 $r[3] = $y; # no push here
1235 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1237 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1238 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1244 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1245 my ($self,$x,$y) = @_;
1247 # NaN if x == NaN or y == NaN or x==y==0
1248 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1249 if (($x->is_nan() || $y->is_nan()) ||
1250 ($x->is_zero() && $y->is_zero()));
1252 # +-inf / +-inf == NaN, reminder also NaN
1253 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1255 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1257 # x / +-inf => 0, remainder x (works even if x == 0)
1258 if ($y->{sign} =~ /^[+-]inf$/)
1260 my $t = $x->copy(); # binf clobbers up $x
1261 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1264 # 5 / 0 => +inf, -6 / 0 => -inf
1265 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1266 # exception: -8 / 0 has remainder -8, not 8
1267 # exception: -inf / 0 has remainder -inf, not inf
1270 # +-inf / 0 => special case for -inf
1271 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1272 if (!$x->is_zero() && !$x->is_inf())
1274 my $t = $x->copy(); # binf clobbers up $x
1276 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1280 # last case: +-inf / ordinary number
1282 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1284 return wantarray ? ($x,$self->bzero()) : $x;
1289 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1290 # (BINT,BINT) (quo,rem) or BINT (only rem)
1291 my ($self,$x,$y,@r) = objectify(2,@_);
1293 return $x if $x->modify('bdiv');
1295 return $self->_div_inf($x,$y)
1296 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1298 return $upgrade->bdiv($upgrade->new($x),$y,@r)
1299 if defined $upgrade && $y->isa($upgrade);
1301 $r[3] = $y; # no push!
1305 wantarray ? ($x->round(@r),$self->bzero(@r)):$x->round(@r) if $x->is_zero();
1307 # Is $x in the interval [0, $y) (aka $x <= $y) ?
1308 my $cmp = $CALC->_acmp($x->{value},$y->{value});
1309 if (($cmp < 0) and (($x->{sign} eq $y->{sign}) or !wantarray))
1311 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1312 if defined $upgrade;
1314 return $x->bzero()->round(@r) unless wantarray;
1315 my $t = $x->copy(); # make copy first, because $x->bzero() clobbers $x
1316 return ($x->bzero()->round(@r),$t);
1320 # shortcut, both are the same, so set to +/- 1
1321 $x->__one( ($x->{sign} ne $y->{sign} ? '-' : '+') );
1322 return $x unless wantarray;
1323 return ($x->round(@r),$self->bzero(@r));
1325 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1326 if defined $upgrade;
1328 # calc new sign and in case $y == +/- 1, return $x
1329 my $xsign = $x->{sign}; # keep
1330 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1331 # check for / +-1 (cant use $y->is_one due to '-'
1332 if ($CALC->_is_one($y->{value}))
1334 return wantarray ? ($x->round(@r),$self->bzero(@r)) : $x->round(@r);
1339 my $rem = $self->bzero();
1340 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1341 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1343 if (! $CALC->_is_zero($rem->{value}))
1345 $rem->{sign} = $y->{sign};
1346 $rem = $y-$rem if $xsign ne $y->{sign}; # one of them '-'
1350 $rem->{sign} = '+'; # dont leave -0
1356 $x->{value} = $CALC->_div($x->{value},$y->{value});
1357 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1363 # modulus (or remainder)
1364 # (BINT or num_str, BINT or num_str) return BINT
1365 my ($self,$x,$y,@r) = objectify(2,@_);
1367 return $x if $x->modify('bmod');
1368 $r[3] = $y; # no push!
1369 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1371 my ($d,$r) = $self->_div_inf($x,$y);
1372 return $r->round(@r);
1375 if ($CALC->can('_mod'))
1377 # calc new sign and in case $y == +/- 1, return $x
1378 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1379 if (!$CALC->_is_zero($x->{value}))
1381 my $xsign = $x->{sign};
1382 $x->{sign} = $y->{sign};
1383 $x = $y-$x if $xsign ne $y->{sign}; # one of them '-'
1387 $x->{sign} = '+'; # dont leave -0
1389 return $x->round(@r);
1391 my ($t,$rem) = $self->bdiv($x->copy(),$y,@r); # slow way (also rounds)
1393 foreach (qw/value sign _a _p/)
1395 $x->{$_} = $rem->{$_};
1402 # (BINT or num_str, BINT or num_str) return BINT
1403 # compute factorial numbers
1404 # modifies first argument
1405 my ($self,$x,@r) = objectify(1,@_);
1407 return $x if $x->modify('bfac');
1409 return $x->bnan() if $x->{sign} ne '+'; # inf, NnN, <0 etc => NaN
1410 return $x->bone(@r) if $x->is_zero() || $x->is_one(); # 0 or 1 => 1
1412 if ($CALC->can('_fac'))
1414 $x->{value} = $CALC->_fac($x->{value});
1415 return $x->round(@r);
1420 my $f = $self->new(2);
1421 while ($f->bacmp($n) < 0)
1423 $x->bmul($f); $f->binc();
1425 $x->bmul($f); # last step
1426 $x->round(@r); # round
1431 # (BINT or num_str, BINT or num_str) return BINT
1432 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1433 # modifies first argument
1434 my ($self,$x,$y,@r) = objectify(2,@_);
1436 return $x if $x->modify('bpow');
1438 return $upgrade->bpow($upgrade->new($x),$y,@r)
1439 if defined $upgrade && $y->isa($upgrade);
1441 $r[3] = $y; # no push!
1442 return $x if $x->{sign} =~ /^[+-]inf$/; # -inf/+inf ** x
1443 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1444 return $x->bone(@r) if $y->is_zero();
1445 return $x->round(@r) if $x->is_one() || $y->is_one();
1446 if ($x->{sign} eq '-' && $CALC->_is_one($x->{value}))
1448 # if $x == -1 and odd/even y => +1/-1
1449 return $y->is_odd() ? $x->round(@r) : $x->babs()->round(@r);
1450 # my Casio FX-5500L has a bug here: -1 ** 2 is -1, but -1 * -1 is 1;
1452 # 1 ** -y => 1 / (1 ** |y|)
1453 # so do test for negative $y after above's clause
1454 return $x->bnan() if $y->{sign} eq '-';
1455 return $x->round(@r) if $x->is_zero(); # 0**y => 0 (if not y <= 0)
1457 if ($CALC->can('_pow'))
1459 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1460 return $x->round(@r);
1463 # based on the assumption that shifting in base 10 is fast, and that mul
1464 # works faster if numbers are small: we count trailing zeros (this step is
1465 # O(1)..O(N), but in case of O(N) we save much more time due to this),
1466 # stripping them out of the multiplication, and add $count * $y zeros
1467 # afterwards like this:
1468 # 300 ** 3 == 300*300*300 == 3*3*3 . '0' x 2 * 3 == 27 . '0' x 6
1469 # creates deep recursion?
1470 # my $zeros = $x->_trailing_zeros();
1473 # $x->brsft($zeros,10); # remove zeros
1474 # $x->bpow($y); # recursion (will not branch into here again)
1475 # $zeros = $y * $zeros; # real number of zeros to add
1476 # $x->blsft($zeros,10);
1477 # return $x->round($a,$p,$r);
1480 my $pow2 = $self->__one();
1481 my $y1 = $class->new($y);
1482 my $two = $self->new(2);
1483 while (!$y1->is_one())
1485 $pow2->bmul($x) if $y1->is_odd();
1489 $x->bmul($pow2) unless $pow2->is_one();
1490 return $x->round(@r);
1495 # (BINT or num_str, BINT or num_str) return BINT
1496 # compute x << y, base n, y >= 0
1497 my ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_);
1499 return $x if $x->modify('blsft');
1500 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1501 return $x->round($a,$p,$r) if $y->is_zero();
1503 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1505 my $t; $t = $CALC->_lsft($x->{value},$y->{value},$n) if $CALC->can('_lsft');
1508 $x->{value} = $t; return $x->round($a,$p,$r);
1511 return $x->bmul( $self->bpow($n, $y, $a, $p, $r), $a, $p, $r );
1516 # (BINT or num_str, BINT or num_str) return BINT
1517 # compute x >> y, base n, y >= 0
1518 my ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_);
1520 return $x if $x->modify('brsft');
1521 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1522 return $x->round($a,$p,$r) if $y->is_zero();
1523 return $x->bzero($a,$p,$r) if $x->is_zero(); # 0 => 0
1525 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1527 # this only works for negative numbers when shifting in base 2
1528 if (($x->{sign} eq '-') && ($n == 2))
1530 return $x->round($a,$p,$r) if $x->is_one('-'); # -1 => -1
1533 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1534 # but perhaps there is a better emulation for two's complement shift...
1535 # if $y != 1, we must simulate it by doing:
1536 # convert to bin, flip all bits, shift, and be done
1537 $x->binc(); # -3 => -2
1538 my $bin = $x->as_bin();
1539 $bin =~ s/^-0b//; # strip '-0b' prefix
1540 $bin =~ tr/10/01/; # flip bits
1542 if (length($bin) <= $y)
1544 $bin = '0'; # shifting to far right creates -1
1545 # 0, because later increment makes
1546 # that 1, attached '-' makes it '-1'
1547 # because -1 >> x == -1 !
1551 $bin =~ s/.{$y}$//; # cut off at the right side
1552 $bin = '1' . $bin; # extend left side by one dummy '1'
1553 $bin =~ tr/10/01/; # flip bits back
1555 my $res = $self->new('0b'.$bin); # add prefix and convert back
1556 $res->binc(); # remember to increment
1557 $x->{value} = $res->{value}; # take over value
1558 return $x->round($a,$p,$r); # we are done now, magic, isn't?
1560 $x->bdec(); # n == 2, but $y == 1: this fixes it
1563 my $t; $t = $CALC->_rsft($x->{value},$y->{value},$n) if $CALC->can('_rsft');
1567 return $x->round($a,$p,$r);
1570 $x->bdiv($self->bpow($n,$y, $a,$p,$r), $a,$p,$r);
1576 #(BINT or num_str, BINT or num_str) return BINT
1578 my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
1580 return $x if $x->modify('band');
1582 local $Math::BigInt::upgrade = undef;
1584 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1585 return $x->bzero() if $y->is_zero() || $x->is_zero();
1587 my $sign = 0; # sign of result
1588 $sign = 1 if ($x->{sign} eq '-') && ($y->{sign} eq '-');
1589 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1590 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1592 if ($CALC->can('_and') && $sx == 1 && $sy == 1)
1594 $x->{value} = $CALC->_and($x->{value},$y->{value});
1595 return $x->round($a,$p,$r);
1598 my $m = $self->bone(); my ($xr,$yr);
1599 my $x10000 = $self->new (0x1000);
1600 my $y1 = copy(ref($x),$y); # make copy
1601 $y1->babs(); # and positive
1602 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1603 use integer; # need this for negative bools
1604 while (!$x1->is_zero() && !$y1->is_zero())
1606 ($x1, $xr) = bdiv($x1, $x10000);
1607 ($y1, $yr) = bdiv($y1, $x10000);
1608 # make both op's numbers!
1609 $x->badd( bmul( $class->new(
1610 abs($sx*int($xr->numify()) & $sy*int($yr->numify()))),
1614 $x->bneg() if $sign;
1615 return $x->round($a,$p,$r);
1620 #(BINT or num_str, BINT or num_str) return BINT
1622 my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
1624 return $x if $x->modify('bior');
1626 local $Math::BigInt::upgrade = undef;
1628 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1629 return $x if $y->is_zero();
1631 my $sign = 0; # sign of result
1632 $sign = 1 if ($x->{sign} eq '-') || ($y->{sign} eq '-');
1633 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1634 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1636 # don't use lib for negative values
1637 if ($CALC->can('_or') && $sx == 1 && $sy == 1)
1639 $x->{value} = $CALC->_or($x->{value},$y->{value});
1640 return $x->round($a,$p,$r);
1643 my $m = $self->bone(); my ($xr,$yr);
1644 my $x10000 = $self->new(0x10000);
1645 my $y1 = copy(ref($x),$y); # make copy
1646 $y1->babs(); # and positive
1647 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1648 use integer; # need this for negative bools
1649 while (!$x1->is_zero() || !$y1->is_zero())
1651 ($x1, $xr) = bdiv($x1,$x10000);
1652 ($y1, $yr) = bdiv($y1,$x10000);
1653 # make both op's numbers!
1654 $x->badd( bmul( $class->new(
1655 abs($sx*int($xr->numify()) | $sy*int($yr->numify()))),
1659 $x->bneg() if $sign;
1660 return $x->round($a,$p,$r);
1665 #(BINT or num_str, BINT or num_str) return BINT
1667 my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
1669 return $x if $x->modify('bxor');
1671 local $Math::BigInt::upgrade = undef;
1673 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1674 return $x if $y->is_zero();
1676 my $sign = 0; # sign of result
1677 $sign = 1 if $x->{sign} ne $y->{sign};
1678 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1679 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1681 # don't use lib for negative values
1682 if ($CALC->can('_xor') && $sx == 1 && $sy == 1)
1684 $x->{value} = $CALC->_xor($x->{value},$y->{value});
1685 return $x->round($a,$p,$r);
1688 my $m = $self->bone(); my ($xr,$yr);
1689 my $x10000 = $self->new(0x10000);
1690 my $y1 = copy(ref($x),$y); # make copy
1691 $y1->babs(); # and positive
1692 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1693 use integer; # need this for negative bools
1694 while (!$x1->is_zero() || !$y1->is_zero())
1696 ($x1, $xr) = bdiv($x1, $x10000);
1697 ($y1, $yr) = bdiv($y1, $x10000);
1698 # make both op's numbers!
1699 $x->badd( bmul( $class->new(
1700 abs($sx*int($xr->numify()) ^ $sy*int($yr->numify()))),
1704 $x->bneg() if $sign;
1705 return $x->round($a,$p,$r);
1710 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1712 my $e = $CALC->_len($x->{value});
1713 return wantarray ? ($e,0) : $e;
1718 # return the nth decimal digit, negative values count backward, 0 is right
1722 return $CALC->_digit($x->{value},$n);
1727 # return the amount of trailing zeros in $x
1729 $x = $class->new($x) unless ref $x;
1731 return 0 if $x->is_zero() || $x->is_odd() || $x->{sign} !~ /^[+-]$/;
1733 return $CALC->_zeros($x->{value}) if $CALC->can('_zeros');
1735 # if not: since we do not know underlying internal representation:
1736 my $es = "$x"; $es =~ /([0]*)$/;
1737 return 0 if !defined $1; # no zeros
1738 return CORE::length("$1"); # as string, not as +0!
1743 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1745 return $x if $x->modify('bsqrt');
1747 return $x->bnan() if $x->{sign} ne '+'; # -x or inf or NaN => NaN
1748 return $x->bzero($a,$p) if $x->is_zero(); # 0 => 0
1749 return $x->round($a,$p,$r) if $x->is_one(); # 1 => 1
1751 return $upgrade->bsqrt($x,$a,$p,$r) if defined $upgrade;
1753 if ($CALC->can('_sqrt'))
1755 $x->{value} = $CALC->_sqrt($x->{value});
1756 return $x->round($a,$p,$r);
1759 return $x->bone($a,$p) if $x < 4; # 2,3 => 1
1761 my $l = int($x->length()/2);
1763 $x->bone(); # keep ref($x), but modify it
1766 my $last = $self->bzero();
1767 my $two = $self->new(2);
1768 my $lastlast = $x+$two;
1769 while ($last != $x && $lastlast != $x)
1771 $lastlast = $last; $last = $x;
1775 $x-- if $x * $x > $y; # overshot?
1776 $x->round($a,$p,$r);
1781 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
1782 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1784 if ($x->{sign} !~ /^[+-]$/)
1786 my $s = $x->{sign}; $s =~ s/^[+-]//;
1787 return $self->new($s); # -inf,+inf => inf
1789 my $e = $class->bzero();
1790 return $e->binc() if $x->is_zero();
1791 $e += $x->_trailing_zeros();
1797 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
1798 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1800 if ($x->{sign} !~ /^[+-]$/)
1802 return $self->new($x->{sign}); # keep + or - sign
1805 # that's inefficient
1806 my $zeros = $m->_trailing_zeros();
1807 $m /= 10 ** $zeros if $zeros != 0;
1813 # return a copy of both the exponent and the mantissa
1814 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1816 return ($x->mantissa(),$x->exponent());
1819 ##############################################################################
1820 # rounding functions
1824 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
1825 # $n == 0 || $n == 1 => round to integer
1826 my $x = shift; $x = $class->new($x) unless ref $x;
1827 my ($scale,$mode) = $x->_scale_p($x->precision(),$x->round_mode(),@_);
1828 return $x if !defined $scale; # no-op
1829 return $x if $x->modify('bfround');
1831 # no-op for BigInts if $n <= 0
1834 $x->{_a} = undef; # clear an eventual set A
1835 $x->{_p} = $scale; return $x;
1838 $x->bround( $x->length()-$scale, $mode);
1839 $x->{_a} = undef; # bround sets {_a}
1840 $x->{_p} = $scale; # so correct it
1844 sub _scan_for_nonzero
1850 my $len = $x->length();
1851 return 0 if $len == 1; # '5' is trailed by invisible zeros
1852 my $follow = $pad - 1;
1853 return 0 if $follow > $len || $follow < 1;
1855 # since we do not know underlying represention of $x, use decimal string
1856 #my $r = substr ($$xs,-$follow);
1857 my $r = substr ("$x",-$follow);
1858 return 1 if $r =~ /[^0]/; return 0;
1863 # to make life easier for switch between MBF and MBI (autoload fxxx()
1864 # like MBF does for bxxx()?)
1866 return $x->bround(@_);
1871 # accuracy: +$n preserve $n digits from left,
1872 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
1874 # and overwrite the rest with 0's, return normalized number
1875 # do not return $x->bnorm(), but $x
1877 my $x = shift; $x = $class->new($x) unless ref $x;
1878 my ($scale,$mode) = $x->_scale_a($x->accuracy(),$x->round_mode(),@_);
1879 return $x if !defined $scale; # no-op
1880 return $x if $x->modify('bround');
1882 if ($x->is_zero() || $scale == 0)
1884 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
1887 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
1889 # we have fewer digits than we want to scale to
1890 my $len = $x->length();
1891 # scale < 0, but > -len (not >=!)
1892 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
1894 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
1898 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
1899 my ($pad,$digit_round,$digit_after);
1900 $pad = $len - $scale;
1901 $pad = abs($scale-1) if $scale < 0;
1903 # do not use digit(), it is costly for binary => decimal
1905 my $xs = $CALC->_str($x->{value});
1908 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
1909 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
1910 $digit_round = '0'; $digit_round = substr($$xs,$pl,1) if $pad <= $len;
1911 $pl++; $pl ++ if $pad >= $len;
1912 $digit_after = '0'; $digit_after = substr($$xs,$pl,1) if $pad > 0;
1914 # print "$pad $pl $$xs dr $digit_round da $digit_after\n";
1916 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
1917 # closer at the remaining digits of the original $x, remember decision
1918 my $round_up = 1; # default round up
1920 ($mode eq 'trunc') || # trunc by round down
1921 ($digit_after =~ /[01234]/) || # round down anyway,
1923 ($digit_after eq '5') && # not 5000...0000
1924 ($x->_scan_for_nonzero($pad,$xs) == 0) &&
1926 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
1927 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
1928 ($mode eq '+inf') && ($x->{sign} eq '-') ||
1929 ($mode eq '-inf') && ($x->{sign} eq '+') ||
1930 ($mode eq 'zero') # round down if zero, sign adjusted below
1932 my $put_back = 0; # not yet modified
1934 # old code, depend on internal representation
1935 # split mantissa at $pad and then pad with zeros
1936 #my $s5 = int($pad / 5);
1940 # $x->{value}->[$i++] = 0; # replace with 5 x 0
1942 #$x->{value}->[$s5] = '00000'.$x->{value}->[$s5]; # pad with 0
1943 #my $rem = $pad % 5; # so much left over
1946 # #print "remainder $rem\n";
1947 ## #print "elem $x->{value}->[$s5]\n";
1948 # substr($x->{value}->[$s5],-$rem,$rem) = '0' x $rem; # stamp w/ '0'
1950 #$x->{value}->[$s5] = int ($x->{value}->[$s5]); # str '05' => int '5'
1951 #print ${$CALC->_str($pad->{value})}," $len\n";
1953 if (($pad > 0) && ($pad <= $len))
1955 substr($$xs,-$pad,$pad) = '0' x $pad;
1960 $x->bzero(); # round to '0'
1963 if ($round_up) # what gave test above?
1966 $pad = $len, $$xs = '0'x$pad if $scale < 0; # tlr: whack 0.51=>1.0
1968 # we modify directly the string variant instead of creating a number and
1970 my $c = 0; $pad ++; # for $pad == $len case
1971 while ($pad <= $len)
1973 $c = substr($$xs,-$pad,1) + 1; $c = '0' if $c eq '10';
1974 substr($$xs,-$pad,1) = $c; $pad++;
1975 last if $c != 0; # no overflow => early out
1977 $$xs = '1'.$$xs if $c == 0;
1979 # $x->badd( Math::BigInt->new($x->{sign}.'1'. '0' x $pad) );
1981 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back in
1983 $x->{_a} = $scale if $scale >= 0;
1986 $x->{_a} = $len+$scale;
1987 $x->{_a} = 0 if $scale < -$len;
1994 # return integer less or equal then number, since it is already integer,
1995 # always returns $self
1996 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1998 # not needed: return $x if $x->modify('bfloor');
1999 return $x->round($a,$p,$r);
2004 # return integer greater or equal then number, since it is already integer,
2005 # always returns $self
2006 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2008 # not needed: return $x if $x->modify('bceil');
2009 return $x->round($a,$p,$r);
2012 ##############################################################################
2013 # private stuff (internal use only)
2017 # internal speedup, set argument to 1, or create a +/- 1
2019 my $x = $self->bone(); # $x->{value} = $CALC->_one();
2020 $x->{sign} = shift || '+';
2026 # Overload will swap params if first one is no object ref so that the first
2027 # one is always an object ref. In this case, third param is true.
2028 # This routine is to overcome the effect of scalar,$object creating an object
2029 # of the class of this package, instead of the second param $object. This
2030 # happens inside overload, when the overload section of this package is
2031 # inherited by sub classes.
2032 # For overload cases (and this is used only there), we need to preserve the
2033 # args, hence the copy().
2034 # You can override this method in a subclass, the overload section will call
2035 # $object->_swap() to make sure it arrives at the proper subclass, with some
2036 # exceptions like '+' and '-'. To make '+' and '-' work, you also need to
2037 # specify your own overload for them.
2039 # object, (object|scalar) => preserve first and make copy
2040 # scalar, object => swapped, re-swap and create new from first
2041 # (using class of second object, not $class!!)
2042 my $self = shift; # for override in subclass
2045 my $c = ref ($_[0]) || $class; # fallback $class should not happen
2046 return ( $c->new($_[1]), $_[0] );
2048 return ( $_[0]->copy(), $_[1] );
2053 # check for strings, if yes, return objects instead
2055 # the first argument is number of args objectify() should look at it will
2056 # return $count+1 elements, the first will be a classname. This is because
2057 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2058 # useless objects beeing created and thrown away. So we cannot simple loop
2059 # over @_. If the given count is 0, all arguments will be used.
2061 # If the second arg is a ref, use it as class.
2062 # If not, try to use it as classname, unless undef, then use $class
2063 # (aka Math::BigInt). The latter shouldn't happen,though.
2066 # $x->badd(1); => ref x, scalar y
2067 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2068 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2069 # Math::BigInt::badd(1,2); => scalar x, scalar y
2070 # In the last case we check number of arguments to turn it silently into
2071 # $class,1,2. (We can not take '1' as class ;o)
2072 # badd($class,1) is not supported (it should, eventually, try to add undef)
2073 # currently it tries 'Math::BigInt' + 1, which will not work.
2075 # some shortcut for the common cases
2078 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2080 my $count = abs(shift || 0);
2082 my (@a,$k,$d); # resulting array, temp, and downgrade
2085 # okay, got object as first
2090 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2092 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2095 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2096 if (defined ${"$a[0]::downgrade"})
2098 $d = ${"$a[0]::downgrade"};
2099 ${"$a[0]::downgrade"} = undef;
2102 # print "Now in objectify, my class is today $a[0]\n";
2110 $k = $a[0]->new($k);
2112 elsif (ref($k) ne $a[0])
2114 # foreign object, try to convert to integer
2115 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2128 $k = $a[0]->new($k);
2130 elsif (ref($k) ne $a[0])
2132 # foreign object, try to convert to integer
2133 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2137 push @a,@_; # return other params, too
2139 die "$class objectify needs list context" unless wantarray;
2140 ${"$a[0]::downgrade"} = $d;
2149 my @a = @_; my $l = scalar @_; my $j = 0;
2150 for ( my $i = 0; $i < $l ; $i++,$j++ )
2152 if ($_[$i] eq ':constant')
2154 # this causes overlord er load to step in
2155 overload::constant integer => sub { $self->new(shift) };
2156 splice @a, $j, 1; $j --;
2158 elsif ($_[$i] eq 'upgrade')
2160 # this causes upgrading
2161 $upgrade = $_[$i+1]; # or undef to disable
2162 my $s = 2; $s = 1 if @a-$j < 2; # avoid "can not modify non-existant..."
2163 splice @a, $j, $s; $j -= $s;
2165 elsif ($_[$i] =~ /^lib$/i)
2167 # this causes a different low lib to take care...
2168 $CALC = $_[$i+1] || '';
2169 my $s = 2; $s = 1 if @a-$j < 2; # avoid "can not modify non-existant..."
2170 splice @a, $j, $s; $j -= $s;
2173 # any non :constant stuff is handled by our parent, Exporter
2174 # even if @_ is empty, to give it a chance
2175 $self->SUPER::import(@a); # need it for subclasses
2176 $self->export_to_level(1,$self,@a); # need it for MBF
2178 # try to load core math lib
2179 my @c = split /\s*,\s*/,$CALC;
2180 push @c,'Calc'; # if all fail, try this
2181 $CALC = ''; # signal error
2182 foreach my $lib (@c)
2184 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2188 # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
2189 # used in the same script, or eval inside import().
2190 (my $mod = $lib . '.pm') =~ s!::!/!g;
2191 # require does not automatically :: => /, so portability problems arise
2192 eval { require $mod; $lib->import( @c ); }
2196 eval "use $lib qw/@c/;";
2198 $CALC = $lib, last if $@ eq ''; # no error in loading lib?
2200 die "Couldn't load any math lib, not even the default" if $CALC eq '';
2205 # convert a (ref to) big hex string to BigInt, return undef for error
2208 my $x = Math::BigInt->bzero();
2211 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2212 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2214 return $x->bnan() if $$hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2216 my $sign = '+'; $sign = '-' if ($$hs =~ /^-/);
2218 $$hs =~ s/^[+-]//; # strip sign
2219 if ($CALC->can('_from_hex'))
2221 $x->{value} = $CALC->_from_hex($hs);
2225 # fallback to pure perl
2226 my $mul = Math::BigInt->bzero(); $mul++;
2227 my $x65536 = Math::BigInt->new(65536);
2228 my $len = CORE::length($$hs)-2;
2229 $len = int($len/4); # 4-digit parts, w/o '0x'
2230 my $val; my $i = -4;
2233 $val = substr($$hs,$i,4);
2234 $val =~ s/^[+-]?0x// if $len == 0; # for last part only because
2235 $val = hex($val); # hex does not like wrong chars
2237 $x += $mul * $val if $val != 0;
2238 $mul *= $x65536 if $len >= 0; # skip last mul
2241 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2247 # convert a (ref to) big binary string to BigInt, return undef for error
2250 my $x = Math::BigInt->bzero();
2252 $$bs =~ s/([01])_([01])/$1$2/g;
2253 $$bs =~ s/([01])_([01])/$1$2/g;
2254 return $x->bnan() if $$bs !~ /^[+-]?0b[01]+$/;
2256 my $sign = '+'; $sign = '-' if ($$bs =~ /^\-/);
2257 $$bs =~ s/^[+-]//; # strip sign
2258 if ($CALC->can('_from_bin'))
2260 $x->{value} = $CALC->_from_bin($bs);
2264 my $mul = Math::BigInt->bzero(); $mul++;
2265 my $x256 = Math::BigInt->new(256);
2266 my $len = CORE::length($$bs)-2;
2267 $len = int($len/8); # 8-digit parts, w/o '0b'
2268 my $val; my $i = -8;
2271 $val = substr($$bs,$i,8);
2272 $val =~ s/^[+-]?0b// if $len == 0; # for last part only
2273 #$val = oct('0b'.$val); # does not work on Perl prior to 5.6.0
2275 # $val = ('0' x (8-CORE::length($val))).$val if CORE::length($val) < 8;
2276 $val = ord(pack('B8',substr('00000000'.$val,-8,8)));
2278 $x += $mul * $val if $val != 0;
2279 $mul *= $x256 if $len >= 0; # skip last mul
2282 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2288 # (ref to num_str) return num_str
2289 # internal, take apart a string and return the pieces
2290 # strip leading/trailing whitespace, leading zeros, underscore and reject
2294 # strip white space at front, also extranous leading zeros
2295 $$x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2296 $$x =~ s/^\s+//; # but this will
2297 $$x =~ s/\s+$//g; # strip white space at end
2299 # shortcut, if nothing to split, return early
2300 if ($$x =~ /^[+-]?\d+$/)
2302 $$x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2303 return (\$sign, $x, \'', \'', \0);
2306 # invalid starting char?
2307 return if $$x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2309 return __from_hex($x) if $$x =~ /^[\-\+]?0x/; # hex string
2310 return __from_bin($x) if $$x =~ /^[\-\+]?0b/; # binary string
2312 # strip underscores between digits
2313 $$x =~ s/(\d)_(\d)/$1$2/g;
2314 $$x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3
2316 # some possible inputs:
2317 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2318 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2
2320 return if $$x =~ /[Ee].*[Ee]/; # more than one E => error
2322 my ($m,$e) = split /[Ee]/,$$x;
2323 $e = '0' if !defined $e || $e eq "";
2324 # sign,value for exponent,mantint,mantfrac
2325 my ($es,$ev,$mis,$miv,$mfv);
2327 if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2331 return if $m eq '.' || $m eq '';
2332 my ($mi,$mf) = split /\./,$m;
2333 $mi = '0' if !defined $mi;
2334 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2335 $mf = '0' if !defined $mf || $mf eq '';
2336 if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2338 $mis = $1||'+'; $miv = $2;
2339 return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
2341 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2344 return; # NaN, not a number
2349 # an object might be asked to return itself as bigint on certain overloaded
2350 # operations, this does exactly this, so that sub classes can simple inherit
2351 # it or override with their own integer conversion routine
2359 # return as hex string, with prefixed 0x
2360 my $x = shift; $x = $class->new($x) if !ref($x);
2362 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2363 return '0x0' if $x->is_zero();
2365 my $es = ''; my $s = '';
2366 $s = $x->{sign} if $x->{sign} eq '-';
2367 if ($CALC->can('_as_hex'))
2369 $es = ${$CALC->_as_hex($x->{value})};
2373 my $x1 = $x->copy()->babs(); my $xr;
2374 my $x10000 = Math::BigInt->new (0x10000);
2375 while (!$x1->is_zero())
2377 ($x1, $xr) = bdiv($x1,$x10000);
2378 $es .= unpack('h4',pack('v',$xr->numify()));
2381 $es =~ s/^[0]+//; # strip leading zeros
2389 # return as binary string, with prefixed 0b
2390 my $x = shift; $x = $class->new($x) if !ref($x);
2392 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2393 return '0b0' if $x->is_zero();
2395 my $es = ''; my $s = '';
2396 $s = $x->{sign} if $x->{sign} eq '-';
2397 if ($CALC->can('_as_bin'))
2399 $es = ${$CALC->_as_bin($x->{value})};
2403 my $x1 = $x->copy()->babs(); my $xr;
2404 my $x10000 = Math::BigInt->new (0x10000);
2405 while (!$x1->is_zero())
2407 ($x1, $xr) = bdiv($x1,$x10000);
2408 $es .= unpack('b16',pack('v',$xr->numify()));
2411 $es =~ s/^[0]+//; # strip leading zeros
2417 ##############################################################################
2418 # internal calculation routines (others are in Math::BigInt::Calc etc)
2422 # (BINT or num_str, BINT or num_str) return BINT
2423 # does modify first argument
2426 my $x = shift; my $ty = shift;
2427 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2428 return $x * $ty / bgcd($x,$ty);
2433 # (BINT or num_str, BINT or num_str) return BINT
2434 # does modify both arguments
2435 # GCD -- Euclids algorithm E, Knuth Vol 2 pg 296
2438 return $x->bnan() if $x->{sign} !~ /^[+-]$/ || $ty->{sign} !~ /^[+-]$/;
2440 while (!$ty->is_zero())
2442 ($x, $ty) = ($ty,bmod($x,$ty));
2447 ###############################################################################
2448 # this method return 0 if the object can be modified, or 1 for not
2449 # We use a fast use constant statement here, to avoid costly calls. Subclasses
2450 # may override it with special code (f.i. Math::BigInt::Constant does so)
2452 sub modify () { 0; }
2459 Math::BigInt - Arbitrary size integer math package
2466 $x = Math::BigInt->new($str); # defaults to 0
2467 $nan = Math::BigInt->bnan(); # create a NotANumber
2468 $zero = Math::BigInt->bzero(); # create a +0
2469 $inf = Math::BigInt->binf(); # create a +inf
2470 $inf = Math::BigInt->binf('-'); # create a -inf
2471 $one = Math::BigInt->bone(); # create a +1
2472 $one = Math::BigInt->bone('-'); # create a -1
2475 $x->is_zero(); # true if arg is +0
2476 $x->is_nan(); # true if arg is NaN
2477 $x->is_one(); # true if arg is +1
2478 $x->is_one('-'); # true if arg is -1
2479 $x->is_odd(); # true if odd, false for even
2480 $x->is_even(); # true if even, false for odd
2481 $x->is_positive(); # true if >= 0
2482 $x->is_negative(); # true if < 0
2483 $x->is_inf(sign); # true if +inf, or -inf (sign is default '+')
2484 $x->is_int(); # true if $x is an integer (not a float)
2486 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2487 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2488 $x->sign(); # return the sign, either +,- or NaN
2489 $x->digit($n); # return the nth digit, counting from right
2490 $x->digit(-$n); # return the nth digit, counting from left
2492 # The following all modify their first argument:
2495 $x->bzero(); # set $x to 0
2496 $x->bnan(); # set $x to NaN
2497 $x->bone(); # set $x to +1
2498 $x->bone('-'); # set $x to -1
2499 $x->binf(); # set $x to inf
2500 $x->binf('-'); # set $x to -inf
2502 $x->bneg(); # negation
2503 $x->babs(); # absolute value
2504 $x->bnorm(); # normalize (no-op)
2505 $x->bnot(); # two's complement (bit wise not)
2506 $x->binc(); # increment x by 1
2507 $x->bdec(); # decrement x by 1
2509 $x->badd($y); # addition (add $y to $x)
2510 $x->bsub($y); # subtraction (subtract $y from $x)
2511 $x->bmul($y); # multiplication (multiply $x by $y)
2512 $x->bdiv($y); # divide, set $x to quotient
2513 # return (quo,rem) or quo if scalar
2515 $x->bmod($y); # modulus (x % y)
2516 $x->bpow($y); # power of arguments (x ** y)
2517 $x->blsft($y); # left shift
2518 $x->brsft($y); # right shift
2519 $x->blsft($y,$n); # left shift, by base $n (like 10)
2520 $x->brsft($y,$n); # right shift, by base $n (like 10)
2522 $x->band($y); # bitwise and
2523 $x->bior($y); # bitwise inclusive or
2524 $x->bxor($y); # bitwise exclusive or
2525 $x->bnot(); # bitwise not (two's complement)
2527 $x->bsqrt(); # calculate square-root
2528 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2530 $x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r
2531 $x->bround($N); # accuracy: preserve $N digits
2532 $x->bfround($N); # round to $Nth digit, no-op for BigInts
2534 # The following do not modify their arguments in BigInt, but do in BigFloat:
2535 $x->bfloor(); # return integer less or equal than $x
2536 $x->bceil(); # return integer greater or equal than $x
2538 # The following do not modify their arguments:
2540 bgcd(@values); # greatest common divisor (no OO style)
2541 blcm(@values); # lowest common multiplicator (no OO style)
2543 $x->length(); # return number of digits in number
2544 ($x,$f) = $x->length(); # length of number and length of fraction part,
2545 # latter is always 0 digits long for BigInt's
2547 $x->exponent(); # return exponent as BigInt
2548 $x->mantissa(); # return (signed) mantissa as BigInt
2549 $x->parts(); # return (mantissa,exponent) as BigInt
2550 $x->copy(); # make a true copy of $x (unlike $y = $x;)
2551 $x->as_number(); # return as BigInt (in BigInt: same as copy())
2553 # conversation to string
2554 $x->bstr(); # normalized string
2555 $x->bsstr(); # normalized string in scientific notation
2556 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
2557 $x->as_bin(); # as signed binary string with prefixed 0b
2561 All operators (inlcuding basic math operations) are overloaded if you
2562 declare your big integers as
2564 $i = new Math::BigInt '123_456_789_123_456_789';
2566 Operations with overloaded operators preserve the arguments which is
2567 exactly what you expect.
2571 =item Canonical notation
2573 Big integer values are strings of the form C</^[+-]\d+$/> with leading
2576 '-0' canonical value '-0', normalized '0'
2577 ' -123_123_123' canonical value '-123123123'
2578 '1_23_456_7890' canonical value '1234567890'
2582 Input values to these routines may be either Math::BigInt objects or
2583 strings of the form C</^\s*[+-]?[\d]+\.?[\d]*E?[+-]?[\d]*$/>.
2585 You can include one underscore between any two digits.
2587 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
2588 Non integer values result in NaN.
2590 Math::BigInt::new() defaults to 0, while Math::BigInt::new('') results
2593 bnorm() on a BigInt object is now effectively a no-op, since the numbers
2594 are always stored in normalized form. On a string, it creates a BigInt
2599 Output values are BigInt objects (normalized), except for bstr(), which
2600 returns a string in normalized form.
2601 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
2602 C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
2603 return either undef, <0, 0 or >0 and are suited for sort.
2609 Each of the methods below accepts three additional parameters. These arguments
2610 $A, $P and $R are accuracy, precision and round_mode. Please see more in the
2611 section about ACCURACY and ROUNDIND.
2615 $x->accuracy(5); # local for $x
2616 $class->accuracy(5); # global for all members of $class
2618 Set or get the global or local accuracy, aka how many significant digits the
2619 results have. Please see the section about L<ACCURACY AND PRECISION> for
2622 Value must be greater than zero. Pass an undef value to disable it:
2624 $x->accuracy(undef);
2625 Math::BigInt->accuracy(undef);
2627 Returns the current accuracy. For C<$x->accuracy()> it will return either the
2628 local accuracy, or if not defined, the global. This means the return value
2629 represents the accuracy that will be in effect for $x:
2631 $y = Math::BigInt->new(1234567); # unrounded
2632 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
2633 $x = Math::BigInt->new(123456); # will be automatically rounded
2634 print "$x $y\n"; # '123500 1234567'
2635 print $x->accuracy(),"\n"; # will be 4
2636 print $y->accuracy(),"\n"; # also 4, since global is 4
2637 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
2638 print $x->accuracy(),"\n"; # still 4
2639 print $y->accuracy(),"\n"; # 5, since global is 5
2645 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
2646 2, but others work, too.
2648 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
2652 $x = Math::BigInt->new(10);
2653 $x->brsft(1); # same as $x >> 1: 5
2654 $x = Math::BigInt->new(1234);
2655 $x->brsft(2,10); # result 12
2657 There is one exception, and that is base 2 with negative $x:
2660 $x = Math::BigInt->new(-5);
2663 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
2668 $x = Math::BigInt->new($str,$A,$P,$R);
2670 Creates a new BigInt object from a string or another BigInt object. The
2671 input is accepted as decimal, hex (with leading '0x') or binary (with leading
2676 $x = Math::BigInt->bnan();
2678 Creates a new BigInt object representing NaN (Not A Number).
2679 If used on an object, it will set it to NaN:
2685 $x = Math::BigInt->bzero();
2687 Creates a new BigInt object representing zero.
2688 If used on an object, it will set it to zero:
2694 $x = Math::BigInt->binf($sign);
2696 Creates a new BigInt object representing infinity. The optional argument is
2697 either '-' or '+', indicating whether you want infinity or minus infinity.
2698 If used on an object, it will set it to infinity:
2705 $x = Math::BigInt->binf($sign);
2707 Creates a new BigInt object representing one. The optional argument is
2708 either '-' or '+', indicating whether you want one or minus one.
2709 If used on an object, it will set it to one:
2714 =head2 is_one() / is_zero() / is_nan() / is_positive() / is_negative() /
2715 is_inf() / is_odd() / is_even() / is_int()
2717 $x->is_zero(); # true if arg is +0
2718 $x->is_nan(); # true if arg is NaN
2719 $x->is_one(); # true if arg is +1
2720 $x->is_one('-'); # true if arg is -1
2721 $x->is_odd(); # true if odd, false for even
2722 $x->is_even(); # true if even, false for odd
2723 $x->is_positive(); # true if >= 0
2724 $x->is_negative(); # true if < 0
2725 $x->is_inf(); # true if +inf
2726 $x->is_inf('-'); # true if -inf (sign is default '+')
2727 $x->is_int(); # true if $x is an integer
2729 These methods all test the BigInt for one condition and return true or false
2730 depending on the input.
2734 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2738 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2742 $x->sign(); # return the sign, either +,- or NaN
2746 $x->digit($n); # return the nth digit, counting from right
2752 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
2753 and '-inf', respectively. Does nothing for NaN or zero.
2759 Set the number to it's absolute value, e.g. change the sign from '-' to '+'
2760 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
2765 $x->bnorm(); # normalize (no-op)
2769 $x->bnot(); # two's complement (bit wise not)
2773 $x->binc(); # increment x by 1
2777 $x->bdec(); # decrement x by 1
2781 $x->badd($y); # addition (add $y to $x)
2785 $x->bsub($y); # subtraction (subtract $y from $x)
2789 $x->bmul($y); # multiplication (multiply $x by $y)
2793 $x->bdiv($y); # divide, set $x to quotient
2794 # return (quo,rem) or quo if scalar
2798 $x->bmod($y); # modulus (x % y)
2802 $x->bpow($y); # power of arguments (x ** y)
2806 $x->blsft($y); # left shift
2807 $x->blsft($y,$n); # left shift, by base $n (like 10)
2811 $x->brsft($y); # right shift
2812 $x->brsft($y,$n); # right shift, by base $n (like 10)
2816 $x->band($y); # bitwise and
2820 $x->bior($y); # bitwise inclusive or
2824 $x->bxor($y); # bitwise exclusive or
2828 $x->bnot(); # bitwise not (two's complement)
2832 $x->bsqrt(); # calculate square-root
2836 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2840 $x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r
2844 $x->bround($N); # accuracy: preserve $N digits
2848 $x->bfround($N); # round to $Nth digit, no-op for BigInts
2854 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
2855 does change $x in BigFloat.
2861 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
2862 does change $x in BigFloat.
2866 bgcd(@values); # greatest common divisor (no OO style)
2870 blcm(@values); # lowest common multiplicator (no OO style)
2875 ($xl,$fl) = $x->length();
2877 Returns the number of digits in the decimal representation of the number.
2878 In list context, returns the length of the integer and fraction part. For
2879 BigInt's, the length of the fraction part will always be 0.
2885 Return the exponent of $x as BigInt.
2891 Return the signed mantissa of $x as BigInt.
2895 $x->parts(); # return (mantissa,exponent) as BigInt
2899 $x->copy(); # make a true copy of $x (unlike $y = $x;)
2903 $x->as_number(); # return as BigInt (in BigInt: same as copy())
2907 $x->bstr(); # normalized string
2911 $x->bsstr(); # normalized string in scientific notation
2915 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
2919 $x->as_bin(); # as signed binary string with prefixed 0b
2921 =head1 ACCURACY and PRECISION
2923 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
2924 accuracy and precision based rounding, both automatically after every
2925 operation as well as manually.
2927 This section describes the accuracy/precision handling in Math::Big* as it
2928 used to be and as it is now, complete with an explanation of all terms and
2931 Not yet implemented things (but with correct description) are marked with '!',
2932 things that need to be answered are marked with '?'.
2934 In the next paragraph follows a short description of terms used here (because
2935 these may differ from terms used by others people or documentation).
2937 During the rest of this document, the shortcuts A (for accuracy), P (for
2938 precision), F (fallback) and R (rounding mode) will be used.
2942 A fixed number of digits before (positive) or after (negative)
2943 the decimal point. For example, 123.45 has a precision of -2. 0 means an
2944 integer like 123 (or 120). A precision of 2 means two digits to the left
2945 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
2946 numbers with zeros before the decimal point may have different precisions,
2947 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
2948 was). It could also have p < 0, when the digits after the decimal point
2951 The string output (of floating point numbers) will be padded with zeros:
2953 Initial value P A Result String
2954 ------------------------------------------------------------
2955 1234.01 -3 1000 1000
2958 1234.001 1 1234 1234.0
2960 1234.01 2 1234.01 1234.01
2961 1234.01 5 1234.01 1234.01000
2963 For BigInts, no padding occurs.
2967 Number of significant digits. Leading zeros are not counted. A
2968 number may have an accuracy greater than the non-zero digits
2969 when there are zeros in it or trailing zeros. For example, 123.456 has
2970 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
2972 The string output (of floating point numbers) will be padded with zeros:
2974 Initial value P A Result String
2975 ------------------------------------------------------------
2977 1234.01 6 1234.01 1234.01
2978 1234.1 8 1234.1 1234.1000
2980 For BigInts, no padding occurs.
2984 When both A and P are undefined, this is used as a fallback accuracy when
2987 =head2 Rounding mode R
2989 When rounding a number, different 'styles' or 'kinds'
2990 of rounding are possible. (Note that random rounding, as in
2991 Math::Round, is not implemented.)
2997 truncation invariably removes all digits following the
2998 rounding place, replacing them with zeros. Thus, 987.65 rounded
2999 to tens (P=1) becomes 980, and rounded to the fourth sigdig
3000 becomes 987.6 (A=4). 123.456 rounded to the second place after the
3001 decimal point (P=-2) becomes 123.46.
3003 All other implemented styles of rounding attempt to round to the
3004 "nearest digit." If the digit D immediately to the right of the
3005 rounding place (skipping the decimal point) is greater than 5, the
3006 number is incremented at the rounding place (possibly causing a
3007 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3008 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3009 truncated at the rounding place: e.g. when rounding to units, 0.4
3010 rounds to 0, and -19.4 rounds to -19.
3012 However the results of other styles of rounding differ if the
3013 digit immediately to the right of the rounding place (skipping the
3014 decimal point) is 5 and if there are no digits, or no digits other
3015 than 0, after that 5. In such cases:
3019 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3020 if it is not already. E.g., when rounding to the first sigdig, 0.45
3021 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3025 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3026 it is not already. E.g., when rounding to the first sigdig, 0.45
3027 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3031 round to plus infinity, i.e. always round up. E.g., when
3032 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3033 and 0.4501 also becomes 0.5.
3037 round to minus infinity, i.e. always round down. E.g., when
3038 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3039 but 0.4501 becomes 0.5.
3043 round to zero, i.e. positive numbers down, negative ones up.
3044 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3045 becomes -0.5, but 0.4501 becomes 0.5.
3049 The handling of A & P in MBI/MBF (the old core code shipped with Perl
3050 versions <= 5.7.2) is like this:
3056 * ffround($p) is able to round to $p number of digits after the decimal
3058 * otherwise P is unused
3060 =item Accuracy (significant digits)
3062 * fround($a) rounds to $a significant digits
3063 * only fdiv() and fsqrt() take A as (optional) paramater
3064 + other operations simply create the same number (fneg etc), or more (fmul)
3066 + rounding/truncating is only done when explicitly calling one of fround
3067 or ffround, and never for BigInt (not implemented)
3068 * fsqrt() simply hands its accuracy argument over to fdiv.
3069 * the documentation and the comment in the code indicate two different ways
3070 on how fdiv() determines the maximum number of digits it should calculate,
3071 and the actual code does yet another thing
3073 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3075 result has at most max(scale, length(dividend), length(divisor)) digits
3077 scale = max(scale, length(dividend)-1,length(divisor)-1);
3078 scale += length(divisior) - length(dividend);
3079 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3080 Actually, the 'difference' added to the scale is calculated from the
3081 number of "significant digits" in dividend and divisor, which is derived
3082 by looking at the length of the mantissa. Which is wrong, since it includes
3083 the + sign (oups) and actually gets 2 for '+100' and 4 for '+101'. Oups
3084 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3085 assumption that 124 has 3 significant digits, while 120/7 will get you
3086 '17', not '17.1' since 120 is thought to have 2 significant digits.
3087 The rounding after the division then uses the remainder and $y to determine
3088 wether it must round up or down.
3089 ? I have no idea which is the right way. That's why I used a slightly more
3090 ? simple scheme and tweaked the few failing testcases to match it.
3094 This is how it works now:
3098 =item Setting/Accessing
3100 * You can set the A global via Math::BigInt->accuracy() or
3101 Math::BigFloat->accuracy() or whatever class you are using.
3102 * You can also set P globally by using Math::SomeClass->precision() likewise.
3103 * Globals are classwide, and not inherited by subclasses.
3104 * to undefine A, use Math::SomeCLass->accuracy(undef);
3105 * to undefine P, use Math::SomeClass->precision(undef);
3106 * Setting Math::SomeClass->accuracy() clears automatically
3107 Math::SomeClass->precision(), and vice versa.
3108 * To be valid, A must be > 0, P can have any value.
3109 * If P is negative, this means round to the P'th place to the right of the
3110 decimal point; positive values mean to the left of the decimal point.
3111 P of 0 means round to integer.
3112 * to find out the current global A, take Math::SomeClass->accuracy()
3113 * to find out the current global P, take Math::SomeClass->precision()
3114 * use $x->accuracy() respective $x->precision() for the local setting of $x.
3115 * Please note that $x->accuracy() respecive $x->precision() fall back to the
3116 defined globals, when $x's A or P is not set.
3118 =item Creating numbers
3120 * When you create a number, you can give it's desired A or P via:
3121 $x = Math::BigInt->new($number,$A,$P);
3122 * Only one of A or P can be defined, otherwise the result is NaN
3123 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
3124 globals (if set) will be used. Thus changing the global defaults later on
3125 will not change the A or P of previously created numbers (i.e., A and P of
3126 $x will be what was in effect when $x was created)
3127 * If given undef for A and P, B<no> rounding will occur, and the globals will
3128 B<not> be used. This is used by subclasses to create numbers without
3129 suffering rounding in the parent. Thus a subclass is able to have it's own
3130 globals enforced upon creation of a number by using
3131 $x = Math::BigInt->new($number,undef,undef):
3133 use Math::Bigint::SomeSubclass;
3136 Math::BigInt->accuracy(2);
3137 Math::BigInt::SomeSubClass->accuracy(3);
3138 $x = Math::BigInt::SomeSubClass->new(1234);
3140 $x is now 1230, and not 1200. A subclass might choose to implement
3141 this otherwise, e.g. falling back to the parent's A and P.
3145 * If A or P are enabled/defined, they are used to round the result of each
3146 operation according to the rules below
3147 * Negative P is ignored in Math::BigInt, since BigInts never have digits
3148 after the decimal point
3149 * Math::BigFloat uses Math::BigInts internally, but setting A or P inside
3150 Math::BigInt as globals should not tamper with the parts of a BigFloat.
3151 Thus a flag is used to mark all Math::BigFloat numbers as 'never round'
3155 * It only makes sense that a number has only one of A or P at a time.
3156 Since you can set/get both A and P, there is a rule that will practically
3157 enforce only A or P to be in effect at a time, even if both are set.
3158 This is called precedence.
3159 * If two objects are involved in an operation, and one of them has A in
3160 effect, and the other P, this results in an error (NaN).
3161 * A takes precendence over P (Hint: A comes before P). If A is defined, it
3162 is used, otherwise P is used. If neither of them is defined, nothing is
3163 used, i.e. the result will have as many digits as it can (with an
3164 exception for fdiv/fsqrt) and will not be rounded.
3165 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
3166 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
3167 If either the dividend's or the divisor's mantissa has more digits than
3168 the value of F, the higher value will be used instead of F.
3169 This is to limit the digits (A) of the result (just consider what would
3170 happen with unlimited A and P in the case of 1/3 :-)
3171 * fdiv will calculate (at least) 4 more digits than required (determined by
3172 A, P or F), and, if F is not used, round the result
3173 (this will still fail in the case of a result like 0.12345000000001 with A
3174 or P of 5, but this can not be helped - or can it?)
3175 * Thus you can have the math done by on Math::Big* class in three modes:
3176 + never round (this is the default):
3177 This is done by setting A and P to undef. No math operation
3178 will round the result, with fdiv() and fsqrt() as exceptions to guard
3179 against overflows. You must explicitely call bround(), bfround() or
3180 round() (the latter with parameters).
3181 Note: Once you have rounded a number, the settings will 'stick' on it
3182 and 'infect' all other numbers engaged in math operations with it, since
3183 local settings have the highest precedence. So, to get SaferRound[tm],
3184 use a copy() before rounding like this:
3186 $x = Math::BigFloat->new(12.34);
3187 $y = Math::BigFloat->new(98.76);
3188 $z = $x * $y; # 1218.6984
3189 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
3190 $z = $x * $y; # still 1218.6984, without
3191 # copy would have been 1210!
3193 + round after each op:
3194 After each single operation (except for testing like is_zero()), the
3195 method round() is called and the result is rounded appropriately. By
3196 setting proper values for A and P, you can have all-the-same-A or
3197 all-the-same-P modes. For example, Math::Currency might set A to undef,
3198 and P to -2, globally.
3200 ?Maybe an extra option that forbids local A & P settings would be in order,
3201 ?so that intermediate rounding does not 'poison' further math?
3203 =item Overriding globals
3205 * you will be able to give A, P and R as an argument to all the calculation
3206 routines; the second parameter is A, the third one is P, and the fourth is
3207 R (shift right by one for binary operations like badd). P is used only if
3208 the first parameter (A) is undefined. These three parameters override the
3209 globals in the order detailed as follows, i.e. the first defined value
3211 (local: per object, global: global default, parameter: argument to sub)
3214 + local A (if defined on both of the operands: smaller one is taken)
3215 + local P (if defined on both of the operands: bigger one is taken)
3219 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
3220 arguments (A and P) instead of one
3222 =item Local settings
3224 * You can set A and P locally by using $x->accuracy() and $x->precision()
3225 and thus force different A and P for different objects/numbers.
3226 * Setting A or P this way immediately rounds $x to the new value.
3227 * $x->accuracy() clears $x->precision(), and vice versa.
3231 * the rounding routines will use the respective global or local settings.
3232 fround()/bround() is for accuracy rounding, while ffround()/bfround()
3234 * the two rounding functions take as the second parameter one of the
3235 following rounding modes (R):
3236 'even', 'odd', '+inf', '-inf', 'zero', 'trunc'
3237 * you can set and get the global R by using Math::SomeClass->round_mode()
3238 or by setting $Math::SomeClass::round_mode
3239 * after each operation, $result->round() is called, and the result may
3240 eventually be rounded (that is, if A or P were set either locally,
3241 globally or as parameter to the operation)
3242 * to manually round a number, call $x->round($A,$P,$round_mode);
3243 this will round the number by using the appropriate rounding function
3244 and then normalize it.
3245 * rounding modifies the local settings of the number:
3247 $x = Math::BigFloat->new(123.456);
3251 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
3252 will be 4 from now on.
3254 =item Default values
3263 * The defaults are set up so that the new code gives the same results as
3264 the old code (except in a few cases on fdiv):
3265 + Both A and P are undefined and thus will not be used for rounding
3266 after each operation.
3267 + round() is thus a no-op, unless given extra parameters A and P
3273 The actual numbers are stored as unsigned big integers (with seperate sign).
3274 You should neither care about nor depend on the internal representation; it
3275 might change without notice. Use only method calls like C<< $x->sign(); >>
3276 instead relying on the internal hash keys like in C<< $x->{sign}; >>.
3280 Math with the numbers is done (by default) by a module called
3281 Math::BigInt::Calc. This is equivalent to saying:
3283 use Math::BigInt lib => 'Calc';
3285 You can change this by using:
3287 use Math::BigInt lib => 'BitVect';
3289 The following would first try to find Math::BigInt::Foo, then
3290 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
3292 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
3294 Calc.pm uses as internal format an array of elements of some decimal base
3295 (usually 1e5 or 1e7) with the least significant digit first, while BitVect.pm
3296 uses a bit vector of base 2, most significant bit first. Other modules might
3297 use even different means of representing the numbers. See the respective
3298 module documentation for further details.
3302 The sign is either '+', '-', 'NaN', '+inf' or '-inf' and stored seperately.
3304 A sign of 'NaN' is used to represent the result when input arguments are not
3305 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
3306 minus infinity. You will get '+inf' when dividing a positive number by 0, and
3307 '-inf' when dividing any negative number by 0.
3309 =head2 mantissa(), exponent() and parts()
3311 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
3314 $m = $x->mantissa();
3315 $e = $x->exponent();
3316 $y = $m * ( 10 ** $e );
3317 print "ok\n" if $x == $y;
3319 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
3320 in one go. Both the returned mantissa and exponent have a sign.
3322 Currently, for BigInts C<$e> will be always 0, except for NaN, +inf and -inf,
3323 where it will be NaN; and for $x == 0, where it will be 1
3324 (to be compatible with Math::BigFloat's internal representation of a zero as
3327 C<$m> will always be a copy of the original number. The relation between $e
3328 and $m might change in the future, but will always be equivalent in a
3329 numerical sense, e.g. $m might get minimized.
3335 sub bint { Math::BigInt->new(shift); }
3337 $x = Math::BigInt->bstr("1234") # string "1234"
3338 $x = "$x"; # same as bstr()
3339 $x = Math::BigInt->bneg("1234"); # Bigint "-1234"
3340 $x = Math::BigInt->babs("-12345"); # Bigint "12345"
3341 $x = Math::BigInt->bnorm("-0 00"); # BigInt "0"
3342 $x = bint(1) + bint(2); # BigInt "3"
3343 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
3344 $x = bint(1); # BigInt "1"
3345 $x = $x + 5 / 2; # BigInt "3"
3346 $x = $x ** 3; # BigInt "27"
3347 $x *= 2; # BigInt "54"
3348 $x = Math::BigInt->new(0); # BigInt "0"
3350 $x = Math::BigInt->badd(4,5) # BigInt "9"
3351 print $x->bsstr(); # 9e+0
3353 Examples for rounding:
3358 $x = Math::BigFloat->new(123.4567);
3359 $y = Math::BigFloat->new(123.456789);
3360 Math::BigFloat->accuracy(4); # no more A than 4
3362 ok ($x->copy()->fround(),123.4); # even rounding
3363 print $x->copy()->fround(),"\n"; # 123.4
3364 Math::BigFloat->round_mode('odd'); # round to odd
3365 print $x->copy()->fround(),"\n"; # 123.5
3366 Math::BigFloat->accuracy(5); # no more A than 5
3367 Math::BigFloat->round_mode('odd'); # round to odd
3368 print $x->copy()->fround(),"\n"; # 123.46
3369 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
3370 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
3372 Math::BigFloat->accuracy(undef); # A not important now
3373 Math::BigFloat->precision(2); # P important
3374 print $x->copy()->bnorm(),"\n"; # 123.46
3375 print $x->copy()->fround(),"\n"; # 123.46
3377 Examples for converting:
3379 my $x = Math::BigInt->new('0b1'.'01' x 123);
3380 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
3382 =head1 Autocreating constants
3384 After C<use Math::BigInt ':constant'> all the B<integer> decimal constants
3385 in the given scope are converted to C<Math::BigInt>. This conversion
3386 happens at compile time.
3390 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
3392 prints the integer value of C<2**100>. Note that without conversion of
3393 constants the expression 2**100 will be calculated as perl scalar.
3395 Please note that strings and floating point constants are not affected,
3398 use Math::BigInt qw/:constant/;
3400 $x = 1234567890123456789012345678901234567890
3401 + 123456789123456789;
3402 $y = '1234567890123456789012345678901234567890'
3403 + '123456789123456789';
3405 do not work. You need an explicit Math::BigInt->new() around one of the
3406 operands. You should also quote large constants to protect loss of precision:
3410 $x = Math::BigInt->new('1234567889123456789123456789123456789');
3412 Without the quotes Perl would convert the large number to a floating point
3413 constant at compile time and then hand the result to BigInt, which results in
3414 an truncated result or a NaN.
3418 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
3419 must be made in the second case. For long numbers, the copy can eat up to 20%
3420 of the work (in the case of addition/subtraction, less for
3421 multiplication/division). If $y is very small compared to $x, the form
3422 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
3423 more time then the actual addition.
3425 With a technique called copy-on-write, the cost of copying with overload could
3426 be minimized or even completely avoided. A test implementation of COW did show
3427 performance gains for overloaded math, but introduced a performance loss due
3428 to a constant overhead for all other operatons.
3430 The rewritten version of this module is slower on certain operations, like
3431 new(), bstr() and numify(). The reason are that it does now more work and
3432 handles more cases. The time spent in these operations is usually gained in
3433 the other operations so that programs on the average should get faster. If
3434 they don't, please contect the author.
3436 Some operations may be slower for small numbers, but are significantly faster
3437 for big numbers. Other operations are now constant (O(1), like bneg(), babs()
3438 etc), instead of O(N) and thus nearly always take much less time. These
3439 optimizations were done on purpose.
3441 If you find the Calc module to slow, try to install any of the replacement
3442 modules and see if they help you.
3444 =head2 Alternative math libraries
3446 You can use an alternative library to drive Math::BigInt via:
3448 use Math::BigInt lib => 'Module';
3450 See L<MATH LIBRARY> for more information.
3452 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
3456 =head1 Subclassing Math::BigInt
3458 The basic design of Math::BigInt allows simple subclasses with very little
3459 work, as long as a few simple rules are followed:
3465 The public API must remain consistent, i.e. if a sub-class is overloading
3466 addition, the sub-class must use the same name, in this case badd(). The
3467 reason for this is that Math::BigInt is optimized to call the object methods
3472 The private object hash keys like C<$x->{sign}> may not be changed, but
3473 additional keys can be added, like C<$x->{_custom}>.
3477 Accessor functions are available for all existing object hash keys and should
3478 be used instead of directly accessing the internal hash keys. The reason for
3479 this is that Math::BigInt itself has a pluggable interface which permits it
3480 to support different storage methods.
3484 More complex sub-classes may have to replicate more of the logic internal of
3485 Math::BigInt if they need to change more basic behaviors. A subclass that
3486 needs to merely change the output only needs to overload C<bstr()>.
3488 All other object methods and overloaded functions can be directly inherited
3489 from the parent class.
3491 At the very minimum, any subclass will need to provide it's own C<new()> and can
3492 store additional hash keys in the object. There are also some package globals
3493 that must be defined, e.g.:
3497 $precision = -2; # round to 2 decimal places
3498 $round_mode = 'even';
3501 Additionally, you might want to provide the following two globals to allow
3502 auto-upgrading and auto-downgrading to work correctly:
3507 This allows Math::BigInt to correctly retrieve package globals from the
3508 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
3509 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
3515 in your subclass to automatically inherit the overloading from the parent. If
3516 you like, you can change part of the overloading, look at Math::String for an
3521 When used like this:
3523 use Math::BigInt upgrade => 'Foo::Bar';
3525 certain operations will 'upgrade' their calculation and thus the result to
3526 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
3528 use Math::BigInt upgrade => 'Math::BigFloat';
3530 As a shortcut, you can use the module C<bignum>:
3534 Also good for oneliners:
3536 perl -Mbignum -le 'print 2 ** 255'
3538 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
3539 as well es preserve accuracy (as in sqrt(3)).
3541 Beware: This feature is not fully implemented yet.
3545 The following methods upgrade themselves unconditionally; that is if upgrade
3546 is in effect, they will always hand up their work:
3558 Beware: This list is not complete.
3560 All other methods upgrade themselves only when one (or all) of their
3561 arguments are of the class mentioned in $upgrade (This might change in later
3562 versions to a more sophisticated scheme):
3568 =item Out of Memory!
3570 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
3571 C<eval()> in your code will crash with "Out of memory". This is probably an
3572 overload/exporter bug. You can workaround by not having C<eval()>
3573 and ':constant' at the same time or upgrade your Perl to a newer version.
3575 =item Fails to load Calc on Perl prior 5.6.0
3577 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
3578 will fall back to eval { require ... } when loading the math lib on Perls
3579 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
3580 filesystems using a different seperator.
3586 Some things might not work as you expect them. Below is documented what is
3587 known to be troublesome:
3591 =item stringify, bstr(), bsstr() and 'cmp'
3593 Both stringify and bstr() now drop the leading '+'. The old code would return
3594 '+3', the new returns '3'. This is to be consistent with Perl and to make
3595 cmp (especially with overloading) to work as you expect. It also solves
3596 problems with Test.pm, it's ok() uses 'eq' internally.
3598 Mark said, when asked about to drop the '+' altogether, or make only cmp work:
3600 I agree (with the first alternative), don't add the '+' on positive
3601 numbers. It's not as important anymore with the new internal
3602 form for numbers. It made doing things like abs and neg easier,
3603 but those have to be done differently now anyway.
3605 So, the following examples will now work all as expected:
3608 BEGIN { plan tests => 1 }
3611 my $x = new Math::BigInt 3*3;
3612 my $y = new Math::BigInt 3*3;
3615 print "$x eq 9" if $x eq $y;
3616 print "$x eq 9" if $x eq '9';
3617 print "$x eq 9" if $x eq 3*3;
3619 Additionally, the following still works:
3621 print "$x == 9" if $x == $y;
3622 print "$x == 9" if $x == 9;
3623 print "$x == 9" if $x == 3*3;
3625 There is now a C<bsstr()> method to get the string in scientific notation aka
3626 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
3627 for comparisation, but Perl will represent some numbers as 100 and others
3628 as 1e+308. If in doubt, convert both arguments to Math::BigInt before doing eq:
3631 BEGIN { plan tests => 3 }
3634 $x = Math::BigInt->new('1e56'); $y = 1e56;
3635 ok ($x,$y); # will fail
3636 ok ($x->bsstr(),$y); # okay
3637 $y = Math::BigInt->new($y);
3640 Alternatively, simple use <=> for comparisations, that will get it always
3641 right. There is not yet a way to get a number automatically represented as
3642 a string that matches exactly the way Perl represents it.
3646 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
3649 $x = Math::BigInt->new(123);
3650 $y = int($x); # BigInt 123
3651 $x = Math::BigFloat->new(123.45);
3652 $y = int($x); # BigInt 123
3654 In all Perl versions you can use C<as_number()> for the same effect:
3656 $x = Math::BigFloat->new(123.45);
3657 $y = $x->as_number(); # BigInt 123
3659 This also works for other subclasses, like Math::String.
3661 It is yet unlcear whether overloaded int() should return a scalar or a BigInt.
3665 The following will probably not do what you expect:
3667 $c = Math::BigInt->new(123);
3668 print $c->length(),"\n"; # prints 30
3670 It prints both the number of digits in the number and in the fraction part
3671 since print calls C<length()> in list context. Use something like:
3673 print scalar $c->length(),"\n"; # prints 3
3677 The following will probably not do what you expect:
3679 print $c->bdiv(10000),"\n";
3681 It prints both quotient and remainder since print calls C<bdiv()> in list
3682 context. Also, C<bdiv()> will modify $c, so be carefull. You probably want
3685 print $c / 10000,"\n";
3686 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
3690 The quotient is always the greatest integer less than or equal to the
3691 real-valued quotient of the two operands, and the remainder (when it is
3692 nonzero) always has the same sign as the second operand; so, for
3702 As a consequence, the behavior of the operator % agrees with the
3703 behavior of Perl's built-in % operator (as documented in the perlop
3704 manpage), and the equation
3706 $x == ($x / $y) * $y + ($x % $y)
3708 holds true for any $x and $y, which justifies calling the two return
3709 values of bdiv() the quotient and remainder. The only exception to this rule
3710 are when $y == 0 and $x is negative, then the remainder will also be
3711 negative. See below under "infinity handling" for the reasoning behing this.
3713 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
3714 not change BigInt's way to do things. This is because under 'use integer' Perl
3715 will do what the underlying C thinks is right and this is different for each
3716 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
3717 the author to implement it ;)
3719 =item infinity handling
3721 Here are some examples that explain the reasons why certain results occur while
3724 The following table shows the result of the division and the remainder, so that
3725 the equation above holds true. Some "ordinary" cases are strewn in to show more
3726 clearly the reasoning:
3728 A / B = C, R so that C * B + R = A
3729 =========================================================
3730 5 / 8 = 0, 5 0 * 8 + 5 = 5
3731 0 / 8 = 0, 0 0 * 8 + 0 = 0
3732 0 / inf = 0, 0 0 * inf + 0 = 0
3733 0 /-inf = 0, 0 0 * -inf + 0 = 0
3734 5 / inf = 0, 5 0 * inf + 5 = 5
3735 5 /-inf = 0, 5 0 * -inf + 5 = 5
3736 -5/ inf = 0, -5 0 * inf + -5 = -5
3737 -5/-inf = 0, -5 0 * -inf + -5 = -5
3738 inf/ 5 = inf, 0 inf * 5 + 0 = inf
3739 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
3740 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
3741 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
3742 5/ 5 = 1, 0 1 * 5 + 0 = 5
3743 -5/ -5 = 1, 0 1 * -5 + 0 = -5
3744 inf/ inf = 1, 0 1 * inf + 0 = inf
3745 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
3746 inf/-inf = -1, 0 -1 * -inf + 0 = inf
3747 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
3748 8/ 0 = inf, 8 inf * 0 + 8 = 8
3749 inf/ 0 = inf, inf inf * 0 + inf = inf
3752 These cases below violate the "remainder has the sign of the second of the two
3753 arguments", since they wouldn't match up otherwise.
3755 A / B = C, R so that C * B + R = A
3756 ========================================================
3757 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
3758 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
3760 =item Modifying and =
3764 $x = Math::BigFloat->new(5);
3767 It will not do what you think, e.g. making a copy of $x. Instead it just makes
3768 a second reference to the B<same> object and stores it in $y. Thus anything
3769 that modifies $x (except overloaded operators) will modify $y, and vice versa.
3770 Or in other words, C<=> is only safe if you modify your BigInts only via
3771 overloaded math. As soon as you use a method call it breaks:
3774 print "$x, $y\n"; # prints '10, 10'
3776 If you want a true copy of $x, use:
3780 You can also chain the calls like this, this will make first a copy and then
3783 $y = $x->copy()->bmul(2);
3785 See also the documentation for overload.pm regarding C<=>.
3789 C<bpow()> (and the rounding functions) now modifies the first argument and
3790 returns it, unlike the old code which left it alone and only returned the
3791 result. This is to be consistent with C<badd()> etc. The first three will
3792 modify $x, the last one won't:
3794 print bpow($x,$i),"\n"; # modify $x
3795 print $x->bpow($i),"\n"; # ditto
3796 print $x **= $i,"\n"; # the same
3797 print $x ** $i,"\n"; # leave $x alone
3799 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
3801 =item Overloading -$x
3811 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
3812 needs to preserve $x since it does not know that it later will get overwritten.
3813 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
3815 With Copy-On-Write, this issue would be gone, but C-o-W is not implemented
3816 since it is slower for all other things.
3818 =item Mixing different object types
3820 In Perl you will get a floating point value if you do one of the following:
3826 With overloaded math, only the first two variants will result in a BigFloat:
3831 $mbf = Math::BigFloat->new(5);
3832 $mbi2 = Math::BigInteger->new(5);
3833 $mbi = Math::BigInteger->new(2);
3835 # what actually gets called:
3836 $float = $mbf + $mbi; # $mbf->badd()
3837 $float = $mbf / $mbi; # $mbf->bdiv()
3838 $integer = $mbi + $mbf; # $mbi->badd()
3839 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
3840 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
3842 This is because math with overloaded operators follows the first (dominating)
3843 operand, and the operation of that is called and returns thus the result. So,
3844 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
3845 the result should be a Math::BigFloat or the second operant is one.
3847 To get a Math::BigFloat you either need to call the operation manually,
3848 make sure the operands are already of the proper type or casted to that type
3849 via Math::BigFloat->new():
3851 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
3853 Beware of simple "casting" the entire expression, this would only convert
3854 the already computed result:
3856 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
3858 Beware also of the order of more complicated expressions like:
3860 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
3861 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
3863 If in doubt, break the expression into simpler terms, or cast all operands
3864 to the desired resulting type.
3866 Scalar values are a bit different, since:
3871 will both result in the proper type due to the way the overloaded math works.
3873 This section also applies to other overloaded math packages, like Math::String.
3875 One solution to you problem might be L<autoupgrading|upgrading>.
3879 C<bsqrt()> works only good if the result is a big integer, e.g. the square
3880 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
3883 If you want a better approximation of the square root, then use:
3885 $x = Math::BigFloat->new(12);
3886 Math::BigFloat->precision(0);
3887 Math::BigFloat->round_mode('even');
3888 print $x->copy->bsqrt(),"\n"; # 4
3890 Math::BigFloat->precision(2);
3891 print $x->bsqrt(),"\n"; # 3.46
3892 print $x->bsqrt(3),"\n"; # 3.464
3896 For negative numbers in base see also L<brsft|brsft>.
3902 This program is free software; you may redistribute it and/or modify it under
3903 the same terms as Perl itself.
3907 L<Math::BigFloat> and L<Math::Big> as well as L<Math::BigInt::BitVect>,
3908 L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
3911 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
3912 more documentation including a full version history, testcases, empty
3913 subclass files and benchmarks.
3917 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
3918 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001.