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 $class = "Math::BigInt";
23 @ISA = qw( Exporter );
24 @EXPORT_OK = qw( objectify bgcd blcm);
25 use vars qw/$round_mode $accuracy $precision $div_scale $rnd_mode/;
26 use vars qw/$upgrade $downgrade/;
27 # the following are internal and should never be accessed from the outside
28 use vars qw/$_trap_nan $_trap_inf/;
31 # Inside overload, the first arg is always an object. If the original code had
32 # it reversed (like $x = 2 * $y), then the third paramater is true.
33 # In some cases (like add, $x = $x + 2 is the same as $x = 2 + $x) this makes
34 # no difference, but in some cases it does.
36 # For overloaded ops with only one argument we simple use $_[0]->copy() to
37 # preserve the argument.
39 # Thus inheritance of overload operators becomes possible and transparent for
40 # our subclasses without the need to repeat the entire overload section there.
43 '=' => sub { $_[0]->copy(); },
45 # some shortcuts for speed (assumes that reversed order of arguments is routed
46 # to normal '+' and we thus can always modify first arg. If this is changed,
47 # this breaks and must be adjusted.)
48 '+=' => sub { $_[0]->badd($_[1]); },
49 '-=' => sub { $_[0]->bsub($_[1]); },
50 '*=' => sub { $_[0]->bmul($_[1]); },
51 '/=' => sub { scalar $_[0]->bdiv($_[1]); },
52 '%=' => sub { $_[0]->bmod($_[1]); },
53 '^=' => sub { $_[0]->bxor($_[1]); },
54 '&=' => sub { $_[0]->band($_[1]); },
55 '|=' => sub { $_[0]->bior($_[1]); },
56 '**=' => sub { $_[0]->bpow($_[1]); },
58 # not supported by Perl yet
59 '..' => \&_pointpoint,
61 '<=>' => sub { $_[2] ?
62 ref($_[0])->bcmp($_[1],$_[0]) :
66 "$_[1]" cmp $_[0]->bstr() :
67 $_[0]->bstr() cmp "$_[1]" },
71 # return Math::Big::cos($_[0], ref($_[0])->accuracy());
74 # make cos()/sin()/exp() "work" with BigInt's or subclasses
75 'cos' => sub { cos($_[0]->numify()) },
76 'sin' => sub { sin($_[0]->numify()) },
77 'exp' => sub { exp($_[0]->numify()) },
78 'atan2' => sub { atan2($_[0]->numify(),$_[1]) },
80 'log' => sub { $_[0]->copy()->blog($_[1]); },
81 'int' => sub { $_[0]->copy(); },
82 'neg' => sub { $_[0]->copy()->bneg(); },
83 'abs' => sub { $_[0]->copy()->babs(); },
84 'sqrt' => sub { $_[0]->copy()->bsqrt(); },
85 '~' => sub { $_[0]->copy()->bnot(); },
87 # for sub it is a bit tricky to keep b: b-a => -a+b
88 '-' => sub { my $c = $_[0]->copy; $_[2] ?
89 $c->bneg()->badd($_[1]) :
91 '+' => sub { $_[0]->copy()->badd($_[1]); },
92 '*' => sub { $_[0]->copy()->bmul($_[1]); },
95 $_[2] ? ref($_[0])->new($_[1])->bdiv($_[0]) : $_[0]->copy->bdiv($_[1]);
98 $_[2] ? ref($_[0])->new($_[1])->bmod($_[0]) : $_[0]->copy->bmod($_[1]);
101 $_[2] ? ref($_[0])->new($_[1])->bpow($_[0]) : $_[0]->copy->bpow($_[1]);
104 $_[2] ? ref($_[0])->new($_[1])->blsft($_[0]) : $_[0]->copy->blsft($_[1]);
107 $_[2] ? ref($_[0])->new($_[1])->brsft($_[0]) : $_[0]->copy->brsft($_[1]);
110 $_[2] ? ref($_[0])->new($_[1])->band($_[0]) : $_[0]->copy->band($_[1]);
113 $_[2] ? ref($_[0])->new($_[1])->bior($_[0]) : $_[0]->copy->bior($_[1]);
116 $_[2] ? ref($_[0])->new($_[1])->bxor($_[0]) : $_[0]->copy->bxor($_[1]);
119 # can modify arg of ++ and --, so avoid a copy() for speed, but don't
120 # use $_[0]->bone(), it would modify $_[0] to be 1!
121 '++' => sub { $_[0]->binc() },
122 '--' => sub { $_[0]->bdec() },
124 # if overloaded, O(1) instead of O(N) and twice as fast for small numbers
126 # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/
127 # v5.6.1 dumps on this: return !$_[0]->is_zero() || undef; :-(
128 my $t = !$_[0]->is_zero();
133 # the original qw() does not work with the TIESCALAR below, why?
134 # Order of arguments unsignificant
135 '""' => sub { $_[0]->bstr(); },
136 '0+' => sub { $_[0]->numify(); }
139 ##############################################################################
140 # global constants, flags and accessory
142 # these are public, but their usage is not recommended, use the accessor
145 $round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
150 $upgrade = undef; # default is no upgrade
151 $downgrade = undef; # default is no downgrade
153 # these are internally, and not to be used from the outside
155 use constant MB_NEVER_ROUND => 0x0001;
157 $_trap_nan = 0; # are NaNs ok? set w/ config()
158 $_trap_inf = 0; # are infs ok? set w/ config()
159 my $nan = 'NaN'; # constants for easier life
161 my $CALC = 'Math::BigInt::Calc'; # module to do the low level math
163 my %CAN; # cache for $CALC->can(...)
164 my $IMPORT = 0; # was import() called yet?
165 # used to make require work
167 ##############################################################################
168 # the old code had $rnd_mode, so we need to support it, too
171 sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; }
172 sub FETCH { return $round_mode; }
173 sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
175 BEGIN { tie $rnd_mode, 'Math::BigInt'; }
177 ##############################################################################
182 # make Class->round_mode() work
184 my $class = ref($self) || $self || __PACKAGE__;
188 if ($m !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
190 require Carp; Carp::croak ("Unknown round mode '$m'");
192 return ${"${class}::round_mode"} = $m;
194 ${"${class}::round_mode"};
200 # make Class->upgrade() work
202 my $class = ref($self) || $self || __PACKAGE__;
203 # need to set new value?
207 return ${"${class}::upgrade"} = $u;
209 ${"${class}::upgrade"};
215 # make Class->downgrade() work
217 my $class = ref($self) || $self || __PACKAGE__;
218 # need to set new value?
222 return ${"${class}::downgrade"} = $u;
224 ${"${class}::downgrade"};
230 # make Class->div_scale() work
232 my $class = ref($self) || $self || __PACKAGE__;
237 require Carp; Carp::croak ('div_scale must be greater than zero');
239 ${"${class}::div_scale"} = shift;
241 ${"${class}::div_scale"};
246 # $x->accuracy($a); ref($x) $a
247 # $x->accuracy(); ref($x)
248 # Class->accuracy(); class
249 # Class->accuracy($a); class $a
252 my $class = ref($x) || $x || __PACKAGE__;
255 # need to set new value?
259 # convert objects to scalars to avoid deep recursion. If object doesn't
260 # have numify(), then hopefully it will have overloading for int() and
261 # boolean test without wandering into a deep recursion path...
262 $a = $a->numify() if ref($a) && $a->can('numify');
266 # also croak on non-numerical
270 Carp::croak ('Argument to accuracy must be greater than zero');
274 require Carp; Carp::croak ('Argument to accuracy must be an integer');
279 # $object->accuracy() or fallback to global
280 $x->bround($a) if $a; # not for undef, 0
281 $x->{_a} = $a; # set/overwrite, even if not rounded
282 $x->{_p} = undef; # clear P
283 $a = ${"${class}::accuracy"} unless defined $a; # proper return value
288 ${"${class}::accuracy"} = $a;
289 ${"${class}::precision"} = undef; # clear P
291 return $a; # shortcut
295 # $object->accuracy() or fallback to global
296 $r = $x->{_a} if ref($x);
297 # but don't return global undef, when $x's accuracy is 0!
298 $r = ${"${class}::accuracy"} if !defined $r;
304 # $x->precision($p); ref($x) $p
305 # $x->precision(); ref($x)
306 # Class->precision(); class
307 # Class->precision($p); class $p
310 my $class = ref($x) || $x || __PACKAGE__;
316 # convert objects to scalars to avoid deep recursion. If object doesn't
317 # have numify(), then hopefully it will have overloading for int() and
318 # boolean test without wandering into a deep recursion path...
319 $p = $p->numify() if ref($p) && $p->can('numify');
320 if ((defined $p) && (int($p) != $p))
322 require Carp; Carp::croak ('Argument to precision must be an integer');
326 # $object->precision() or fallback to global
327 $x->bfround($p) if $p; # not for undef, 0
328 $x->{_p} = $p; # set/overwrite, even if not rounded
329 $x->{_a} = undef; # clear A
330 $p = ${"${class}::precision"} unless defined $p; # proper return value
335 ${"${class}::precision"} = $p;
336 ${"${class}::accuracy"} = undef; # clear A
338 return $p; # shortcut
342 # $object->precision() or fallback to global
343 $r = $x->{_p} if ref($x);
344 # but don't return global undef, when $x's precision is 0!
345 $r = ${"${class}::precision"} if !defined $r;
351 # return (or set) configuration data as hash ref
352 my $class = shift || 'Math::BigInt';
357 # try to set given options as arguments from hash
360 if (ref($args) ne 'HASH')
364 # these values can be "set"
368 upgrade downgrade precision accuracy round_mode div_scale/
371 $set_args->{$key} = $args->{$key} if exists $args->{$key};
372 delete $args->{$key};
377 Carp::croak ("Illegal key(s) '",
378 join("','",keys %$args),"' passed to $class\->config()");
380 foreach my $key (keys %$set_args)
382 if ($key =~ /^trap_(inf|nan)\z/)
384 ${"${class}::_trap_$1"} = ($set_args->{"trap_$1"} ? 1 : 0);
387 # use a call instead of just setting the $variable to check argument
388 $class->$key($set_args->{$key});
392 # now return actual configuration
396 lib_version => ${"${CALC}::VERSION"},
398 trap_nan => ${"${class}::_trap_nan"},
399 trap_inf => ${"${class}::_trap_inf"},
400 version => ${"${class}::VERSION"},
403 upgrade downgrade precision accuracy round_mode div_scale
406 $cfg->{$key} = ${"${class}::$key"};
413 # select accuracy parameter based on precedence,
414 # used by bround() and bfround(), may return undef for scale (means no op)
415 my ($x,$s,$m,$scale,$mode) = @_;
416 $scale = $x->{_a} if !defined $scale;
417 $scale = $s if (!defined $scale);
418 $mode = $m if !defined $mode;
419 return ($scale,$mode);
424 # select precision parameter based on precedence,
425 # used by bround() and bfround(), may return undef for scale (means no op)
426 my ($x,$s,$m,$scale,$mode) = @_;
427 $scale = $x->{_p} if !defined $scale;
428 $scale = $s if (!defined $scale);
429 $mode = $m if !defined $mode;
430 return ($scale,$mode);
433 ##############################################################################
441 # if two arguments, the first one is the class to "swallow" subclasses
449 return unless ref($x); # only for objects
451 my $self = {}; bless $self,$c;
453 foreach my $k (keys %$x)
457 $self->{value} = $CALC->_copy($x->{value}); next;
459 if (!($r = ref($x->{$k})))
461 $self->{$k} = $x->{$k}; next;
465 $self->{$k} = \${$x->{$k}};
467 elsif ($r eq 'ARRAY')
469 $self->{$k} = [ @{$x->{$k}} ];
473 # only one level deep!
474 foreach my $h (keys %{$x->{$k}})
476 $self->{$k}->{$h} = $x->{$k}->{$h};
482 if ($xk->can('copy'))
484 $self->{$k} = $xk->copy();
488 $self->{$k} = $xk->new($xk);
497 # create a new BigInt object from a string or another BigInt object.
498 # see hash keys documented at top
500 # the argument could be an object, so avoid ||, && etc on it, this would
501 # cause costly overloaded code to be called. The only allowed ops are
504 my ($class,$wanted,$a,$p,$r) = @_;
506 # avoid numify-calls by not using || on $wanted!
507 return $class->bzero($a,$p) if !defined $wanted; # default to 0
508 return $class->copy($wanted,$a,$p,$r)
509 if ref($wanted) && $wanted->isa($class); # MBI or subclass
511 $class->import() if $IMPORT == 0; # make require work
513 my $self = bless {}, $class;
515 # shortcut for "normal" numbers
516 if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*\z/))
518 $self->{sign} = $1 || '+';
520 if ($wanted =~ /^[+-]/)
522 # remove sign without touching wanted to make it work with constants
523 my $t = $wanted; $t =~ s/^[+-]//; $ref = \$t;
525 # force to string version (otherwise Pari is unhappy about overflowed
526 # constants, for instance)
527 # not good, BigInt shouldn't need to know about alternative libs:
528 # $ref = \"$$ref" if $CALC eq 'Math::BigInt::Pari';
529 $self->{value} = $CALC->_new($ref);
531 if ( (defined $a) || (defined $p)
532 || (defined ${"${class}::precision"})
533 || (defined ${"${class}::accuracy"})
536 $self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p);
541 # handle '+inf', '-inf' first
542 if ($wanted =~ /^[+-]?inf$/)
544 $self->{value} = $CALC->_zero();
545 $self->{sign} = $wanted; $self->{sign} = '+inf' if $self->{sign} eq 'inf';
548 # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign
549 my ($mis,$miv,$mfv,$es,$ev) = _split(\$wanted);
554 require Carp; Carp::croak("$wanted is not a number in $class");
556 $self->{value} = $CALC->_zero();
557 $self->{sign} = $nan;
562 # _from_hex or _from_bin
563 $self->{value} = $mis->{value};
564 $self->{sign} = $mis->{sign};
565 return $self; # throw away $mis
567 # make integer from mantissa by adjusting exp, then convert to bigint
568 $self->{sign} = $$mis; # store sign
569 $self->{value} = $CALC->_zero(); # for all the NaN cases
570 my $e = int("$$es$$ev"); # exponent (avoid recursion)
573 my $diff = $e - CORE::length($$mfv);
574 if ($diff < 0) # Not integer
578 require Carp; Carp::croak("$wanted not an integer in $class");
581 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
582 $self->{sign} = $nan;
586 # adjust fraction and add it to value
587 #print "diff > 0 $$miv\n";
588 $$miv = $$miv . ($$mfv . '0' x $diff);
593 if ($$mfv ne '') # e <= 0
595 # fraction and negative/zero E => NOI
598 require Carp; Carp::croak("$wanted not an integer in $class");
600 #print "NOI 2 \$\$mfv '$$mfv'\n";
601 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
602 $self->{sign} = $nan;
606 # xE-y, and empty mfv
609 if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
613 require Carp; Carp::croak("$wanted not an integer in $class");
616 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
617 $self->{sign} = $nan;
621 $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
622 $self->{value} = $CALC->_new($miv) if $self->{sign} =~ /^[+-]$/;
623 # if any of the globals is set, use them to round and store them inside $self
624 # do not round for new($x,undef,undef) since that is used by MBF to signal
626 $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p;
632 # create a bigint 'NaN', if given a BigInt, set it to 'NaN'
634 $self = $class if !defined $self;
637 my $c = $self; $self = {}; bless $self, $c;
640 if (${"${class}::_trap_nan"})
643 Carp::croak ("Tried to set $self to NaN in $class\::bnan()");
645 $self->import() if $IMPORT == 0; # make require work
646 return if $self->modify('bnan');
647 if ($self->can('_bnan'))
649 # use subclass to initialize
654 # otherwise do our own thing
655 $self->{value} = $CALC->_zero();
657 $self->{sign} = $nan;
658 delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly
664 # create a bigint '+-inf', if given a BigInt, set it to '+-inf'
665 # the sign is either '+', or if given, used from there
667 my $sign = shift; $sign = '+' if !defined $sign || $sign !~ /^-(inf)?$/;
668 $self = $class if !defined $self;
671 my $c = $self; $self = {}; bless $self, $c;
674 if (${"${class}::_trap_inf"})
677 Carp::croak ("Tried to set $self to +-inf in $class\::binfn()");
679 $self->import() if $IMPORT == 0; # make require work
680 return if $self->modify('binf');
681 if ($self->can('_binf'))
683 # use subclass to initialize
688 # otherwise do our own thing
689 $self->{value} = $CALC->_zero();
691 $sign = $sign . 'inf' if $sign !~ /inf$/; # - => -inf
692 $self->{sign} = $sign;
693 ($self->{_a},$self->{_p}) = @_; # take over requested rounding
699 # create a bigint '+0', if given a BigInt, set it to 0
701 $self = $class if !defined $self;
705 my $c = $self; $self = {}; bless $self, $c;
707 $self->import() if $IMPORT == 0; # make require work
708 return if $self->modify('bzero');
710 if ($self->can('_bzero'))
712 # use subclass to initialize
717 # otherwise do our own thing
718 $self->{value} = $CALC->_zero();
725 # call like: $x->bzero($a,$p,$r,$y);
726 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
731 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
733 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
741 # create a bigint '+1' (or -1 if given sign '-'),
742 # if given a BigInt, set it to +1 or -1, respecively
744 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
745 $self = $class if !defined $self;
749 my $c = $self; $self = {}; bless $self, $c;
751 $self->import() if $IMPORT == 0; # make require work
752 return if $self->modify('bone');
754 if ($self->can('_bone'))
756 # use subclass to initialize
761 # otherwise do our own thing
762 $self->{value} = $CALC->_one();
764 $self->{sign} = $sign;
769 # call like: $x->bone($sign,$a,$p,$r,$y);
770 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
774 # call like: $x->bone($sign,$a,$p,$r);
776 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
778 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
784 ##############################################################################
785 # string conversation
789 # (ref to BFLOAT or num_str ) return num_str
790 # Convert number from internal format to scientific string format.
791 # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
792 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
793 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
795 if ($x->{sign} !~ /^[+-]$/)
797 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
800 my ($m,$e) = $x->parts();
801 my $sign = 'e+'; # e can only be positive
802 return $m->bstr().$sign.$e->bstr();
807 # make a string from bigint object
808 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
809 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
811 if ($x->{sign} !~ /^[+-]$/)
813 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
816 my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
817 return $es.${$CALC->_str($x->{value})};
822 # Make a "normal" scalar from a BigInt object
823 my $x = shift; $x = $class->new($x) unless ref $x;
825 return $x->bstr() if $x->{sign} !~ /^[+-]$/;
826 my $num = $CALC->_num($x->{value});
827 return -$num if $x->{sign} eq '-';
831 ##############################################################################
832 # public stuff (usually prefixed with "b")
836 # return the sign of the number: +/-/-inf/+inf/NaN
837 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
842 sub _find_round_parameters
844 # After any operation or when calling round(), the result is rounded by
845 # regarding the A & P from arguments, local parameters, or globals.
847 # !!!!!!! If you change this, remember to change round(), too! !!!!!!!!!!
849 # This procedure finds the round parameters, but it is for speed reasons
850 # duplicated in round. Otherwise, it is tested by the testsuite and used
853 # returns ($self) or ($self,$a,$p,$r) - sets $self to NaN of both A and P
854 # were requested/defined (locally or globally or both)
856 my ($self,$a,$p,$r,@args) = @_;
857 # $a accuracy, if given by caller
858 # $p precision, if given by caller
859 # $r round_mode, if given by caller
860 # @args all 'other' arguments (0 for unary, 1 for binary ops)
862 # leave bigfloat parts alone
863 return ($self) if exists $self->{_f} && ($self->{_f} & MB_NEVER_ROUND) != 0;
865 my $c = ref($self); # find out class of argument(s)
868 # now pick $a or $p, but only if we have got "arguments"
871 foreach ($self,@args)
873 # take the defined one, or if both defined, the one that is smaller
874 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
879 # even if $a is defined, take $p, to signal error for both defined
880 foreach ($self,@args)
882 # take the defined one, or if both defined, the one that is bigger
884 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
887 # if still none defined, use globals (#2)
888 $a = ${"$c\::accuracy"} unless defined $a;
889 $p = ${"$c\::precision"} unless defined $p;
891 # A == 0 is useless, so undef it to signal no rounding
892 $a = undef if defined $a && $a == 0;
895 return ($self) unless defined $a || defined $p; # early out
897 # set A and set P is an fatal error
898 return ($self->bnan()) if defined $a && defined $p; # error
900 $r = ${"$c\::round_mode"} unless defined $r;
901 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
903 require Carp; Carp::croak ("Unknown round mode '$r'");
911 # Round $self according to given parameters, or given second argument's
912 # parameters or global defaults
914 # for speed reasons, _find_round_parameters is embeded here:
916 my ($self,$a,$p,$r,@args) = @_;
917 # $a accuracy, if given by caller
918 # $p precision, if given by caller
919 # $r round_mode, if given by caller
920 # @args all 'other' arguments (0 for unary, 1 for binary ops)
922 # leave bigfloat parts alone
923 return ($self) if exists $self->{_f} && ($self->{_f} & MB_NEVER_ROUND) != 0;
925 my $c = ref($self); # find out class of argument(s)
928 # now pick $a or $p, but only if we have got "arguments"
931 foreach ($self,@args)
933 # take the defined one, or if both defined, the one that is smaller
934 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
939 # even if $a is defined, take $p, to signal error for both defined
940 foreach ($self,@args)
942 # take the defined one, or if both defined, the one that is bigger
944 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
947 # if still none defined, use globals (#2)
948 $a = ${"$c\::accuracy"} unless defined $a;
949 $p = ${"$c\::precision"} unless defined $p;
951 # A == 0 is useless, so undef it to signal no rounding
952 $a = undef if defined $a && $a == 0;
955 return $self unless defined $a || defined $p; # early out
957 # set A and set P is an fatal error
958 return $self->bnan() if defined $a && defined $p;
960 $r = ${"$c\::round_mode"} unless defined $r;
961 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
966 # now round, by calling either fround or ffround:
969 $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a;
971 else # both can't be undefined due to early out
973 $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p;
975 $self->bnorm(); # after round, normalize
980 # (numstr or BINT) return BINT
981 # Normalize number -- no-op here
982 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
988 # (BINT or num_str) return BINT
989 # make number absolute, or return absolute BINT from string
990 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
992 return $x if $x->modify('babs');
993 # post-normalized abs for internal use (does nothing for NaN)
994 $x->{sign} =~ s/^-/+/;
1000 # (BINT or num_str) return BINT
1001 # negate number or make a negated number from string
1002 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1004 return $x if $x->modify('bneg');
1006 # for +0 dont negate (to have always normalized)
1007 $x->{sign} =~ tr/+-/-+/ if !$x->is_zero(); # does nothing for NaN
1013 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
1014 # (BINT or num_str, BINT or num_str) return cond_code
1017 my ($self,$x,$y) = (ref($_[0]),@_);
1019 # objectify is costly, so avoid it
1020 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1022 ($self,$x,$y) = objectify(2,@_);
1025 return $upgrade->bcmp($x,$y) if defined $upgrade &&
1026 ((!$x->isa($self)) || (!$y->isa($self)));
1028 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1030 # handle +-inf and NaN
1031 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1032 return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
1033 return +1 if $x->{sign} eq '+inf';
1034 return -1 if $x->{sign} eq '-inf';
1035 return -1 if $y->{sign} eq '+inf';
1038 # check sign for speed first
1039 return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
1040 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
1042 # have same sign, so compare absolute values. Don't make tests for zero here
1043 # because it's actually slower than testin in Calc (especially w/ Pari et al)
1045 # post-normalized compare for internal use (honors signs)
1046 if ($x->{sign} eq '+')
1048 # $x and $y both > 0
1049 return $CALC->_acmp($x->{value},$y->{value});
1053 $CALC->_acmp($y->{value},$x->{value}); # swaped (lib returns 0,1,-1)
1058 # Compares 2 values, ignoring their signs.
1059 # Returns one of undef, <0, =0, >0. (suitable for sort)
1060 # (BINT, BINT) return cond_code
1063 my ($self,$x,$y) = (ref($_[0]),@_);
1064 # objectify is costly, so avoid it
1065 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1067 ($self,$x,$y) = objectify(2,@_);
1070 return $upgrade->bacmp($x,$y) if defined $upgrade &&
1071 ((!$x->isa($self)) || (!$y->isa($self)));
1073 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1075 # handle +-inf and NaN
1076 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1077 return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
1078 return +1; # inf is always bigger
1080 $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
1085 # add second arg (BINT or string) to first (BINT) (modifies first)
1086 # return result as BINT
1089 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1090 # objectify is costly, so avoid it
1091 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1093 ($self,$x,$y,@r) = objectify(2,@_);
1096 return $x if $x->modify('badd');
1097 return $upgrade->badd($upgrade->new($x),$upgrade->new($y),@r) if defined $upgrade &&
1098 ((!$x->isa($self)) || (!$y->isa($self)));
1100 $r[3] = $y; # no push!
1101 # inf and NaN handling
1102 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1105 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1107 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1109 # +inf++inf or -inf+-inf => same, rest is NaN
1110 return $x if $x->{sign} eq $y->{sign};
1113 # +-inf + something => +inf
1114 # something +-inf => +-inf
1115 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
1119 my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
1123 $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
1128 my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
1131 $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
1136 # speedup, if equal, set result to 0
1137 $x->{value} = $CALC->_zero();
1142 $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
1146 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1152 # (BINT or num_str, BINT or num_str) return BINT
1153 # subtract second arg from first, modify first
1156 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1157 # objectify is costly, so avoid it
1158 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1160 ($self,$x,$y,@r) = objectify(2,@_);
1163 return $x if $x->modify('bsub');
1165 # upgrade done by badd():
1166 # return $upgrade->badd($x,$y,@r) if defined $upgrade &&
1167 # ((!$x->isa($self)) || (!$y->isa($self)));
1171 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1175 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
1176 $x->badd($y,@r); # badd does not leave internal zeros
1177 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
1178 $x; # already rounded by badd() or no round necc.
1183 # increment arg by one
1184 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1185 return $x if $x->modify('binc');
1187 if ($x->{sign} eq '+')
1189 $x->{value} = $CALC->_inc($x->{value});
1190 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1193 elsif ($x->{sign} eq '-')
1195 $x->{value} = $CALC->_dec($x->{value});
1196 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1197 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1200 # inf, nan handling etc
1201 $x->badd($self->bone(),$a,$p,$r); # badd does round
1206 # decrement arg by one
1207 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1208 return $x if $x->modify('bdec');
1210 my $zero = $CALC->_is_zero($x->{value}) && $x->{sign} eq '+';
1212 if (($x->{sign} eq '-') || $zero)
1214 $x->{value} = $CALC->_inc($x->{value});
1215 $x->{sign} = '-' if $zero; # 0 => 1 => -1
1216 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1217 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1221 elsif ($x->{sign} eq '+')
1223 $x->{value} = $CALC->_dec($x->{value});
1224 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1227 # inf, nan handling etc
1228 $x->badd($self->bone('-'),$a,$p,$r); # badd does round
1233 # calculate $x = $a ** $base + $b and return $a (e.g. the log() to base
1237 my ($self,$x,$base,@r) = (ref($_[0]),@_);
1238 # objectify is costly, so avoid it
1239 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1241 ($self,$x,$base,@r) = objectify(2,@_);
1244 # inf, -inf, NaN, <0 => NaN
1246 if $x->{sign} ne '+' || $base->{sign} ne '+';
1248 return $upgrade->blog($upgrade->new($x),$base,@r) if
1249 defined $upgrade && (ref($x) ne $upgrade || ref($base) ne $upgrade);
1253 my $rc = $CALC->_log_int($x->{value},$base->{value});
1254 return $x->bnan() unless defined $rc;
1256 return $x->round(@r);
1259 return $x->bnan() if $x->is_zero() || $base->is_zero() || $base->is_one();
1261 my $acmp = $x->bacmp($base);
1262 return $x->bone('+',@r) if $acmp == 0;
1263 return $x->bzero(@r) if $acmp < 0 || $x->is_one();
1265 # blog($x,$base) ** $base + $y = $x
1267 # this trial multiplication is very fast, even for large counts (like for
1268 # 2 ** 1024, since this still requires only 1024 very fast steps
1269 # (multiplication of a large number by a very small number is very fast))
1270 # See Calc for an even faster algorightmn
1271 my $x_org = $x->copy(); # preserve orgx
1272 $x->bzero(); # keep ref to $x
1273 my $trial = $base->copy();
1274 while ($trial->bacmp($x_org) <= 0)
1276 $trial->bmul($base); $x->binc();
1283 # (BINT or num_str, BINT or num_str) return BINT
1284 # does not modify arguments, but returns new object
1285 # Lowest Common Multiplicator
1287 my $y = shift; my ($x);
1294 $x = $class->new($y);
1296 while (@_) { $x = __lcm($x,shift); }
1302 # (BINT or num_str, BINT or num_str) return BINT
1303 # does not modify arguments, but returns new object
1304 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1307 $y = __PACKAGE__->new($y) if !ref($y);
1309 my $x = $y->copy(); # keep arguments
1314 $y = shift; $y = $self->new($y) if !ref($y);
1315 next if $y->is_zero();
1316 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1317 $x->{value} = $CALC->_gcd($x->{value},$y->{value}); last if $x->is_one();
1324 $y = shift; $y = $self->new($y) if !ref($y);
1325 $x = __gcd($x,$y->copy()); last if $x->is_one(); # _gcd handles NaN
1333 # (num_str or BINT) return BINT
1334 # represent ~x as twos-complement number
1335 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1336 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1338 return $x if $x->modify('bnot');
1339 $x->binc()->bneg(); # binc already does round
1342 ##############################################################################
1343 # is_foo test routines
1344 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1348 # return true if arg (BINT or num_str) is zero (array '+', '0')
1349 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1351 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1352 $CALC->_is_zero($x->{value});
1357 # return true if arg (BINT or num_str) is NaN
1358 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1360 $x->{sign} eq $nan ? 1 : 0;
1365 # return true if arg (BINT or num_str) is +-inf
1366 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1370 $sign = '[+-]inf' if $sign eq ''; # +- doesn't matter, only that's inf
1371 $sign = "[$1]inf" if $sign =~ /^([+-])(inf)?$/; # extract '+' or '-'
1372 return $x->{sign} =~ /^$sign$/ ? 1 : 0;
1374 $x->{sign} =~ /^[+-]inf$/ ? 1 : 0; # only +-inf is infinity
1379 # return true if arg (BINT or num_str) is +1, or -1 if sign is given
1380 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1382 $sign = '+' if !defined $sign || $sign ne '-';
1384 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1385 $CALC->_is_one($x->{value});
1390 # return true when arg (BINT or num_str) is odd, false for even
1391 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1393 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1394 $CALC->_is_odd($x->{value});
1399 # return true when arg (BINT or num_str) is even, false for odd
1400 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1402 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1403 $CALC->_is_even($x->{value});
1408 # return true when arg (BINT or num_str) is positive (>= 0)
1409 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1411 $x->{sign} =~ /^\+/ ? 1 : 0; # +inf is also positive, but NaN not
1416 # return true when arg (BINT or num_str) is negative (< 0)
1417 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1419 $x->{sign} =~ /^-/ ? 1 : 0; # -inf is also negative, but NaN not
1424 # return true when arg (BINT or num_str) is an integer
1425 # always true for BigInt, but different for BigFloats
1426 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1428 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1431 ###############################################################################
1435 # multiply two numbers -- stolen from Knuth Vol 2 pg 233
1436 # (BINT or num_str, BINT or num_str) return BINT
1439 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1440 # objectify is costly, so avoid it
1441 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1443 ($self,$x,$y,@r) = objectify(2,@_);
1446 return $x if $x->modify('bmul');
1448 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1451 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1453 return $x->bnan() if $x->is_zero() || $y->is_zero();
1454 # result will always be +-inf:
1455 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1456 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1457 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1458 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1459 return $x->binf('-');
1462 return $upgrade->bmul($x,$y,@r)
1463 if defined $upgrade && $y->isa($upgrade);
1465 $r[3] = $y; # no push here
1467 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1469 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1470 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1472 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1478 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1479 my ($self,$x,$y) = @_;
1481 # NaN if x == NaN or y == NaN or x==y==0
1482 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1483 if (($x->is_nan() || $y->is_nan()) ||
1484 ($x->is_zero() && $y->is_zero()));
1486 # +-inf / +-inf == NaN, reminder also NaN
1487 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1489 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1491 # x / +-inf => 0, remainder x (works even if x == 0)
1492 if ($y->{sign} =~ /^[+-]inf$/)
1494 my $t = $x->copy(); # bzero clobbers up $x
1495 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1498 # 5 / 0 => +inf, -6 / 0 => -inf
1499 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1500 # exception: -8 / 0 has remainder -8, not 8
1501 # exception: -inf / 0 has remainder -inf, not inf
1504 # +-inf / 0 => special case for -inf
1505 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1506 if (!$x->is_zero() && !$x->is_inf())
1508 my $t = $x->copy(); # binf clobbers up $x
1510 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1514 # last case: +-inf / ordinary number
1516 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1518 return wantarray ? ($x,$self->bzero()) : $x;
1523 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1524 # (BINT,BINT) (quo,rem) or BINT (only rem)
1527 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1528 # objectify is costly, so avoid it
1529 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1531 ($self,$x,$y,@r) = objectify(2,@_);
1534 return $x if $x->modify('bdiv');
1536 return $self->_div_inf($x,$y)
1537 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1539 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1540 if defined $upgrade;
1542 $r[3] = $y; # no push!
1544 # calc new sign and in case $y == +/- 1, return $x
1545 my $xsign = $x->{sign}; # keep
1546 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1550 my $rem = $self->bzero();
1551 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1552 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1553 $rem->{_a} = $x->{_a};
1554 $rem->{_p} = $x->{_p};
1555 $x->round(@r) if !exists $x->{_f} || ($x->{_f} & MB_NEVER_ROUND) == 0;
1556 if (! $CALC->_is_zero($rem->{value}))
1558 $rem->{sign} = $y->{sign};
1559 $rem = $y->copy()->bsub($rem) if $xsign ne $y->{sign}; # one of them '-'
1563 $rem->{sign} = '+'; # dont leave -0
1565 $rem->round(@r) if !exists $rem->{_f} || ($rem->{_f} & MB_NEVER_ROUND) == 0;
1569 $x->{value} = $CALC->_div($x->{value},$y->{value});
1570 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1572 $x->round(@r) if !exists $x->{_f} || ($x->{_f} & MB_NEVER_ROUND) == 0;
1576 ###############################################################################
1581 # modulus (or remainder)
1582 # (BINT or num_str, BINT or num_str) return BINT
1585 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1586 # objectify is costly, so avoid it
1587 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1589 ($self,$x,$y,@r) = objectify(2,@_);
1592 return $x if $x->modify('bmod');
1593 $r[3] = $y; # no push!
1594 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1596 my ($d,$r) = $self->_div_inf($x,$y);
1597 $x->{sign} = $r->{sign};
1598 $x->{value} = $r->{value};
1599 return $x->round(@r);
1604 # calc new sign and in case $y == +/- 1, return $x
1605 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1606 if (!$CALC->_is_zero($x->{value}))
1608 my $xsign = $x->{sign};
1609 $x->{sign} = $y->{sign};
1610 if ($xsign ne $y->{sign})
1612 my $t = $CALC->_copy($x->{value}); # copy $x
1613 $x->{value} = $CALC->_sub($y->{value},$t,1); # $y-$x
1618 $x->{sign} = '+'; # dont leave -0
1620 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1623 # disable upgrade temporarily, otherwise endless loop due to bdiv()
1624 local $upgrade = undef;
1625 my ($t,$rem) = $self->bdiv($x->copy(),$y,@r); # slow way (also rounds)
1627 foreach (qw/value sign _a _p/)
1629 $x->{$_} = $rem->{$_};
1636 # Modular inverse. given a number which is (hopefully) relatively
1637 # prime to the modulus, calculate its inverse using Euclid's
1638 # alogrithm. If the number is not relatively prime to the modulus
1639 # (i.e. their gcd is not one) then NaN is returned.
1642 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1643 # objectify is costly, so avoid it
1644 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1646 ($self,$x,$y,@r) = objectify(2,@_);
1649 return $x if $x->modify('bmodinv');
1652 if ($y->{sign} ne '+' # -, NaN, +inf, -inf
1653 || $x->is_zero() # or num == 0
1654 || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1657 # put least residue into $x if $x was negative, and thus make it positive
1658 $x->bmod($y) if $x->{sign} eq '-';
1663 ($x->{value},$sign) = $CALC->_modinv($x->{value},$y->{value});
1664 $x->bnan() if !defined $x->{value}; # in case no GCD found
1665 return $x if !defined $sign; # already real result
1666 $x->{sign} = $sign; # flip/flop see below
1667 $x->bmod($y); # calc real result
1670 my ($u, $u1) = ($self->bzero(), $self->bone());
1671 my ($a, $b) = ($y->copy(), $x->copy());
1673 # first step need always be done since $num (and thus $b) is never 0
1674 # Note that the loop is aligned so that the check occurs between #2 and #1
1675 # thus saving us one step #2 at the loop end. Typical loop count is 1. Even
1676 # a case with 28 loops still gains about 3% with this layout.
1678 ($a, $q, $b) = ($b, $a->bdiv($b)); # step #1
1679 # Euclid's Algorithm (calculate GCD of ($a,$b) in $a and also calculate
1680 # two values in $u and $u1, we use only $u1 afterwards)
1681 my $sign = 1; # flip-flop
1682 while (!$b->is_zero()) # found GCD if $b == 0
1684 # the original algorithm had:
1685 # ($u, $u1) = ($u1, $u->bsub($u1->copy()->bmul($q))); # step #2
1686 # The following creates exact the same sequence of numbers in $u1,
1687 # except for the sign ($u1 is now always positive). Since formerly
1688 # the sign of $u1 was alternating between '-' and '+', the $sign
1689 # flip-flop will take care of that, so that at the end of the loop
1690 # we have the real sign of $u1. Keeping numbers positive gains us
1691 # speed since badd() is faster than bsub() and makes it possible
1692 # to have the algorithmn in Calc for even more speed.
1694 ($u, $u1) = ($u1, $u->badd($u1->copy()->bmul($q))); # step #2
1695 $sign = - $sign; # flip sign
1697 ($a, $q, $b) = ($b, $a->bdiv($b)); # step #1 again
1700 # If the gcd is not 1, then return NaN! It would be pointless to
1701 # have called bgcd to check this first, because we would then be
1702 # performing the same Euclidean Algorithm *twice*.
1703 return $x->bnan() unless $a->is_one();
1705 $u1->bneg() if $sign != 1; # need to flip?
1707 $u1->bmod($y); # calc result
1708 $x->{value} = $u1->{value}; # and copy over to $x
1709 $x->{sign} = $u1->{sign}; # to modify in place
1715 # takes a very large number to a very large exponent in a given very
1716 # large modulus, quickly, thanks to binary exponentation. supports
1717 # negative exponents.
1718 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1720 return $num if $num->modify('bmodpow');
1722 # check modulus for valid values
1723 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1724 || $mod->is_zero());
1726 # check exponent for valid values
1727 if ($exp->{sign} =~ /\w/)
1729 # i.e., if it's NaN, +inf, or -inf...
1730 return $num->bnan();
1733 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
1735 # check num for valid values (also NaN if there was no inverse but $exp < 0)
1736 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1740 # $mod is positive, sign on $exp is ignored, result also positive
1741 $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
1745 # in the trivial case,
1746 return $num->bzero(@r) if $mod->is_one();
1747 return $num->bone('+',@r) if $num->is_zero() or $num->is_one();
1749 # $num->bmod($mod); # if $x is large, make it smaller first
1750 my $acc = $num->copy(); # but this is not really faster...
1752 $num->bone(); # keep ref to $num
1754 my $expbin = $exp->as_bin(); $expbin =~ s/^[-]?0b//; # ignore sign and prefix
1755 my $len = CORE::length($expbin);
1758 $num->bmul($acc)->bmod($mod) if substr($expbin,$len,1) eq '1';
1759 $acc->bmul($acc)->bmod($mod);
1765 ###############################################################################
1769 # (BINT or num_str, BINT or num_str) return BINT
1770 # compute factorial number from $x, modify $x in place
1771 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1773 return $x if $x->modify('bfac');
1775 return $x->bnan() if $x->{sign} ne '+'; # inf, NnN, <0 etc => NaN
1779 $x->{value} = $CALC->_fac($x->{value});
1780 return $x->round(@r);
1783 return $x->bone('+',@r) if $x->is_zero() || $x->is_one(); # 0 or 1 => 1
1787 # seems we need not to temp. clear A/P of $x since the result is the same
1788 my $f = $self->new(2);
1789 while ($f->bacmp($n) < 0)
1791 $x->bmul($f); $f->binc();
1793 $x->bmul($f,@r); # last step and also round
1798 # (BINT or num_str, BINT or num_str) return BINT
1799 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1800 # modifies first argument
1803 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1804 # objectify is costly, so avoid it
1805 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1807 ($self,$x,$y,@r) = objectify(2,@_);
1810 return $x if $x->modify('bpow');
1812 return $upgrade->bpow($upgrade->new($x),$y,@r)
1813 if defined $upgrade && !$y->isa($self);
1815 $r[3] = $y; # no push!
1816 return $x if $x->{sign} =~ /^[+-]inf$/; # -inf/+inf ** x
1817 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1818 return $x->bone('+',@r) if $y->is_zero();
1819 return $x->round(@r) if $x->is_one() || $y->is_one();
1820 if ($x->{sign} eq '-' && $CALC->_is_one($x->{value}))
1822 # if $x == -1 and odd/even y => +1/-1
1823 return $y->is_odd() ? $x->round(@r) : $x->babs()->round(@r);
1824 # my Casio FX-5500L has a bug here: -1 ** 2 is -1, but -1 * -1 is 1;
1826 # 1 ** -y => 1 / (1 ** |y|)
1827 # so do test for negative $y after above's clause
1828 return $x->bnan() if $y->{sign} eq '-';
1829 return $x->round(@r) if $x->is_zero(); # 0**y => 0 (if not y <= 0)
1833 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1834 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1838 # based on the assumption that shifting in base 10 is fast, and that mul
1839 # works faster if numbers are small: we count trailing zeros (this step is
1840 # O(1)..O(N), but in case of O(N) we save much more time due to this),
1841 # stripping them out of the multiplication, and add $count * $y zeros
1842 # afterwards like this:
1843 # 300 ** 3 == 300*300*300 == 3*3*3 . '0' x 2 * 3 == 27 . '0' x 6
1844 # creates deep recursion since brsft/blsft use bpow sometimes.
1845 # my $zeros = $x->_trailing_zeros();
1848 # $x->brsft($zeros,10); # remove zeros
1849 # $x->bpow($y); # recursion (will not branch into here again)
1850 # $zeros = $y * $zeros; # real number of zeros to add
1851 # $x->blsft($zeros,10);
1852 # return $x->round(@r);
1855 my $pow2 = $self->bone();
1856 my $y_bin = $y->as_bin(); $y_bin =~ s/^0b//;
1857 my $len = CORE::length($y_bin);
1860 $pow2->bmul($x) if substr($y_bin,$len,1) eq '1'; # is odd?
1864 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1870 # (BINT or num_str, BINT or num_str) return BINT
1871 # compute x << y, base n, y >= 0
1874 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1875 # objectify is costly, so avoid it
1876 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1878 ($self,$x,$y,$n,@r) = objectify(2,@_);
1881 return $x if $x->modify('blsft');
1882 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1883 return $x->round(@r) if $y->is_zero();
1885 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1887 my $t; $t = $CALC->_lsft($x->{value},$y->{value},$n) if $CAN{lsft};
1890 $x->{value} = $t; return $x->round(@r);
1893 return $x->bmul( $self->bpow($n, $y, @r), @r );
1898 # (BINT or num_str, BINT or num_str) return BINT
1899 # compute x >> y, base n, y >= 0
1902 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1903 # objectify is costly, so avoid it
1904 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1906 ($self,$x,$y,$n,@r) = objectify(2,@_);
1909 return $x if $x->modify('brsft');
1910 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1911 return $x->round(@r) if $y->is_zero();
1912 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
1914 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1916 # this only works for negative numbers when shifting in base 2
1917 if (($x->{sign} eq '-') && ($n == 2))
1919 return $x->round(@r) if $x->is_one('-'); # -1 => -1
1922 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1923 # but perhaps there is a better emulation for two's complement shift...
1924 # if $y != 1, we must simulate it by doing:
1925 # convert to bin, flip all bits, shift, and be done
1926 $x->binc(); # -3 => -2
1927 my $bin = $x->as_bin();
1928 $bin =~ s/^-0b//; # strip '-0b' prefix
1929 $bin =~ tr/10/01/; # flip bits
1931 if (CORE::length($bin) <= $y)
1933 $bin = '0'; # shifting to far right creates -1
1934 # 0, because later increment makes
1935 # that 1, attached '-' makes it '-1'
1936 # because -1 >> x == -1 !
1940 $bin =~ s/.{$y}$//; # cut off at the right side
1941 $bin = '1' . $bin; # extend left side by one dummy '1'
1942 $bin =~ tr/10/01/; # flip bits back
1944 my $res = $self->new('0b'.$bin); # add prefix and convert back
1945 $res->binc(); # remember to increment
1946 $x->{value} = $res->{value}; # take over value
1947 return $x->round(@r); # we are done now, magic, isn't?
1949 $x->bdec(); # n == 2, but $y == 1: this fixes it
1952 my $t; $t = $CALC->_rsft($x->{value},$y->{value},$n) if $CAN{rsft};
1956 return $x->round(@r);
1959 $x->bdiv($self->bpow($n,$y, @r), @r);
1965 #(BINT or num_str, BINT or num_str) return BINT
1969 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1970 # objectify is costly, so avoid it
1971 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1973 ($self,$x,$y,@r) = objectify(2,@_);
1976 return $x if $x->modify('band');
1978 $r[3] = $y; # no push!
1979 local $Math::BigInt::upgrade = undef;
1981 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1983 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1984 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1986 if ($CAN{and} && $sx == 1 && $sy == 1)
1988 $x->{value} = $CALC->_and($x->{value},$y->{value});
1989 return $x->round(@r);
1992 if ($CAN{signed_and})
1994 $x->{value} = $CALC->_signed_and($x->{value},$y->{value},$sx,$sy);
1995 return $x->round(@r);
1998 return $x->bzero(@r) if $y->is_zero() || $x->is_zero();
2000 my $sign = 0; # sign of result
2001 $sign = 1 if ($x->{sign} eq '-') && ($y->{sign} eq '-');
2005 if ($sx == -1) # if x is negative
2007 # two's complement: inc and flip all "bits" in $bx
2008 $bx = $x->binc()->as_hex(); # -1 => 0, -2 => 1, -3 => 2 etc
2010 $bx =~ tr/0123456789abcdef/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/;
2014 $bx = $x->as_hex(); # get binary representation
2016 $bx =~ tr/fedcba9876543210/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/;
2018 if ($sy == -1) # if y is negative
2020 # two's complement: inc and flip all "bits" in $by
2021 $by = $y->copy()->binc()->as_hex(); # -1 => 0, -2 => 1, -3 => 2 etc
2023 $by =~ tr/0123456789abcdef/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/;
2027 $by = $y->as_hex(); # get binary representation
2029 $by =~ tr/fedcba9876543210/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/;
2031 # now we have bit-strings from X and Y, reverse them for padding
2035 # cut the longer string to the length of the shorter one (the result would
2036 # be 0 due to AND anyway)
2037 my $diff = CORE::length($bx) - CORE::length($by);
2040 $bx = substr($bx,0,CORE::length($by));
2044 $by = substr($by,0,CORE::length($bx));
2047 # and the strings together
2050 # and reverse the result again
2053 # one of $x or $y was negative, so need to flip bits in the result
2054 # in both cases (one or two of them negative, or both positive) we need
2055 # to get the characters back.
2058 $bx =~ tr/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/0123456789abcdef/;
2062 $bx =~ tr/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/fedcba9876543210/;
2068 $x->{value} = $CALC->_from_hex( \$bx );
2072 $r = $self->new($bx);
2073 $x->{value} = $r->{value};
2076 # calculate sign of result
2078 $x->{sign} = '-' if $sx == $sy && $sx == -1 && !$x->is_zero();
2080 $x->bdec() if $sign == 1;
2087 #(BINT or num_str, BINT or num_str) return BINT
2091 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2092 # objectify is costly, so avoid it
2093 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2095 ($self,$x,$y,@r) = objectify(2,@_);
2098 return $x if $x->modify('bior');
2099 $r[3] = $y; # no push!
2101 local $Math::BigInt::upgrade = undef;
2103 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
2105 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
2106 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
2108 # the sign of X follows the sign of X, e.g. sign of Y irrelevant for bior()
2110 # don't use lib for negative values
2111 if ($CAN{or} && $sx == 1 && $sy == 1)
2113 $x->{value} = $CALC->_or($x->{value},$y->{value});
2114 return $x->round(@r);
2117 # if lib can do negatvie values, so use it
2118 if ($CAN{signed_or})
2120 $x->{value} = $CALC->_signed_or($x->{value},$y->{value},$sx,$sy);
2121 return $x->round(@r);
2124 return $x->round(@r) if $y->is_zero();
2126 my $sign = 0; # sign of result
2127 $sign = 1 if ($x->{sign} eq '-') || ($y->{sign} eq '-');
2131 if ($sx == -1) # if x is negative
2133 # two's complement: inc and flip all "bits" in $bx
2134 $bx = $x->binc()->as_hex(); # -1 => 0, -2 => 1, -3 => 2 etc
2136 $bx =~ tr/0123456789abcdef/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/;
2140 $bx = $x->as_hex(); # get binary representation
2142 $bx =~ tr/fedcba9876543210/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/;
2144 if ($sy == -1) # if y is negative
2146 # two's complement: inc and flip all "bits" in $by
2147 $by = $y->copy()->binc()->as_hex(); # -1 => 0, -2 => 1, -3 => 2 etc
2149 $by =~ tr/0123456789abcdef/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/;
2153 $by = $y->as_hex(); # get binary representation
2155 $by =~ tr/fedcba9876543210/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/;
2157 # now we have bit-strings from X and Y, reverse them for padding
2161 # padd the shorter string
2162 my $xx = "\x00"; $xx = "\x0f" if $sx == -1;
2163 my $yy = "\x00"; $yy = "\x0f" if $sy == -1;
2164 my $diff = CORE::length($bx) - CORE::length($by);
2171 $bx .= $xx x abs($diff);
2174 # or the strings together
2177 # and reverse the result again
2180 # one of $x or $y was negative, so need to flip bits in the result
2181 # in both cases (one or two of them negative, or both positive) we need
2182 # to get the characters back.
2185 $bx =~ tr/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/0123456789abcdef/;
2189 $bx =~ tr/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/fedcba9876543210/;
2195 $x->{value} = $CALC->_from_hex( \$bx );
2199 $r = $self->new($bx);
2200 $x->{value} = $r->{value};
2203 # if one of X or Y was negative, we need to decrement result
2204 $x->bdec() if $sign == 1;
2211 #(BINT or num_str, BINT or num_str) return BINT
2215 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2216 # objectify is costly, so avoid it
2217 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2219 ($self,$x,$y,@r) = objectify(2,@_);
2222 return $x if $x->modify('bxor');
2223 $r[3] = $y; # no push!
2225 local $Math::BigInt::upgrade = undef;
2227 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
2229 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
2230 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
2232 # don't use lib for negative values
2233 if ($CAN{xor} && $sx == 1 && $sy == 1)
2235 $x->{value} = $CALC->_xor($x->{value},$y->{value});
2236 return $x->round(@r);
2239 # if lib can do negatvie values, so use it
2240 if ($CAN{signed_xor})
2242 $x->{value} = $CALC->_signed_xor($x->{value},$y->{value},$sx,$sy);
2243 return $x->round(@r);
2246 return $x->round(@r) if $y->is_zero();
2248 my $sign = 0; # sign of result
2249 $sign = 1 if $x->{sign} ne $y->{sign};
2253 if ($sx == -1) # if x is negative
2255 # two's complement: inc and flip all "bits" in $bx
2256 $bx = $x->binc()->as_hex(); # -1 => 0, -2 => 1, -3 => 2 etc
2258 $bx =~ tr/0123456789abcdef/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/;
2262 $bx = $x->as_hex(); # get binary representation
2264 $bx =~ tr/fedcba9876543210/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/;
2266 if ($sy == -1) # if y is negative
2268 # two's complement: inc and flip all "bits" in $by
2269 $by = $y->copy()->binc()->as_hex(); # -1 => 0, -2 => 1, -3 => 2 etc
2271 $by =~ tr/0123456789abcdef/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/;
2275 $by = $y->as_hex(); # get binary representation
2277 $by =~ tr/fedcba9876543210/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/;
2279 # now we have bit-strings from X and Y, reverse them for padding
2283 # padd the shorter string
2284 my $xx = "\x00"; $xx = "\x0f" if $sx == -1;
2285 my $yy = "\x00"; $yy = "\x0f" if $sy == -1;
2286 my $diff = CORE::length($bx) - CORE::length($by);
2293 $bx .= $xx x abs($diff);
2296 # xor the strings together
2299 # and reverse the result again
2302 # one of $x or $y was negative, so need to flip bits in the result
2303 # in both cases (one or two of them negative, or both positive) we need
2304 # to get the characters back.
2307 $bx =~ tr/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/0123456789abcdef/;
2311 $bx =~ tr/\x0f\x0e\x0d\x0c\x0b\x0a\x09\x08\x07\x06\x05\x04\x03\x02\x01\x00/fedcba9876543210/;
2317 $x->{value} = $CALC->_from_hex( \$bx );
2321 $r = $self->new($bx);
2322 $x->{value} = $r->{value};
2325 # calculate sign of result
2327 $x->{sign} = '-' if $sx != $sy && !$x->is_zero();
2329 $x->bdec() if $sign == 1;
2336 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2338 my $e = $CALC->_len($x->{value});
2339 wantarray ? ($e,0) : $e;
2344 # return the nth decimal digit, negative values count backward, 0 is right
2345 my ($self,$x,$n) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2347 $CALC->_digit($x->{value},$n||0);
2352 # return the amount of trailing zeros in $x
2354 $x = $class->new($x) unless ref $x;
2356 return 0 if $x->is_zero() || $x->is_odd() || $x->{sign} !~ /^[+-]$/;
2358 return $CALC->_zeros($x->{value}) if $CAN{zeros};
2360 # if not: since we do not know underlying internal representation:
2361 my $es = "$x"; $es =~ /([0]*)$/;
2362 return 0 if !defined $1; # no zeros
2363 CORE::length("$1"); # as string, not as +0!
2368 # calculate square root of $x
2369 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2371 return $x if $x->modify('bsqrt');
2373 return $x->bnan() if $x->{sign} !~ /^\+/; # -x or -inf or NaN => NaN
2374 return $x if $x->{sign} eq '+inf'; # sqrt(+inf) == inf
2376 return $upgrade->bsqrt($x,@r) if defined $upgrade;
2380 $x->{value} = $CALC->_sqrt($x->{value});
2381 return $x->round(@r);
2385 return $x->round(@r) if $x->is_zero(); # 0,1 => 0,1
2387 return $x->bone('+',@r) if $x < 4; # 1,2,3 => 1
2389 my $l = int($x->length()/2);
2391 $x->bone(); # keep ref($x), but modify it
2392 $x->blsft($l,10) if $l != 0; # first guess: 1.('0' x (l/2))
2394 my $last = $self->bzero();
2395 my $two = $self->new(2);
2396 my $lastlast = $self->bzero();
2397 #my $lastlast = $x+$two;
2398 while ($last != $x && $lastlast != $x)
2400 $lastlast = $last; $last = $x->copy();
2404 $x->bdec() if $x * $x > $y; # overshot?
2410 # calculate $y'th root of $x
2413 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2415 $y = $self->new(2) unless defined $y;
2417 # objectify is costly, so avoid it
2418 if ((!ref($x)) || (ref($x) ne ref($y)))
2420 ($self,$x,$y,@r) = $self->objectify(2,@_);
2423 return $x if $x->modify('broot');
2425 # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
2426 return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
2427 $y->{sign} !~ /^\+$/;
2429 return $x->round(@r)
2430 if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
2432 return $upgrade->new($x)->broot($upgrade->new($y),@r) if defined $upgrade;
2436 $x->{value} = $CALC->_root($x->{value},$y->{value});
2437 return $x->round(@r);
2440 return $x->bsqrt() if $y->bacmp(2) == 0; # 2 => square root
2442 # since we take at least a cubic root, and only 8 ** 1/3 >= 2 (==2):
2443 return $x->bone('+',@r) if $x < 8; # $x=2..7 => 1
2445 my $num = $x->numify();
2447 if ($num <= 1000000)
2449 $x = $self->new( int($num ** (1 / $y->numify()) ));
2450 return $x->round(@r);
2453 # if $n is a power of two, we can repeatedly take sqrt($X) and find the
2454 # proper result, because sqrt(sqrt($x)) == root($x,4)
2455 # See Calc.pm for more details
2456 my $b = $y->as_bin();
2457 if ($b =~ /0b1(0+)/)
2459 my $count = CORE::length($1); # 0b100 => len('00') => 2
2460 my $cnt = $count; # counter for loop
2461 my $shift = $self->new(6);
2462 $x->blsft($shift); # add some zeros (even amount)
2465 # 'inflate' $X by adding more zeros
2467 # calculate sqrt($x), $x is now a bit too big, again. In the next
2468 # round we make even bigger, again.
2471 # $x is still to big, so truncate result
2476 # Should compute a guess of the result (by rule of thumb), then improve it
2477 # via Newton's method or something similiar.
2479 warn ('broot() not fully implemented in BigInt.');
2481 return $x->round(@r);
2486 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
2487 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2489 if ($x->{sign} !~ /^[+-]$/)
2491 my $s = $x->{sign}; $s =~ s/^[+-]//;
2492 return $self->new($s); # -inf,+inf => inf
2494 my $e = $class->bzero();
2495 return $e->binc() if $x->is_zero();
2496 $e += $x->_trailing_zeros();
2502 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
2503 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2505 if ($x->{sign} !~ /^[+-]$/)
2507 return $self->new($x->{sign}); # keep + or - sign
2510 # that's inefficient
2511 my $zeros = $m->_trailing_zeros();
2512 $m->brsft($zeros,10) if $zeros != 0;
2518 # return a copy of both the exponent and the mantissa
2519 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2521 ($x->mantissa(),$x->exponent());
2524 ##############################################################################
2525 # rounding functions
2529 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
2530 # $n == 0 || $n == 1 => round to integer
2531 my $x = shift; $x = $class->new($x) unless ref $x;
2532 my ($scale,$mode) = $x->_scale_p($x->precision(),$x->round_mode(),@_);
2533 return $x if !defined $scale; # no-op
2534 return $x if $x->modify('bfround');
2536 # no-op for BigInts if $n <= 0
2539 $x->{_a} = undef; # clear an eventual set A
2540 $x->{_p} = $scale; return $x;
2543 $x->bround( $x->length()-$scale, $mode);
2544 $x->{_a} = undef; # bround sets {_a}
2545 $x->{_p} = $scale; # so correct it
2549 sub _scan_for_nonzero
2551 # internal, used by bround()
2556 my $len = $x->length();
2557 return 0 if $len == 1; # '5' is trailed by invisible zeros
2558 my $follow = $pad - 1;
2559 return 0 if $follow > $len || $follow < 1;
2561 # since we do not know underlying represention of $x, use decimal string
2562 my $r = substr ("$x",-$follow);
2563 $r =~ /[^0]/ ? 1 : 0;
2568 # Exists to make life easier for switch between MBF and MBI (should we
2569 # autoload fxxx() like MBF does for bxxx()?)
2576 # accuracy: +$n preserve $n digits from left,
2577 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
2579 # and overwrite the rest with 0's, return normalized number
2580 # do not return $x->bnorm(), but $x
2582 my $x = shift; $x = $class->new($x) unless ref $x;
2583 my ($scale,$mode) = $x->_scale_a($x->accuracy(),$x->round_mode(),@_);
2584 return $x if !defined $scale; # no-op
2585 return $x if $x->modify('bround');
2587 if ($x->is_zero() || $scale == 0)
2589 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2592 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2594 # we have fewer digits than we want to scale to
2595 my $len = $x->length();
2596 # convert $scale to a scalar in case it is an object (put's a limit on the
2597 # number length, but this would already limited by memory constraints), makes
2599 $scale = $scale->numify() if ref ($scale);
2601 # scale < 0, but > -len (not >=!)
2602 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2604 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2608 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2609 my ($pad,$digit_round,$digit_after);
2610 $pad = $len - $scale;
2611 $pad = abs($scale-1) if $scale < 0;
2613 # do not use digit(), it is costly for binary => decimal
2615 my $xs = $CALC->_str($x->{value});
2618 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2619 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2620 $digit_round = '0'; $digit_round = substr($$xs,$pl,1) if $pad <= $len;
2621 $pl++; $pl ++ if $pad >= $len;
2622 $digit_after = '0'; $digit_after = substr($$xs,$pl,1) if $pad > 0;
2624 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2625 # closer at the remaining digits of the original $x, remember decision
2626 my $round_up = 1; # default round up
2628 ($mode eq 'trunc') || # trunc by round down
2629 ($digit_after =~ /[01234]/) || # round down anyway,
2631 ($digit_after eq '5') && # not 5000...0000
2632 ($x->_scan_for_nonzero($pad,$xs) == 0) &&
2634 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2635 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2636 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2637 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2638 ($mode eq 'zero') # round down if zero, sign adjusted below
2640 my $put_back = 0; # not yet modified
2642 if (($pad > 0) && ($pad <= $len))
2644 substr($$xs,-$pad,$pad) = '0' x $pad;
2649 $x->bzero(); # round to '0'
2652 if ($round_up) # what gave test above?
2655 $pad = $len, $$xs = '0' x $pad if $scale < 0; # tlr: whack 0.51=>1.0
2657 # we modify directly the string variant instead of creating a number and
2658 # adding it, since that is faster (we already have the string)
2659 my $c = 0; $pad ++; # for $pad == $len case
2660 while ($pad <= $len)
2662 $c = substr($$xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2663 substr($$xs,-$pad,1) = $c; $pad++;
2664 last if $c != 0; # no overflow => early out
2666 $$xs = '1'.$$xs if $c == 0;
2669 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back in if needed
2671 $x->{_a} = $scale if $scale >= 0;
2674 $x->{_a} = $len+$scale;
2675 $x->{_a} = 0 if $scale < -$len;
2682 # return integer less or equal then number; no-op since it's already integer
2683 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2690 # return integer greater or equal then number; no-op since it's already int
2691 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2698 # An object might be asked to return itself as bigint on certain overloaded
2699 # operations, this does exactly this, so that sub classes can simple inherit
2700 # it or override with their own integer conversion routine.
2706 # return as hex string, with prefixed 0x
2707 my $x = shift; $x = $class->new($x) if !ref($x);
2709 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2711 my $es = ''; my $s = '';
2712 $s = $x->{sign} if $x->{sign} eq '-';
2715 $es = ${$CALC->_as_hex($x->{value})};
2719 return '0x0' if $x->is_zero();
2721 my $x1 = $x->copy()->babs(); my ($xr,$x10000,$h);
2724 $x10000 = Math::BigInt->new (0x10000); $h = 'h4';
2728 $x10000 = Math::BigInt->new (0x1000); $h = 'h3';
2730 while (!$x1->is_zero())
2732 ($x1, $xr) = bdiv($x1,$x10000);
2733 $es .= unpack($h,pack('v',$xr->numify()));
2736 $es =~ s/^[0]+//; # strip leading zeros
2744 # return as binary string, with prefixed 0b
2745 my $x = shift; $x = $class->new($x) if !ref($x);
2747 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2749 my $es = ''; my $s = '';
2750 $s = $x->{sign} if $x->{sign} eq '-';
2753 $es = ${$CALC->_as_bin($x->{value})};
2757 return '0b0' if $x->is_zero();
2758 my $x1 = $x->copy()->babs(); my ($xr,$x10000,$b);
2761 $x10000 = Math::BigInt->new (0x10000); $b = 'b16';
2765 $x10000 = Math::BigInt->new (0x1000); $b = 'b12';
2767 while (!$x1->is_zero())
2769 ($x1, $xr) = bdiv($x1,$x10000);
2770 $es .= unpack($b,pack('v',$xr->numify()));
2773 $es =~ s/^[0]+//; # strip leading zeros
2779 ##############################################################################
2780 # private stuff (internal use only)
2784 # check for strings, if yes, return objects instead
2786 # the first argument is number of args objectify() should look at it will
2787 # return $count+1 elements, the first will be a classname. This is because
2788 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2789 # useless objects beeing created and thrown away. So we cannot simple loop
2790 # over @_. If the given count is 0, all arguments will be used.
2792 # If the second arg is a ref, use it as class.
2793 # If not, try to use it as classname, unless undef, then use $class
2794 # (aka Math::BigInt). The latter shouldn't happen,though.
2797 # $x->badd(1); => ref x, scalar y
2798 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2799 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2800 # Math::BigInt::badd(1,2); => scalar x, scalar y
2801 # In the last case we check number of arguments to turn it silently into
2802 # $class,1,2. (We can not take '1' as class ;o)
2803 # badd($class,1) is not supported (it should, eventually, try to add undef)
2804 # currently it tries 'Math::BigInt' + 1, which will not work.
2806 # some shortcut for the common cases
2808 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2810 my $count = abs(shift || 0);
2812 my (@a,$k,$d); # resulting array, temp, and downgrade
2815 # okay, got object as first
2820 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2822 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2826 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2827 if (defined ${"$a[0]::downgrade"})
2829 $d = ${"$a[0]::downgrade"};
2830 ${"$a[0]::downgrade"} = undef;
2833 my $up = ${"$a[0]::upgrade"};
2834 #print "Now in objectify, my class is today $a[0], count = $count\n";
2842 $k = $a[0]->new($k);
2844 elsif (!defined $up && ref($k) ne $a[0])
2846 # foreign object, try to convert to integer
2847 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2860 $k = $a[0]->new($k);
2862 elsif (!defined $up && ref($k) ne $a[0])
2864 # foreign object, try to convert to integer
2865 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2869 push @a,@_; # return other params, too
2873 require Carp; Carp::croak ("$class objectify needs list context");
2875 ${"$a[0]::downgrade"} = $d;
2883 $IMPORT++; # remember we did import()
2884 my @a; my $l = scalar @_;
2885 for ( my $i = 0; $i < $l ; $i++ )
2887 if ($_[$i] eq ':constant')
2889 # this causes overlord er load to step in
2891 integer => sub { $self->new(shift) },
2892 binary => sub { $self->new(shift) };
2894 elsif ($_[$i] eq 'upgrade')
2896 # this causes upgrading
2897 $upgrade = $_[$i+1]; # or undef to disable
2900 elsif ($_[$i] =~ /^lib$/i)
2902 # this causes a different low lib to take care...
2903 $CALC = $_[$i+1] || '';
2911 # any non :constant stuff is handled by our parent, Exporter
2912 # even if @_ is empty, to give it a chance
2913 $self->SUPER::import(@a); # need it for subclasses
2914 $self->export_to_level(1,$self,@a); # need it for MBF
2916 # try to load core math lib
2917 my @c = split /\s*,\s*/,$CALC;
2918 push @c,'Calc'; # if all fail, try this
2919 $CALC = ''; # signal error
2920 foreach my $lib (@c)
2922 next if ($lib || '') eq '';
2923 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2927 # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
2928 # used in the same script, or eval inside import().
2929 my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
2930 my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
2932 $file = File::Spec->catfile (@parts, $file);
2933 eval { require "$file"; $lib->import( @c ); }
2937 eval "use $lib qw/@c/;";
2939 $CALC = $lib, last if $@ eq ''; # no error in loading lib?
2944 Carp::croak ("Couldn't load any math lib, not even 'Calc.pm'");
2951 # fill $CAN with the results of $CALC->can(...)
2954 for my $method (qw/gcd mod modinv modpow fac pow lsft rsft
2955 and signed_and or signed_or xor signed_xor
2956 from_hex as_hex from_bin as_bin
2957 zeros sqrt root log_int log
2960 $CAN{$method} = $CALC->can("_$method") ? 1 : 0;
2966 # convert a (ref to) big hex string to BigInt, return undef for error
2969 my $x = Math::BigInt->bzero();
2972 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2973 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2975 return $x->bnan() if $$hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2977 my $sign = '+'; $sign = '-' if ($$hs =~ /^-/);
2979 $$hs =~ s/^[+-]//; # strip sign
2980 if ($CAN{'_from_hex'})
2982 $x->{value} = $CALC->_from_hex($hs);
2986 # fallback to pure perl
2987 my $mul = Math::BigInt->bzero(); $mul++;
2988 my $x65536 = Math::BigInt->new(65536);
2989 my $len = CORE::length($$hs)-2;
2990 $len = int($len/4); # 4-digit parts, w/o '0x'
2991 my $val; my $i = -4;
2994 $val = substr($$hs,$i,4);
2995 $val =~ s/^[+-]?0x// if $len == 0; # for last part only because
2996 $val = hex($val); # hex does not like wrong chars
2998 $x += $mul * $val if $val != 0;
2999 $mul *= $x65536 if $len >= 0; # skip last mul
3002 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
3008 # convert a (ref to) big binary string to BigInt, return undef for error
3011 my $x = Math::BigInt->bzero();
3013 $$bs =~ s/([01])_([01])/$1$2/g;
3014 $$bs =~ s/([01])_([01])/$1$2/g;
3015 return $x->bnan() if $$bs !~ /^[+-]?0b[01]+$/;
3017 my $sign = '+'; $sign = '-' if ($$bs =~ /^\-/);
3018 $$bs =~ s/^[+-]//; # strip sign
3019 if ($CAN{'_from_bin'})
3021 $x->{value} = $CALC->_from_bin($bs);
3025 my $mul = Math::BigInt->bzero(); $mul++;
3026 my $x256 = Math::BigInt->new(256);
3027 my $len = CORE::length($$bs)-2;
3028 $len = int($len/8); # 8-digit parts, w/o '0b'
3029 my $val; my $i = -8;
3032 $val = substr($$bs,$i,8);
3033 $val =~ s/^[+-]?0b// if $len == 0; # for last part only
3034 #$val = oct('0b'.$val); # does not work on Perl prior to 5.6.0
3036 # $val = ('0' x (8-CORE::length($val))).$val if CORE::length($val) < 8;
3037 $val = ord(pack('B8',substr('00000000'.$val,-8,8)));
3039 $x += $mul * $val if $val != 0;
3040 $mul *= $x256 if $len >= 0; # skip last mul
3043 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
3049 # (ref to num_str) return num_str
3050 # internal, take apart a string and return the pieces
3051 # strip leading/trailing whitespace, leading zeros, underscore and reject
3055 # strip white space at front, also extranous leading zeros
3056 $$x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
3057 $$x =~ s/^\s+//; # but this will
3058 $$x =~ s/\s+$//g; # strip white space at end
3060 # shortcut, if nothing to split, return early
3061 if ($$x =~ /^[+-]?\d+\z/)
3063 $$x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
3064 return (\$sign, $x, \'', \'', \0);
3067 # invalid starting char?
3068 return if $$x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
3070 return __from_hex($x) if $$x =~ /^[\-\+]?0x/; # hex string
3071 return __from_bin($x) if $$x =~ /^[\-\+]?0b/; # binary string
3073 # strip underscores between digits
3074 $$x =~ s/(\d)_(\d)/$1$2/g;
3075 $$x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3
3077 # some possible inputs:
3078 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
3079 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 # 0e999
3081 #return if $$x =~ /[Ee].*[Ee]/; # more than one E => error
3083 my ($m,$e,$last) = split /[Ee]/,$$x;
3084 return if defined $last; # last defined => 1e2E3 or others
3085 $e = '0' if !defined $e || $e eq "";
3087 # sign,value for exponent,mantint,mantfrac
3088 my ($es,$ev,$mis,$miv,$mfv);
3090 if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
3094 return if $m eq '.' || $m eq '';
3095 my ($mi,$mf,$lastf) = split /\./,$m;
3096 return if defined $lastf; # last defined => 1.2.3 or others
3097 $mi = '0' if !defined $mi;
3098 $mi .= '0' if $mi =~ /^[\-\+]?$/;
3099 $mf = '0' if !defined $mf || $mf eq '';
3100 if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
3102 $mis = $1||'+'; $miv = $2;
3103 return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
3105 # handle the 0e999 case here
3106 $ev = 0 if $miv eq '0' && $mfv eq '';
3107 return (\$mis,\$miv,\$mfv,\$es,\$ev);
3110 return; # NaN, not a number
3113 ##############################################################################
3114 # internal calculation routines (others are in Math::BigInt::Calc etc)
3118 # (BINT or num_str, BINT or num_str) return BINT
3119 # does modify first argument
3122 my $x = shift; my $ty = shift;
3123 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
3124 return $x * $ty / bgcd($x,$ty);
3129 # (BINT or num_str, BINT or num_str) return BINT
3130 # does modify both arguments
3131 # GCD -- Euclids algorithm E, Knuth Vol 2 pg 296
3134 return $x->bnan() if $x->{sign} !~ /^[+-]$/ || $ty->{sign} !~ /^[+-]$/;
3136 while (!$ty->is_zero())
3138 ($x, $ty) = ($ty,bmod($x,$ty));
3143 ###############################################################################
3144 # this method return 0 if the object can be modified, or 1 for not
3145 # We use a fast use constant statement here, to avoid costly calls. Subclasses
3146 # may override it with special code (f.i. Math::BigInt::Constant does so)
3148 sub modify () { 0; }
3155 Math::BigInt - Arbitrary size integer math package
3161 # or make it faster: install (optional) Math::BigInt::GMP
3162 # and always use (it will fall back to pure Perl if the
3163 # GMP library is not installed):
3165 use Math::BigInt lib => 'GMP';
3168 $x = Math::BigInt->new($str); # defaults to 0
3169 $nan = Math::BigInt->bnan(); # create a NotANumber
3170 $zero = Math::BigInt->bzero(); # create a +0
3171 $inf = Math::BigInt->binf(); # create a +inf
3172 $inf = Math::BigInt->binf('-'); # create a -inf
3173 $one = Math::BigInt->bone(); # create a +1
3174 $one = Math::BigInt->bone('-'); # create a -1
3176 # Testing (don't modify their arguments)
3177 # (return true if the condition is met, otherwise false)
3179 $x->is_zero(); # if $x is +0
3180 $x->is_nan(); # if $x is NaN
3181 $x->is_one(); # if $x is +1
3182 $x->is_one('-'); # if $x is -1
3183 $x->is_odd(); # if $x is odd
3184 $x->is_even(); # if $x is even
3185 $x->is_positive(); # if $x >= 0
3186 $x->is_negative(); # if $x < 0
3187 $x->is_inf(sign); # if $x is +inf, or -inf (sign is default '+')
3188 $x->is_int(); # if $x is an integer (not a float)
3190 # comparing and digit/sign extration
3191 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
3192 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
3193 $x->sign(); # return the sign, either +,- or NaN
3194 $x->digit($n); # return the nth digit, counting from right
3195 $x->digit(-$n); # return the nth digit, counting from left
3197 # The following all modify their first argument. If you want to preserve
3198 # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
3199 # neccessary when mixing $a = $b assigments with non-overloaded math.
3201 $x->bzero(); # set $x to 0
3202 $x->bnan(); # set $x to NaN
3203 $x->bone(); # set $x to +1
3204 $x->bone('-'); # set $x to -1
3205 $x->binf(); # set $x to inf
3206 $x->binf('-'); # set $x to -inf
3208 $x->bneg(); # negation
3209 $x->babs(); # absolute value
3210 $x->bnorm(); # normalize (no-op in BigInt)
3211 $x->bnot(); # two's complement (bit wise not)
3212 $x->binc(); # increment $x by 1
3213 $x->bdec(); # decrement $x by 1
3215 $x->badd($y); # addition (add $y to $x)
3216 $x->bsub($y); # subtraction (subtract $y from $x)
3217 $x->bmul($y); # multiplication (multiply $x by $y)
3218 $x->bdiv($y); # divide, set $x to quotient
3219 # return (quo,rem) or quo if scalar
3221 $x->bmod($y); # modulus (x % y)
3222 $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
3223 $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
3225 $x->bpow($y); # power of arguments (x ** y)
3226 $x->blsft($y); # left shift
3227 $x->brsft($y); # right shift
3228 $x->blsft($y,$n); # left shift, by base $n (like 10)
3229 $x->brsft($y,$n); # right shift, by base $n (like 10)
3231 $x->band($y); # bitwise and
3232 $x->bior($y); # bitwise inclusive or
3233 $x->bxor($y); # bitwise exclusive or
3234 $x->bnot(); # bitwise not (two's complement)
3236 $x->bsqrt(); # calculate square-root
3237 $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
3238 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3240 $x->round($A,$P,$mode); # round to accuracy or precision using mode $mode
3241 $x->bround($N); # accuracy: preserve $N digits
3242 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3244 # The following do not modify their arguments in BigInt (are no-ops),
3245 # but do so in BigFloat:
3247 $x->bfloor(); # return integer less or equal than $x
3248 $x->bceil(); # return integer greater or equal than $x
3250 # The following do not modify their arguments:
3252 bgcd(@values); # greatest common divisor (no OO style)
3253 blcm(@values); # lowest common multiplicator (no OO style)
3255 $x->length(); # return number of digits in number
3256 ($x,$f) = $x->length(); # length of number and length of fraction part,
3257 # latter is always 0 digits long for BigInt's
3259 $x->exponent(); # return exponent as BigInt
3260 $x->mantissa(); # return (signed) mantissa as BigInt
3261 $x->parts(); # return (mantissa,exponent) as BigInt
3262 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3263 $x->as_number(); # return as BigInt (in BigInt: same as copy())
3265 # conversation to string (do not modify their argument)
3266 $x->bstr(); # normalized string
3267 $x->bsstr(); # normalized string in scientific notation
3268 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3269 $x->as_bin(); # as signed binary string with prefixed 0b
3272 # precision and accuracy (see section about rounding for more)
3273 $x->precision(); # return P of $x (or global, if P of $x undef)
3274 $x->precision($n); # set P of $x to $n
3275 $x->accuracy(); # return A of $x (or global, if A of $x undef)
3276 $x->accuracy($n); # set A $x to $n
3279 Math::BigInt->precision(); # get/set global P for all BigInt objects
3280 Math::BigInt->accuracy(); # get/set global A for all BigInt objects
3281 Math::BigInt->config(); # return hash containing configuration
3285 All operators (inlcuding basic math operations) are overloaded if you
3286 declare your big integers as
3288 $i = new Math::BigInt '123_456_789_123_456_789';
3290 Operations with overloaded operators preserve the arguments which is
3291 exactly what you expect.
3297 Input values to these routines may be any string, that looks like a number
3298 and results in an integer, including hexadecimal and binary numbers.
3300 Scalars holding numbers may also be passed, but note that non-integer numbers
3301 may already have lost precision due to the conversation to float. Quote
3302 your input if you want BigInt to see all the digits:
3304 $x = Math::BigInt->new(12345678890123456789); # bad
3305 $x = Math::BigInt->new('12345678901234567890'); # good
3307 You can include one underscore between any two digits.
3309 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
3310 Non-integer values result in NaN.
3312 Currently, Math::BigInt::new() defaults to 0, while Math::BigInt::new('')
3313 results in 'NaN'. This might change in the future, so use always the following
3314 explicit forms to get a zero or NaN:
3316 $zero = Math::BigInt->bzero();
3317 $nan = Math::BigInt->bnan();
3319 C<bnorm()> on a BigInt object is now effectively a no-op, since the numbers
3320 are always stored in normalized form. If passed a string, creates a BigInt
3321 object from the input.
3325 Output values are BigInt objects (normalized), except for bstr(), which
3326 returns a string in normalized form.
3327 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
3328 C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
3329 return either undef, <0, 0 or >0 and are suited for sort.
3335 Each of the methods below (except config(), accuracy() and precision())
3336 accepts three additional parameters. These arguments $A, $P and $R are
3337 accuracy, precision and round_mode. Please see the section about
3338 L<ACCURACY and PRECISION> for more information.
3344 print Dumper ( Math::BigInt->config() );
3345 print Math::BigInt->config()->{lib},"\n";
3347 Returns a hash containing the configuration, e.g. the version number, lib
3348 loaded etc. The following hash keys are currently filled in with the
3349 appropriate information.
3353 ============================================================
3354 lib Name of the low-level math library
3356 lib_version Version of low-level math library (see 'lib')
3358 class The class name of config() you just called
3360 upgrade To which class math operations might be upgraded
3362 downgrade To which class math operations might be downgraded
3364 precision Global precision
3366 accuracy Global accuracy
3368 round_mode Global round mode
3370 version version number of the class you used
3372 div_scale Fallback acccuracy for div
3374 trap_nan If true, traps creation of NaN via croak()
3376 trap_inf If true, traps creation of +inf/-inf via croak()
3379 The following values can be set by passing C<config()> a reference to a hash:
3382 upgrade downgrade precision accuracy round_mode div_scale
3386 $new_cfg = Math::BigInt->config( { trap_inf => 1, precision => 5 } );
3390 $x->accuracy(5); # local for $x
3391 CLASS->accuracy(5); # global for all members of CLASS
3392 $A = $x->accuracy(); # read out
3393 $A = CLASS->accuracy(); # read out
3395 Set or get the global or local accuracy, aka how many significant digits the
3398 Please see the section about L<ACCURACY AND PRECISION> for further details.
3400 Value must be greater than zero. Pass an undef value to disable it:
3402 $x->accuracy(undef);
3403 Math::BigInt->accuracy(undef);
3405 Returns the current accuracy. For C<$x->accuracy()> it will return either the
3406 local accuracy, or if not defined, the global. This means the return value
3407 represents the accuracy that will be in effect for $x:
3409 $y = Math::BigInt->new(1234567); # unrounded
3410 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
3411 $x = Math::BigInt->new(123456); # will be automatically rounded
3412 print "$x $y\n"; # '123500 1234567'
3413 print $x->accuracy(),"\n"; # will be 4
3414 print $y->accuracy(),"\n"; # also 4, since global is 4
3415 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
3416 print $x->accuracy(),"\n"; # still 4
3417 print $y->accuracy(),"\n"; # 5, since global is 5
3419 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
3420 globals separated from Math::BigInt, but it is possible to subclass
3421 Math::BigInt and make the globals of the subclass aliases to the ones from
3426 $x->precision(-2); # local for $x, round right of the dot
3427 $x->precision(2); # ditto, but round left of the dot
3428 CLASS->accuracy(5); # global for all members of CLASS
3429 CLASS->precision(-5); # ditto
3430 $P = CLASS->precision(); # read out
3431 $P = $x->precision(); # read out
3433 Set or get the global or local precision, aka how many digits the result has
3434 after the dot (or where to round it when passing a positive number). In
3435 Math::BigInt, passing a negative number precision has no effect since no
3436 numbers have digits after the dot.
3438 Please see the section about L<ACCURACY AND PRECISION> for further details.
3440 Value must be greater than zero. Pass an undef value to disable it:
3442 $x->precision(undef);
3443 Math::BigInt->precision(undef);
3445 Returns the current precision. For C<$x->precision()> it will return either the
3446 local precision of $x, or if not defined, the global. This means the return
3447 value represents the accuracy that will be in effect for $x:
3449 $y = Math::BigInt->new(1234567); # unrounded
3450 print Math::BigInt->precision(4),"\n"; # set 4, print 4
3451 $x = Math::BigInt->new(123456); # will be automatically rounded
3453 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
3454 globals separated from Math::BigInt, but it is possible to subclass
3455 Math::BigInt and make the globals of the subclass aliases to the ones from
3462 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
3463 2, but others work, too.
3465 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
3469 $x = Math::BigInt->new(10);
3470 $x->brsft(1); # same as $x >> 1: 5
3471 $x = Math::BigInt->new(1234);
3472 $x->brsft(2,10); # result 12
3474 There is one exception, and that is base 2 with negative $x:
3477 $x = Math::BigInt->new(-5);
3480 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
3485 $x = Math::BigInt->new($str,$A,$P,$R);
3487 Creates a new BigInt object from a scalar or another BigInt object. The
3488 input is accepted as decimal, hex (with leading '0x') or binary (with leading
3491 See L<Input> for more info on accepted input formats.
3495 $x = Math::BigInt->bnan();
3497 Creates a new BigInt object representing NaN (Not A Number).
3498 If used on an object, it will set it to NaN:
3504 $x = Math::BigInt->bzero();
3506 Creates a new BigInt object representing zero.
3507 If used on an object, it will set it to zero:
3513 $x = Math::BigInt->binf($sign);
3515 Creates a new BigInt object representing infinity. The optional argument is
3516 either '-' or '+', indicating whether you want infinity or minus infinity.
3517 If used on an object, it will set it to infinity:
3524 $x = Math::BigInt->binf($sign);
3526 Creates a new BigInt object representing one. The optional argument is
3527 either '-' or '+', indicating whether you want one or minus one.
3528 If used on an object, it will set it to one:
3533 =head2 is_one()/is_zero()/is_nan()/is_inf()
3536 $x->is_zero(); # true if arg is +0
3537 $x->is_nan(); # true if arg is NaN
3538 $x->is_one(); # true if arg is +1
3539 $x->is_one('-'); # true if arg is -1
3540 $x->is_inf(); # true if +inf
3541 $x->is_inf('-'); # true if -inf (sign is default '+')
3543 These methods all test the BigInt for beeing one specific value and return
3544 true or false depending on the input. These are faster than doing something
3549 =head2 is_positive()/is_negative()
3551 $x->is_positive(); # true if >= 0
3552 $x->is_negative(); # true if < 0
3554 The methods return true if the argument is positive or negative, respectively.
3555 C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
3556 C<-inf> is negative. A C<zero> is positive.
3558 These methods are only testing the sign, and not the value.
3560 =head2 is_odd()/is_even()/is_int()
3562 $x->is_odd(); # true if odd, false for even
3563 $x->is_even(); # true if even, false for odd
3564 $x->is_int(); # true if $x is an integer
3566 The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
3567 C<-inf> are not integers and are neither odd nor even.
3569 In BigInt, all numbers except C<NaN>, C<+inf> and C<-inf> are integers.
3575 Compares $x with $y and takes the sign into account.
3576 Returns -1, 0, 1 or undef.
3582 Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
3588 Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
3592 $x->digit($n); # return the nth digit, counting from right
3594 If C<$n> is negative, returns the digit counting from left.
3600 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
3601 and '-inf', respectively. Does nothing for NaN or zero.
3607 Set the number to it's absolute value, e.g. change the sign from '-' to '+'
3608 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
3613 $x->bnorm(); # normalize (no-op)
3619 Two's complement (bit wise not). This is equivalent to
3627 $x->binc(); # increment x by 1
3631 $x->bdec(); # decrement x by 1
3635 $x->badd($y); # addition (add $y to $x)
3639 $x->bsub($y); # subtraction (subtract $y from $x)
3643 $x->bmul($y); # multiplication (multiply $x by $y)
3647 $x->bdiv($y); # divide, set $x to quotient
3648 # return (quo,rem) or quo if scalar
3652 $x->bmod($y); # modulus (x % y)
3656 num->bmodinv($mod); # modular inverse
3658 Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
3659 returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
3660 C<bgcd($num, $mod)==1>.
3664 $num->bmodpow($exp,$mod); # modular exponentation
3665 # ($num**$exp % $mod)
3667 Returns the value of C<$num> taken to the power C<$exp> in the modulus
3668 C<$mod> using binary exponentation. C<bmodpow> is far superior to
3673 because it is much faster - it reduces internal variables into
3674 the modulus whenever possible, so it operates on smaller numbers.
3676 C<bmodpow> also supports negative exponents.
3678 bmodpow($num, -1, $mod)
3680 is exactly equivalent to
3686 $x->bpow($y); # power of arguments (x ** y)
3690 $x->blsft($y); # left shift
3691 $x->blsft($y,$n); # left shift, in base $n (like 10)
3695 $x->brsft($y); # right shift
3696 $x->brsft($y,$n); # right shift, in base $n (like 10)
3700 $x->band($y); # bitwise and
3704 $x->bior($y); # bitwise inclusive or
3708 $x->bxor($y); # bitwise exclusive or
3712 $x->bnot(); # bitwise not (two's complement)
3716 $x->bsqrt(); # calculate square-root
3720 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3724 $x->round($A,$P,$round_mode);
3726 Round $x to accuracy C<$A> or precision C<$P> using the round mode
3731 $x->bround($N); # accuracy: preserve $N digits
3735 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3741 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3742 does change $x in BigFloat.
3748 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3749 does change $x in BigFloat.
3753 bgcd(@values); # greatest common divisor (no OO style)
3757 blcm(@values); # lowest common multiplicator (no OO style)
3762 ($xl,$fl) = $x->length();
3764 Returns the number of digits in the decimal representation of the number.
3765 In list context, returns the length of the integer and fraction part. For
3766 BigInt's, the length of the fraction part will always be 0.
3772 Return the exponent of $x as BigInt.
3778 Return the signed mantissa of $x as BigInt.
3782 $x->parts(); # return (mantissa,exponent) as BigInt
3786 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3790 $x->as_number(); # return as BigInt (in BigInt: same as copy())
3794 $x->bstr(); # return normalized string
3798 $x->bsstr(); # normalized string in scientific notation
3802 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3806 $x->as_bin(); # as signed binary string with prefixed 0b
3808 =head1 ACCURACY and PRECISION
3810 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3811 accuracy and precision based rounding, both automatically after every
3812 operation, as well as manually.
3814 This section describes the accuracy/precision handling in Math::Big* as it
3815 used to be and as it is now, complete with an explanation of all terms and
3818 Not yet implemented things (but with correct description) are marked with '!',
3819 things that need to be answered are marked with '?'.
3821 In the next paragraph follows a short description of terms used here (because
3822 these may differ from terms used by others people or documentation).
3824 During the rest of this document, the shortcuts A (for accuracy), P (for
3825 precision), F (fallback) and R (rounding mode) will be used.
3829 A fixed number of digits before (positive) or after (negative)
3830 the decimal point. For example, 123.45 has a precision of -2. 0 means an
3831 integer like 123 (or 120). A precision of 2 means two digits to the left
3832 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3833 numbers with zeros before the decimal point may have different precisions,
3834 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3835 was). It could also have p < 0, when the digits after the decimal point
3838 The string output (of floating point numbers) will be padded with zeros:
3840 Initial value P A Result String
3841 ------------------------------------------------------------
3842 1234.01 -3 1000 1000
3845 1234.001 1 1234 1234.0
3847 1234.01 2 1234.01 1234.01
3848 1234.01 5 1234.01 1234.01000
3850 For BigInts, no padding occurs.
3854 Number of significant digits. Leading zeros are not counted. A
3855 number may have an accuracy greater than the non-zero digits
3856 when there are zeros in it or trailing zeros. For example, 123.456 has
3857 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3859 The string output (of floating point numbers) will be padded with zeros:
3861 Initial value P A Result String
3862 ------------------------------------------------------------
3864 1234.01 6 1234.01 1234.01
3865 1234.1 8 1234.1 1234.1000
3867 For BigInts, no padding occurs.
3871 When both A and P are undefined, this is used as a fallback accuracy when
3874 =head2 Rounding mode R
3876 When rounding a number, different 'styles' or 'kinds'
3877 of rounding are possible. (Note that random rounding, as in
3878 Math::Round, is not implemented.)
3884 truncation invariably removes all digits following the
3885 rounding place, replacing them with zeros. Thus, 987.65 rounded
3886 to tens (P=1) becomes 980, and rounded to the fourth sigdig
3887 becomes 987.6 (A=4). 123.456 rounded to the second place after the
3888 decimal point (P=-2) becomes 123.46.
3890 All other implemented styles of rounding attempt to round to the
3891 "nearest digit." If the digit D immediately to the right of the
3892 rounding place (skipping the decimal point) is greater than 5, the
3893 number is incremented at the rounding place (possibly causing a
3894 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3895 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3896 truncated at the rounding place: e.g. when rounding to units, 0.4
3897 rounds to 0, and -19.4 rounds to -19.
3899 However the results of other styles of rounding differ if the
3900 digit immediately to the right of the rounding place (skipping the
3901 decimal point) is 5 and if there are no digits, or no digits other
3902 than 0, after that 5. In such cases:
3906 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3907 if it is not already. E.g., when rounding to the first sigdig, 0.45
3908 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3912 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3913 it is not already. E.g., when rounding to the first sigdig, 0.45
3914 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3918 round to plus infinity, i.e. always round up. E.g., when
3919 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3920 and 0.4501 also becomes 0.5.
3924 round to minus infinity, i.e. always round down. E.g., when
3925 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3926 but 0.4501 becomes 0.5.
3930 round to zero, i.e. positive numbers down, negative ones up.
3931 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3932 becomes -0.5, but 0.4501 becomes 0.5.
3936 The handling of A & P in MBI/MBF (the old core code shipped with Perl
3937 versions <= 5.7.2) is like this:
3943 * ffround($p) is able to round to $p number of digits after the decimal
3945 * otherwise P is unused
3947 =item Accuracy (significant digits)
3949 * fround($a) rounds to $a significant digits
3950 * only fdiv() and fsqrt() take A as (optional) paramater
3951 + other operations simply create the same number (fneg etc), or more (fmul)
3953 + rounding/truncating is only done when explicitly calling one of fround
3954 or ffround, and never for BigInt (not implemented)
3955 * fsqrt() simply hands its accuracy argument over to fdiv.
3956 * the documentation and the comment in the code indicate two different ways
3957 on how fdiv() determines the maximum number of digits it should calculate,
3958 and the actual code does yet another thing
3960 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3962 result has at most max(scale, length(dividend), length(divisor)) digits
3964 scale = max(scale, length(dividend)-1,length(divisor)-1);
3965 scale += length(divisior) - length(dividend);
3966 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3967 Actually, the 'difference' added to the scale is calculated from the
3968 number of "significant digits" in dividend and divisor, which is derived
3969 by looking at the length of the mantissa. Which is wrong, since it includes
3970 the + sign (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops
3971 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3972 assumption that 124 has 3 significant digits, while 120/7 will get you
3973 '17', not '17.1' since 120 is thought to have 2 significant digits.
3974 The rounding after the division then uses the remainder and $y to determine
3975 wether it must round up or down.
3976 ? I have no idea which is the right way. That's why I used a slightly more
3977 ? simple scheme and tweaked the few failing testcases to match it.
3981 This is how it works now:
3985 =item Setting/Accessing
3987 * You can set the A global via C<< Math::BigInt->accuracy() >> or
3988 C<< Math::BigFloat->accuracy() >> or whatever class you are using.
3989 * You can also set P globally by using C<< Math::SomeClass->precision() >>
3991 * Globals are classwide, and not inherited by subclasses.
3992 * to undefine A, use C<< Math::SomeCLass->accuracy(undef); >>
3993 * to undefine P, use C<< Math::SomeClass->precision(undef); >>
3994 * Setting C<< Math::SomeClass->accuracy() >> clears automatically
3995 C<< Math::SomeClass->precision() >>, and vice versa.
3996 * To be valid, A must be > 0, P can have any value.
3997 * If P is negative, this means round to the P'th place to the right of the
3998 decimal point; positive values mean to the left of the decimal point.
3999 P of 0 means round to integer.
4000 * to find out the current global A, use C<< Math::SomeClass->accuracy() >>
4001 * to find out the current global P, use C<< Math::SomeClass->precision() >>
4002 * use C<< $x->accuracy() >> respective C<< $x->precision() >> for the local
4003 setting of C<< $x >>.
4004 * Please note that C<< $x->accuracy() >> respecive C<< $x->precision() >>
4005 return eventually defined global A or P, when C<< $x >>'s A or P is not
4008 =item Creating numbers
4010 * When you create a number, you can give it's desired A or P via:
4011 $x = Math::BigInt->new($number,$A,$P);
4012 * Only one of A or P can be defined, otherwise the result is NaN
4013 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
4014 globals (if set) will be used. Thus changing the global defaults later on
4015 will not change the A or P of previously created numbers (i.e., A and P of
4016 $x will be what was in effect when $x was created)
4017 * If given undef for A and P, B<no> rounding will occur, and the globals will
4018 B<not> be used. This is used by subclasses to create numbers without
4019 suffering rounding in the parent. Thus a subclass is able to have it's own
4020 globals enforced upon creation of a number by using
4021 C<< $x = Math::BigInt->new($number,undef,undef) >>:
4023 use Math::BigInt::SomeSubclass;
4026 Math::BigInt->accuracy(2);
4027 Math::BigInt::SomeSubClass->accuracy(3);
4028 $x = Math::BigInt::SomeSubClass->new(1234);
4030 $x is now 1230, and not 1200. A subclass might choose to implement
4031 this otherwise, e.g. falling back to the parent's A and P.
4035 * If A or P are enabled/defined, they are used to round the result of each
4036 operation according to the rules below
4037 * Negative P is ignored in Math::BigInt, since BigInts never have digits
4038 after the decimal point
4039 * Math::BigFloat uses Math::BigInt internally, but setting A or P inside
4040 Math::BigInt as globals does not tamper with the parts of a BigFloat.
4041 A flag is used to mark all Math::BigFloat numbers as 'never round'.
4045 * It only makes sense that a number has only one of A or P at a time.
4046 If you set either A or P on one object, or globally, the other one will
4047 be automatically cleared.
4048 * If two objects are involved in an operation, and one of them has A in
4049 effect, and the other P, this results in an error (NaN).
4050 * A takes precendence over P (Hint: A comes before P).
4051 If neither of them is defined, nothing is used, i.e. the result will have
4052 as many digits as it can (with an exception for fdiv/fsqrt) and will not
4054 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
4055 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
4056 If either the dividend's or the divisor's mantissa has more digits than
4057 the value of F, the higher value will be used instead of F.
4058 This is to limit the digits (A) of the result (just consider what would
4059 happen with unlimited A and P in the case of 1/3 :-)
4060 * fdiv will calculate (at least) 4 more digits than required (determined by
4061 A, P or F), and, if F is not used, round the result
4062 (this will still fail in the case of a result like 0.12345000000001 with A
4063 or P of 5, but this can not be helped - or can it?)
4064 * Thus you can have the math done by on Math::Big* class in two modi:
4065 + never round (this is the default):
4066 This is done by setting A and P to undef. No math operation
4067 will round the result, with fdiv() and fsqrt() as exceptions to guard
4068 against overflows. You must explicitely call bround(), bfround() or
4069 round() (the latter with parameters).
4070 Note: Once you have rounded a number, the settings will 'stick' on it
4071 and 'infect' all other numbers engaged in math operations with it, since
4072 local settings have the highest precedence. So, to get SaferRound[tm],
4073 use a copy() before rounding like this:
4075 $x = Math::BigFloat->new(12.34);
4076 $y = Math::BigFloat->new(98.76);
4077 $z = $x * $y; # 1218.6984
4078 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
4079 $z = $x * $y; # still 1218.6984, without
4080 # copy would have been 1210!
4082 + round after each op:
4083 After each single operation (except for testing like is_zero()), the
4084 method round() is called and the result is rounded appropriately. By
4085 setting proper values for A and P, you can have all-the-same-A or
4086 all-the-same-P modes. For example, Math::Currency might set A to undef,
4087 and P to -2, globally.
4089 ?Maybe an extra option that forbids local A & P settings would be in order,
4090 ?so that intermediate rounding does not 'poison' further math?
4092 =item Overriding globals
4094 * you will be able to give A, P and R as an argument to all the calculation
4095 routines; the second parameter is A, the third one is P, and the fourth is
4096 R (shift right by one for binary operations like badd). P is used only if
4097 the first parameter (A) is undefined. These three parameters override the
4098 globals in the order detailed as follows, i.e. the first defined value
4100 (local: per object, global: global default, parameter: argument to sub)
4103 + local A (if defined on both of the operands: smaller one is taken)
4104 + local P (if defined on both of the operands: bigger one is taken)
4108 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
4109 arguments (A and P) instead of one
4111 =item Local settings
4113 * You can set A or P locally by using C<< $x->accuracy() >> or
4114 C<< $x->precision() >>
4115 and thus force different A and P for different objects/numbers.
4116 * Setting A or P this way immediately rounds $x to the new value.
4117 * C<< $x->accuracy() >> clears C<< $x->precision() >>, and vice versa.
4121 * the rounding routines will use the respective global or local settings.
4122 fround()/bround() is for accuracy rounding, while ffround()/bfround()
4124 * the two rounding functions take as the second parameter one of the
4125 following rounding modes (R):
4126 'even', 'odd', '+inf', '-inf', 'zero', 'trunc'
4127 * you can set/get the global R by using C<< Math::SomeClass->round_mode() >>
4128 or by setting C<< $Math::SomeClass::round_mode >>
4129 * after each operation, C<< $result->round() >> is called, and the result may
4130 eventually be rounded (that is, if A or P were set either locally,
4131 globally or as parameter to the operation)
4132 * to manually round a number, call C<< $x->round($A,$P,$round_mode); >>
4133 this will round the number by using the appropriate rounding function
4134 and then normalize it.
4135 * rounding modifies the local settings of the number:
4137 $x = Math::BigFloat->new(123.456);
4141 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
4142 will be 4 from now on.
4144 =item Default values
4153 * The defaults are set up so that the new code gives the same results as
4154 the old code (except in a few cases on fdiv):
4155 + Both A and P are undefined and thus will not be used for rounding
4156 after each operation.
4157 + round() is thus a no-op, unless given extra parameters A and P
4163 The actual numbers are stored as unsigned big integers (with seperate sign).
4164 You should neither care about nor depend on the internal representation; it
4165 might change without notice. Use only method calls like C<< $x->sign(); >>
4166 instead relying on the internal hash keys like in C<< $x->{sign}; >>.
4170 Math with the numbers is done (by default) by a module called
4171 C<Math::BigInt::Calc>. This is equivalent to saying:
4173 use Math::BigInt lib => 'Calc';
4175 You can change this by using:
4177 use Math::BigInt lib => 'BitVect';
4179 The following would first try to find Math::BigInt::Foo, then
4180 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
4182 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
4184 Since Math::BigInt::GMP is in almost all cases faster than Calc (especially in
4185 cases involving really big numbers, where it is B<much> faster), and there is
4186 no penalty if Math::BigInt::GMP is not installed, it is a good idea to always
4189 use Math::BigInt lib => 'GMP';
4191 Different low-level libraries use different formats to store the
4192 numbers. You should not depend on the number having a specific format.
4194 See the respective math library module documentation for further details.
4198 The sign is either '+', '-', 'NaN', '+inf' or '-inf' and stored seperately.
4200 A sign of 'NaN' is used to represent the result when input arguments are not
4201 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
4202 minus infinity. You will get '+inf' when dividing a positive number by 0, and
4203 '-inf' when dividing any negative number by 0.
4205 =head2 mantissa(), exponent() and parts()
4207 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
4210 $m = $x->mantissa();
4211 $e = $x->exponent();
4212 $y = $m * ( 10 ** $e );
4213 print "ok\n" if $x == $y;
4215 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
4216 in one go. Both the returned mantissa and exponent have a sign.
4218 Currently, for BigInts C<$e> is always 0, except for NaN, +inf and -inf,
4219 where it is C<NaN>; and for C<$x == 0>, where it is C<1> (to be compatible
4220 with Math::BigFloat's internal representation of a zero as C<0E1>).
4222 C<$m> is currently just a copy of the original number. The relation between
4223 C<$e> and C<$m> will stay always the same, though their real values might
4230 sub bint { Math::BigInt->new(shift); }
4232 $x = Math::BigInt->bstr("1234") # string "1234"
4233 $x = "$x"; # same as bstr()
4234 $x = Math::BigInt->bneg("1234"); # BigInt "-1234"
4235 $x = Math::BigInt->babs("-12345"); # BigInt "12345"
4236 $x = Math::BigInt->bnorm("-0 00"); # BigInt "0"
4237 $x = bint(1) + bint(2); # BigInt "3"
4238 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
4239 $x = bint(1); # BigInt "1"
4240 $x = $x + 5 / 2; # BigInt "3"
4241 $x = $x ** 3; # BigInt "27"
4242 $x *= 2; # BigInt "54"
4243 $x = Math::BigInt->new(0); # BigInt "0"
4245 $x = Math::BigInt->badd(4,5) # BigInt "9"
4246 print $x->bsstr(); # 9e+0
4248 Examples for rounding:
4253 $x = Math::BigFloat->new(123.4567);
4254 $y = Math::BigFloat->new(123.456789);
4255 Math::BigFloat->accuracy(4); # no more A than 4
4257 ok ($x->copy()->fround(),123.4); # even rounding
4258 print $x->copy()->fround(),"\n"; # 123.4
4259 Math::BigFloat->round_mode('odd'); # round to odd
4260 print $x->copy()->fround(),"\n"; # 123.5
4261 Math::BigFloat->accuracy(5); # no more A than 5
4262 Math::BigFloat->round_mode('odd'); # round to odd
4263 print $x->copy()->fround(),"\n"; # 123.46
4264 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
4265 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
4267 Math::BigFloat->accuracy(undef); # A not important now
4268 Math::BigFloat->precision(2); # P important
4269 print $x->copy()->bnorm(),"\n"; # 123.46
4270 print $x->copy()->fround(),"\n"; # 123.46
4272 Examples for converting:
4274 my $x = Math::BigInt->new('0b1'.'01' x 123);
4275 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
4277 =head1 Autocreating constants
4279 After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
4280 and binary constants in the given scope are converted to C<Math::BigInt>.
4281 This conversion happens at compile time.
4285 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
4287 prints the integer value of C<2**100>. Note that without conversion of
4288 constants the expression 2**100 will be calculated as perl scalar.
4290 Please note that strings and floating point constants are not affected,
4293 use Math::BigInt qw/:constant/;
4295 $x = 1234567890123456789012345678901234567890
4296 + 123456789123456789;
4297 $y = '1234567890123456789012345678901234567890'
4298 + '123456789123456789';
4300 do not work. You need an explicit Math::BigInt->new() around one of the
4301 operands. You should also quote large constants to protect loss of precision:
4305 $x = Math::BigInt->new('1234567889123456789123456789123456789');
4307 Without the quotes Perl would convert the large number to a floating point
4308 constant at compile time and then hand the result to BigInt, which results in
4309 an truncated result or a NaN.
4311 This also applies to integers that look like floating point constants:
4313 use Math::BigInt ':constant';
4315 print ref(123e2),"\n";
4316 print ref(123.2e2),"\n";
4318 will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
4319 to get this to work.
4323 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
4324 must be made in the second case. For long numbers, the copy can eat up to 20%
4325 of the work (in the case of addition/subtraction, less for
4326 multiplication/division). If $y is very small compared to $x, the form
4327 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
4328 more time then the actual addition.
4330 With a technique called copy-on-write, the cost of copying with overload could
4331 be minimized or even completely avoided. A test implementation of COW did show
4332 performance gains for overloaded math, but introduced a performance loss due
4333 to a constant overhead for all other operatons. So Math::BigInt does currently
4336 The rewritten version of this module (vs. v0.01) is slower on certain
4337 operations, like C<new()>, C<bstr()> and C<numify()>. The reason are that it
4338 does now more work and handles much more cases. The time spent in these
4339 operations is usually gained in the other math operations so that code on
4340 the average should get (much) faster. If they don't, please contact the author.
4342 Some operations may be slower for small numbers, but are significantly faster
4343 for big numbers. Other operations are now constant (O(1), like C<bneg()>,
4344 C<babs()> etc), instead of O(N) and thus nearly always take much less time.
4345 These optimizations were done on purpose.
4347 If you find the Calc module to slow, try to install any of the replacement
4348 modules and see if they help you.
4350 =head2 Alternative math libraries
4352 You can use an alternative library to drive Math::BigInt via:
4354 use Math::BigInt lib => 'Module';
4356 See L<MATH LIBRARY> for more information.
4358 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
4362 =head1 Subclassing Math::BigInt
4364 The basic design of Math::BigInt allows simple subclasses with very little
4365 work, as long as a few simple rules are followed:
4371 The public API must remain consistent, i.e. if a sub-class is overloading
4372 addition, the sub-class must use the same name, in this case badd(). The
4373 reason for this is that Math::BigInt is optimized to call the object methods
4378 The private object hash keys like C<$x->{sign}> may not be changed, but
4379 additional keys can be added, like C<$x->{_custom}>.
4383 Accessor functions are available for all existing object hash keys and should
4384 be used instead of directly accessing the internal hash keys. The reason for
4385 this is that Math::BigInt itself has a pluggable interface which permits it
4386 to support different storage methods.
4390 More complex sub-classes may have to replicate more of the logic internal of
4391 Math::BigInt if they need to change more basic behaviors. A subclass that
4392 needs to merely change the output only needs to overload C<bstr()>.
4394 All other object methods and overloaded functions can be directly inherited
4395 from the parent class.
4397 At the very minimum, any subclass will need to provide it's own C<new()> and can
4398 store additional hash keys in the object. There are also some package globals
4399 that must be defined, e.g.:
4403 $precision = -2; # round to 2 decimal places
4404 $round_mode = 'even';
4407 Additionally, you might want to provide the following two globals to allow
4408 auto-upgrading and auto-downgrading to work correctly:
4413 This allows Math::BigInt to correctly retrieve package globals from the
4414 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
4415 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
4421 in your subclass to automatically inherit the overloading from the parent. If
4422 you like, you can change part of the overloading, look at Math::String for an
4427 When used like this:
4429 use Math::BigInt upgrade => 'Foo::Bar';
4431 certain operations will 'upgrade' their calculation and thus the result to
4432 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
4434 use Math::BigInt upgrade => 'Math::BigFloat';
4436 As a shortcut, you can use the module C<bignum>:
4440 Also good for oneliners:
4442 perl -Mbignum -le 'print 2 ** 255'
4444 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
4445 as well es preserve accuracy (as in sqrt(3)).
4447 Beware: This feature is not fully implemented yet.
4451 The following methods upgrade themselves unconditionally; that is if upgrade
4452 is in effect, they will always hand up their work:
4464 Beware: This list is not complete.
4466 All other methods upgrade themselves only when one (or all) of their
4467 arguments are of the class mentioned in $upgrade (This might change in later
4468 versions to a more sophisticated scheme):
4474 =item broot() does not work
4476 The broot() function in BigInt may only work for small values. This will be
4477 fixed in a later version.
4479 =item Out of Memory!
4481 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
4482 C<eval()> in your code will crash with "Out of memory". This is probably an
4483 overload/exporter bug. You can workaround by not having C<eval()>
4484 and ':constant' at the same time or upgrade your Perl to a newer version.
4486 =item Fails to load Calc on Perl prior 5.6.0
4488 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
4489 will fall back to eval { require ... } when loading the math lib on Perls
4490 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
4491 filesystems using a different seperator.
4497 Some things might not work as you expect them. Below is documented what is
4498 known to be troublesome:
4502 =item bstr(), bsstr() and 'cmp'
4504 Both C<bstr()> and C<bsstr()> as well as automated stringify via overload now
4505 drop the leading '+'. The old code would return '+3', the new returns '3'.
4506 This is to be consistent with Perl and to make C<cmp> (especially with
4507 overloading) to work as you expect. It also solves problems with C<Test.pm>,
4508 because it's C<ok()> uses 'eq' internally.
4510 Mark Biggar said, when asked about to drop the '+' altogether, or make only
4513 I agree (with the first alternative), don't add the '+' on positive
4514 numbers. It's not as important anymore with the new internal
4515 form for numbers. It made doing things like abs and neg easier,
4516 but those have to be done differently now anyway.
4518 So, the following examples will now work all as expected:
4521 BEGIN { plan tests => 1 }
4524 my $x = new Math::BigInt 3*3;
4525 my $y = new Math::BigInt 3*3;
4528 print "$x eq 9" if $x eq $y;
4529 print "$x eq 9" if $x eq '9';
4530 print "$x eq 9" if $x eq 3*3;
4532 Additionally, the following still works:
4534 print "$x == 9" if $x == $y;
4535 print "$x == 9" if $x == 9;
4536 print "$x == 9" if $x == 3*3;
4538 There is now a C<bsstr()> method to get the string in scientific notation aka
4539 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
4540 for comparisation, but Perl will represent some numbers as 100 and others
4541 as 1e+308. If in doubt, convert both arguments to Math::BigInt before
4542 comparing them as strings:
4545 BEGIN { plan tests => 3 }
4548 $x = Math::BigInt->new('1e56'); $y = 1e56;
4549 ok ($x,$y); # will fail
4550 ok ($x->bsstr(),$y); # okay
4551 $y = Math::BigInt->new($y);
4554 Alternatively, simple use C<< <=> >> for comparisations, this will get it
4555 always right. There is not yet a way to get a number automatically represented
4556 as a string that matches exactly the way Perl represents it.
4560 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
4563 $x = Math::BigInt->new(123);
4564 $y = int($x); # BigInt 123
4565 $x = Math::BigFloat->new(123.45);
4566 $y = int($x); # BigInt 123
4568 In all Perl versions you can use C<as_number()> for the same effect:
4570 $x = Math::BigFloat->new(123.45);
4571 $y = $x->as_number(); # BigInt 123
4573 This also works for other subclasses, like Math::String.
4575 It is yet unlcear whether overloaded int() should return a scalar or a BigInt.
4579 The following will probably not do what you expect:
4581 $c = Math::BigInt->new(123);
4582 print $c->length(),"\n"; # prints 30
4584 It prints both the number of digits in the number and in the fraction part
4585 since print calls C<length()> in list context. Use something like:
4587 print scalar $c->length(),"\n"; # prints 3
4591 The following will probably not do what you expect:
4593 print $c->bdiv(10000),"\n";
4595 It prints both quotient and remainder since print calls C<bdiv()> in list
4596 context. Also, C<bdiv()> will modify $c, so be carefull. You probably want
4599 print $c / 10000,"\n";
4600 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
4604 The quotient is always the greatest integer less than or equal to the
4605 real-valued quotient of the two operands, and the remainder (when it is
4606 nonzero) always has the same sign as the second operand; so, for
4616 As a consequence, the behavior of the operator % agrees with the
4617 behavior of Perl's built-in % operator (as documented in the perlop
4618 manpage), and the equation
4620 $x == ($x / $y) * $y + ($x % $y)
4622 holds true for any $x and $y, which justifies calling the two return
4623 values of bdiv() the quotient and remainder. The only exception to this rule
4624 are when $y == 0 and $x is negative, then the remainder will also be
4625 negative. See below under "infinity handling" for the reasoning behing this.
4627 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
4628 not change BigInt's way to do things. This is because under 'use integer' Perl
4629 will do what the underlying C thinks is right and this is different for each
4630 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
4631 the author to implement it ;)
4633 =item infinity handling
4635 Here are some examples that explain the reasons why certain results occur while
4638 The following table shows the result of the division and the remainder, so that
4639 the equation above holds true. Some "ordinary" cases are strewn in to show more
4640 clearly the reasoning:
4642 A / B = C, R so that C * B + R = A
4643 =========================================================
4644 5 / 8 = 0, 5 0 * 8 + 5 = 5
4645 0 / 8 = 0, 0 0 * 8 + 0 = 0
4646 0 / inf = 0, 0 0 * inf + 0 = 0
4647 0 /-inf = 0, 0 0 * -inf + 0 = 0
4648 5 / inf = 0, 5 0 * inf + 5 = 5
4649 5 /-inf = 0, 5 0 * -inf + 5 = 5
4650 -5/ inf = 0, -5 0 * inf + -5 = -5
4651 -5/-inf = 0, -5 0 * -inf + -5 = -5
4652 inf/ 5 = inf, 0 inf * 5 + 0 = inf
4653 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
4654 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
4655 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
4656 5/ 5 = 1, 0 1 * 5 + 0 = 5
4657 -5/ -5 = 1, 0 1 * -5 + 0 = -5
4658 inf/ inf = 1, 0 1 * inf + 0 = inf
4659 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
4660 inf/-inf = -1, 0 -1 * -inf + 0 = inf
4661 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
4662 8/ 0 = inf, 8 inf * 0 + 8 = 8
4663 inf/ 0 = inf, inf inf * 0 + inf = inf
4666 These cases below violate the "remainder has the sign of the second of the two
4667 arguments", since they wouldn't match up otherwise.
4669 A / B = C, R so that C * B + R = A
4670 ========================================================
4671 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
4672 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
4674 =item Modifying and =
4678 $x = Math::BigFloat->new(5);
4681 It will not do what you think, e.g. making a copy of $x. Instead it just makes
4682 a second reference to the B<same> object and stores it in $y. Thus anything
4683 that modifies $x (except overloaded operators) will modify $y, and vice versa.
4684 Or in other words, C<=> is only safe if you modify your BigInts only via
4685 overloaded math. As soon as you use a method call it breaks:
4688 print "$x, $y\n"; # prints '10, 10'
4690 If you want a true copy of $x, use:
4694 You can also chain the calls like this, this will make first a copy and then
4697 $y = $x->copy()->bmul(2);
4699 See also the documentation for overload.pm regarding C<=>.
4703 C<bpow()> (and the rounding functions) now modifies the first argument and
4704 returns it, unlike the old code which left it alone and only returned the
4705 result. This is to be consistent with C<badd()> etc. The first three will
4706 modify $x, the last one won't:
4708 print bpow($x,$i),"\n"; # modify $x
4709 print $x->bpow($i),"\n"; # ditto
4710 print $x **= $i,"\n"; # the same
4711 print $x ** $i,"\n"; # leave $x alone
4713 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4715 =item Overloading -$x
4725 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4726 needs to preserve $x since it does not know that it later will get overwritten.
4727 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4729 With Copy-On-Write, this issue would be gone, but C-o-W is not implemented
4730 since it is slower for all other things.
4732 =item Mixing different object types
4734 In Perl you will get a floating point value if you do one of the following:
4740 With overloaded math, only the first two variants will result in a BigFloat:
4745 $mbf = Math::BigFloat->new(5);
4746 $mbi2 = Math::BigInteger->new(5);
4747 $mbi = Math::BigInteger->new(2);
4749 # what actually gets called:
4750 $float = $mbf + $mbi; # $mbf->badd()
4751 $float = $mbf / $mbi; # $mbf->bdiv()
4752 $integer = $mbi + $mbf; # $mbi->badd()
4753 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4754 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4756 This is because math with overloaded operators follows the first (dominating)
4757 operand, and the operation of that is called and returns thus the result. So,
4758 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4759 the result should be a Math::BigFloat or the second operant is one.
4761 To get a Math::BigFloat you either need to call the operation manually,
4762 make sure the operands are already of the proper type or casted to that type
4763 via Math::BigFloat->new():
4765 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4767 Beware of simple "casting" the entire expression, this would only convert
4768 the already computed result:
4770 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4772 Beware also of the order of more complicated expressions like:
4774 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4775 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4777 If in doubt, break the expression into simpler terms, or cast all operands
4778 to the desired resulting type.
4780 Scalar values are a bit different, since:
4785 will both result in the proper type due to the way the overloaded math works.
4787 This section also applies to other overloaded math packages, like Math::String.
4789 One solution to you problem might be autoupgrading|upgrading. See the
4790 pragmas L<bignum>, L<bigint> and L<bigrat> for an easy way to do this.
4794 C<bsqrt()> works only good if the result is a big integer, e.g. the square
4795 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4796 mode. The reason is that the result is always truncated to an integer.
4798 If you want a better approximation of the square root, then use:
4800 $x = Math::BigFloat->new(12);
4801 Math::BigFloat->precision(0);
4802 Math::BigFloat->round_mode('even');
4803 print $x->copy->bsqrt(),"\n"; # 4
4805 Math::BigFloat->precision(2);
4806 print $x->bsqrt(),"\n"; # 3.46
4807 print $x->bsqrt(3),"\n"; # 3.464
4811 For negative numbers in base see also L<brsft|brsft>.
4817 This program is free software; you may redistribute it and/or modify it under
4818 the same terms as Perl itself.
4822 L<Math::BigFloat>, L<Math::BigRat> and L<Math::Big> as well as
4823 L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4825 The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest
4826 because they solve the autoupgrading/downgrading issue, at least partly.
4829 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4830 more documentation including a full version history, testcases, empty
4831 subclass files and benchmarks.
4835 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4836 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001, 2002
4837 and still at it in 2003.
4839 Many people contributed in one or more ways to the final beast, see the file
4840 CREDITS for an (uncomplete) list. If you miss your name, please drop me a