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 $class = "Math::BigInt";
23 @ISA = qw( Exporter );
24 @EXPORT_OK = qw( objectify bgcd blcm);
25 # _trap_inf and _trap_nan are internal and should never be accessed from the
27 use vars qw/$round_mode $accuracy $precision $div_scale $rnd_mode
28 $upgrade $downgrade $_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]" },
69 # make cos()/sin()/exp() "work" with BigInt's or subclasses
70 'cos' => sub { cos($_[0]->numify()) },
71 'sin' => sub { sin($_[0]->numify()) },
72 'exp' => sub { exp($_[0]->numify()) },
73 'atan2' => sub { atan2($_[0]->numify(),$_[1]) },
75 'log' => sub { $_[0]->copy()->blog($_[1]); },
76 'int' => sub { $_[0]->copy(); },
77 'neg' => sub { $_[0]->copy()->bneg(); },
78 'abs' => sub { $_[0]->copy()->babs(); },
79 'sqrt' => sub { $_[0]->copy()->bsqrt(); },
80 '~' => sub { $_[0]->copy()->bnot(); },
82 # for sub it is a bit tricky to keep b: b-a => -a+b
83 '-' => sub { my $c = $_[0]->copy; $_[2] ?
84 $c->bneg()->badd($_[1]) :
86 '+' => sub { $_[0]->copy()->badd($_[1]); },
87 '*' => sub { $_[0]->copy()->bmul($_[1]); },
90 $_[2] ? ref($_[0])->new($_[1])->bdiv($_[0]) : $_[0]->copy->bdiv($_[1]);
93 $_[2] ? ref($_[0])->new($_[1])->bmod($_[0]) : $_[0]->copy->bmod($_[1]);
96 $_[2] ? ref($_[0])->new($_[1])->bpow($_[0]) : $_[0]->copy->bpow($_[1]);
99 $_[2] ? ref($_[0])->new($_[1])->blsft($_[0]) : $_[0]->copy->blsft($_[1]);
102 $_[2] ? ref($_[0])->new($_[1])->brsft($_[0]) : $_[0]->copy->brsft($_[1]);
105 $_[2] ? ref($_[0])->new($_[1])->band($_[0]) : $_[0]->copy->band($_[1]);
108 $_[2] ? ref($_[0])->new($_[1])->bior($_[0]) : $_[0]->copy->bior($_[1]);
111 $_[2] ? ref($_[0])->new($_[1])->bxor($_[0]) : $_[0]->copy->bxor($_[1]);
114 # can modify arg of ++ and --, so avoid a copy() for speed, but don't
115 # use $_[0]->bone(), it would modify $_[0] to be 1!
116 '++' => sub { $_[0]->binc() },
117 '--' => sub { $_[0]->bdec() },
119 # if overloaded, O(1) instead of O(N) and twice as fast for small numbers
121 # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/
122 # v5.6.1 dumps on this: return !$_[0]->is_zero() || undef; :-(
124 $t = 1 if !$_[0]->is_zero();
128 # the original qw() does not work with the TIESCALAR below, why?
129 # Order of arguments unsignificant
130 '""' => sub { $_[0]->bstr(); },
131 '0+' => sub { $_[0]->numify(); }
134 ##############################################################################
135 # global constants, flags and accessory
137 # these are public, but their usage is not recommended, use the accessor
140 $round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
145 $upgrade = undef; # default is no upgrade
146 $downgrade = undef; # default is no downgrade
148 # these are internally, and not to be used from the outside
150 sub MB_NEVER_ROUND () { 0x0001; }
152 $_trap_nan = 0; # are NaNs ok? set w/ config()
153 $_trap_inf = 0; # are infs ok? set w/ config()
154 my $nan = 'NaN'; # constants for easier life
156 my $CALC = 'Math::BigInt::Calc'; # module to do the low level math
158 my %CAN; # cache for $CALC->can(...)
159 my $IMPORT = 0; # was import() called yet?
160 # used to make require work
162 my $EMU_LIB = 'Math/BigInt/CalcEmu.pm'; # emulate low-level math
163 my $EMU = 'Math::BigInt::CalcEmu'; # emulate low-level math
165 ##############################################################################
166 # the old code had $rnd_mode, so we need to support it, too
169 sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; }
170 sub FETCH { return $round_mode; }
171 sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
175 # tie to enable $rnd_mode to work transparently
176 tie $rnd_mode, 'Math::BigInt';
178 # set up some handy alias names
179 *as_int = \&as_number;
180 *is_pos = \&is_positive;
181 *is_neg = \&is_negative;
184 ##############################################################################
189 # make Class->round_mode() work
191 my $class = ref($self) || $self || __PACKAGE__;
195 if ($m !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
197 require Carp; Carp::croak ("Unknown round mode '$m'");
199 return ${"${class}::round_mode"} = $m;
201 ${"${class}::round_mode"};
207 # make Class->upgrade() work
209 my $class = ref($self) || $self || __PACKAGE__;
210 # need to set new value?
214 return ${"${class}::upgrade"} = $u;
216 ${"${class}::upgrade"};
222 # make Class->downgrade() work
224 my $class = ref($self) || $self || __PACKAGE__;
225 # need to set new value?
229 return ${"${class}::downgrade"} = $u;
231 ${"${class}::downgrade"};
237 # make Class->div_scale() work
239 my $class = ref($self) || $self || __PACKAGE__;
244 require Carp; Carp::croak ('div_scale must be greater than zero');
246 ${"${class}::div_scale"} = shift;
248 ${"${class}::div_scale"};
253 # $x->accuracy($a); ref($x) $a
254 # $x->accuracy(); ref($x)
255 # Class->accuracy(); class
256 # Class->accuracy($a); class $a
259 my $class = ref($x) || $x || __PACKAGE__;
262 # need to set new value?
266 # convert objects to scalars to avoid deep recursion. If object doesn't
267 # have numify(), then hopefully it will have overloading for int() and
268 # boolean test without wandering into a deep recursion path...
269 $a = $a->numify() if ref($a) && $a->can('numify');
273 # also croak on non-numerical
277 Carp::croak ('Argument to accuracy must be greater than zero');
281 require Carp; Carp::croak ('Argument to accuracy must be an integer');
286 # $object->accuracy() or fallback to global
287 $x->bround($a) if $a; # not for undef, 0
288 $x->{_a} = $a; # set/overwrite, even if not rounded
289 delete $x->{_p}; # clear P
290 $a = ${"${class}::accuracy"} unless defined $a; # proper return value
294 ${"${class}::accuracy"} = $a; # set global A
295 ${"${class}::precision"} = undef; # clear global P
297 return $a; # shortcut
301 # $object->accuracy() or fallback to global
302 $r = $x->{_a} if ref($x);
303 # but don't return global undef, when $x's accuracy is 0!
304 $r = ${"${class}::accuracy"} if !defined $r;
310 # $x->precision($p); ref($x) $p
311 # $x->precision(); ref($x)
312 # Class->precision(); class
313 # Class->precision($p); class $p
316 my $class = ref($x) || $x || __PACKAGE__;
322 # convert objects to scalars to avoid deep recursion. If object doesn't
323 # have numify(), then hopefully it will have overloading for int() and
324 # boolean test without wandering into a deep recursion path...
325 $p = $p->numify() if ref($p) && $p->can('numify');
326 if ((defined $p) && (int($p) != $p))
328 require Carp; Carp::croak ('Argument to precision must be an integer');
332 # $object->precision() or fallback to global
333 $x->bfround($p) if $p; # not for undef, 0
334 $x->{_p} = $p; # set/overwrite, even if not rounded
335 delete $x->{_a}; # clear A
336 $p = ${"${class}::precision"} unless defined $p; # proper return value
340 ${"${class}::precision"} = $p; # set global P
341 ${"${class}::accuracy"} = undef; # clear global A
343 return $p; # shortcut
347 # $object->precision() or fallback to global
348 $r = $x->{_p} if ref($x);
349 # but don't return global undef, when $x's precision is 0!
350 $r = ${"${class}::precision"} if !defined $r;
356 # return (or set) configuration data as hash ref
357 my $class = shift || 'Math::BigInt';
362 # try to set given options as arguments from hash
365 if (ref($args) ne 'HASH')
369 # these values can be "set"
373 upgrade downgrade precision accuracy round_mode div_scale/
376 $set_args->{$key} = $args->{$key} if exists $args->{$key};
377 delete $args->{$key};
382 Carp::croak ("Illegal key(s) '",
383 join("','",keys %$args),"' passed to $class\->config()");
385 foreach my $key (keys %$set_args)
387 if ($key =~ /^trap_(inf|nan)\z/)
389 ${"${class}::_trap_$1"} = ($set_args->{"trap_$1"} ? 1 : 0);
392 # use a call instead of just setting the $variable to check argument
393 $class->$key($set_args->{$key});
397 # now return actual configuration
401 lib_version => ${"${CALC}::VERSION"},
403 trap_nan => ${"${class}::_trap_nan"},
404 trap_inf => ${"${class}::_trap_inf"},
405 version => ${"${class}::VERSION"},
408 upgrade downgrade precision accuracy round_mode div_scale
411 $cfg->{$key} = ${"${class}::$key"};
418 # select accuracy parameter based on precedence,
419 # used by bround() and bfround(), may return undef for scale (means no op)
420 my ($x,$s,$m,$scale,$mode) = @_;
421 $scale = $x->{_a} if !defined $scale;
422 $scale = $s if (!defined $scale);
423 $mode = $m if !defined $mode;
424 return ($scale,$mode);
429 # select precision parameter based on precedence,
430 # used by bround() and bfround(), may return undef for scale (means no op)
431 my ($x,$s,$m,$scale,$mode) = @_;
432 $scale = $x->{_p} if !defined $scale;
433 $scale = $s if (!defined $scale);
434 $mode = $m if !defined $mode;
435 return ($scale,$mode);
438 ##############################################################################
446 # if two arguments, the first one is the class to "swallow" subclasses
454 return unless ref($x); # only for objects
456 my $self = {}; bless $self,$c;
458 foreach my $k (keys %$x)
462 $self->{value} = $CALC->_copy($x->{value}); next;
464 if (!($r = ref($x->{$k})))
466 $self->{$k} = $x->{$k}; next;
470 $self->{$k} = \${$x->{$k}};
472 elsif ($r eq 'ARRAY')
474 $self->{$k} = [ @{$x->{$k}} ];
478 # only one level deep!
479 foreach my $h (keys %{$x->{$k}})
481 $self->{$k}->{$h} = $x->{$k}->{$h};
487 if ($xk->can('copy'))
489 $self->{$k} = $xk->copy();
493 $self->{$k} = $xk->new($xk);
502 # create a new BigInt object from a string or another BigInt object.
503 # see hash keys documented at top
505 # the argument could be an object, so avoid ||, && etc on it, this would
506 # cause costly overloaded code to be called. The only allowed ops are
509 my ($class,$wanted,$a,$p,$r) = @_;
511 # avoid numify-calls by not using || on $wanted!
512 return $class->bzero($a,$p) if !defined $wanted; # default to 0
513 return $class->copy($wanted,$a,$p,$r)
514 if ref($wanted) && $wanted->isa($class); # MBI or subclass
516 $class->import() if $IMPORT == 0; # make require work
518 my $self = bless {}, $class;
520 # shortcut for "normal" numbers
521 if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*\z/))
523 $self->{sign} = $1 || '+';
525 if ($wanted =~ /^[+-]/)
527 # remove sign without touching wanted to make it work with constants
528 my $t = $wanted; $t =~ s/^[+-]//; $ref = \$t;
530 # force to string version (otherwise Pari is unhappy about overflowed
531 # constants, for instance)
532 # not good, BigInt shouldn't need to know about alternative libs:
533 # $ref = \"$$ref" if $CALC eq 'Math::BigInt::Pari';
534 $self->{value} = $CALC->_new($ref);
536 if ( (defined $a) || (defined $p)
537 || (defined ${"${class}::precision"})
538 || (defined ${"${class}::accuracy"})
541 $self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p);
546 # handle '+inf', '-inf' first
547 if ($wanted =~ /^[+-]?inf$/)
549 $self->{value} = $CALC->_zero();
550 $self->{sign} = $wanted; $self->{sign} = '+inf' if $self->{sign} eq 'inf';
553 # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign
554 my ($mis,$miv,$mfv,$es,$ev) = _split(\$wanted);
559 require Carp; Carp::croak("$wanted is not a number in $class");
561 $self->{value} = $CALC->_zero();
562 $self->{sign} = $nan;
567 # _from_hex or _from_bin
568 $self->{value} = $mis->{value};
569 $self->{sign} = $mis->{sign};
570 return $self; # throw away $mis
572 # make integer from mantissa by adjusting exp, then convert to bigint
573 $self->{sign} = $$mis; # store sign
574 $self->{value} = $CALC->_zero(); # for all the NaN cases
575 my $e = int("$$es$$ev"); # exponent (avoid recursion)
578 my $diff = $e - CORE::length($$mfv);
579 if ($diff < 0) # Not integer
583 require Carp; Carp::croak("$wanted not an integer in $class");
586 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
587 $self->{sign} = $nan;
591 # adjust fraction and add it to value
592 #print "diff > 0 $$miv\n";
593 $$miv = $$miv . ($$mfv . '0' x $diff);
598 if ($$mfv ne '') # e <= 0
600 # fraction and negative/zero E => NOI
603 require Carp; Carp::croak("$wanted not an integer in $class");
605 #print "NOI 2 \$\$mfv '$$mfv'\n";
606 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
607 $self->{sign} = $nan;
611 # xE-y, and empty mfv
614 if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
618 require Carp; Carp::croak("$wanted not an integer in $class");
621 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
622 $self->{sign} = $nan;
626 $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
627 $self->{value} = $CALC->_new($miv) if $self->{sign} =~ /^[+-]$/;
628 # if any of the globals is set, use them to round and store them inside $self
629 # do not round for new($x,undef,undef) since that is used by MBF to signal
631 $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p;
637 # create a bigint 'NaN', if given a BigInt, set it to 'NaN'
639 $self = $class if !defined $self;
642 my $c = $self; $self = {}; bless $self, $c;
645 if (${"${class}::_trap_nan"})
648 Carp::croak ("Tried to set $self to NaN in $class\::bnan()");
650 $self->import() if $IMPORT == 0; # make require work
651 return if $self->modify('bnan');
652 if ($self->can('_bnan'))
654 # use subclass to initialize
659 # otherwise do our own thing
660 $self->{value} = $CALC->_zero();
662 $self->{sign} = $nan;
663 delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly
669 # create a bigint '+-inf', if given a BigInt, set it to '+-inf'
670 # the sign is either '+', or if given, used from there
672 my $sign = shift; $sign = '+' if !defined $sign || $sign !~ /^-(inf)?$/;
673 $self = $class if !defined $self;
676 my $c = $self; $self = {}; bless $self, $c;
679 if (${"${class}::_trap_inf"})
682 Carp::croak ("Tried to set $self to +-inf in $class\::binfn()");
684 $self->import() if $IMPORT == 0; # make require work
685 return if $self->modify('binf');
686 if ($self->can('_binf'))
688 # use subclass to initialize
693 # otherwise do our own thing
694 $self->{value} = $CALC->_zero();
696 $sign = $sign . 'inf' if $sign !~ /inf$/; # - => -inf
697 $self->{sign} = $sign;
698 ($self->{_a},$self->{_p}) = @_; # take over requested rounding
704 # create a bigint '+0', if given a BigInt, set it to 0
706 $self = $class if !defined $self;
710 my $c = $self; $self = {}; bless $self, $c;
712 $self->import() if $IMPORT == 0; # make require work
713 return if $self->modify('bzero');
715 if ($self->can('_bzero'))
717 # use subclass to initialize
722 # otherwise do our own thing
723 $self->{value} = $CALC->_zero();
730 # call like: $x->bzero($a,$p,$r,$y);
731 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
736 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
738 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
746 # create a bigint '+1' (or -1 if given sign '-'),
747 # if given a BigInt, set it to +1 or -1, respecively
749 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
750 $self = $class if !defined $self;
754 my $c = $self; $self = {}; bless $self, $c;
756 $self->import() if $IMPORT == 0; # make require work
757 return if $self->modify('bone');
759 if ($self->can('_bone'))
761 # use subclass to initialize
766 # otherwise do our own thing
767 $self->{value} = $CALC->_one();
769 $self->{sign} = $sign;
774 # call like: $x->bone($sign,$a,$p,$r,$y);
775 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
779 # call like: $x->bone($sign,$a,$p,$r);
781 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
783 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
789 ##############################################################################
790 # string conversation
794 # (ref to BFLOAT or num_str ) return num_str
795 # Convert number from internal format to scientific string format.
796 # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
797 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
798 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
800 if ($x->{sign} !~ /^[+-]$/)
802 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
805 my ($m,$e) = $x->parts();
806 #$m->bstr() . 'e+' . $e->bstr(); # e can only be positive in BigInt
807 # 'e+' because E can only be positive in BigInt
808 $m->bstr() . 'e+' . ${$CALC->_str($e->{value})};
813 # make a string from bigint object
814 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
815 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
817 if ($x->{sign} !~ /^[+-]$/)
819 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
822 my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
823 $es.${$CALC->_str($x->{value})};
828 # Make a "normal" scalar from a BigInt object
829 my $x = shift; $x = $class->new($x) unless ref $x;
831 return $x->bstr() if $x->{sign} !~ /^[+-]$/;
832 my $num = $CALC->_num($x->{value});
833 return -$num if $x->{sign} eq '-';
837 ##############################################################################
838 # public stuff (usually prefixed with "b")
842 # return the sign of the number: +/-/-inf/+inf/NaN
843 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
848 sub _find_round_parameters
850 # After any operation or when calling round(), the result is rounded by
851 # regarding the A & P from arguments, local parameters, or globals.
853 # !!!!!!! If you change this, remember to change round(), too! !!!!!!!!!!
855 # This procedure finds the round parameters, but it is for speed reasons
856 # duplicated in round. Otherwise, it is tested by the testsuite and used
859 # returns ($self) or ($self,$a,$p,$r) - sets $self to NaN of both A and P
860 # were requested/defined (locally or globally or both)
862 my ($self,$a,$p,$r,@args) = @_;
863 # $a accuracy, if given by caller
864 # $p precision, if given by caller
865 # $r round_mode, if given by caller
866 # @args all 'other' arguments (0 for unary, 1 for binary ops)
868 # leave bigfloat parts alone
869 return ($self) if exists $self->{_f} && ($self->{_f} & MB_NEVER_ROUND) != 0;
871 my $c = ref($self); # find out class of argument(s)
874 # now pick $a or $p, but only if we have got "arguments"
877 foreach ($self,@args)
879 # take the defined one, or if both defined, the one that is smaller
880 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
885 # even if $a is defined, take $p, to signal error for both defined
886 foreach ($self,@args)
888 # take the defined one, or if both defined, the one that is bigger
890 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
893 # if still none defined, use globals (#2)
894 $a = ${"$c\::accuracy"} unless defined $a;
895 $p = ${"$c\::precision"} unless defined $p;
897 # A == 0 is useless, so undef it to signal no rounding
898 $a = undef if defined $a && $a == 0;
901 return ($self) unless defined $a || defined $p; # early out
903 # set A and set P is an fatal error
904 return ($self->bnan()) if defined $a && defined $p; # error
906 $r = ${"$c\::round_mode"} unless defined $r;
907 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
909 require Carp; Carp::croak ("Unknown round mode '$r'");
917 # Round $self according to given parameters, or given second argument's
918 # parameters or global defaults
920 # for speed reasons, _find_round_parameters is embeded here:
922 my ($self,$a,$p,$r,@args) = @_;
923 # $a accuracy, if given by caller
924 # $p precision, if given by caller
925 # $r round_mode, if given by caller
926 # @args all 'other' arguments (0 for unary, 1 for binary ops)
928 # leave bigfloat parts alone
929 return ($self) if exists $self->{_f} && ($self->{_f} & MB_NEVER_ROUND) != 0;
931 my $c = ref($self); # find out class of argument(s)
934 # now pick $a or $p, but only if we have got "arguments"
937 foreach ($self,@args)
939 # take the defined one, or if both defined, the one that is smaller
940 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
945 # even if $a is defined, take $p, to signal error for both defined
946 foreach ($self,@args)
948 # take the defined one, or if both defined, the one that is bigger
950 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
953 # if still none defined, use globals (#2)
954 $a = ${"$c\::accuracy"} unless defined $a;
955 $p = ${"$c\::precision"} unless defined $p;
957 # A == 0 is useless, so undef it to signal no rounding
958 $a = undef if defined $a && $a == 0;
961 return $self unless defined $a || defined $p; # early out
963 # set A and set P is an fatal error
964 return $self->bnan() if defined $a && defined $p;
966 $r = ${"$c\::round_mode"} unless defined $r;
967 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
969 require Carp; Carp::croak ("Unknown round mode '$r'");
972 # now round, by calling either fround or ffround:
975 $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a;
977 else # both can't be undefined due to early out
979 $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p;
981 $self->bnorm(); # after round, normalize
986 # (numstr or BINT) return BINT
987 # Normalize number -- no-op here
988 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
994 # (BINT or num_str) return BINT
995 # make number absolute, or return absolute BINT from string
996 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
998 return $x if $x->modify('babs');
999 # post-normalized abs for internal use (does nothing for NaN)
1000 $x->{sign} =~ s/^-/+/;
1006 # (BINT or num_str) return BINT
1007 # negate number or make a negated number from string
1008 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1010 return $x if $x->modify('bneg');
1012 # for +0 dont negate (to have always normalized)
1013 $x->{sign} =~ tr/+-/-+/ if !$x->is_zero(); # does nothing for NaN
1019 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
1020 # (BINT or num_str, BINT or num_str) return cond_code
1023 my ($self,$x,$y) = (ref($_[0]),@_);
1025 # objectify is costly, so avoid it
1026 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1028 ($self,$x,$y) = objectify(2,@_);
1031 return $upgrade->bcmp($x,$y) if defined $upgrade &&
1032 ((!$x->isa($self)) || (!$y->isa($self)));
1034 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1036 # handle +-inf and NaN
1037 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1038 return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
1039 return +1 if $x->{sign} eq '+inf';
1040 return -1 if $x->{sign} eq '-inf';
1041 return -1 if $y->{sign} eq '+inf';
1044 # check sign for speed first
1045 return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
1046 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
1048 # have same sign, so compare absolute values. Don't make tests for zero here
1049 # because it's actually slower than testin in Calc (especially w/ Pari et al)
1051 # post-normalized compare for internal use (honors signs)
1052 if ($x->{sign} eq '+')
1054 # $x and $y both > 0
1055 return $CALC->_acmp($x->{value},$y->{value});
1059 $CALC->_acmp($y->{value},$x->{value}); # swaped acmp (lib returns 0,1,-1)
1064 # Compares 2 values, ignoring their signs.
1065 # Returns one of undef, <0, =0, >0. (suitable for sort)
1066 # (BINT, BINT) return cond_code
1069 my ($self,$x,$y) = (ref($_[0]),@_);
1070 # objectify is costly, so avoid it
1071 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1073 ($self,$x,$y) = objectify(2,@_);
1076 return $upgrade->bacmp($x,$y) if defined $upgrade &&
1077 ((!$x->isa($self)) || (!$y->isa($self)));
1079 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1081 # handle +-inf and NaN
1082 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1083 return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
1084 return 1 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} !~ /^[+-]inf$/;
1087 $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
1092 # add second arg (BINT or string) to first (BINT) (modifies first)
1093 # return result as BINT
1096 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1097 # objectify is costly, so avoid it
1098 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1100 ($self,$x,$y,@r) = objectify(2,@_);
1103 return $x if $x->modify('badd');
1104 return $upgrade->badd($upgrade->new($x),$upgrade->new($y),@r) if defined $upgrade &&
1105 ((!$x->isa($self)) || (!$y->isa($self)));
1107 $r[3] = $y; # no push!
1108 # inf and NaN handling
1109 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1112 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1114 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1116 # +inf++inf or -inf+-inf => same, rest is NaN
1117 return $x if $x->{sign} eq $y->{sign};
1120 # +-inf + something => +inf
1121 # something +-inf => +-inf
1122 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
1126 my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
1130 $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
1134 my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
1137 $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
1142 # speedup, if equal, set result to 0
1143 $x->{value} = $CALC->_zero();
1148 $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
1151 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1157 # (BINT or num_str, BINT or num_str) return BINT
1158 # subtract second arg from first, modify first
1161 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1162 # objectify is costly, so avoid it
1163 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1165 ($self,$x,$y,@r) = objectify(2,@_);
1168 return $x if $x->modify('bsub');
1170 # upgrade done by badd():
1171 # return $upgrade->badd($x,$y,@r) if defined $upgrade &&
1172 # ((!$x->isa($self)) || (!$y->isa($self)));
1176 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1180 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
1181 $x->badd($y,@r); # badd does not leave internal zeros
1182 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
1183 $x; # already rounded by badd() or no round necc.
1188 # increment arg by one
1189 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1190 return $x if $x->modify('binc');
1192 if ($x->{sign} eq '+')
1194 $x->{value} = $CALC->_inc($x->{value});
1195 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1198 elsif ($x->{sign} eq '-')
1200 $x->{value} = $CALC->_dec($x->{value});
1201 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1202 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1205 # inf, nan handling etc
1206 $x->badd($self->bone(),$a,$p,$r); # badd does round
1211 # decrement arg by one
1212 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1213 return $x if $x->modify('bdec');
1215 if ($x->{sign} eq '-')
1218 $x->{value} = $CALC->_inc($x->{value});
1222 return $x->badd($self->bone('-'),@r) unless $x->{sign} eq '+'; # inf/NaN
1224 if ($CALC->_is_zero($x->{value}))
1227 $x->{value} = $CALC->_one(); $x->{sign} = '-'; # 0 => -1
1232 $x->{value} = $CALC->_dec($x->{value});
1235 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1241 # calculate $x = $a ** $base + $b and return $a (e.g. the log() to base
1245 my ($self,$x,$base,@r) = (ref($_[0]),@_);
1246 # objectify is costly, so avoid it
1247 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1249 ($self,$x,$base,@r) = objectify(2,$class,@_);
1252 return $x if $x->modify('blog');
1254 # inf, -inf, NaN, <0 => NaN
1256 if $x->{sign} ne '+' || $base->{sign} ne '+';
1258 return $upgrade->blog($upgrade->new($x),$base,@r) if
1259 defined $upgrade && (ref($x) ne $upgrade || ref($base) ne $upgrade);
1263 my ($rc,$exact) = $CALC->_log_int($x->{value},$base->{value});
1264 return $x->bnan() unless defined $rc;
1266 return $x->round(@r);
1270 __emu_blog($self,$x,$base,@r);
1275 # (BINT or num_str, BINT or num_str) return BINT
1276 # does not modify arguments, but returns new object
1277 # Lowest Common Multiplicator
1279 my $y = shift; my ($x);
1286 $x = $class->new($y);
1288 while (@_) { $x = __lcm($x,shift); }
1294 # (BINT or num_str, BINT or num_str) return BINT
1295 # does not modify arguments, but returns new object
1296 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1299 $y = __PACKAGE__->new($y) if !ref($y);
1301 my $x = $y->copy(); # keep arguments
1306 $y = shift; $y = $self->new($y) if !ref($y);
1307 next if $y->is_zero();
1308 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1309 $x->{value} = $CALC->_gcd($x->{value},$y->{value}); last if $x->is_one();
1316 $y = shift; $y = $self->new($y) if !ref($y);
1317 $x = __gcd($x,$y->copy()); last if $x->is_one(); # _gcd handles NaN
1325 # (num_str or BINT) return BINT
1326 # represent ~x as twos-complement number
1327 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1328 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1330 return $x if $x->modify('bnot');
1331 $x->binc()->bneg(); # binc already does round
1334 ##############################################################################
1335 # is_foo test routines
1336 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1340 # return true if arg (BINT or num_str) is zero (array '+', '0')
1341 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1343 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1344 $CALC->_is_zero($x->{value});
1349 # return true if arg (BINT or num_str) is NaN
1350 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1352 $x->{sign} eq $nan ? 1 : 0;
1357 # return true if arg (BINT or num_str) is +-inf
1358 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1362 $sign = '[+-]inf' if $sign eq ''; # +- doesn't matter, only that's inf
1363 $sign = "[$1]inf" if $sign =~ /^([+-])(inf)?$/; # extract '+' or '-'
1364 return $x->{sign} =~ /^$sign$/ ? 1 : 0;
1366 $x->{sign} =~ /^[+-]inf$/ ? 1 : 0; # only +-inf is infinity
1371 # return true if arg (BINT or num_str) is +1, or -1 if sign is given
1372 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1374 $sign = '+' if !defined $sign || $sign ne '-';
1376 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1377 $CALC->_is_one($x->{value});
1382 # return true when arg (BINT or num_str) is odd, false for even
1383 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1385 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1386 $CALC->_is_odd($x->{value});
1391 # return true when arg (BINT or num_str) is even, false for odd
1392 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1394 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1395 $CALC->_is_even($x->{value});
1400 # return true when arg (BINT or num_str) is positive (>= 0)
1401 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1403 $x->{sign} =~ /^\+/ ? 1 : 0; # +inf is also positive, but NaN not
1408 # return true when arg (BINT or num_str) is negative (< 0)
1409 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1411 $x->{sign} =~ /^-/ ? 1 : 0; # -inf is also negative, but NaN not
1416 # return true when arg (BINT or num_str) is an integer
1417 # always true for BigInt, but different for BigFloats
1418 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1420 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1423 ###############################################################################
1427 # multiply two numbers -- stolen from Knuth Vol 2 pg 233
1428 # (BINT or num_str, BINT or num_str) return BINT
1431 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1432 # objectify is costly, so avoid it
1433 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1435 ($self,$x,$y,@r) = objectify(2,@_);
1438 return $x if $x->modify('bmul');
1440 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1443 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1445 return $x->bnan() if $x->is_zero() || $y->is_zero();
1446 # result will always be +-inf:
1447 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1448 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1449 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1450 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1451 return $x->binf('-');
1454 return $upgrade->bmul($x,$y,@r)
1455 if defined $upgrade && $y->isa($upgrade);
1457 $r[3] = $y; # no push here
1459 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1461 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1462 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1464 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1470 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1471 my ($self,$x,$y) = @_;
1473 # NaN if x == NaN or y == NaN or x==y==0
1474 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1475 if (($x->is_nan() || $y->is_nan()) ||
1476 ($x->is_zero() && $y->is_zero()));
1478 # +-inf / +-inf == NaN, reminder also NaN
1479 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1481 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1483 # x / +-inf => 0, remainder x (works even if x == 0)
1484 if ($y->{sign} =~ /^[+-]inf$/)
1486 my $t = $x->copy(); # bzero clobbers up $x
1487 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1490 # 5 / 0 => +inf, -6 / 0 => -inf
1491 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1492 # exception: -8 / 0 has remainder -8, not 8
1493 # exception: -inf / 0 has remainder -inf, not inf
1496 # +-inf / 0 => special case for -inf
1497 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1498 if (!$x->is_zero() && !$x->is_inf())
1500 my $t = $x->copy(); # binf clobbers up $x
1502 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1506 # last case: +-inf / ordinary number
1508 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1510 return wantarray ? ($x,$self->bzero()) : $x;
1515 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1516 # (BINT,BINT) (quo,rem) or BINT (only rem)
1519 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1520 # objectify is costly, so avoid it
1521 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1523 ($self,$x,$y,@r) = objectify(2,@_);
1526 return $x if $x->modify('bdiv');
1528 return $self->_div_inf($x,$y)
1529 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1531 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1532 if defined $upgrade;
1534 $r[3] = $y; # no push!
1536 # calc new sign and in case $y == +/- 1, return $x
1537 my $xsign = $x->{sign}; # keep
1538 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1542 my $rem = $self->bzero();
1543 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1544 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1545 $rem->{_a} = $x->{_a};
1546 $rem->{_p} = $x->{_p};
1547 $x->round(@r) if !exists $x->{_f} || ($x->{_f} & MB_NEVER_ROUND) == 0;
1548 if (! $CALC->_is_zero($rem->{value}))
1550 $rem->{sign} = $y->{sign};
1551 $rem = $y->copy()->bsub($rem) if $xsign ne $y->{sign}; # one of them '-'
1555 $rem->{sign} = '+'; # dont leave -0
1557 $rem->round(@r) if !exists $rem->{_f} || ($rem->{_f} & MB_NEVER_ROUND) == 0;
1561 $x->{value} = $CALC->_div($x->{value},$y->{value});
1562 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1564 $x->round(@r) if !exists $x->{_f} || ($x->{_f} & MB_NEVER_ROUND) == 0;
1568 ###############################################################################
1573 # modulus (or remainder)
1574 # (BINT or num_str, BINT or num_str) return BINT
1577 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1578 # objectify is costly, so avoid it
1579 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1581 ($self,$x,$y,@r) = objectify(2,@_);
1584 return $x if $x->modify('bmod');
1585 $r[3] = $y; # no push!
1586 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1588 my ($d,$r) = $self->_div_inf($x,$y);
1589 $x->{sign} = $r->{sign};
1590 $x->{value} = $r->{value};
1591 return $x->round(@r);
1596 # calc new sign and in case $y == +/- 1, return $x
1597 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1598 if (!$CALC->_is_zero($x->{value}))
1600 my $xsign = $x->{sign};
1601 $x->{sign} = $y->{sign};
1602 if ($xsign ne $y->{sign})
1604 my $t = $CALC->_copy($x->{value}); # copy $x
1605 $x->{value} = $CALC->_sub($y->{value},$t,1); # $y-$x
1610 $x->{sign} = '+'; # dont leave -0
1612 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1615 # disable upgrade temporarily, otherwise endless loop due to bdiv()
1616 local $upgrade = undef;
1617 my ($t,$rem) = $self->bdiv($x->copy(),$y,@r); # slow way (also rounds)
1619 foreach (qw/value sign _a _p/)
1621 $x->{$_} = $rem->{$_};
1628 # Modular inverse. given a number which is (hopefully) relatively
1629 # prime to the modulus, calculate its inverse using Euclid's
1630 # alogrithm. If the number is not relatively prime to the modulus
1631 # (i.e. their gcd is not one) then NaN is returned.
1634 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1635 # objectify is costly, so avoid it
1636 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1638 ($self,$x,$y,@r) = objectify(2,@_);
1641 return $x if $x->modify('bmodinv');
1644 if ($y->{sign} ne '+' # -, NaN, +inf, -inf
1645 || $x->is_zero() # or num == 0
1646 || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1649 # put least residue into $x if $x was negative, and thus make it positive
1650 $x->bmod($y) if $x->{sign} eq '-';
1655 ($x->{value},$sign) = $CALC->_modinv($x->{value},$y->{value});
1656 return $x->bnan() if !defined $x->{value}; # in case no GCD found
1657 return $x if !defined $sign; # already real result
1658 $x->{sign} = $sign; # flip/flop see below
1659 $x->bmod($y); # calc real result
1664 __emu_bmodinv($self,$x,$y,@r);
1669 # takes a very large number to a very large exponent in a given very
1670 # large modulus, quickly, thanks to binary exponentation. supports
1671 # negative exponents.
1672 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1674 return $num if $num->modify('bmodpow');
1676 # check modulus for valid values
1677 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1678 || $mod->is_zero());
1680 # check exponent for valid values
1681 if ($exp->{sign} =~ /\w/)
1683 # i.e., if it's NaN, +inf, or -inf...
1684 return $num->bnan();
1687 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
1689 # check num for valid values (also NaN if there was no inverse but $exp < 0)
1690 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1694 # $mod is positive, sign on $exp is ignored, result also positive
1695 $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
1700 __emu_bmodpow($self,$num,$exp,$mod,@r);
1703 ###############################################################################
1707 # (BINT or num_str, BINT or num_str) return BINT
1708 # compute factorial number from $x, modify $x in place
1709 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1711 return $x if $x->modify('bfac');
1713 return $x if $x->{sign} eq '+inf'; # inf => inf
1714 return $x->bnan() if $x->{sign} ne '+'; # NaN, <0 etc => NaN
1718 $x->{value} = $CALC->_fac($x->{value});
1719 return $x->round(@r);
1723 __emu_bfac($self,$x,@r);
1728 # (BINT or num_str, BINT or num_str) return BINT
1729 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1730 # modifies first argument
1733 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1734 # objectify is costly, so avoid it
1735 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1737 ($self,$x,$y,@r) = objectify(2,@_);
1740 return $x if $x->modify('bpow');
1742 return $upgrade->bpow($upgrade->new($x),$y,@r)
1743 if defined $upgrade && !$y->isa($self);
1745 $r[3] = $y; # no push!
1746 return $x if $x->{sign} =~ /^[+-]inf$/; # -inf/+inf ** x
1747 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1749 # cases 0 ** Y, X ** 0, X ** 1, 1 ** Y are handled by Calc or Emu
1751 if ($x->{sign} eq '-' && $CALC->_is_one($x->{value}))
1753 # if $x == -1 and odd/even y => +1/-1
1754 return $y->is_odd() ? $x->round(@r) : $x->babs()->round(@r);
1755 # my Casio FX-5500L has a bug here: -1 ** 2 is -1, but -1 * -1 is 1;
1757 # 1 ** -y => 1 / (1 ** |y|)
1758 # so do test for negative $y after above's clause
1759 return $x->bnan() if $y->{sign} eq '-' && !$x->is_one();
1763 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1764 $x->{sign} = '+' if $CALC->_is_zero($y->{value});
1765 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1770 __emu_bpow($self,$x,$y,@r);
1775 # (BINT or num_str, BINT or num_str) return BINT
1776 # compute x << y, base n, y >= 0
1779 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1780 # objectify is costly, so avoid it
1781 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1783 ($self,$x,$y,$n,@r) = objectify(2,@_);
1786 return $x if $x->modify('blsft');
1787 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1788 return $x->round(@r) if $y->is_zero();
1790 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1792 my $t; $t = $CALC->_lsft($x->{value},$y->{value},$n) if $CAN{lsft};
1795 $x->{value} = $t; return $x->round(@r);
1798 $x->bmul( $self->bpow($n, $y, @r), @r );
1803 # (BINT or num_str, BINT or num_str) return BINT
1804 # compute x >> y, base n, y >= 0
1807 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1808 # objectify is costly, so avoid it
1809 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1811 ($self,$x,$y,$n,@r) = objectify(2,@_);
1814 return $x if $x->modify('brsft');
1815 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1816 return $x->round(@r) if $y->is_zero();
1817 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
1819 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1821 # this only works for negative numbers when shifting in base 2
1822 if (($x->{sign} eq '-') && ($n == 2))
1824 return $x->round(@r) if $x->is_one('-'); # -1 => -1
1827 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1828 # but perhaps there is a better emulation for two's complement shift...
1829 # if $y != 1, we must simulate it by doing:
1830 # convert to bin, flip all bits, shift, and be done
1831 $x->binc(); # -3 => -2
1832 my $bin = $x->as_bin();
1833 $bin =~ s/^-0b//; # strip '-0b' prefix
1834 $bin =~ tr/10/01/; # flip bits
1836 if (CORE::length($bin) <= $y)
1838 $bin = '0'; # shifting to far right creates -1
1839 # 0, because later increment makes
1840 # that 1, attached '-' makes it '-1'
1841 # because -1 >> x == -1 !
1845 $bin =~ s/.{$y}$//; # cut off at the right side
1846 $bin = '1' . $bin; # extend left side by one dummy '1'
1847 $bin =~ tr/10/01/; # flip bits back
1849 my $res = $self->new('0b'.$bin); # add prefix and convert back
1850 $res->binc(); # remember to increment
1851 $x->{value} = $res->{value}; # take over value
1852 return $x->round(@r); # we are done now, magic, isn't?
1854 # x < 0, n == 2, y == 1
1855 $x->bdec(); # n == 2, but $y == 1: this fixes it
1858 my $t; $t = $CALC->_rsft($x->{value},$y->{value},$n) if $CAN{rsft};
1862 return $x->round(@r);
1865 $x->bdiv($self->bpow($n,$y, @r), @r);
1871 #(BINT or num_str, BINT or num_str) return BINT
1875 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1876 # objectify is costly, so avoid it
1877 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1879 ($self,$x,$y,@r) = objectify(2,@_);
1882 return $x if $x->modify('band');
1884 $r[3] = $y; # no push!
1886 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1888 my $sx = $x->{sign} eq '+' ? 1 : -1;
1889 my $sy = $y->{sign} eq '+' ? 1 : -1;
1891 if ($CAN{and} && $sx == 1 && $sy == 1)
1893 $x->{value} = $CALC->_and($x->{value},$y->{value});
1894 return $x->round(@r);
1897 if ($CAN{signed_and})
1899 $x->{value} = $CALC->_signed_and($x->{value},$y->{value},$sx,$sy);
1900 return $x->round(@r);
1904 __emu_band($self,$x,$y,$sx,$sy,@r);
1909 #(BINT or num_str, BINT or num_str) return BINT
1913 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1914 # objectify is costly, so avoid it
1915 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1917 ($self,$x,$y,@r) = objectify(2,@_);
1920 return $x if $x->modify('bior');
1921 $r[3] = $y; # no push!
1923 local $Math::BigInt::upgrade = undef;
1925 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1927 my $sx = $x->{sign} eq '+' ? 1 : -1;
1928 my $sy = $y->{sign} eq '+' ? 1 : -1;
1930 # the sign of X follows the sign of X, e.g. sign of Y irrelevant for bior()
1932 # don't use lib for negative values
1933 if ($CAN{or} && $sx == 1 && $sy == 1)
1935 $x->{value} = $CALC->_or($x->{value},$y->{value});
1936 return $x->round(@r);
1939 # if lib can do negative values, let it handle this
1940 if ($CAN{signed_or})
1942 $x->{value} = $CALC->_signed_or($x->{value},$y->{value},$sx,$sy);
1943 return $x->round(@r);
1947 __emu_bior($self,$x,$y,$sx,$sy,@r);
1952 #(BINT or num_str, BINT or num_str) return BINT
1956 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1957 # objectify is costly, so avoid it
1958 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1960 ($self,$x,$y,@r) = objectify(2,@_);
1963 return $x if $x->modify('bxor');
1964 $r[3] = $y; # no push!
1966 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1968 my $sx = $x->{sign} eq '+' ? 1 : -1;
1969 my $sy = $y->{sign} eq '+' ? 1 : -1;
1971 # don't use lib for negative values
1972 if ($CAN{xor} && $sx == 1 && $sy == 1)
1974 $x->{value} = $CALC->_xor($x->{value},$y->{value});
1975 return $x->round(@r);
1978 # if lib can do negative values, let it handle this
1979 if ($CAN{signed_xor})
1981 $x->{value} = $CALC->_signed_xor($x->{value},$y->{value},$sx,$sy);
1982 return $x->round(@r);
1986 __emu_bxor($self,$x,$y,$sx,$sy,@r);
1991 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1993 my $e = $CALC->_len($x->{value});
1994 wantarray ? ($e,0) : $e;
1999 # return the nth decimal digit, negative values count backward, 0 is right
2000 my ($self,$x,$n) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2002 $n = $n->numify() if ref($n);
2003 $CALC->_digit($x->{value},$n||0);
2008 # return the amount of trailing zeros in $x (as scalar)
2010 $x = $class->new($x) unless ref $x;
2012 return 0 if $x->is_zero() || $x->is_odd() || $x->{sign} !~ /^[+-]$/;
2014 return $CALC->_zeros($x->{value}) if $CAN{zeros};
2016 # if not: since we do not know underlying internal representation:
2017 my $es = "$x"; $es =~ /([0]*)$/;
2018 return 0 if !defined $1; # no zeros
2019 CORE::length("$1"); # $1 as string, not as +0!
2024 # calculate square root of $x
2025 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2027 return $x if $x->modify('bsqrt');
2029 return $x->bnan() if $x->{sign} !~ /^\+/; # -x or -inf or NaN => NaN
2030 return $x if $x->{sign} eq '+inf'; # sqrt(+inf) == inf
2032 return $upgrade->bsqrt($x,@r) if defined $upgrade;
2036 $x->{value} = $CALC->_sqrt($x->{value});
2037 return $x->round(@r);
2041 __emu_bsqrt($self,$x,@r);
2046 # calculate $y'th root of $x
2049 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2051 $y = $self->new(2) unless defined $y;
2053 # objectify is costly, so avoid it
2054 if ((!ref($x)) || (ref($x) ne ref($y)))
2056 ($self,$x,$y,@r) = objectify(2,$self || $class,@_);
2059 return $x if $x->modify('broot');
2061 # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
2062 return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
2063 $y->{sign} !~ /^\+$/;
2065 return $x->round(@r)
2066 if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
2068 return $upgrade->new($x)->broot($upgrade->new($y),@r) if defined $upgrade;
2072 $x->{value} = $CALC->_root($x->{value},$y->{value});
2073 return $x->round(@r);
2077 __emu_broot($self,$x,$y,@r);
2082 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
2083 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2085 if ($x->{sign} !~ /^[+-]$/)
2087 my $s = $x->{sign}; $s =~ s/^[+-]//; # NaN, -inf,+inf => NaN or inf
2088 return $self->new($s);
2090 return $self->bone() if $x->is_zero();
2092 $self->new($x->_trailing_zeros());
2097 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
2098 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2100 if ($x->{sign} !~ /^[+-]$/)
2102 # for NaN, +inf, -inf: keep the sign
2103 return $self->new($x->{sign});
2105 my $m = $x->copy(); delete $m->{_p}; delete $m->{_a};
2106 # that's a bit inefficient:
2107 my $zeros = $m->_trailing_zeros();
2108 $m->brsft($zeros,10) if $zeros != 0;
2114 # return a copy of both the exponent and the mantissa
2115 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2117 ($x->mantissa(),$x->exponent());
2120 ##############################################################################
2121 # rounding functions
2125 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
2126 # $n == 0 || $n == 1 => round to integer
2127 my $x = shift; my $self = ref($x) || $x; $x = $self->new($x) unless ref $x;
2129 my ($scale,$mode) = $x->_scale_p($x->precision(),$x->round_mode(),@_);
2131 return $x if !defined $scale || $x->modify('bfround'); # no-op
2133 # no-op for BigInts if $n <= 0
2134 $x->bround( $x->length()-$scale, $mode) if $scale > 0;
2136 delete $x->{_a}; # delete to save memory
2137 $x->{_p} = $scale; # store new _p
2141 sub _scan_for_nonzero
2143 # internal, used by bround()
2144 my ($x,$pad,$xs) = @_;
2146 my $len = $x->length();
2147 return 0 if $len == 1; # '5' is trailed by invisible zeros
2148 my $follow = $pad - 1;
2149 return 0 if $follow > $len || $follow < 1;
2151 # since we do not know underlying represention of $x, use decimal string
2152 my $r = substr ("$x",-$follow);
2153 $r =~ /[^0]/ ? 1 : 0;
2158 # Exists to make life easier for switch between MBF and MBI (should we
2159 # autoload fxxx() like MBF does for bxxx()?)
2166 # accuracy: +$n preserve $n digits from left,
2167 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
2169 # and overwrite the rest with 0's, return normalized number
2170 # do not return $x->bnorm(), but $x
2172 my $x = shift; $x = $class->new($x) unless ref $x;
2173 my ($scale,$mode) = $x->_scale_a($x->accuracy(),$x->round_mode(),@_);
2174 return $x if !defined $scale; # no-op
2175 return $x if $x->modify('bround');
2177 if ($x->is_zero() || $scale == 0)
2179 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2182 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2184 # we have fewer digits than we want to scale to
2185 my $len = $x->length();
2186 # convert $scale to a scalar in case it is an object (put's a limit on the
2187 # number length, but this would already limited by memory constraints), makes
2189 $scale = $scale->numify() if ref ($scale);
2191 # scale < 0, but > -len (not >=!)
2192 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2194 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2198 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2199 my ($pad,$digit_round,$digit_after);
2200 $pad = $len - $scale;
2201 $pad = abs($scale-1) if $scale < 0;
2203 # do not use digit(), it is costly for binary => decimal
2205 my $xs = $CALC->_str($x->{value});
2208 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2209 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2210 $digit_round = '0'; $digit_round = substr($$xs,$pl,1) if $pad <= $len;
2211 $pl++; $pl ++ if $pad >= $len;
2212 $digit_after = '0'; $digit_after = substr($$xs,$pl,1) if $pad > 0;
2214 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2215 # closer at the remaining digits of the original $x, remember decision
2216 my $round_up = 1; # default round up
2218 ($mode eq 'trunc') || # trunc by round down
2219 ($digit_after =~ /[01234]/) || # round down anyway,
2221 ($digit_after eq '5') && # not 5000...0000
2222 ($x->_scan_for_nonzero($pad,$xs) == 0) &&
2224 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2225 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2226 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2227 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2228 ($mode eq 'zero') # round down if zero, sign adjusted below
2230 my $put_back = 0; # not yet modified
2232 if (($pad > 0) && ($pad <= $len))
2234 substr($$xs,-$pad,$pad) = '0' x $pad;
2239 $x->bzero(); # round to '0'
2242 if ($round_up) # what gave test above?
2245 $pad = $len, $$xs = '0' x $pad if $scale < 0; # tlr: whack 0.51=>1.0
2247 # we modify directly the string variant instead of creating a number and
2248 # adding it, since that is faster (we already have the string)
2249 my $c = 0; $pad ++; # for $pad == $len case
2250 while ($pad <= $len)
2252 $c = substr($$xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2253 substr($$xs,-$pad,1) = $c; $pad++;
2254 last if $c != 0; # no overflow => early out
2256 $$xs = '1'.$$xs if $c == 0;
2259 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back in if needed
2261 $x->{_a} = $scale if $scale >= 0;
2264 $x->{_a} = $len+$scale;
2265 $x->{_a} = 0 if $scale < -$len;
2272 # return integer less or equal then number; no-op since it's already integer
2273 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2280 # return integer greater or equal then number; no-op since it's already int
2281 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2288 # An object might be asked to return itself as bigint on certain overloaded
2289 # operations, this does exactly this, so that sub classes can simple inherit
2290 # it or override with their own integer conversion routine.
2296 # return as hex string, with prefixed 0x
2297 my $x = shift; $x = $class->new($x) if !ref($x);
2299 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2302 $s = $x->{sign} if $x->{sign} eq '-';
2305 return $s . ${$CALC->_as_hex($x->{value})};
2309 __emu_as_hex(ref($x),$x,$s);
2314 # return as binary string, with prefixed 0b
2315 my $x = shift; $x = $class->new($x) if !ref($x);
2317 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2319 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2322 return $s . ${$CALC->_as_bin($x->{value})};
2326 __emu_as_bin(ref($x),$x,$s);
2330 ##############################################################################
2331 # private stuff (internal use only)
2335 # check for strings, if yes, return objects instead
2337 # the first argument is number of args objectify() should look at it will
2338 # return $count+1 elements, the first will be a classname. This is because
2339 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2340 # useless objects beeing created and thrown away. So we cannot simple loop
2341 # over @_. If the given count is 0, all arguments will be used.
2343 # If the second arg is a ref, use it as class.
2344 # If not, try to use it as classname, unless undef, then use $class
2345 # (aka Math::BigInt). The latter shouldn't happen,though.
2348 # $x->badd(1); => ref x, scalar y
2349 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2350 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2351 # Math::BigInt::badd(1,2); => scalar x, scalar y
2352 # In the last case we check number of arguments to turn it silently into
2353 # $class,1,2. (We can not take '1' as class ;o)
2354 # badd($class,1) is not supported (it should, eventually, try to add undef)
2355 # currently it tries 'Math::BigInt' + 1, which will not work.
2357 # some shortcut for the common cases
2359 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2361 my $count = abs(shift || 0);
2363 my (@a,$k,$d); # resulting array, temp, and downgrade
2366 # okay, got object as first
2371 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2373 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2377 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2378 if (defined ${"$a[0]::downgrade"})
2380 $d = ${"$a[0]::downgrade"};
2381 ${"$a[0]::downgrade"} = undef;
2384 my $up = ${"$a[0]::upgrade"};
2385 #print "Now in objectify, my class is today $a[0], count = $count\n";
2393 $k = $a[0]->new($k);
2395 elsif (!defined $up && ref($k) ne $a[0])
2397 # foreign object, try to convert to integer
2398 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2411 $k = $a[0]->new($k);
2413 elsif (!defined $up && ref($k) ne $a[0])
2415 # foreign object, try to convert to integer
2416 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2420 push @a,@_; # return other params, too
2424 require Carp; Carp::croak ("$class objectify needs list context");
2426 ${"$a[0]::downgrade"} = $d;
2434 $IMPORT++; # remember we did import()
2435 my @a; my $l = scalar @_;
2436 for ( my $i = 0; $i < $l ; $i++ )
2438 if ($_[$i] eq ':constant')
2440 # this causes overlord er load to step in
2442 integer => sub { $self->new(shift) },
2443 binary => sub { $self->new(shift) };
2445 elsif ($_[$i] eq 'upgrade')
2447 # this causes upgrading
2448 $upgrade = $_[$i+1]; # or undef to disable
2451 elsif ($_[$i] =~ /^lib$/i)
2453 # this causes a different low lib to take care...
2454 $CALC = $_[$i+1] || '';
2462 # any non :constant stuff is handled by our parent, Exporter
2463 # even if @_ is empty, to give it a chance
2464 $self->SUPER::import(@a); # need it for subclasses
2465 $self->export_to_level(1,$self,@a); # need it for MBF
2467 # try to load core math lib
2468 my @c = split /\s*,\s*/,$CALC;
2469 push @c,'Calc'; # if all fail, try this
2470 $CALC = ''; # signal error
2471 foreach my $lib (@c)
2473 next if ($lib || '') eq '';
2474 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2478 # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
2479 # used in the same script, or eval inside import().
2480 my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
2481 my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
2483 $file = File::Spec->catfile (@parts, $file);
2484 eval { require "$file"; $lib->import( @c ); }
2488 eval "use $lib qw/@c/;";
2490 $CALC = $lib, last if $@ eq ''; # no error in loading lib?
2495 Carp::croak ("Couldn't load any math lib, not even 'Calc.pm'");
2502 # fill $CAN with the results of $CALC->can(...)
2505 for my $method (qw/gcd mod modinv modpow fac pow lsft rsft
2506 and signed_and or signed_or xor signed_xor
2507 from_hex as_hex from_bin as_bin
2508 zeros sqrt root log_int log
2511 $CAN{$method} = $CALC->can("_$method") ? 1 : 0;
2517 # convert a (ref to) big hex string to BigInt, return undef for error
2520 my $x = Math::BigInt->bzero();
2523 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2524 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2526 return $x->bnan() if $$hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2528 my $sign = '+'; $sign = '-' if ($$hs =~ /^-/);
2530 $$hs =~ s/^[+-]//; # strip sign
2531 if ($CAN{'from_hex'})
2533 $x->{value} = $CALC->_from_hex($hs);
2537 # fallback to pure perl
2538 my $mul = Math::BigInt->bone();
2539 my $x65536 = Math::BigInt->new(65536);
2540 my $len = CORE::length($$hs)-2; # minus 2 for 0x
2541 $len = int($len/4); # 4-digit parts, w/o '0x'
2542 my $val; my $i = -4;
2545 $val = substr($$hs,$i,4);
2546 $val =~ s/^[+-]?0x// if $len == 0; # for last part only because
2547 $val = hex($val); # hex does not like wrong chars
2549 $x += $mul * $val if $val != 0;
2550 $mul *= $x65536 if $len >= 0; # skip last mul
2553 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2559 # convert a (ref to) big binary string to BigInt, return undef for error
2562 my $x = Math::BigInt->bzero();
2564 $$bs =~ s/([01])_([01])/$1$2/g;
2565 $$bs =~ s/([01])_([01])/$1$2/g;
2566 return $x->bnan() if $$bs !~ /^[+-]?0b[01]+$/;
2568 my $sign = '+'; $sign = '-' if ($$bs =~ /^\-/);
2569 $$bs =~ s/^[+-]//; # strip sign
2570 if ($CAN{'from_bin'})
2572 $x->{value} = $CALC->_from_bin($bs);
2576 my $mul = Math::BigInt->bone();
2577 my $x256 = Math::BigInt->new(256);
2578 my $len = CORE::length($$bs)-2; # minus 2 for 0b
2579 $len = int($len/8); # 8-digit parts, w/o '0b'
2580 my $val; my $i = -8;
2583 $val = substr($$bs,$i,8);
2584 $val =~ s/^[+-]?0b// if $len == 0; # for last part only
2585 #$val = oct('0b'.$val); # does not work on Perl prior to 5.6.0
2587 # $val = ('0' x (8-CORE::length($val))).$val if CORE::length($val) < 8;
2588 $val = ord(pack('B8',substr('00000000'.$val,-8,8)));
2590 $x += $mul * $val if $val != 0;
2591 $mul *= $x256 if $len >= 0; # skip last mul
2594 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2600 # (ref to num_str) return num_str
2601 # internal, take apart a string and return the pieces
2602 # strip leading/trailing whitespace, leading zeros, underscore and reject
2606 # strip white space at front, also extranous leading zeros
2607 $$x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2608 $$x =~ s/^\s+//; # but this will
2609 $$x =~ s/\s+$//g; # strip white space at end
2611 # shortcut, if nothing to split, return early
2612 if ($$x =~ /^[+-]?\d+\z/)
2614 $$x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2615 return (\$sign, $x, \'', \'', \0);
2618 # invalid starting char?
2619 return if $$x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2621 return __from_hex($x) if $$x =~ /^[\-\+]?0x/; # hex string
2622 return __from_bin($x) if $$x =~ /^[\-\+]?0b/; # binary string
2624 # strip underscores between digits
2625 $$x =~ s/(\d)_(\d)/$1$2/g;
2626 $$x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3
2628 # some possible inputs:
2629 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2630 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 # 0e999
2632 #return if $$x =~ /[Ee].*[Ee]/; # more than one E => error
2634 my ($m,$e,$last) = split /[Ee]/,$$x;
2635 return if defined $last; # last defined => 1e2E3 or others
2636 $e = '0' if !defined $e || $e eq "";
2638 # sign,value for exponent,mantint,mantfrac
2639 my ($es,$ev,$mis,$miv,$mfv);
2641 if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2645 return if $m eq '.' || $m eq '';
2646 my ($mi,$mf,$lastf) = split /\./,$m;
2647 return if defined $lastf; # lastf defined => 1.2.3 or others
2648 $mi = '0' if !defined $mi;
2649 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2650 $mf = '0' if !defined $mf || $mf eq '';
2651 if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2653 $mis = $1||'+'; $miv = $2;
2654 return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
2656 # handle the 0e999 case here
2657 $ev = 0 if $miv eq '0' && $mfv eq '';
2658 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2661 return; # NaN, not a number
2664 ##############################################################################
2665 # internal calculation routines (others are in Math::BigInt::Calc etc)
2669 # (BINT or num_str, BINT or num_str) return BINT
2670 # does modify first argument
2673 my $x = shift; my $ty = shift;
2674 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2675 return $x * $ty / bgcd($x,$ty);
2680 # (BINT or num_str, BINT or num_str) return BINT
2681 # does modify both arguments
2682 # GCD -- Euclids algorithm E, Knuth Vol 2 pg 296
2685 return $x->bnan() if $x->{sign} !~ /^[+-]$/ || $ty->{sign} !~ /^[+-]$/;
2687 while (!$ty->is_zero())
2689 ($x, $ty) = ($ty,bmod($x,$ty));
2694 ###############################################################################
2695 # this method return 0 if the object can be modified, or 1 for not
2696 # We use a fast constant sub() here, to avoid costly calls. Subclasses
2697 # may override it with special code (f.i. Math::BigInt::Constant does so)
2699 sub modify () { 0; }
2706 Math::BigInt - Arbitrary size integer math package
2712 # or make it faster: install (optional) Math::BigInt::GMP
2713 # and always use (it will fall back to pure Perl if the
2714 # GMP library is not installed):
2716 use Math::BigInt lib => 'GMP';
2719 $x = Math::BigInt->new($str); # defaults to 0
2720 $nan = Math::BigInt->bnan(); # create a NotANumber
2721 $zero = Math::BigInt->bzero(); # create a +0
2722 $inf = Math::BigInt->binf(); # create a +inf
2723 $inf = Math::BigInt->binf('-'); # create a -inf
2724 $one = Math::BigInt->bone(); # create a +1
2725 $one = Math::BigInt->bone('-'); # create a -1
2727 # Testing (don't modify their arguments)
2728 # (return true if the condition is met, otherwise false)
2730 $x->is_zero(); # if $x is +0
2731 $x->is_nan(); # if $x is NaN
2732 $x->is_one(); # if $x is +1
2733 $x->is_one('-'); # if $x is -1
2734 $x->is_odd(); # if $x is odd
2735 $x->is_even(); # if $x is even
2736 $x->is_pos(); # if $x >= 0
2737 $x->is_neg(); # if $x < 0
2738 $x->is_inf(sign); # if $x is +inf, or -inf (sign is default '+')
2739 $x->is_int(); # if $x is an integer (not a float)
2741 # comparing and digit/sign extration
2742 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2743 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2744 $x->sign(); # return the sign, either +,- or NaN
2745 $x->digit($n); # return the nth digit, counting from right
2746 $x->digit(-$n); # return the nth digit, counting from left
2748 # The following all modify their first argument. If you want to preserve
2749 # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
2750 # neccessary when mixing $a = $b assigments with non-overloaded math.
2752 $x->bzero(); # set $x to 0
2753 $x->bnan(); # set $x to NaN
2754 $x->bone(); # set $x to +1
2755 $x->bone('-'); # set $x to -1
2756 $x->binf(); # set $x to inf
2757 $x->binf('-'); # set $x to -inf
2759 $x->bneg(); # negation
2760 $x->babs(); # absolute value
2761 $x->bnorm(); # normalize (no-op in BigInt)
2762 $x->bnot(); # two's complement (bit wise not)
2763 $x->binc(); # increment $x by 1
2764 $x->bdec(); # decrement $x by 1
2766 $x->badd($y); # addition (add $y to $x)
2767 $x->bsub($y); # subtraction (subtract $y from $x)
2768 $x->bmul($y); # multiplication (multiply $x by $y)
2769 $x->bdiv($y); # divide, set $x to quotient
2770 # return (quo,rem) or quo if scalar
2772 $x->bmod($y); # modulus (x % y)
2773 $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
2774 $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
2776 $x->bpow($y); # power of arguments (x ** y)
2777 $x->blsft($y); # left shift
2778 $x->brsft($y); # right shift
2779 $x->blsft($y,$n); # left shift, by base $n (like 10)
2780 $x->brsft($y,$n); # right shift, by base $n (like 10)
2782 $x->band($y); # bitwise and
2783 $x->bior($y); # bitwise inclusive or
2784 $x->bxor($y); # bitwise exclusive or
2785 $x->bnot(); # bitwise not (two's complement)
2787 $x->bsqrt(); # calculate square-root
2788 $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
2789 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2791 $x->round($A,$P,$mode); # round to accuracy or precision using mode $mode
2792 $x->bround($N); # accuracy: preserve $N digits
2793 $x->bfround($N); # round to $Nth digit, no-op for BigInts
2795 # The following do not modify their arguments in BigInt (are no-ops),
2796 # but do so in BigFloat:
2798 $x->bfloor(); # return integer less or equal than $x
2799 $x->bceil(); # return integer greater or equal than $x
2801 # The following do not modify their arguments:
2803 bgcd(@values); # greatest common divisor (no OO style)
2804 blcm(@values); # lowest common multiplicator (no OO style)
2806 $x->length(); # return number of digits in number
2807 ($x,$f) = $x->length(); # length of number and length of fraction part,
2808 # latter is always 0 digits long for BigInt's
2810 $x->exponent(); # return exponent as BigInt
2811 $x->mantissa(); # return (signed) mantissa as BigInt
2812 $x->parts(); # return (mantissa,exponent) as BigInt
2813 $x->copy(); # make a true copy of $x (unlike $y = $x;)
2814 $x->as_int(); # return as BigInt (in BigInt: same as copy())
2815 $x->numify(); # return as scalar (might overflow!)
2817 # conversation to string (do not modify their argument)
2818 $x->bstr(); # normalized string
2819 $x->bsstr(); # normalized string in scientific notation
2820 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
2821 $x->as_bin(); # as signed binary string with prefixed 0b
2824 # precision and accuracy (see section about rounding for more)
2825 $x->precision(); # return P of $x (or global, if P of $x undef)
2826 $x->precision($n); # set P of $x to $n
2827 $x->accuracy(); # return A of $x (or global, if A of $x undef)
2828 $x->accuracy($n); # set A $x to $n
2831 Math::BigInt->precision(); # get/set global P for all BigInt objects
2832 Math::BigInt->accuracy(); # get/set global A for all BigInt objects
2833 Math::BigInt->config(); # return hash containing configuration
2837 All operators (inlcuding basic math operations) are overloaded if you
2838 declare your big integers as
2840 $i = new Math::BigInt '123_456_789_123_456_789';
2842 Operations with overloaded operators preserve the arguments which is
2843 exactly what you expect.
2849 Input values to these routines may be any string, that looks like a number
2850 and results in an integer, including hexadecimal and binary numbers.
2852 Scalars holding numbers may also be passed, but note that non-integer numbers
2853 may already have lost precision due to the conversation to float. Quote
2854 your input if you want BigInt to see all the digits:
2856 $x = Math::BigInt->new(12345678890123456789); # bad
2857 $x = Math::BigInt->new('12345678901234567890'); # good
2859 You can include one underscore between any two digits.
2861 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
2862 Non-integer values result in NaN.
2864 Currently, Math::BigInt::new() defaults to 0, while Math::BigInt::new('')
2865 results in 'NaN'. This might change in the future, so use always the following
2866 explicit forms to get a zero or NaN:
2868 $zero = Math::BigInt->bzero();
2869 $nan = Math::BigInt->bnan();
2871 C<bnorm()> on a BigInt object is now effectively a no-op, since the numbers
2872 are always stored in normalized form. If passed a string, creates a BigInt
2873 object from the input.
2877 Output values are BigInt objects (normalized), except for bstr(), which
2878 returns a string in normalized form.
2879 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
2880 C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
2881 return either undef, <0, 0 or >0 and are suited for sort.
2887 Each of the methods below (except config(), accuracy() and precision())
2888 accepts three additional parameters. These arguments $A, $P and $R are
2889 accuracy, precision and round_mode. Please see the section about
2890 L<ACCURACY and PRECISION> for more information.
2896 print Dumper ( Math::BigInt->config() );
2897 print Math::BigInt->config()->{lib},"\n";
2899 Returns a hash containing the configuration, e.g. the version number, lib
2900 loaded etc. The following hash keys are currently filled in with the
2901 appropriate information.
2905 ============================================================
2906 lib Name of the low-level math library
2908 lib_version Version of low-level math library (see 'lib')
2910 class The class name of config() you just called
2912 upgrade To which class math operations might be upgraded
2914 downgrade To which class math operations might be downgraded
2916 precision Global precision
2918 accuracy Global accuracy
2920 round_mode Global round mode
2922 version version number of the class you used
2924 div_scale Fallback acccuracy for div
2926 trap_nan If true, traps creation of NaN via croak()
2928 trap_inf If true, traps creation of +inf/-inf via croak()
2931 The following values can be set by passing C<config()> a reference to a hash:
2934 upgrade downgrade precision accuracy round_mode div_scale
2938 $new_cfg = Math::BigInt->config( { trap_inf => 1, precision => 5 } );
2942 $x->accuracy(5); # local for $x
2943 CLASS->accuracy(5); # global for all members of CLASS
2944 $A = $x->accuracy(); # read out
2945 $A = CLASS->accuracy(); # read out
2947 Set or get the global or local accuracy, aka how many significant digits the
2950 Please see the section about L<ACCURACY AND PRECISION> for further details.
2952 Value must be greater than zero. Pass an undef value to disable it:
2954 $x->accuracy(undef);
2955 Math::BigInt->accuracy(undef);
2957 Returns the current accuracy. For C<$x->accuracy()> it will return either the
2958 local accuracy, or if not defined, the global. This means the return value
2959 represents the accuracy that will be in effect for $x:
2961 $y = Math::BigInt->new(1234567); # unrounded
2962 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
2963 $x = Math::BigInt->new(123456); # will be automatically rounded
2964 print "$x $y\n"; # '123500 1234567'
2965 print $x->accuracy(),"\n"; # will be 4
2966 print $y->accuracy(),"\n"; # also 4, since global is 4
2967 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
2968 print $x->accuracy(),"\n"; # still 4
2969 print $y->accuracy(),"\n"; # 5, since global is 5
2971 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
2972 globals separated from Math::BigInt, but it is possible to subclass
2973 Math::BigInt and make the globals of the subclass aliases to the ones from
2978 $x->precision(-2); # local for $x, round right of the dot
2979 $x->precision(2); # ditto, but round left of the dot
2980 CLASS->accuracy(5); # global for all members of CLASS
2981 CLASS->precision(-5); # ditto
2982 $P = CLASS->precision(); # read out
2983 $P = $x->precision(); # read out
2985 Set or get the global or local precision, aka how many digits the result has
2986 after the dot (or where to round it when passing a positive number). In
2987 Math::BigInt, passing a negative number precision has no effect since no
2988 numbers have digits after the dot.
2990 Please see the section about L<ACCURACY AND PRECISION> for further details.
2992 Value must be greater than zero. Pass an undef value to disable it:
2994 $x->precision(undef);
2995 Math::BigInt->precision(undef);
2997 Returns the current precision. For C<$x->precision()> it will return either the
2998 local precision of $x, or if not defined, the global. This means the return
2999 value represents the accuracy that will be in effect for $x:
3001 $y = Math::BigInt->new(1234567); # unrounded
3002 print Math::BigInt->precision(4),"\n"; # set 4, print 4
3003 $x = Math::BigInt->new(123456); # will be automatically rounded
3005 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
3006 globals separated from Math::BigInt, but it is possible to subclass
3007 Math::BigInt and make the globals of the subclass aliases to the ones from
3014 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
3015 2, but others work, too.
3017 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
3021 $x = Math::BigInt->new(10);
3022 $x->brsft(1); # same as $x >> 1: 5
3023 $x = Math::BigInt->new(1234);
3024 $x->brsft(2,10); # result 12
3026 There is one exception, and that is base 2 with negative $x:
3029 $x = Math::BigInt->new(-5);
3032 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
3037 $x = Math::BigInt->new($str,$A,$P,$R);
3039 Creates a new BigInt object from a scalar or another BigInt object. The
3040 input is accepted as decimal, hex (with leading '0x') or binary (with leading
3043 See L<Input> for more info on accepted input formats.
3047 $x = Math::BigInt->bnan();
3049 Creates a new BigInt object representing NaN (Not A Number).
3050 If used on an object, it will set it to NaN:
3056 $x = Math::BigInt->bzero();
3058 Creates a new BigInt object representing zero.
3059 If used on an object, it will set it to zero:
3065 $x = Math::BigInt->binf($sign);
3067 Creates a new BigInt object representing infinity. The optional argument is
3068 either '-' or '+', indicating whether you want infinity or minus infinity.
3069 If used on an object, it will set it to infinity:
3076 $x = Math::BigInt->binf($sign);
3078 Creates a new BigInt object representing one. The optional argument is
3079 either '-' or '+', indicating whether you want one or minus one.
3080 If used on an object, it will set it to one:
3085 =head2 is_one()/is_zero()/is_nan()/is_inf()
3088 $x->is_zero(); # true if arg is +0
3089 $x->is_nan(); # true if arg is NaN
3090 $x->is_one(); # true if arg is +1
3091 $x->is_one('-'); # true if arg is -1
3092 $x->is_inf(); # true if +inf
3093 $x->is_inf('-'); # true if -inf (sign is default '+')
3095 These methods all test the BigInt for beeing one specific value and return
3096 true or false depending on the input. These are faster than doing something
3101 =head2 is_pos()/is_neg()
3103 $x->is_pos(); # true if >= 0
3104 $x->is_neg(); # true if < 0
3106 The methods return true if the argument is positive or negative, respectively.
3107 C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
3108 C<-inf> is negative. A C<zero> is positive.
3110 These methods are only testing the sign, and not the value.
3112 C<is_positive()> and C<is_negative()> are aliase to C<is_pos()> and
3113 C<is_neg()>, respectively. C<is_positive()> and C<is_negative()> were
3114 introduced in v1.36, while C<is_pos()> and C<is_neg()> were only introduced
3117 =head2 is_odd()/is_even()/is_int()
3119 $x->is_odd(); # true if odd, false for even
3120 $x->is_even(); # true if even, false for odd
3121 $x->is_int(); # true if $x is an integer
3123 The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
3124 C<-inf> are not integers and are neither odd nor even.
3126 In BigInt, all numbers except C<NaN>, C<+inf> and C<-inf> are integers.
3132 Compares $x with $y and takes the sign into account.
3133 Returns -1, 0, 1 or undef.
3139 Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
3145 Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
3149 $x->digit($n); # return the nth digit, counting from right
3151 If C<$n> is negative, returns the digit counting from left.
3157 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
3158 and '-inf', respectively. Does nothing for NaN or zero.
3164 Set the number to it's absolute value, e.g. change the sign from '-' to '+'
3165 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
3170 $x->bnorm(); # normalize (no-op)
3176 Two's complement (bit wise not). This is equivalent to
3184 $x->binc(); # increment x by 1
3188 $x->bdec(); # decrement x by 1
3192 $x->badd($y); # addition (add $y to $x)
3196 $x->bsub($y); # subtraction (subtract $y from $x)
3200 $x->bmul($y); # multiplication (multiply $x by $y)
3204 $x->bdiv($y); # divide, set $x to quotient
3205 # return (quo,rem) or quo if scalar
3209 $x->bmod($y); # modulus (x % y)
3213 num->bmodinv($mod); # modular inverse
3215 Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
3216 returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
3217 C<bgcd($num, $mod)==1>.
3221 $num->bmodpow($exp,$mod); # modular exponentation
3222 # ($num**$exp % $mod)
3224 Returns the value of C<$num> taken to the power C<$exp> in the modulus
3225 C<$mod> using binary exponentation. C<bmodpow> is far superior to
3230 because it is much faster - it reduces internal variables into
3231 the modulus whenever possible, so it operates on smaller numbers.
3233 C<bmodpow> also supports negative exponents.
3235 bmodpow($num, -1, $mod)
3237 is exactly equivalent to
3243 $x->bpow($y); # power of arguments (x ** y)
3247 $x->blsft($y); # left shift
3248 $x->blsft($y,$n); # left shift, in base $n (like 10)
3252 $x->brsft($y); # right shift
3253 $x->brsft($y,$n); # right shift, in base $n (like 10)
3257 $x->band($y); # bitwise and
3261 $x->bior($y); # bitwise inclusive or
3265 $x->bxor($y); # bitwise exclusive or
3269 $x->bnot(); # bitwise not (two's complement)
3273 $x->bsqrt(); # calculate square-root
3277 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3281 $x->round($A,$P,$round_mode);
3283 Round $x to accuracy C<$A> or precision C<$P> using the round mode
3288 $x->bround($N); # accuracy: preserve $N digits
3292 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3298 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3299 does change $x in BigFloat.
3305 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3306 does change $x in BigFloat.
3310 bgcd(@values); # greatest common divisor (no OO style)
3314 blcm(@values); # lowest common multiplicator (no OO style)
3319 ($xl,$fl) = $x->length();
3321 Returns the number of digits in the decimal representation of the number.
3322 In list context, returns the length of the integer and fraction part. For
3323 BigInt's, the length of the fraction part will always be 0.
3329 Return the exponent of $x as BigInt.
3335 Return the signed mantissa of $x as BigInt.
3339 $x->parts(); # return (mantissa,exponent) as BigInt
3343 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3349 Returns $x as a BigInt (truncated towards zero). In BigInt this is the same as
3352 C<as_number()> is an alias to this method. C<as_number> was introduced in
3353 v1.22, while C<as_int()> was only introduced in v1.68.
3359 Returns a normalized string represantation of C<$x>.
3363 $x->bsstr(); # normalized string in scientific notation
3367 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3371 $x->as_bin(); # as signed binary string with prefixed 0b
3373 =head1 ACCURACY and PRECISION
3375 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3376 accuracy and precision based rounding, both automatically after every
3377 operation, as well as manually.
3379 This section describes the accuracy/precision handling in Math::Big* as it
3380 used to be and as it is now, complete with an explanation of all terms and
3383 Not yet implemented things (but with correct description) are marked with '!',
3384 things that need to be answered are marked with '?'.
3386 In the next paragraph follows a short description of terms used here (because
3387 these may differ from terms used by others people or documentation).
3389 During the rest of this document, the shortcuts A (for accuracy), P (for
3390 precision), F (fallback) and R (rounding mode) will be used.
3394 A fixed number of digits before (positive) or after (negative)
3395 the decimal point. For example, 123.45 has a precision of -2. 0 means an
3396 integer like 123 (or 120). A precision of 2 means two digits to the left
3397 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3398 numbers with zeros before the decimal point may have different precisions,
3399 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3400 was). It could also have p < 0, when the digits after the decimal point
3403 The string output (of floating point numbers) will be padded with zeros:
3405 Initial value P A Result String
3406 ------------------------------------------------------------
3407 1234.01 -3 1000 1000
3410 1234.001 1 1234 1234.0
3412 1234.01 2 1234.01 1234.01
3413 1234.01 5 1234.01 1234.01000
3415 For BigInts, no padding occurs.
3419 Number of significant digits. Leading zeros are not counted. A
3420 number may have an accuracy greater than the non-zero digits
3421 when there are zeros in it or trailing zeros. For example, 123.456 has
3422 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3424 The string output (of floating point numbers) will be padded with zeros:
3426 Initial value P A Result String
3427 ------------------------------------------------------------
3429 1234.01 6 1234.01 1234.01
3430 1234.1 8 1234.1 1234.1000
3432 For BigInts, no padding occurs.
3436 When both A and P are undefined, this is used as a fallback accuracy when
3439 =head2 Rounding mode R
3441 When rounding a number, different 'styles' or 'kinds'
3442 of rounding are possible. (Note that random rounding, as in
3443 Math::Round, is not implemented.)
3449 truncation invariably removes all digits following the
3450 rounding place, replacing them with zeros. Thus, 987.65 rounded
3451 to tens (P=1) becomes 980, and rounded to the fourth sigdig
3452 becomes 987.6 (A=4). 123.456 rounded to the second place after the
3453 decimal point (P=-2) becomes 123.46.
3455 All other implemented styles of rounding attempt to round to the
3456 "nearest digit." If the digit D immediately to the right of the
3457 rounding place (skipping the decimal point) is greater than 5, the
3458 number is incremented at the rounding place (possibly causing a
3459 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3460 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3461 truncated at the rounding place: e.g. when rounding to units, 0.4
3462 rounds to 0, and -19.4 rounds to -19.
3464 However the results of other styles of rounding differ if the
3465 digit immediately to the right of the rounding place (skipping the
3466 decimal point) is 5 and if there are no digits, or no digits other
3467 than 0, after that 5. In such cases:
3471 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3472 if it is not already. E.g., when rounding to the first sigdig, 0.45
3473 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3477 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3478 it is not already. E.g., when rounding to the first sigdig, 0.45
3479 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3483 round to plus infinity, i.e. always round up. E.g., when
3484 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3485 and 0.4501 also becomes 0.5.
3489 round to minus infinity, i.e. always round down. E.g., when
3490 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3491 but 0.4501 becomes 0.5.
3495 round to zero, i.e. positive numbers down, negative ones up.
3496 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3497 becomes -0.5, but 0.4501 becomes 0.5.
3501 The handling of A & P in MBI/MBF (the old core code shipped with Perl
3502 versions <= 5.7.2) is like this:
3508 * ffround($p) is able to round to $p number of digits after the decimal
3510 * otherwise P is unused
3512 =item Accuracy (significant digits)
3514 * fround($a) rounds to $a significant digits
3515 * only fdiv() and fsqrt() take A as (optional) paramater
3516 + other operations simply create the same number (fneg etc), or more (fmul)
3518 + rounding/truncating is only done when explicitly calling one of fround
3519 or ffround, and never for BigInt (not implemented)
3520 * fsqrt() simply hands its accuracy argument over to fdiv.
3521 * the documentation and the comment in the code indicate two different ways
3522 on how fdiv() determines the maximum number of digits it should calculate,
3523 and the actual code does yet another thing
3525 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3527 result has at most max(scale, length(dividend), length(divisor)) digits
3529 scale = max(scale, length(dividend)-1,length(divisor)-1);
3530 scale += length(divisior) - length(dividend);
3531 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3532 Actually, the 'difference' added to the scale is calculated from the
3533 number of "significant digits" in dividend and divisor, which is derived
3534 by looking at the length of the mantissa. Which is wrong, since it includes
3535 the + sign (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops
3536 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3537 assumption that 124 has 3 significant digits, while 120/7 will get you
3538 '17', not '17.1' since 120 is thought to have 2 significant digits.
3539 The rounding after the division then uses the remainder and $y to determine
3540 wether it must round up or down.
3541 ? I have no idea which is the right way. That's why I used a slightly more
3542 ? simple scheme and tweaked the few failing testcases to match it.
3546 This is how it works now:
3550 =item Setting/Accessing
3552 * You can set the A global via C<< Math::BigInt->accuracy() >> or
3553 C<< Math::BigFloat->accuracy() >> or whatever class you are using.
3554 * You can also set P globally by using C<< Math::SomeClass->precision() >>
3556 * Globals are classwide, and not inherited by subclasses.
3557 * to undefine A, use C<< Math::SomeCLass->accuracy(undef); >>
3558 * to undefine P, use C<< Math::SomeClass->precision(undef); >>
3559 * Setting C<< Math::SomeClass->accuracy() >> clears automatically
3560 C<< Math::SomeClass->precision() >>, and vice versa.
3561 * To be valid, A must be > 0, P can have any value.
3562 * If P is negative, this means round to the P'th place to the right of the
3563 decimal point; positive values mean to the left of the decimal point.
3564 P of 0 means round to integer.
3565 * to find out the current global A, use C<< Math::SomeClass->accuracy() >>
3566 * to find out the current global P, use C<< Math::SomeClass->precision() >>
3567 * use C<< $x->accuracy() >> respective C<< $x->precision() >> for the local
3568 setting of C<< $x >>.
3569 * Please note that C<< $x->accuracy() >> respecive C<< $x->precision() >>
3570 return eventually defined global A or P, when C<< $x >>'s A or P is not
3573 =item Creating numbers
3575 * When you create a number, you can give it's desired A or P via:
3576 $x = Math::BigInt->new($number,$A,$P);
3577 * Only one of A or P can be defined, otherwise the result is NaN
3578 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
3579 globals (if set) will be used. Thus changing the global defaults later on
3580 will not change the A or P of previously created numbers (i.e., A and P of
3581 $x will be what was in effect when $x was created)
3582 * If given undef for A and P, B<no> rounding will occur, and the globals will
3583 B<not> be used. This is used by subclasses to create numbers without
3584 suffering rounding in the parent. Thus a subclass is able to have it's own
3585 globals enforced upon creation of a number by using
3586 C<< $x = Math::BigInt->new($number,undef,undef) >>:
3588 use Math::BigInt::SomeSubclass;
3591 Math::BigInt->accuracy(2);
3592 Math::BigInt::SomeSubClass->accuracy(3);
3593 $x = Math::BigInt::SomeSubClass->new(1234);
3595 $x is now 1230, and not 1200. A subclass might choose to implement
3596 this otherwise, e.g. falling back to the parent's A and P.
3600 * If A or P are enabled/defined, they are used to round the result of each
3601 operation according to the rules below
3602 * Negative P is ignored in Math::BigInt, since BigInts never have digits
3603 after the decimal point
3604 * Math::BigFloat uses Math::BigInt internally, but setting A or P inside
3605 Math::BigInt as globals does not tamper with the parts of a BigFloat.
3606 A flag is used to mark all Math::BigFloat numbers as 'never round'.
3610 * It only makes sense that a number has only one of A or P at a time.
3611 If you set either A or P on one object, or globally, the other one will
3612 be automatically cleared.
3613 * If two objects are involved in an operation, and one of them has A in
3614 effect, and the other P, this results in an error (NaN).
3615 * A takes precendence over P (Hint: A comes before P).
3616 If neither of them is defined, nothing is used, i.e. the result will have
3617 as many digits as it can (with an exception for fdiv/fsqrt) and will not
3619 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
3620 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
3621 If either the dividend's or the divisor's mantissa has more digits than
3622 the value of F, the higher value will be used instead of F.
3623 This is to limit the digits (A) of the result (just consider what would
3624 happen with unlimited A and P in the case of 1/3 :-)
3625 * fdiv will calculate (at least) 4 more digits than required (determined by
3626 A, P or F), and, if F is not used, round the result
3627 (this will still fail in the case of a result like 0.12345000000001 with A
3628 or P of 5, but this can not be helped - or can it?)
3629 * Thus you can have the math done by on Math::Big* class in two modi:
3630 + never round (this is the default):
3631 This is done by setting A and P to undef. No math operation
3632 will round the result, with fdiv() and fsqrt() as exceptions to guard
3633 against overflows. You must explicitely call bround(), bfround() or
3634 round() (the latter with parameters).
3635 Note: Once you have rounded a number, the settings will 'stick' on it
3636 and 'infect' all other numbers engaged in math operations with it, since
3637 local settings have the highest precedence. So, to get SaferRound[tm],
3638 use a copy() before rounding like this:
3640 $x = Math::BigFloat->new(12.34);
3641 $y = Math::BigFloat->new(98.76);
3642 $z = $x * $y; # 1218.6984
3643 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
3644 $z = $x * $y; # still 1218.6984, without
3645 # copy would have been 1210!
3647 + round after each op:
3648 After each single operation (except for testing like is_zero()), the
3649 method round() is called and the result is rounded appropriately. By
3650 setting proper values for A and P, you can have all-the-same-A or
3651 all-the-same-P modes. For example, Math::Currency might set A to undef,
3652 and P to -2, globally.
3654 ?Maybe an extra option that forbids local A & P settings would be in order,
3655 ?so that intermediate rounding does not 'poison' further math?
3657 =item Overriding globals
3659 * you will be able to give A, P and R as an argument to all the calculation
3660 routines; the second parameter is A, the third one is P, and the fourth is
3661 R (shift right by one for binary operations like badd). P is used only if
3662 the first parameter (A) is undefined. These three parameters override the
3663 globals in the order detailed as follows, i.e. the first defined value
3665 (local: per object, global: global default, parameter: argument to sub)
3668 + local A (if defined on both of the operands: smaller one is taken)
3669 + local P (if defined on both of the operands: bigger one is taken)
3673 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
3674 arguments (A and P) instead of one
3676 =item Local settings
3678 * You can set A or P locally by using C<< $x->accuracy() >> or
3679 C<< $x->precision() >>
3680 and thus force different A and P for different objects/numbers.
3681 * Setting A or P this way immediately rounds $x to the new value.
3682 * C<< $x->accuracy() >> clears C<< $x->precision() >>, and vice versa.
3686 * the rounding routines will use the respective global or local settings.
3687 fround()/bround() is for accuracy rounding, while ffround()/bfround()
3689 * the two rounding functions take as the second parameter one of the
3690 following rounding modes (R):
3691 'even', 'odd', '+inf', '-inf', 'zero', 'trunc'
3692 * you can set/get the global R by using C<< Math::SomeClass->round_mode() >>
3693 or by setting C<< $Math::SomeClass::round_mode >>
3694 * after each operation, C<< $result->round() >> is called, and the result may
3695 eventually be rounded (that is, if A or P were set either locally,
3696 globally or as parameter to the operation)
3697 * to manually round a number, call C<< $x->round($A,$P,$round_mode); >>
3698 this will round the number by using the appropriate rounding function
3699 and then normalize it.
3700 * rounding modifies the local settings of the number:
3702 $x = Math::BigFloat->new(123.456);
3706 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
3707 will be 4 from now on.
3709 =item Default values
3718 * The defaults are set up so that the new code gives the same results as
3719 the old code (except in a few cases on fdiv):
3720 + Both A and P are undefined and thus will not be used for rounding
3721 after each operation.
3722 + round() is thus a no-op, unless given extra parameters A and P
3728 The actual numbers are stored as unsigned big integers (with seperate sign).
3729 You should neither care about nor depend on the internal representation; it
3730 might change without notice. Use only method calls like C<< $x->sign(); >>
3731 instead relying on the internal hash keys like in C<< $x->{sign}; >>.
3735 Math with the numbers is done (by default) by a module called
3736 C<Math::BigInt::Calc>. This is equivalent to saying:
3738 use Math::BigInt lib => 'Calc';
3740 You can change this by using:
3742 use Math::BigInt lib => 'BitVect';
3744 The following would first try to find Math::BigInt::Foo, then
3745 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
3747 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
3749 Since Math::BigInt::GMP is in almost all cases faster than Calc (especially in
3750 cases involving really big numbers, where it is B<much> faster), and there is
3751 no penalty if Math::BigInt::GMP is not installed, it is a good idea to always
3754 use Math::BigInt lib => 'GMP';
3756 Different low-level libraries use different formats to store the
3757 numbers. You should not depend on the number having a specific format.
3759 See the respective math library module documentation for further details.
3763 The sign is either '+', '-', 'NaN', '+inf' or '-inf' and stored seperately.
3765 A sign of 'NaN' is used to represent the result when input arguments are not
3766 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
3767 minus infinity. You will get '+inf' when dividing a positive number by 0, and
3768 '-inf' when dividing any negative number by 0.
3770 =head2 mantissa(), exponent() and parts()
3772 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
3775 $m = $x->mantissa();
3776 $e = $x->exponent();
3777 $y = $m * ( 10 ** $e );
3778 print "ok\n" if $x == $y;
3780 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
3781 in one go. Both the returned mantissa and exponent have a sign.
3783 Currently, for BigInts C<$e> is always 0, except for NaN, +inf and -inf,
3784 where it is C<NaN>; and for C<$x == 0>, where it is C<1> (to be compatible
3785 with Math::BigFloat's internal representation of a zero as C<0E1>).
3787 C<$m> is currently just a copy of the original number. The relation between
3788 C<$e> and C<$m> will stay always the same, though their real values might
3795 sub bint { Math::BigInt->new(shift); }
3797 $x = Math::BigInt->bstr("1234") # string "1234"
3798 $x = "$x"; # same as bstr()
3799 $x = Math::BigInt->bneg("1234"); # BigInt "-1234"
3800 $x = Math::BigInt->babs("-12345"); # BigInt "12345"
3801 $x = Math::BigInt->bnorm("-0 00"); # BigInt "0"
3802 $x = bint(1) + bint(2); # BigInt "3"
3803 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
3804 $x = bint(1); # BigInt "1"
3805 $x = $x + 5 / 2; # BigInt "3"
3806 $x = $x ** 3; # BigInt "27"
3807 $x *= 2; # BigInt "54"
3808 $x = Math::BigInt->new(0); # BigInt "0"
3810 $x = Math::BigInt->badd(4,5) # BigInt "9"
3811 print $x->bsstr(); # 9e+0
3813 Examples for rounding:
3818 $x = Math::BigFloat->new(123.4567);
3819 $y = Math::BigFloat->new(123.456789);
3820 Math::BigFloat->accuracy(4); # no more A than 4
3822 ok ($x->copy()->fround(),123.4); # even rounding
3823 print $x->copy()->fround(),"\n"; # 123.4
3824 Math::BigFloat->round_mode('odd'); # round to odd
3825 print $x->copy()->fround(),"\n"; # 123.5
3826 Math::BigFloat->accuracy(5); # no more A than 5
3827 Math::BigFloat->round_mode('odd'); # round to odd
3828 print $x->copy()->fround(),"\n"; # 123.46
3829 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
3830 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
3832 Math::BigFloat->accuracy(undef); # A not important now
3833 Math::BigFloat->precision(2); # P important
3834 print $x->copy()->bnorm(),"\n"; # 123.46
3835 print $x->copy()->fround(),"\n"; # 123.46
3837 Examples for converting:
3839 my $x = Math::BigInt->new('0b1'.'01' x 123);
3840 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
3842 =head1 Autocreating constants
3844 After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
3845 and binary constants in the given scope are converted to C<Math::BigInt>.
3846 This conversion happens at compile time.
3850 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
3852 prints the integer value of C<2**100>. Note that without conversion of
3853 constants the expression 2**100 will be calculated as perl scalar.
3855 Please note that strings and floating point constants are not affected,
3858 use Math::BigInt qw/:constant/;
3860 $x = 1234567890123456789012345678901234567890
3861 + 123456789123456789;
3862 $y = '1234567890123456789012345678901234567890'
3863 + '123456789123456789';
3865 do not work. You need an explicit Math::BigInt->new() around one of the
3866 operands. You should also quote large constants to protect loss of precision:
3870 $x = Math::BigInt->new('1234567889123456789123456789123456789');
3872 Without the quotes Perl would convert the large number to a floating point
3873 constant at compile time and then hand the result to BigInt, which results in
3874 an truncated result or a NaN.
3876 This also applies to integers that look like floating point constants:
3878 use Math::BigInt ':constant';
3880 print ref(123e2),"\n";
3881 print ref(123.2e2),"\n";
3883 will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
3884 to get this to work.
3888 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
3889 must be made in the second case. For long numbers, the copy can eat up to 20%
3890 of the work (in the case of addition/subtraction, less for
3891 multiplication/division). If $y is very small compared to $x, the form
3892 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
3893 more time then the actual addition.
3895 With a technique called copy-on-write, the cost of copying with overload could
3896 be minimized or even completely avoided. A test implementation of COW did show
3897 performance gains for overloaded math, but introduced a performance loss due
3898 to a constant overhead for all other operatons. So Math::BigInt does currently
3901 The rewritten version of this module (vs. v0.01) is slower on certain
3902 operations, like C<new()>, C<bstr()> and C<numify()>. The reason are that it
3903 does now more work and handles much more cases. The time spent in these
3904 operations is usually gained in the other math operations so that code on
3905 the average should get (much) faster. If they don't, please contact the author.
3907 Some operations may be slower for small numbers, but are significantly faster
3908 for big numbers. Other operations are now constant (O(1), like C<bneg()>,
3909 C<babs()> etc), instead of O(N) and thus nearly always take much less time.
3910 These optimizations were done on purpose.
3912 If you find the Calc module to slow, try to install any of the replacement
3913 modules and see if they help you.
3915 =head2 Alternative math libraries
3917 You can use an alternative library to drive Math::BigInt via:
3919 use Math::BigInt lib => 'Module';
3921 See L<MATH LIBRARY> for more information.
3923 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
3927 =head1 Subclassing Math::BigInt
3929 The basic design of Math::BigInt allows simple subclasses with very little
3930 work, as long as a few simple rules are followed:
3936 The public API must remain consistent, i.e. if a sub-class is overloading
3937 addition, the sub-class must use the same name, in this case badd(). The
3938 reason for this is that Math::BigInt is optimized to call the object methods
3943 The private object hash keys like C<$x->{sign}> may not be changed, but
3944 additional keys can be added, like C<$x->{_custom}>.
3948 Accessor functions are available for all existing object hash keys and should
3949 be used instead of directly accessing the internal hash keys. The reason for
3950 this is that Math::BigInt itself has a pluggable interface which permits it
3951 to support different storage methods.
3955 More complex sub-classes may have to replicate more of the logic internal of
3956 Math::BigInt if they need to change more basic behaviors. A subclass that
3957 needs to merely change the output only needs to overload C<bstr()>.
3959 All other object methods and overloaded functions can be directly inherited
3960 from the parent class.
3962 At the very minimum, any subclass will need to provide it's own C<new()> and can
3963 store additional hash keys in the object. There are also some package globals
3964 that must be defined, e.g.:
3968 $precision = -2; # round to 2 decimal places
3969 $round_mode = 'even';
3972 Additionally, you might want to provide the following two globals to allow
3973 auto-upgrading and auto-downgrading to work correctly:
3978 This allows Math::BigInt to correctly retrieve package globals from the
3979 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
3980 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
3986 in your subclass to automatically inherit the overloading from the parent. If
3987 you like, you can change part of the overloading, look at Math::String for an
3992 When used like this:
3994 use Math::BigInt upgrade => 'Foo::Bar';
3996 certain operations will 'upgrade' their calculation and thus the result to
3997 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
3999 use Math::BigInt upgrade => 'Math::BigFloat';
4001 As a shortcut, you can use the module C<bignum>:
4005 Also good for oneliners:
4007 perl -Mbignum -le 'print 2 ** 255'
4009 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
4010 as well es preserve accuracy (as in sqrt(3)).
4012 Beware: This feature is not fully implemented yet.
4016 The following methods upgrade themselves unconditionally; that is if upgrade
4017 is in effect, they will always hand up their work:
4029 Beware: This list is not complete.
4031 All other methods upgrade themselves only when one (or all) of their
4032 arguments are of the class mentioned in $upgrade (This might change in later
4033 versions to a more sophisticated scheme):
4039 =item broot() does not work
4041 The broot() function in BigInt may only work for small values. This will be
4042 fixed in a later version.
4044 =item Out of Memory!
4046 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
4047 C<eval()> in your code will crash with "Out of memory". This is probably an
4048 overload/exporter bug. You can workaround by not having C<eval()>
4049 and ':constant' at the same time or upgrade your Perl to a newer version.
4051 =item Fails to load Calc on Perl prior 5.6.0
4053 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
4054 will fall back to eval { require ... } when loading the math lib on Perls
4055 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
4056 filesystems using a different seperator.
4062 Some things might not work as you expect them. Below is documented what is
4063 known to be troublesome:
4067 =item bstr(), bsstr() and 'cmp'
4069 Both C<bstr()> and C<bsstr()> as well as automated stringify via overload now
4070 drop the leading '+'. The old code would return '+3', the new returns '3'.
4071 This is to be consistent with Perl and to make C<cmp> (especially with
4072 overloading) to work as you expect. It also solves problems with C<Test.pm>,
4073 because it's C<ok()> uses 'eq' internally.
4075 Mark Biggar said, when asked about to drop the '+' altogether, or make only
4078 I agree (with the first alternative), don't add the '+' on positive
4079 numbers. It's not as important anymore with the new internal
4080 form for numbers. It made doing things like abs and neg easier,
4081 but those have to be done differently now anyway.
4083 So, the following examples will now work all as expected:
4086 BEGIN { plan tests => 1 }
4089 my $x = new Math::BigInt 3*3;
4090 my $y = new Math::BigInt 3*3;
4093 print "$x eq 9" if $x eq $y;
4094 print "$x eq 9" if $x eq '9';
4095 print "$x eq 9" if $x eq 3*3;
4097 Additionally, the following still works:
4099 print "$x == 9" if $x == $y;
4100 print "$x == 9" if $x == 9;
4101 print "$x == 9" if $x == 3*3;
4103 There is now a C<bsstr()> method to get the string in scientific notation aka
4104 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
4105 for comparisation, but Perl will represent some numbers as 100 and others
4106 as 1e+308. If in doubt, convert both arguments to Math::BigInt before
4107 comparing them as strings:
4110 BEGIN { plan tests => 3 }
4113 $x = Math::BigInt->new('1e56'); $y = 1e56;
4114 ok ($x,$y); # will fail
4115 ok ($x->bsstr(),$y); # okay
4116 $y = Math::BigInt->new($y);
4119 Alternatively, simple use C<< <=> >> for comparisations, this will get it
4120 always right. There is not yet a way to get a number automatically represented
4121 as a string that matches exactly the way Perl represents it.
4125 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
4128 $x = Math::BigInt->new(123);
4129 $y = int($x); # BigInt 123
4130 $x = Math::BigFloat->new(123.45);
4131 $y = int($x); # BigInt 123
4133 In all Perl versions you can use C<as_number()> for the same effect:
4135 $x = Math::BigFloat->new(123.45);
4136 $y = $x->as_number(); # BigInt 123
4138 This also works for other subclasses, like Math::String.
4140 It is yet unlcear whether overloaded int() should return a scalar or a BigInt.
4144 The following will probably not do what you expect:
4146 $c = Math::BigInt->new(123);
4147 print $c->length(),"\n"; # prints 30
4149 It prints both the number of digits in the number and in the fraction part
4150 since print calls C<length()> in list context. Use something like:
4152 print scalar $c->length(),"\n"; # prints 3
4156 The following will probably not do what you expect:
4158 print $c->bdiv(10000),"\n";
4160 It prints both quotient and remainder since print calls C<bdiv()> in list
4161 context. Also, C<bdiv()> will modify $c, so be carefull. You probably want
4164 print $c / 10000,"\n";
4165 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
4169 The quotient is always the greatest integer less than or equal to the
4170 real-valued quotient of the two operands, and the remainder (when it is
4171 nonzero) always has the same sign as the second operand; so, for
4181 As a consequence, the behavior of the operator % agrees with the
4182 behavior of Perl's built-in % operator (as documented in the perlop
4183 manpage), and the equation
4185 $x == ($x / $y) * $y + ($x % $y)
4187 holds true for any $x and $y, which justifies calling the two return
4188 values of bdiv() the quotient and remainder. The only exception to this rule
4189 are when $y == 0 and $x is negative, then the remainder will also be
4190 negative. See below under "infinity handling" for the reasoning behing this.
4192 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
4193 not change BigInt's way to do things. This is because under 'use integer' Perl
4194 will do what the underlying C thinks is right and this is different for each
4195 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
4196 the author to implement it ;)
4198 =item infinity handling
4200 Here are some examples that explain the reasons why certain results occur while
4203 The following table shows the result of the division and the remainder, so that
4204 the equation above holds true. Some "ordinary" cases are strewn in to show more
4205 clearly the reasoning:
4207 A / B = C, R so that C * B + R = A
4208 =========================================================
4209 5 / 8 = 0, 5 0 * 8 + 5 = 5
4210 0 / 8 = 0, 0 0 * 8 + 0 = 0
4211 0 / inf = 0, 0 0 * inf + 0 = 0
4212 0 /-inf = 0, 0 0 * -inf + 0 = 0
4213 5 / inf = 0, 5 0 * inf + 5 = 5
4214 5 /-inf = 0, 5 0 * -inf + 5 = 5
4215 -5/ inf = 0, -5 0 * inf + -5 = -5
4216 -5/-inf = 0, -5 0 * -inf + -5 = -5
4217 inf/ 5 = inf, 0 inf * 5 + 0 = inf
4218 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
4219 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
4220 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
4221 5/ 5 = 1, 0 1 * 5 + 0 = 5
4222 -5/ -5 = 1, 0 1 * -5 + 0 = -5
4223 inf/ inf = 1, 0 1 * inf + 0 = inf
4224 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
4225 inf/-inf = -1, 0 -1 * -inf + 0 = inf
4226 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
4227 8/ 0 = inf, 8 inf * 0 + 8 = 8
4228 inf/ 0 = inf, inf inf * 0 + inf = inf
4231 These cases below violate the "remainder has the sign of the second of the two
4232 arguments", since they wouldn't match up otherwise.
4234 A / B = C, R so that C * B + R = A
4235 ========================================================
4236 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
4237 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
4239 =item Modifying and =
4243 $x = Math::BigFloat->new(5);
4246 It will not do what you think, e.g. making a copy of $x. Instead it just makes
4247 a second reference to the B<same> object and stores it in $y. Thus anything
4248 that modifies $x (except overloaded operators) will modify $y, and vice versa.
4249 Or in other words, C<=> is only safe if you modify your BigInts only via
4250 overloaded math. As soon as you use a method call it breaks:
4253 print "$x, $y\n"; # prints '10, 10'
4255 If you want a true copy of $x, use:
4259 You can also chain the calls like this, this will make first a copy and then
4262 $y = $x->copy()->bmul(2);
4264 See also the documentation for overload.pm regarding C<=>.
4268 C<bpow()> (and the rounding functions) now modifies the first argument and
4269 returns it, unlike the old code which left it alone and only returned the
4270 result. This is to be consistent with C<badd()> etc. The first three will
4271 modify $x, the last one won't:
4273 print bpow($x,$i),"\n"; # modify $x
4274 print $x->bpow($i),"\n"; # ditto
4275 print $x **= $i,"\n"; # the same
4276 print $x ** $i,"\n"; # leave $x alone
4278 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4280 =item Overloading -$x
4290 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4291 needs to preserve $x since it does not know that it later will get overwritten.
4292 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4294 With Copy-On-Write, this issue would be gone, but C-o-W is not implemented
4295 since it is slower for all other things.
4297 =item Mixing different object types
4299 In Perl you will get a floating point value if you do one of the following:
4305 With overloaded math, only the first two variants will result in a BigFloat:
4310 $mbf = Math::BigFloat->new(5);
4311 $mbi2 = Math::BigInteger->new(5);
4312 $mbi = Math::BigInteger->new(2);
4314 # what actually gets called:
4315 $float = $mbf + $mbi; # $mbf->badd()
4316 $float = $mbf / $mbi; # $mbf->bdiv()
4317 $integer = $mbi + $mbf; # $mbi->badd()
4318 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4319 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4321 This is because math with overloaded operators follows the first (dominating)
4322 operand, and the operation of that is called and returns thus the result. So,
4323 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4324 the result should be a Math::BigFloat or the second operant is one.
4326 To get a Math::BigFloat you either need to call the operation manually,
4327 make sure the operands are already of the proper type or casted to that type
4328 via Math::BigFloat->new():
4330 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4332 Beware of simple "casting" the entire expression, this would only convert
4333 the already computed result:
4335 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4337 Beware also of the order of more complicated expressions like:
4339 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4340 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4342 If in doubt, break the expression into simpler terms, or cast all operands
4343 to the desired resulting type.
4345 Scalar values are a bit different, since:
4350 will both result in the proper type due to the way the overloaded math works.
4352 This section also applies to other overloaded math packages, like Math::String.
4354 One solution to you problem might be autoupgrading|upgrading. See the
4355 pragmas L<bignum>, L<bigint> and L<bigrat> for an easy way to do this.
4359 C<bsqrt()> works only good if the result is a big integer, e.g. the square
4360 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4361 mode. The reason is that the result is always truncated to an integer.
4363 If you want a better approximation of the square root, then use:
4365 $x = Math::BigFloat->new(12);
4366 Math::BigFloat->precision(0);
4367 Math::BigFloat->round_mode('even');
4368 print $x->copy->bsqrt(),"\n"; # 4
4370 Math::BigFloat->precision(2);
4371 print $x->bsqrt(),"\n"; # 3.46
4372 print $x->bsqrt(3),"\n"; # 3.464
4376 For negative numbers in base see also L<brsft|brsft>.
4382 This program is free software; you may redistribute it and/or modify it under
4383 the same terms as Perl itself.
4387 L<Math::BigFloat>, L<Math::BigRat> and L<Math::Big> as well as
4388 L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4390 The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest
4391 because they solve the autoupgrading/downgrading issue, at least partly.
4394 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4395 more documentation including a full version history, testcases, empty
4396 subclass files and benchmarks.
4400 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4401 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001 - 2003
4402 and still at it in 2004.
4404 Many people contributed in one or more ways to the final beast, see the file
4405 CREDITS for an (uncomplete) list. If you miss your name, please drop me a