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 $class = "Math::BigInt";
24 @EXPORT_OK = qw(objectify bgcd blcm);
26 # _trap_inf and _trap_nan are internal and should never be accessed from the
28 use vars qw/$round_mode $accuracy $precision $div_scale $rnd_mode
29 $upgrade $downgrade $_trap_nan $_trap_inf/;
32 # Inside overload, the first arg is always an object. If the original code had
33 # it reversed (like $x = 2 * $y), then the third paramater is true.
34 # In some cases (like add, $x = $x + 2 is the same as $x = 2 + $x) this makes
35 # no difference, but in some cases it does.
37 # For overloaded ops with only one argument we simple use $_[0]->copy() to
38 # preserve the argument.
40 # Thus inheritance of overload operators becomes possible and transparent for
41 # our subclasses without the need to repeat the entire overload section there.
44 '=' => sub { $_[0]->copy(); },
46 # some shortcuts for speed (assumes that reversed order of arguments is routed
47 # to normal '+' and we thus can always modify first arg. If this is changed,
48 # this breaks and must be adjusted.)
49 '+=' => sub { $_[0]->badd($_[1]); },
50 '-=' => sub { $_[0]->bsub($_[1]); },
51 '*=' => sub { $_[0]->bmul($_[1]); },
52 '/=' => sub { scalar $_[0]->bdiv($_[1]); },
53 '%=' => sub { $_[0]->bmod($_[1]); },
54 '^=' => sub { $_[0]->bxor($_[1]); },
55 '&=' => sub { $_[0]->band($_[1]); },
56 '|=' => sub { $_[0]->bior($_[1]); },
58 '**=' => sub { $_[0]->bpow($_[1]); },
59 '<<=' => sub { $_[0]->blsft($_[1]); },
60 '>>=' => sub { $_[0]->brsft($_[1]); },
62 # not supported by Perl yet
63 '..' => \&_pointpoint,
65 '<=>' => sub { my $rc = $_[2] ?
66 ref($_[0])->bcmp($_[1],$_[0]) :
68 $rc = 1 unless defined $rc;
71 # we need '>=' to get things like "1 >= NaN" right:
72 '>=' => sub { my $rc = $_[2] ?
73 ref($_[0])->bcmp($_[1],$_[0]) :
75 # if there was a NaN involved, return false
76 return '' unless defined $rc;
81 "$_[1]" cmp $_[0]->bstr() :
82 $_[0]->bstr() cmp "$_[1]" },
84 # make cos()/sin()/atan2() "work" with BigInt's or subclasses
85 'cos' => sub { cos($_[0]->numify()) },
86 'sin' => sub { sin($_[0]->numify()) },
87 'atan2' => sub { $_[2] ?
88 atan2($_[1],$_[0]->numify()) :
89 atan2($_[0]->numify(),$_[1]) },
91 # are not yet overloadable
92 #'hex' => sub { print "hex"; $_[0]; },
93 #'oct' => sub { print "oct"; $_[0]; },
95 # log(N) is log(N, e), where e is Euler's number
96 'log' => sub { $_[0]->copy()->blog($_[1], undef); },
97 'exp' => sub { $_[0]->copy()->bexp($_[1]); },
98 'int' => sub { $_[0]->copy(); },
99 'neg' => sub { $_[0]->copy()->bneg(); },
100 'abs' => sub { $_[0]->copy()->babs(); },
101 'sqrt' => sub { $_[0]->copy()->bsqrt(); },
102 '~' => sub { $_[0]->copy()->bnot(); },
104 # for subtract it's a bit tricky to not modify b: b-a => -a+b
105 '-' => sub { my $c = $_[0]->copy; $_[2] ?
106 $c->bneg()->badd( $_[1]) :
108 '+' => sub { $_[0]->copy()->badd($_[1]); },
109 '*' => sub { $_[0]->copy()->bmul($_[1]); },
112 $_[2] ? ref($_[0])->new($_[1])->bdiv($_[0]) : $_[0]->copy->bdiv($_[1]);
115 $_[2] ? ref($_[0])->new($_[1])->bmod($_[0]) : $_[0]->copy->bmod($_[1]);
118 $_[2] ? ref($_[0])->new($_[1])->bpow($_[0]) : $_[0]->copy->bpow($_[1]);
121 $_[2] ? ref($_[0])->new($_[1])->blsft($_[0]) : $_[0]->copy->blsft($_[1]);
124 $_[2] ? ref($_[0])->new($_[1])->brsft($_[0]) : $_[0]->copy->brsft($_[1]);
127 $_[2] ? ref($_[0])->new($_[1])->band($_[0]) : $_[0]->copy->band($_[1]);
130 $_[2] ? ref($_[0])->new($_[1])->bior($_[0]) : $_[0]->copy->bior($_[1]);
133 $_[2] ? ref($_[0])->new($_[1])->bxor($_[0]) : $_[0]->copy->bxor($_[1]);
136 # can modify arg of ++ and --, so avoid a copy() for speed, but don't
137 # use $_[0]->bone(), it would modify $_[0] to be 1!
138 '++' => sub { $_[0]->binc() },
139 '--' => sub { $_[0]->bdec() },
141 # if overloaded, O(1) instead of O(N) and twice as fast for small numbers
143 # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/
144 # v5.6.1 dumps on this: return !$_[0]->is_zero() || undef; :-(
146 $t = 1 if !$_[0]->is_zero();
150 # the original qw() does not work with the TIESCALAR below, why?
151 # Order of arguments unsignificant
152 '""' => sub { $_[0]->bstr(); },
153 '0+' => sub { $_[0]->numify(); }
156 ##############################################################################
157 # global constants, flags and accessory
159 # These vars are public, but their direct usage is not recommended, use the
160 # accessor methods instead
162 $round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'
167 $upgrade = undef; # default is no upgrade
168 $downgrade = undef; # default is no downgrade
170 # These are internally, and not to be used from the outside at all
172 $_trap_nan = 0; # are NaNs ok? set w/ config()
173 $_trap_inf = 0; # are infs ok? set w/ config()
174 my $nan = 'NaN'; # constants for easier life
176 my $CALC = 'Math::BigInt::FastCalc'; # module to do the low level math
177 # default is FastCalc.pm
178 my $IMPORT = 0; # was import() called yet?
179 # used to make require work
180 my %WARN; # warn only once for low-level libs
181 my %CAN; # cache for $CALC->can(...)
182 my %CALLBACKS; # callbacks to notify on lib loads
183 my $EMU_LIB = 'Math/BigInt/CalcEmu.pm'; # emulate low-level math
185 ##############################################################################
186 # the old code had $rnd_mode, so we need to support it, too
189 sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; }
190 sub FETCH { return $round_mode; }
191 sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
195 # tie to enable $rnd_mode to work transparently
196 tie $rnd_mode, 'Math::BigInt';
198 # set up some handy alias names
199 *as_int = \&as_number;
200 *is_pos = \&is_positive;
201 *is_neg = \&is_negative;
204 ##############################################################################
209 # make Class->round_mode() work
211 my $class = ref($self) || $self || __PACKAGE__;
215 if ($m !~ /^(even|odd|\+inf|\-inf|zero|trunc|common)$/)
217 require Carp; Carp::croak ("Unknown round mode '$m'");
219 return ${"${class}::round_mode"} = $m;
221 ${"${class}::round_mode"};
227 # make Class->upgrade() work
229 my $class = ref($self) || $self || __PACKAGE__;
230 # need to set new value?
233 return ${"${class}::upgrade"} = $_[0];
235 ${"${class}::upgrade"};
241 # make Class->downgrade() work
243 my $class = ref($self) || $self || __PACKAGE__;
244 # need to set new value?
247 return ${"${class}::downgrade"} = $_[0];
249 ${"${class}::downgrade"};
255 # make Class->div_scale() work
257 my $class = ref($self) || $self || __PACKAGE__;
262 require Carp; Carp::croak ('div_scale must be greater than zero');
264 ${"${class}::div_scale"} = $_[0];
266 ${"${class}::div_scale"};
271 # $x->accuracy($a); ref($x) $a
272 # $x->accuracy(); ref($x)
273 # Class->accuracy(); class
274 # Class->accuracy($a); class $a
277 my $class = ref($x) || $x || __PACKAGE__;
280 # need to set new value?
284 # convert objects to scalars to avoid deep recursion. If object doesn't
285 # have numify(), then hopefully it will have overloading for int() and
286 # boolean test without wandering into a deep recursion path...
287 $a = $a->numify() if ref($a) && $a->can('numify');
291 # also croak on non-numerical
295 Carp::croak ('Argument to accuracy must be greater than zero');
299 require Carp; Carp::croak ('Argument to accuracy must be an integer');
304 # $object->accuracy() or fallback to global
305 $x->bround($a) if $a; # not for undef, 0
306 $x->{_a} = $a; # set/overwrite, even if not rounded
307 delete $x->{_p}; # clear P
308 $a = ${"${class}::accuracy"} unless defined $a; # proper return value
312 ${"${class}::accuracy"} = $a; # set global A
313 ${"${class}::precision"} = undef; # clear global P
315 return $a; # shortcut
319 # $object->accuracy() or fallback to global
320 $a = $x->{_a} if ref($x);
321 # but don't return global undef, when $x's accuracy is 0!
322 $a = ${"${class}::accuracy"} if !defined $a;
328 # $x->precision($p); ref($x) $p
329 # $x->precision(); ref($x)
330 # Class->precision(); class
331 # Class->precision($p); class $p
334 my $class = ref($x) || $x || __PACKAGE__;
340 # convert objects to scalars to avoid deep recursion. If object doesn't
341 # have numify(), then hopefully it will have overloading for int() and
342 # boolean test without wandering into a deep recursion path...
343 $p = $p->numify() if ref($p) && $p->can('numify');
344 if ((defined $p) && (int($p) != $p))
346 require Carp; Carp::croak ('Argument to precision must be an integer');
350 # $object->precision() or fallback to global
351 $x->bfround($p) if $p; # not for undef, 0
352 $x->{_p} = $p; # set/overwrite, even if not rounded
353 delete $x->{_a}; # clear A
354 $p = ${"${class}::precision"} unless defined $p; # proper return value
358 ${"${class}::precision"} = $p; # set global P
359 ${"${class}::accuracy"} = undef; # clear global A
361 return $p; # shortcut
365 # $object->precision() or fallback to global
366 $p = $x->{_p} if ref($x);
367 # but don't return global undef, when $x's precision is 0!
368 $p = ${"${class}::precision"} if !defined $p;
374 # return (or set) configuration data as hash ref
375 my $class = shift || 'Math::BigInt';
380 # try to set given options as arguments from hash
383 if (ref($args) ne 'HASH')
387 # these values can be "set"
391 upgrade downgrade precision accuracy round_mode div_scale/
394 $set_args->{$key} = $args->{$key} if exists $args->{$key};
395 delete $args->{$key};
400 Carp::croak ("Illegal key(s) '",
401 join("','",keys %$args),"' passed to $class\->config()");
403 foreach my $key (keys %$set_args)
405 if ($key =~ /^trap_(inf|nan)\z/)
407 ${"${class}::_trap_$1"} = ($set_args->{"trap_$1"} ? 1 : 0);
410 # use a call instead of just setting the $variable to check argument
411 $class->$key($set_args->{$key});
415 # now return actual configuration
419 lib_version => ${"${CALC}::VERSION"},
421 trap_nan => ${"${class}::_trap_nan"},
422 trap_inf => ${"${class}::_trap_inf"},
423 version => ${"${class}::VERSION"},
426 upgrade downgrade precision accuracy round_mode div_scale
429 $cfg->{$key} = ${"${class}::$key"};
436 # select accuracy parameter based on precedence,
437 # used by bround() and bfround(), may return undef for scale (means no op)
438 my ($x,$scale,$mode) = @_;
440 $scale = $x->{_a} unless defined $scale;
445 $scale = ${ $class . '::accuracy' } unless defined $scale;
446 $mode = ${ $class . '::round_mode' } unless defined $mode;
453 # select precision parameter based on precedence,
454 # used by bround() and bfround(), may return undef for scale (means no op)
455 my ($x,$scale,$mode) = @_;
457 $scale = $x->{_p} unless defined $scale;
462 $scale = ${ $class . '::precision' } unless defined $scale;
463 $mode = ${ $class . '::round_mode' } unless defined $mode;
468 ##############################################################################
476 # if two arguments, the first one is the class to "swallow" subclasses
484 return unless ref($x); # only for objects
486 my $self = bless {}, $c;
488 $self->{sign} = $x->{sign};
489 $self->{value} = $CALC->_copy($x->{value});
490 $self->{_a} = $x->{_a} if defined $x->{_a};
491 $self->{_p} = $x->{_p} if defined $x->{_p};
497 # create a new BigInt object from a string or another BigInt object.
498 # see hash keys documented at top
500 # the argument could be an object, so avoid ||, && etc on it, this would
501 # cause costly overloaded code to be called. The only allowed ops are
504 my ($class,$wanted,$a,$p,$r) = @_;
506 # avoid numify-calls by not using || on $wanted!
507 return $class->bzero($a,$p) if !defined $wanted; # default to 0
508 return $class->copy($wanted,$a,$p,$r)
509 if ref($wanted) && $wanted->isa($class); # MBI or subclass
511 $class->import() if $IMPORT == 0; # make require work
513 my $self = bless {}, $class;
515 # shortcut for "normal" numbers
516 if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*\z/))
518 $self->{sign} = $1 || '+';
520 if ($wanted =~ /^[+-]/)
522 # remove sign without touching wanted to make it work with constants
523 my $t = $wanted; $t =~ s/^[+-]//;
524 $self->{value} = $CALC->_new($t);
528 $self->{value} = $CALC->_new($wanted);
531 if ( (defined $a) || (defined $p)
532 || (defined ${"${class}::precision"})
533 || (defined ${"${class}::accuracy"})
536 $self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p);
541 # handle '+inf', '-inf' first
542 if ($wanted =~ /^[+-]?inf\z/)
544 $self->{sign} = $wanted; # set a default sign for bstr()
545 return $self->binf($wanted);
547 # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign
548 my ($mis,$miv,$mfv,$es,$ev) = _split($wanted);
553 require Carp; Carp::croak("$wanted is not a number in $class");
555 $self->{value} = $CALC->_zero();
556 $self->{sign} = $nan;
561 # _from_hex or _from_bin
562 $self->{value} = $mis->{value};
563 $self->{sign} = $mis->{sign};
564 return $self; # throw away $mis
566 # make integer from mantissa by adjusting exp, then convert to bigint
567 $self->{sign} = $$mis; # store sign
568 $self->{value} = $CALC->_zero(); # for all the NaN cases
569 my $e = int("$$es$$ev"); # exponent (avoid recursion)
572 my $diff = $e - CORE::length($$mfv);
573 if ($diff < 0) # Not integer
577 require Carp; Carp::croak("$wanted not an integer in $class");
580 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
581 $self->{sign} = $nan;
585 # adjust fraction and add it to value
586 #print "diff > 0 $$miv\n";
587 $$miv = $$miv . ($$mfv . '0' x $diff);
592 if ($$mfv ne '') # e <= 0
594 # fraction and negative/zero E => NOI
597 require Carp; Carp::croak("$wanted not an integer in $class");
599 #print "NOI 2 \$\$mfv '$$mfv'\n";
600 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
601 $self->{sign} = $nan;
605 # xE-y, and empty mfv
608 if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
612 require Carp; Carp::croak("$wanted not an integer in $class");
615 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
616 $self->{sign} = $nan;
620 $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
621 $self->{value} = $CALC->_new($$miv) if $self->{sign} =~ /^[+-]$/;
622 # if any of the globals is set, use them to round and store them inside $self
623 # do not round for new($x,undef,undef) since that is used by MBF to signal
625 $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p;
631 # create a bigint 'NaN', if given a BigInt, set it to 'NaN'
633 $self = $class if !defined $self;
636 my $c = $self; $self = {}; bless $self, $c;
639 if (${"${class}::_trap_nan"})
642 Carp::croak ("Tried to set $self to NaN in $class\::bnan()");
644 $self->import() if $IMPORT == 0; # make require work
645 return if $self->modify('bnan');
646 if ($self->can('_bnan'))
648 # use subclass to initialize
653 # otherwise do our own thing
654 $self->{value} = $CALC->_zero();
656 $self->{sign} = $nan;
657 delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly
663 # create a bigint '+-inf', if given a BigInt, set it to '+-inf'
664 # the sign is either '+', or if given, used from there
666 my $sign = shift; $sign = '+' if !defined $sign || $sign !~ /^-(inf)?$/;
667 $self = $class if !defined $self;
670 my $c = $self; $self = {}; bless $self, $c;
673 if (${"${class}::_trap_inf"})
676 Carp::croak ("Tried to set $self to +-inf in $class\::binf()");
678 $self->import() if $IMPORT == 0; # make require work
679 return if $self->modify('binf');
680 if ($self->can('_binf'))
682 # use subclass to initialize
687 # otherwise do our own thing
688 $self->{value} = $CALC->_zero();
690 $sign = $sign . 'inf' if $sign !~ /inf$/; # - => -inf
691 $self->{sign} = $sign;
692 ($self->{_a},$self->{_p}) = @_; # take over requested rounding
698 # create a bigint '+0', if given a BigInt, set it to 0
700 $self = __PACKAGE__ if !defined $self;
704 my $c = $self; $self = {}; bless $self, $c;
706 $self->import() if $IMPORT == 0; # make require work
707 return if $self->modify('bzero');
709 if ($self->can('_bzero'))
711 # use subclass to initialize
716 # otherwise do our own thing
717 $self->{value} = $CALC->_zero();
724 # call like: $x->bzero($a,$p,$r,$y);
725 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
730 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
732 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
740 # create a bigint '+1' (or -1 if given sign '-'),
741 # if given a BigInt, set it to +1 or -1, respectively
743 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
744 $self = $class if !defined $self;
748 my $c = $self; $self = {}; bless $self, $c;
750 $self->import() if $IMPORT == 0; # make require work
751 return if $self->modify('bone');
753 if ($self->can('_bone'))
755 # use subclass to initialize
760 # otherwise do our own thing
761 $self->{value} = $CALC->_one();
763 $self->{sign} = $sign;
768 # call like: $x->bone($sign,$a,$p,$r,$y);
769 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
773 # call like: $x->bone($sign,$a,$p,$r);
775 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
777 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
783 ##############################################################################
784 # string conversation
788 # (ref to BFLOAT or num_str ) return num_str
789 # Convert number from internal format to scientific string format.
790 # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
791 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
793 if ($x->{sign} !~ /^[+-]$/)
795 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
798 my ($m,$e) = $x->parts();
799 #$m->bstr() . 'e+' . $e->bstr(); # e can only be positive in BigInt
800 # 'e+' because E can only be positive in BigInt
801 $m->bstr() . 'e+' . $CALC->_str($e->{value});
806 # make a string from bigint object
807 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
809 if ($x->{sign} !~ /^[+-]$/)
811 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
814 my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
815 $es.$CALC->_str($x->{value});
820 # Make a "normal" scalar from a BigInt object
821 my $x = shift; $x = $class->new($x) unless ref $x;
823 return $x->bstr() if $x->{sign} !~ /^[+-]$/;
824 my $num = $CALC->_num($x->{value});
825 return -$num if $x->{sign} eq '-';
829 ##############################################################################
830 # public stuff (usually prefixed with "b")
834 # return the sign of the number: +/-/-inf/+inf/NaN
835 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
840 sub _find_round_parameters
842 # After any operation or when calling round(), the result is rounded by
843 # regarding the A & P from arguments, local parameters, or globals.
845 # !!!!!!! If you change this, remember to change round(), too! !!!!!!!!!!
847 # This procedure finds the round parameters, but it is for speed reasons
848 # duplicated in round. Otherwise, it is tested by the testsuite and used
851 # returns ($self) or ($self,$a,$p,$r) - sets $self to NaN of both A and P
852 # were requested/defined (locally or globally or both)
854 my ($self,$a,$p,$r,@args) = @_;
855 # $a accuracy, if given by caller
856 # $p precision, if given by caller
857 # $r round_mode, if given by caller
858 # @args all 'other' arguments (0 for unary, 1 for binary ops)
860 my $c = ref($self); # find out class of argument(s)
863 # now pick $a or $p, but only if we have got "arguments"
866 foreach ($self,@args)
868 # take the defined one, or if both defined, the one that is smaller
869 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
874 # even if $a is defined, take $p, to signal error for both defined
875 foreach ($self,@args)
877 # take the defined one, or if both defined, the one that is bigger
879 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
882 # if still none defined, use globals (#2)
883 $a = ${"$c\::accuracy"} unless defined $a;
884 $p = ${"$c\::precision"} unless defined $p;
886 # A == 0 is useless, so undef it to signal no rounding
887 $a = undef if defined $a && $a == 0;
890 return ($self) unless defined $a || defined $p; # early out
892 # set A and set P is an fatal error
893 return ($self->bnan()) if defined $a && defined $p; # error
895 $r = ${"$c\::round_mode"} unless defined $r;
896 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc|common)$/)
898 require Carp; Carp::croak ("Unknown round mode '$r'");
906 # Round $self according to given parameters, or given second argument's
907 # parameters or global defaults
909 # for speed reasons, _find_round_parameters is embeded here:
911 my ($self,$a,$p,$r,@args) = @_;
912 # $a accuracy, if given by caller
913 # $p precision, if given by caller
914 # $r round_mode, if given by caller
915 # @args all 'other' arguments (0 for unary, 1 for binary ops)
917 my $c = ref($self); # find out class of argument(s)
920 # now pick $a or $p, but only if we have got "arguments"
923 foreach ($self,@args)
925 # take the defined one, or if both defined, the one that is smaller
926 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
931 # even if $a is defined, take $p, to signal error for both defined
932 foreach ($self,@args)
934 # take the defined one, or if both defined, the one that is bigger
936 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
939 # if still none defined, use globals (#2)
940 $a = ${"$c\::accuracy"} unless defined $a;
941 $p = ${"$c\::precision"} unless defined $p;
943 # A == 0 is useless, so undef it to signal no rounding
944 $a = undef if defined $a && $a == 0;
947 return $self unless defined $a || defined $p; # early out
949 # set A and set P is an fatal error
950 return $self->bnan() if defined $a && defined $p;
952 $r = ${"$c\::round_mode"} unless defined $r;
953 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc|common)$/)
955 require Carp; Carp::croak ("Unknown round mode '$r'");
958 # now round, by calling either fround or ffround:
961 $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a;
963 else # both can't be undefined due to early out
965 $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p;
967 # bround() or bfround() already callled bnorm() if nec.
973 # (numstr or BINT) return BINT
974 # Normalize number -- no-op here
975 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
981 # (BINT or num_str) return BINT
982 # make number absolute, or return absolute BINT from string
983 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
985 return $x if $x->modify('babs');
986 # post-normalized abs for internal use (does nothing for NaN)
987 $x->{sign} =~ s/^-/+/;
993 # (BINT or num_str) return BINT
994 # negate number or make a negated number from string
995 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
997 return $x if $x->modify('bneg');
999 # for +0 dont negate (to have always normalized +0). Does nothing for 'NaN'
1000 $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $CALC->_is_zero($x->{value}));
1006 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
1007 # (BINT or num_str, BINT or num_str) return cond_code
1010 my ($self,$x,$y) = (ref($_[0]),@_);
1012 # objectify is costly, so avoid it
1013 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1015 ($self,$x,$y) = objectify(2,@_);
1018 return $upgrade->bcmp($x,$y) if defined $upgrade &&
1019 ((!$x->isa($self)) || (!$y->isa($self)));
1021 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1023 # handle +-inf and NaN
1024 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1025 return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
1026 return +1 if $x->{sign} eq '+inf';
1027 return -1 if $x->{sign} eq '-inf';
1028 return -1 if $y->{sign} eq '+inf';
1031 # check sign for speed first
1032 return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
1033 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
1035 # have same sign, so compare absolute values. Don't make tests for zero here
1036 # because it's actually slower than testin in Calc (especially w/ Pari et al)
1038 # post-normalized compare for internal use (honors signs)
1039 if ($x->{sign} eq '+')
1041 # $x and $y both > 0
1042 return $CALC->_acmp($x->{value},$y->{value});
1046 $CALC->_acmp($y->{value},$x->{value}); # swaped acmp (lib returns 0,1,-1)
1051 # Compares 2 values, ignoring their signs.
1052 # Returns one of undef, <0, =0, >0. (suitable for sort)
1053 # (BINT, BINT) return cond_code
1056 my ($self,$x,$y) = (ref($_[0]),@_);
1057 # objectify is costly, so avoid it
1058 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1060 ($self,$x,$y) = objectify(2,@_);
1063 return $upgrade->bacmp($x,$y) if defined $upgrade &&
1064 ((!$x->isa($self)) || (!$y->isa($self)));
1066 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1068 # handle +-inf and NaN
1069 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1070 return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
1071 return 1 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} !~ /^[+-]inf$/;
1074 $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
1079 # add second arg (BINT or string) to first (BINT) (modifies first)
1080 # return result as BINT
1083 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1084 # objectify is costly, so avoid it
1085 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1087 ($self,$x,$y,@r) = objectify(2,@_);
1090 return $x if $x->modify('badd');
1091 return $upgrade->badd($upgrade->new($x),$upgrade->new($y),@r) if defined $upgrade &&
1092 ((!$x->isa($self)) || (!$y->isa($self)));
1094 $r[3] = $y; # no push!
1095 # inf and NaN handling
1096 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1099 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1101 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1103 # +inf++inf or -inf+-inf => same, rest is NaN
1104 return $x if $x->{sign} eq $y->{sign};
1107 # +-inf + something => +inf
1108 # something +-inf => +-inf
1109 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
1113 my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
1117 $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
1121 my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
1124 $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
1129 # speedup, if equal, set result to 0
1130 $x->{value} = $CALC->_zero();
1135 $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
1143 # (BINT or num_str, BINT or num_str) return BINT
1144 # subtract second arg from first, modify first
1147 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1149 # objectify is costly, so avoid it
1150 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1152 ($self,$x,$y,@r) = objectify(2,@_);
1155 return $x if $x->modify('bsub');
1157 return $upgrade->new($x)->bsub($upgrade->new($y),@r) if defined $upgrade &&
1158 ((!$x->isa($self)) || (!$y->isa($self)));
1160 return $x->round(@r) if $y->is_zero();
1162 # To correctly handle the lone special case $x->bsub($x), we note the sign
1163 # of $x, then flip the sign from $y, and if the sign of $x did change, too,
1164 # then we caught the special case:
1165 my $xsign = $x->{sign};
1166 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
1167 if ($xsign ne $x->{sign})
1169 # special case of $x->bsub($x) results in 0
1170 return $x->bzero(@r) if $xsign =~ /^[+-]$/;
1171 return $x->bnan(); # NaN, -inf, +inf
1173 $x->badd($y,@r); # badd does not leave internal zeros
1174 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
1175 $x; # already rounded by badd() or no round nec.
1180 # increment arg by one
1181 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1182 return $x if $x->modify('binc');
1184 if ($x->{sign} eq '+')
1186 $x->{value} = $CALC->_inc($x->{value});
1187 return $x->round($a,$p,$r);
1189 elsif ($x->{sign} eq '-')
1191 $x->{value} = $CALC->_dec($x->{value});
1192 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1193 return $x->round($a,$p,$r);
1195 # inf, nan handling etc
1196 $x->badd($self->bone(),$a,$p,$r); # badd does round
1201 # decrement arg by one
1202 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1203 return $x if $x->modify('bdec');
1205 if ($x->{sign} eq '-')
1208 $x->{value} = $CALC->_inc($x->{value});
1212 return $x->badd($self->bone('-'),@r) unless $x->{sign} eq '+'; # inf or NaN
1214 if ($CALC->_is_zero($x->{value}))
1217 $x->{value} = $CALC->_one(); $x->{sign} = '-'; # 0 => -1
1222 $x->{value} = $CALC->_dec($x->{value});
1230 # calculate $x = $a ** $base + $b and return $a (e.g. the log() to base
1234 my ($self,$x,$base,@r) = (undef,@_);
1235 # objectify is costly, so avoid it
1236 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1238 ($self,$x,$base,@r) = objectify(1,ref($x),@_);
1241 return $x if $x->modify('blog');
1243 # inf, -inf, NaN, <0 => NaN
1245 if $x->{sign} ne '+' || (defined $base && $base->{sign} ne '+');
1247 return $upgrade->blog($upgrade->new($x),$base,@r) if
1250 # fix for bug #24969:
1251 # the default base is e (Euler's number) which is not an integer
1254 require Math::BigFloat;
1255 my $u = Math::BigFloat->blog(Math::BigFloat->new($x))->as_int();
1256 # modify $x in place
1257 $x->{value} = $u->{value};
1258 $x->{sign} = $u->{sign};
1262 my ($rc,$exact) = $CALC->_log_int($x->{value},$base->{value});
1263 return $x->bnan() unless defined $rc; # not possible to take log?
1270 # Calculate n over k (binomial coefficient or "choose" function) as integer.
1272 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1274 # objectify is costly, so avoid it
1275 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1277 ($self,$x,$y,@r) = objectify(2,@_);
1280 return $x if $x->modify('bnok');
1281 return $x->bnan() if $x->{sign} eq 'NaN' || $y->{sign} eq 'NaN';
1282 return $x->binf() if $x->{sign} eq '+inf';
1284 # k > n or k < 0 => 0
1285 my $cmp = $x->bacmp($y);
1286 return $x->bzero() if $cmp < 0 || $y->{sign} =~ /^-/;
1288 return $x->bone(@r) if $cmp == 0;
1290 if ($CALC->can('_nok'))
1292 $x->{value} = $CALC->_nok($x->{value},$y->{value});
1296 # ( 7 ) 7! 7*6*5 * 4*3*2*1 7 * 6 * 5
1297 # ( - ) = --------- = --------------- = ---------
1298 # ( 3 ) 3! (7-3)! 3*2*1 * 4*3*2*1 3 * 2 * 1
1300 # compute n - k + 2 (so we start with 5 in the example above)
1305 my $r = $z->copy(); $z->binc();
1306 my $d = $self->new(2);
1307 while ($z->bacmp($x) <= 0) # f < x ?
1309 $r->bmul($z); $r->bdiv($d);
1310 $z->binc(); $d->binc();
1312 $x->{value} = $r->{value}; $x->{sign} = '+';
1314 else { $x->bone(); }
1321 # Calculate e ** $x (Euler's number to the power of X), truncated to
1323 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1324 return $x if $x->modify('bexp');
1326 # inf, -inf, NaN, <0 => NaN
1327 return $x->bnan() if $x->{sign} eq 'NaN';
1328 return $x->bone() if $x->is_zero();
1329 return $x if $x->{sign} eq '+inf';
1330 return $x->bzero() if $x->{sign} eq '-inf';
1334 # run through Math::BigFloat unless told otherwise
1335 require Math::BigFloat unless defined $upgrade;
1336 local $upgrade = 'Math::BigFloat' unless defined $upgrade;
1337 # calculate result, truncate it to integer
1338 $u = $upgrade->bexp($upgrade->new($x),@r);
1341 if (!defined $upgrade)
1344 # modify $x in place
1345 $x->{value} = $u->{value};
1353 # (BINT or num_str, BINT or num_str) return BINT
1354 # does not modify arguments, but returns new object
1355 # Lowest Common Multiplicator
1357 my $y = shift; my ($x);
1364 $x = $class->new($y);
1369 my $y = shift; $y = $self->new($y) if !ref ($y);
1377 # (BINT or num_str, BINT or num_str) return BINT
1378 # does not modify arguments, but returns new object
1379 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1382 $y = $class->new($y) if !ref($y);
1384 my $x = $y->copy()->babs(); # keep arguments
1385 return $x->bnan() if $x->{sign} !~ /^[+-]$/; # x NaN?
1389 $y = shift; $y = $self->new($y) if !ref($y);
1390 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1391 $x->{value} = $CALC->_gcd($x->{value},$y->{value});
1392 last if $CALC->_is_one($x->{value});
1399 # (num_str or BINT) return BINT
1400 # represent ~x as twos-complement number
1401 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1402 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1404 return $x if $x->modify('bnot');
1405 $x->binc()->bneg(); # binc already does round
1408 ##############################################################################
1409 # is_foo test routines
1410 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1414 # return true if arg (BINT or num_str) is zero (array '+', '0')
1415 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1417 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1418 $CALC->_is_zero($x->{value});
1423 # return true if arg (BINT or num_str) is NaN
1424 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1426 $x->{sign} eq $nan ? 1 : 0;
1431 # return true if arg (BINT or num_str) is +-inf
1432 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1436 $sign = '[+-]inf' if $sign eq ''; # +- doesn't matter, only that's inf
1437 $sign = "[$1]inf" if $sign =~ /^([+-])(inf)?$/; # extract '+' or '-'
1438 return $x->{sign} =~ /^$sign$/ ? 1 : 0;
1440 $x->{sign} =~ /^[+-]inf$/ ? 1 : 0; # only +-inf is infinity
1445 # return true if arg (BINT or num_str) is +1, or -1 if sign is given
1446 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1448 $sign = '+' if !defined $sign || $sign ne '-';
1450 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1451 $CALC->_is_one($x->{value});
1456 # return true when arg (BINT or num_str) is odd, false for even
1457 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1459 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1460 $CALC->_is_odd($x->{value});
1465 # return true when arg (BINT or num_str) is even, false for odd
1466 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1468 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1469 $CALC->_is_even($x->{value});
1474 # return true when arg (BINT or num_str) is positive (>= 0)
1475 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1477 return 1 if $x->{sign} eq '+inf'; # +inf is positive
1479 # 0+ is neither positive nor negative
1480 ($x->{sign} eq '+' && !$x->is_zero()) ? 1 : 0;
1485 # return true when arg (BINT or num_str) is negative (< 0)
1486 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1488 $x->{sign} =~ /^-/ ? 1 : 0; # -inf is negative, but NaN is not
1493 # return true when arg (BINT or num_str) is an integer
1494 # always true for BigInt, but different for BigFloats
1495 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1497 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1500 ###############################################################################
1504 # multiply two numbers -- stolen from Knuth Vol 2 pg 233
1505 # (BINT or num_str, BINT or num_str) return BINT
1508 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1509 # objectify is costly, so avoid it
1510 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1512 ($self,$x,$y,@r) = objectify(2,@_);
1515 return $x if $x->modify('bmul');
1517 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1520 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1522 return $x->bnan() if $x->is_zero() || $y->is_zero();
1523 # result will always be +-inf:
1524 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1525 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1526 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1527 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1528 return $x->binf('-');
1531 return $upgrade->bmul($x,$upgrade->new($y),@r)
1532 if defined $upgrade && !$y->isa($self);
1534 $r[3] = $y; # no push here
1536 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1538 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1539 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1546 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1547 my ($self,$x,$y) = @_;
1549 # NaN if x == NaN or y == NaN or x==y==0
1550 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1551 if (($x->is_nan() || $y->is_nan()) ||
1552 ($x->is_zero() && $y->is_zero()));
1554 # +-inf / +-inf == NaN, reminder also NaN
1555 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1557 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1559 # x / +-inf => 0, remainder x (works even if x == 0)
1560 if ($y->{sign} =~ /^[+-]inf$/)
1562 my $t = $x->copy(); # bzero clobbers up $x
1563 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1566 # 5 / 0 => +inf, -6 / 0 => -inf
1567 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1568 # exception: -8 / 0 has remainder -8, not 8
1569 # exception: -inf / 0 has remainder -inf, not inf
1572 # +-inf / 0 => special case for -inf
1573 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1574 if (!$x->is_zero() && !$x->is_inf())
1576 my $t = $x->copy(); # binf clobbers up $x
1578 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1582 # last case: +-inf / ordinary number
1584 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1586 return wantarray ? ($x,$self->bzero()) : $x;
1591 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1592 # (BINT,BINT) (quo,rem) or BINT (only rem)
1595 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1596 # objectify is costly, so avoid it
1597 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1599 ($self,$x,$y,@r) = objectify(2,@_);
1602 return $x if $x->modify('bdiv');
1604 return $self->_div_inf($x,$y)
1605 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1607 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1608 if defined $upgrade;
1610 $r[3] = $y; # no push!
1612 # calc new sign and in case $y == +/- 1, return $x
1613 my $xsign = $x->{sign}; # keep
1614 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1618 my $rem = $self->bzero();
1619 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1620 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1621 $rem->{_a} = $x->{_a};
1622 $rem->{_p} = $x->{_p};
1624 if (! $CALC->_is_zero($rem->{value}))
1626 $rem->{sign} = $y->{sign};
1627 $rem = $y->copy()->bsub($rem) if $xsign ne $y->{sign}; # one of them '-'
1631 $rem->{sign} = '+'; # dont leave -0
1637 $x->{value} = $CALC->_div($x->{value},$y->{value});
1638 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1643 ###############################################################################
1648 # modulus (or remainder)
1649 # (BINT or num_str, BINT or num_str) return BINT
1652 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1653 # objectify is costly, so avoid it
1654 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1656 ($self,$x,$y,@r) = objectify(2,@_);
1659 return $x if $x->modify('bmod');
1660 $r[3] = $y; # no push!
1661 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1663 my ($d,$r) = $self->_div_inf($x,$y);
1664 $x->{sign} = $r->{sign};
1665 $x->{value} = $r->{value};
1666 return $x->round(@r);
1669 # calc new sign and in case $y == +/- 1, return $x
1670 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1671 if (!$CALC->_is_zero($x->{value}))
1673 $x->{value} = $CALC->_sub($y->{value},$x->{value},1) # $y-$x
1674 if ($x->{sign} ne $y->{sign});
1675 $x->{sign} = $y->{sign};
1679 $x->{sign} = '+'; # dont leave -0
1686 # Modular inverse. given a number which is (hopefully) relatively
1687 # prime to the modulus, calculate its inverse using Euclid's
1688 # alogrithm. If the number is not relatively prime to the modulus
1689 # (i.e. their gcd is not one) then NaN is returned.
1692 my ($self,$x,$y,@r) = (undef,@_);
1693 # objectify is costly, so avoid it
1694 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1696 ($self,$x,$y,@r) = objectify(2,@_);
1699 return $x if $x->modify('bmodinv');
1702 if ($y->{sign} ne '+' # -, NaN, +inf, -inf
1703 || $x->is_zero() # or num == 0
1704 || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1707 # put least residue into $x if $x was negative, and thus make it positive
1708 $x->bmod($y) if $x->{sign} eq '-';
1711 ($x->{value},$sign) = $CALC->_modinv($x->{value},$y->{value});
1712 return $x->bnan() if !defined $x->{value}; # in case no GCD found
1713 return $x if !defined $sign; # already real result
1714 $x->{sign} = $sign; # flip/flop see below
1715 $x->bmod($y); # calc real result
1721 # takes a very large number to a very large exponent in a given very
1722 # large modulus, quickly, thanks to binary exponentation. supports
1723 # negative exponents.
1724 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1726 return $num if $num->modify('bmodpow');
1728 # check modulus for valid values
1729 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1730 || $mod->is_zero());
1732 # check exponent for valid values
1733 if ($exp->{sign} =~ /\w/)
1735 # i.e., if it's NaN, +inf, or -inf...
1736 return $num->bnan();
1739 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
1741 # check num for valid values (also NaN if there was no inverse but $exp < 0)
1742 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1744 # $mod is positive, sign on $exp is ignored, result also positive
1745 $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
1749 ###############################################################################
1753 # (BINT or num_str, BINT or num_str) return BINT
1754 # compute factorial number from $x, modify $x in place
1755 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1757 return $x if $x->modify('bfac') || $x->{sign} eq '+inf'; # inf => inf
1758 return $x->bnan() if $x->{sign} ne '+'; # NaN, <0 etc => NaN
1760 $x->{value} = $CALC->_fac($x->{value});
1766 # (BINT or num_str, BINT or num_str) return BINT
1767 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1768 # modifies first argument
1771 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1772 # objectify is costly, so avoid it
1773 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1775 ($self,$x,$y,@r) = objectify(2,@_);
1778 return $x if $x->modify('bpow');
1780 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1783 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1785 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1791 if ($x->{sign} =~ /^[+-]inf/)
1794 return $x->bnan() if $y->is_zero();
1795 # -inf ** -1 => 1/inf => 0
1796 return $x->bzero() if $y->is_one('-') && $x->is_negative();
1799 return $x if $x->{sign} eq '+inf';
1801 # -inf ** Y => -inf if Y is odd
1802 return $x if $y->is_odd();
1808 return $x if $x->is_one();
1811 return $x if $x->is_zero() && $y->{sign} =~ /^[+]/;
1814 return $x->binf() if $x->is_zero();
1817 return $x->bnan() if $x->is_one('-') && $y->{sign} =~ /^[-]/;
1820 return $x->bzero() if $x->{sign} eq '-' && $y->{sign} =~ /^[-]/;
1823 return $x->bnan() if $x->{sign} eq '-';
1826 return $x->binf() if $y->{sign} =~ /^[+]/;
1831 return $upgrade->bpow($upgrade->new($x),$y,@r)
1832 if defined $upgrade && (!$y->isa($self) || $y->{sign} eq '-');
1834 $r[3] = $y; # no push!
1836 # cases 0 ** Y, X ** 0, X ** 1, 1 ** Y are handled by Calc or Emu
1839 $new_sign = $y->is_odd() ? '-' : '+' if ($x->{sign} ne '+');
1841 # 0 ** -7 => ( 1 / (0 ** 7)) => 1 / 0 => +inf
1843 if $y->{sign} eq '-' && $x->{sign} eq '+' && $CALC->_is_zero($x->{value});
1844 # 1 ** -y => 1 / (1 ** |y|)
1845 # so do test for negative $y after above's clause
1846 return $x->bnan() if $y->{sign} eq '-' && !$CALC->_is_one($x->{value});
1848 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1849 $x->{sign} = $new_sign;
1850 $x->{sign} = '+' if $CALC->_is_zero($y->{value});
1856 # (BINT or num_str, BINT or num_str) return BINT
1857 # compute x << y, base n, y >= 0
1860 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1861 # objectify is costly, so avoid it
1862 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1864 ($self,$x,$y,$n,@r) = objectify(2,@_);
1867 return $x if $x->modify('blsft');
1868 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1869 return $x->round(@r) if $y->is_zero();
1871 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1873 $x->{value} = $CALC->_lsft($x->{value},$y->{value},$n);
1879 # (BINT or num_str, BINT or num_str) return BINT
1880 # compute x >> y, base n, y >= 0
1883 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1884 # objectify is costly, so avoid it
1885 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1887 ($self,$x,$y,$n,@r) = objectify(2,@_);
1890 return $x if $x->modify('brsft');
1891 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1892 return $x->round(@r) if $y->is_zero();
1893 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
1895 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1897 # this only works for negative numbers when shifting in base 2
1898 if (($x->{sign} eq '-') && ($n == 2))
1900 return $x->round(@r) if $x->is_one('-'); # -1 => -1
1903 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1904 # but perhaps there is a better emulation for two's complement shift...
1905 # if $y != 1, we must simulate it by doing:
1906 # convert to bin, flip all bits, shift, and be done
1907 $x->binc(); # -3 => -2
1908 my $bin = $x->as_bin();
1909 $bin =~ s/^-0b//; # strip '-0b' prefix
1910 $bin =~ tr/10/01/; # flip bits
1912 if ($y >= CORE::length($bin))
1914 $bin = '0'; # shifting to far right creates -1
1915 # 0, because later increment makes
1916 # that 1, attached '-' makes it '-1'
1917 # because -1 >> x == -1 !
1921 $bin =~ s/.{$y}$//; # cut off at the right side
1922 $bin = '1' . $bin; # extend left side by one dummy '1'
1923 $bin =~ tr/10/01/; # flip bits back
1925 my $res = $self->new('0b'.$bin); # add prefix and convert back
1926 $res->binc(); # remember to increment
1927 $x->{value} = $res->{value}; # take over value
1928 return $x->round(@r); # we are done now, magic, isn't?
1930 # x < 0, n == 2, y == 1
1931 $x->bdec(); # n == 2, but $y == 1: this fixes it
1934 $x->{value} = $CALC->_rsft($x->{value},$y->{value},$n);
1940 #(BINT or num_str, BINT or num_str) return BINT
1944 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1945 # objectify is costly, so avoid it
1946 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1948 ($self,$x,$y,@r) = objectify(2,@_);
1951 return $x if $x->modify('band');
1953 $r[3] = $y; # no push!
1955 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1957 my $sx = $x->{sign} eq '+' ? 1 : -1;
1958 my $sy = $y->{sign} eq '+' ? 1 : -1;
1960 if ($sx == 1 && $sy == 1)
1962 $x->{value} = $CALC->_and($x->{value},$y->{value});
1963 return $x->round(@r);
1966 if ($CAN{signed_and})
1968 $x->{value} = $CALC->_signed_and($x->{value},$y->{value},$sx,$sy);
1969 return $x->round(@r);
1973 __emu_band($self,$x,$y,$sx,$sy,@r);
1978 #(BINT or num_str, BINT or num_str) return BINT
1982 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1983 # objectify is costly, so avoid it
1984 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1986 ($self,$x,$y,@r) = objectify(2,@_);
1989 return $x if $x->modify('bior');
1990 $r[3] = $y; # no push!
1992 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1994 my $sx = $x->{sign} eq '+' ? 1 : -1;
1995 my $sy = $y->{sign} eq '+' ? 1 : -1;
1997 # the sign of X follows the sign of X, e.g. sign of Y irrelevant for bior()
1999 # don't use lib for negative values
2000 if ($sx == 1 && $sy == 1)
2002 $x->{value} = $CALC->_or($x->{value},$y->{value});
2003 return $x->round(@r);
2006 # if lib can do negative values, let it handle this
2007 if ($CAN{signed_or})
2009 $x->{value} = $CALC->_signed_or($x->{value},$y->{value},$sx,$sy);
2010 return $x->round(@r);
2014 __emu_bior($self,$x,$y,$sx,$sy,@r);
2019 #(BINT or num_str, BINT or num_str) return BINT
2023 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2024 # objectify is costly, so avoid it
2025 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2027 ($self,$x,$y,@r) = objectify(2,@_);
2030 return $x if $x->modify('bxor');
2031 $r[3] = $y; # no push!
2033 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
2035 my $sx = $x->{sign} eq '+' ? 1 : -1;
2036 my $sy = $y->{sign} eq '+' ? 1 : -1;
2038 # don't use lib for negative values
2039 if ($sx == 1 && $sy == 1)
2041 $x->{value} = $CALC->_xor($x->{value},$y->{value});
2042 return $x->round(@r);
2045 # if lib can do negative values, let it handle this
2046 if ($CAN{signed_xor})
2048 $x->{value} = $CALC->_signed_xor($x->{value},$y->{value},$sx,$sy);
2049 return $x->round(@r);
2053 __emu_bxor($self,$x,$y,$sx,$sy,@r);
2058 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2060 my $e = $CALC->_len($x->{value});
2061 wantarray ? ($e,0) : $e;
2066 # return the nth decimal digit, negative values count backward, 0 is right
2067 my ($self,$x,$n) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2069 $n = $n->numify() if ref($n);
2070 $CALC->_digit($x->{value},$n||0);
2075 # return the amount of trailing zeros in $x (as scalar)
2077 $x = $class->new($x) unless ref $x;
2079 return 0 if $x->{sign} !~ /^[+-]$/; # NaN, inf, -inf etc
2081 $CALC->_zeros($x->{value}); # must handle odd values, 0 etc
2086 # calculate square root of $x
2087 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2089 return $x if $x->modify('bsqrt');
2091 return $x->bnan() if $x->{sign} !~ /^\+/; # -x or -inf or NaN => NaN
2092 return $x if $x->{sign} eq '+inf'; # sqrt(+inf) == inf
2094 return $upgrade->bsqrt($x,@r) if defined $upgrade;
2096 $x->{value} = $CALC->_sqrt($x->{value});
2102 # calculate $y'th root of $x
2105 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2107 $y = $self->new(2) unless defined $y;
2109 # objectify is costly, so avoid it
2110 if ((!ref($x)) || (ref($x) ne ref($y)))
2112 ($self,$x,$y,@r) = objectify(2,$self || $class,@_);
2115 return $x if $x->modify('broot');
2117 # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
2118 return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
2119 $y->{sign} !~ /^\+$/;
2121 return $x->round(@r)
2122 if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
2124 return $upgrade->new($x)->broot($upgrade->new($y),@r) if defined $upgrade;
2126 $x->{value} = $CALC->_root($x->{value},$y->{value});
2132 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
2133 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2135 if ($x->{sign} !~ /^[+-]$/)
2137 my $s = $x->{sign}; $s =~ s/^[+-]//; # NaN, -inf,+inf => NaN or inf
2138 return $self->new($s);
2140 return $self->bone() if $x->is_zero();
2142 # 12300 => 2 trailing zeros => exponent is 2
2143 $self->new( $CALC->_zeros($x->{value}) );
2148 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
2149 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2151 if ($x->{sign} !~ /^[+-]$/)
2153 # for NaN, +inf, -inf: keep the sign
2154 return $self->new($x->{sign});
2156 my $m = $x->copy(); delete $m->{_p}; delete $m->{_a};
2158 # that's a bit inefficient:
2159 my $zeros = $CALC->_zeros($m->{value});
2160 $m->brsft($zeros,10) if $zeros != 0;
2166 # return a copy of both the exponent and the mantissa
2167 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2169 ($x->mantissa(),$x->exponent());
2172 ##############################################################################
2173 # rounding functions
2177 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
2178 # $n == 0 || $n == 1 => round to integer
2179 my $x = shift; my $self = ref($x) || $x; $x = $self->new($x) unless ref $x;
2181 my ($scale,$mode) = $x->_scale_p(@_);
2183 return $x if !defined $scale || $x->modify('bfround'); # no-op
2185 # no-op for BigInts if $n <= 0
2186 $x->bround( $x->length()-$scale, $mode) if $scale > 0;
2188 delete $x->{_a}; # delete to save memory
2189 $x->{_p} = $scale; # store new _p
2193 sub _scan_for_nonzero
2195 # internal, used by bround() to scan for non-zeros after a '5'
2196 my ($x,$pad,$xs,$len) = @_;
2198 return 0 if $len == 1; # "5" is trailed by invisible zeros
2199 my $follow = $pad - 1;
2200 return 0 if $follow > $len || $follow < 1;
2202 # use the string form to check whether only '0's follow or not
2203 substr ($xs,-$follow) =~ /[^0]/ ? 1 : 0;
2208 # Exists to make life easier for switch between MBF and MBI (should we
2209 # autoload fxxx() like MBF does for bxxx()?)
2210 my $x = shift; $x = $class->new($x) unless ref $x;
2216 # accuracy: +$n preserve $n digits from left,
2217 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
2219 # and overwrite the rest with 0's, return normalized number
2220 # do not return $x->bnorm(), but $x
2222 my $x = shift; $x = $class->new($x) unless ref $x;
2223 my ($scale,$mode) = $x->_scale_a(@_);
2224 return $x if !defined $scale || $x->modify('bround'); # no-op
2226 if ($x->is_zero() || $scale == 0)
2228 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2231 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2233 # we have fewer digits than we want to scale to
2234 my $len = $x->length();
2235 # convert $scale to a scalar in case it is an object (put's a limit on the
2236 # number length, but this would already limited by memory constraints), makes
2238 $scale = $scale->numify() if ref ($scale);
2240 # scale < 0, but > -len (not >=!)
2241 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2243 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2247 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2248 my ($pad,$digit_round,$digit_after);
2249 $pad = $len - $scale;
2250 $pad = abs($scale-1) if $scale < 0;
2252 # do not use digit(), it is very costly for binary => decimal
2253 # getting the entire string is also costly, but we need to do it only once
2254 my $xs = $CALC->_str($x->{value});
2257 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2258 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2259 $digit_round = '0'; $digit_round = substr($xs,$pl,1) if $pad <= $len;
2260 $pl++; $pl ++ if $pad >= $len;
2261 $digit_after = '0'; $digit_after = substr($xs,$pl,1) if $pad > 0;
2263 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2264 # closer at the remaining digits of the original $x, remember decision
2265 my $round_up = 1; # default round up
2267 ($mode eq 'trunc') || # trunc by round down
2268 ($digit_after =~ /[01234]/) || # round down anyway,
2270 ($digit_after eq '5') && # not 5000...0000
2271 ($x->_scan_for_nonzero($pad,$xs,$len) == 0) &&
2273 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2274 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2275 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2276 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2277 ($mode eq 'zero') # round down if zero, sign adjusted below
2279 my $put_back = 0; # not yet modified
2281 if (($pad > 0) && ($pad <= $len))
2283 substr($xs,-$pad,$pad) = '0' x $pad; # replace with '00...'
2284 $put_back = 1; # need to put back
2288 $x->bzero(); # round to '0'
2291 if ($round_up) # what gave test above?
2293 $put_back = 1; # need to put back
2294 $pad = $len, $xs = '0' x $pad if $scale < 0; # tlr: whack 0.51=>1.0
2296 # we modify directly the string variant instead of creating a number and
2297 # adding it, since that is faster (we already have the string)
2298 my $c = 0; $pad ++; # for $pad == $len case
2299 while ($pad <= $len)
2301 $c = substr($xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2302 substr($xs,-$pad,1) = $c; $pad++;
2303 last if $c != 0; # no overflow => early out
2305 $xs = '1'.$xs if $c == 0;
2308 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back, if needed
2310 $x->{_a} = $scale if $scale >= 0;
2313 $x->{_a} = $len+$scale;
2314 $x->{_a} = 0 if $scale < -$len;
2321 # return integer less or equal then number; no-op since it's already integer
2322 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2329 # return integer greater or equal then number; no-op since it's already int
2330 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2337 # An object might be asked to return itself as bigint on certain overloaded
2338 # operations. This does exactly this, so that sub classes can simple inherit
2339 # it or override with their own integer conversion routine.
2345 # return as hex string, with prefixed 0x
2346 my $x = shift; $x = $class->new($x) if !ref($x);
2348 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2351 $s = $x->{sign} if $x->{sign} eq '-';
2352 $s . $CALC->_as_hex($x->{value});
2357 # return as binary string, with prefixed 0b
2358 my $x = shift; $x = $class->new($x) if !ref($x);
2360 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2362 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2363 return $s . $CALC->_as_bin($x->{value});
2368 # return as octal string, with prefixed 0
2369 my $x = shift; $x = $class->new($x) if !ref($x);
2371 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2373 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2374 return $s . $CALC->_as_oct($x->{value});
2377 ##############################################################################
2378 # private stuff (internal use only)
2382 # check for strings, if yes, return objects instead
2384 # the first argument is number of args objectify() should look at it will
2385 # return $count+1 elements, the first will be a classname. This is because
2386 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2387 # useless objects being created and thrown away. So we cannot simple loop
2388 # over @_. If the given count is 0, all arguments will be used.
2390 # If the second arg is a ref, use it as class.
2391 # If not, try to use it as classname, unless undef, then use $class
2392 # (aka Math::BigInt). The latter shouldn't happen,though.
2395 # $x->badd(1); => ref x, scalar y
2396 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2397 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2398 # Math::BigInt::badd(1,2); => scalar x, scalar y
2399 # In the last case we check number of arguments to turn it silently into
2400 # $class,1,2. (We can not take '1' as class ;o)
2401 # badd($class,1) is not supported (it should, eventually, try to add undef)
2402 # currently it tries 'Math::BigInt' + 1, which will not work.
2404 # some shortcut for the common cases
2406 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2408 my $count = abs(shift || 0);
2410 my (@a,$k,$d); # resulting array, temp, and downgrade
2413 # okay, got object as first
2418 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2420 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2424 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2425 if (defined ${"$a[0]::downgrade"})
2427 $d = ${"$a[0]::downgrade"};
2428 ${"$a[0]::downgrade"} = undef;
2431 my $up = ${"$a[0]::upgrade"};
2432 # print STDERR "# Now in objectify, my class is today $a[0], count = $count\n";
2440 $k = $a[0]->new($k);
2442 elsif (!defined $up && ref($k) ne $a[0])
2444 # foreign object, try to convert to integer
2445 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2458 $k = $a[0]->new($k);
2460 elsif (!defined $up && ref($k) ne $a[0])
2462 # foreign object, try to convert to integer
2463 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2467 push @a,@_; # return other params, too
2471 require Carp; Carp::croak ("$class objectify needs list context");
2473 ${"$a[0]::downgrade"} = $d;
2477 sub _register_callback
2479 my ($class,$callback) = @_;
2481 if (ref($callback) ne 'CODE')
2484 Carp::croak ("$callback is not a coderef");
2486 $CALLBACKS{$class} = $callback;
2493 $IMPORT++; # remember we did import()
2494 my @a; my $l = scalar @_;
2495 my $warn_or_die = 0; # 0 - no warn, 1 - warn, 2 - die
2496 for ( my $i = 0; $i < $l ; $i++ )
2498 if ($_[$i] eq ':constant')
2500 # this causes overlord er load to step in
2502 integer => sub { $self->new(shift) },
2503 binary => sub { $self->new(shift) };
2505 elsif ($_[$i] eq 'upgrade')
2507 # this causes upgrading
2508 $upgrade = $_[$i+1]; # or undef to disable
2511 elsif ($_[$i] =~ /^(lib|try|only)\z/)
2513 # this causes a different low lib to take care...
2514 $CALC = $_[$i+1] || '';
2515 # lib => 1 (warn on fallback), try => 0 (no warn), only => 2 (die on fallback)
2516 $warn_or_die = 1 if $_[$i] eq 'lib';
2517 $warn_or_die = 2 if $_[$i] eq 'only';
2525 # any non :constant stuff is handled by our parent, Exporter
2530 $self->SUPER::import(@a); # need it for subclasses
2531 $self->export_to_level(1,$self,@a); # need it for MBF
2534 # try to load core math lib
2535 my @c = split /\s*,\s*/,$CALC;
2538 $_ =~ tr/a-zA-Z0-9://cd; # limit to sane characters
2540 push @c, \'FastCalc', \'Calc' # if all fail, try these
2541 if $warn_or_die < 2; # but not for "only"
2542 $CALC = ''; # signal error
2545 # fallback libraries are "marked" as \'string', extract string if nec.
2546 my $lib = $l; $lib = $$l if ref($l);
2548 next if ($lib || '') eq '';
2549 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2553 # Perl < 5.6.0 dies with "out of memory!" when eval("") and ':constant' is
2554 # used in the same script, or eval("") inside import().
2555 my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
2556 my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
2558 $file = File::Spec->catfile (@parts, $file);
2559 eval { require "$file"; $lib->import( @c ); }
2563 eval "use $lib qw/@c/;";
2568 # loaded it ok, see if the api_version() is high enough
2569 if ($lib->can('api_version') && $lib->api_version() >= 1.0)
2572 # api_version matches, check if it really provides anything we need
2576 add mul div sub dec inc
2577 acmp len digit is_one is_zero is_even is_odd
2579 zeros new copy check
2580 from_hex from_oct from_bin as_hex as_bin as_oct
2581 rsft lsft xor and or
2582 mod sqrt root fac pow modinv modpow log_int gcd
2585 if (!$lib->can("_$method"))
2587 if (($WARN{$lib}||0) < 2)
2590 Carp::carp ("$lib is missing method '_$method'");
2591 $WARN{$lib} = 1; # still warn about the lib
2600 if ($warn_or_die > 0 && ref($l))
2603 my $msg = "Math::BigInt: couldn't load specified math lib(s), fallback to $lib";
2604 Carp::carp ($msg) if $warn_or_die == 1;
2605 Carp::croak ($msg) if $warn_or_die == 2;
2607 last; # found a usable one, break
2611 if (($WARN{$lib}||0) < 2)
2613 my $ver = eval "\$$lib\::VERSION" || 'unknown';
2615 Carp::carp ("Cannot load outdated $lib v$ver, please upgrade");
2616 $WARN{$lib} = 2; # never warn again
2624 if ($warn_or_die == 2)
2626 Carp::croak ("Couldn't load specified math lib(s) and fallback disallowed");
2630 Carp::croak ("Couldn't load any math lib(s), not even fallback to Calc.pm");
2635 foreach my $class (keys %CALLBACKS)
2637 &{$CALLBACKS{$class}}($CALC);
2640 # Fill $CAN with the results of $CALC->can(...) for emulating lower math lib
2644 for my $method (qw/ signed_and signed_or signed_xor /)
2646 $CAN{$method} = $CALC->can("_$method") ? 1 : 0;
2654 # create a bigint from a hexadecimal string
2655 my ($self, $hs) = @_;
2657 my $rc = $self->__from_hex($hs);
2659 return $self->bnan() unless defined $rc;
2666 # create a bigint from a hexadecimal string
2667 my ($self, $bs) = @_;
2669 my $rc = $self->__from_bin($bs);
2671 return $self->bnan() unless defined $rc;
2678 # create a bigint from a hexadecimal string
2679 my ($self, $os) = @_;
2681 my $x = $self->bzero();
2684 $os =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2685 $os =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2687 return $x->bnan() if $os !~ /^[\-\+]?0[0-9]+$/;
2689 my $sign = '+'; $sign = '-' if $os =~ /^-/;
2691 $os =~ s/^[+-]//; # strip sign
2692 $x->{value} = $CALC->_from_oct($os);
2693 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2700 # convert a (ref to) big hex string to BigInt, return undef for error
2703 my $x = Math::BigInt->bzero();
2706 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2707 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2709 return $x->bnan() if $hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2711 my $sign = '+'; $sign = '-' if $hs =~ /^-/;
2713 $hs =~ s/^[+-]//; # strip sign
2714 $x->{value} = $CALC->_from_hex($hs);
2715 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2722 # convert a (ref to) big binary string to BigInt, return undef for error
2725 my $x = Math::BigInt->bzero();
2728 $bs =~ s/([01])_([01])/$1$2/g;
2729 $bs =~ s/([01])_([01])/$1$2/g;
2730 return $x->bnan() if $bs !~ /^[+-]?0b[01]+$/;
2732 my $sign = '+'; $sign = '-' if $bs =~ /^\-/;
2733 $bs =~ s/^[+-]//; # strip sign
2735 $x->{value} = $CALC->_from_bin($bs);
2736 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2742 # input: num_str; output: undef for invalid or
2743 # (\$mantissa_sign,\$mantissa_value,\$mantissa_fraction,\$exp_sign,\$exp_value)
2744 # Internal, take apart a string and return the pieces.
2745 # Strip leading/trailing whitespace, leading zeros, underscore and reject
2749 # strip white space at front, also extranous leading zeros
2750 $x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2751 $x =~ s/^\s+//; # but this will
2752 $x =~ s/\s+$//g; # strip white space at end
2754 # shortcut, if nothing to split, return early
2755 if ($x =~ /^[+-]?[0-9]+\z/)
2757 $x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2758 return (\$sign, \$x, \'', \'', \0);
2761 # invalid starting char?
2762 return if $x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2764 return __from_hex($x) if $x =~ /^[\-\+]?0x/; # hex string
2765 return __from_bin($x) if $x =~ /^[\-\+]?0b/; # binary string
2767 # strip underscores between digits
2768 $x =~ s/([0-9])_([0-9])/$1$2/g;
2769 $x =~ s/([0-9])_([0-9])/$1$2/g; # do twice for 1_2_3
2771 # some possible inputs:
2772 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2773 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 # 0e999
2775 my ($m,$e,$last) = split /[Ee]/,$x;
2776 return if defined $last; # last defined => 1e2E3 or others
2777 $e = '0' if !defined $e || $e eq "";
2779 # sign,value for exponent,mantint,mantfrac
2780 my ($es,$ev,$mis,$miv,$mfv);
2782 if ($e =~ /^([+-]?)0*([0-9]+)$/) # strip leading zeros
2786 return if $m eq '.' || $m eq '';
2787 my ($mi,$mf,$lastf) = split /\./,$m;
2788 return if defined $lastf; # lastf defined => 1.2.3 or others
2789 $mi = '0' if !defined $mi;
2790 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2791 $mf = '0' if !defined $mf || $mf eq '';
2792 if ($mi =~ /^([+-]?)0*([0-9]+)$/) # strip leading zeros
2794 $mis = $1||'+'; $miv = $2;
2795 return unless ($mf =~ /^([0-9]*?)0*$/); # strip trailing zeros
2797 # handle the 0e999 case here
2798 $ev = 0 if $miv eq '0' && $mfv eq '';
2799 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2802 return; # NaN, not a number
2805 ##############################################################################
2806 # internal calculation routines (others are in Math::BigInt::Calc etc)
2810 # (BINT or num_str, BINT or num_str) return BINT
2811 # does modify first argument
2815 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2816 my $method = ref($x) . '::bgcd';
2818 $x * $ty / &$method($x,$ty);
2821 ###############################################################################
2822 # this method returns 0 if the object can be modified, or 1 if not.
2823 # We use a fast constant sub() here, to avoid costly calls. Subclasses
2824 # may override it with special code (f.i. Math::BigInt::Constant does so)
2826 sub modify () { 0; }
2835 Math::BigInt - Arbitrary size integer/float math package
2841 # or make it faster: install (optional) Math::BigInt::GMP
2842 # and always use (it will fall back to pure Perl if the
2843 # GMP library is not installed):
2845 # will warn if Math::BigInt::GMP cannot be found
2846 use Math::BigInt lib => 'GMP';
2848 # to supress the warning use this:
2849 # use Math::BigInt try => 'GMP';
2851 my $str = '1234567890';
2852 my @values = (64,74,18);
2853 my $n = 1; my $sign = '-';
2856 $x = Math::BigInt->new($str); # defaults to 0
2857 $y = $x->copy(); # make a true copy
2858 $nan = Math::BigInt->bnan(); # create a NotANumber
2859 $zero = Math::BigInt->bzero(); # create a +0
2860 $inf = Math::BigInt->binf(); # create a +inf
2861 $inf = Math::BigInt->binf('-'); # create a -inf
2862 $one = Math::BigInt->bone(); # create a +1
2863 $one = Math::BigInt->bone('-'); # create a -1
2865 $h = Math::BigInt->new('0x123'); # from hexadecimal
2866 $b = Math::BigInt->new('0b101'); # from binary
2867 $o = Math::BigInt->from_oct('0101'); # from octal
2869 # Testing (don't modify their arguments)
2870 # (return true if the condition is met, otherwise false)
2872 $x->is_zero(); # if $x is +0
2873 $x->is_nan(); # if $x is NaN
2874 $x->is_one(); # if $x is +1
2875 $x->is_one('-'); # if $x is -1
2876 $x->is_odd(); # if $x is odd
2877 $x->is_even(); # if $x is even
2878 $x->is_pos(); # if $x >= 0
2879 $x->is_neg(); # if $x < 0
2880 $x->is_inf($sign); # if $x is +inf, or -inf (sign is default '+')
2881 $x->is_int(); # if $x is an integer (not a float)
2883 # comparing and digit/sign extraction
2884 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2885 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2886 $x->sign(); # return the sign, either +,- or NaN
2887 $x->digit($n); # return the nth digit, counting from right
2888 $x->digit(-$n); # return the nth digit, counting from left
2890 # The following all modify their first argument. If you want to preserve
2891 # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
2892 # necessary when mixing $a = $b assignments with non-overloaded math.
2894 $x->bzero(); # set $x to 0
2895 $x->bnan(); # set $x to NaN
2896 $x->bone(); # set $x to +1
2897 $x->bone('-'); # set $x to -1
2898 $x->binf(); # set $x to inf
2899 $x->binf('-'); # set $x to -inf
2901 $x->bneg(); # negation
2902 $x->babs(); # absolute value
2903 $x->bnorm(); # normalize (no-op in BigInt)
2904 $x->bnot(); # two's complement (bit wise not)
2905 $x->binc(); # increment $x by 1
2906 $x->bdec(); # decrement $x by 1
2908 $x->badd($y); # addition (add $y to $x)
2909 $x->bsub($y); # subtraction (subtract $y from $x)
2910 $x->bmul($y); # multiplication (multiply $x by $y)
2911 $x->bdiv($y); # divide, set $x to quotient
2912 # return (quo,rem) or quo if scalar
2914 $x->bmod($y); # modulus (x % y)
2915 $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
2916 $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
2918 $x->bpow($y); # power of arguments (x ** y)
2919 $x->blsft($y); # left shift in base 2
2920 $x->brsft($y); # right shift in base 2
2921 # returns (quo,rem) or quo if in scalar context
2922 $x->blsft($y,$n); # left shift by $y places in base $n
2923 $x->brsft($y,$n); # right shift by $y places in base $n
2924 # returns (quo,rem) or quo if in scalar context
2926 $x->band($y); # bitwise and
2927 $x->bior($y); # bitwise inclusive or
2928 $x->bxor($y); # bitwise exclusive or
2929 $x->bnot(); # bitwise not (two's complement)
2931 $x->bsqrt(); # calculate square-root
2932 $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
2933 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2935 $x->bnok($y); # x over y (binomial coefficient n over k)
2937 $x->blog(); # logarithm of $x to base e (Euler's number)
2938 $x->blog($base); # logarithm of $x to base $base (f.i. 2)
2939 $x->bexp(); # calculate e ** $x where e is Euler's number
2941 $x->round($A,$P,$mode); # round to accuracy or precision using mode $mode
2942 $x->bround($n); # accuracy: preserve $n digits
2943 $x->bfround($n); # round to $nth digit, no-op for BigInts
2945 # The following do not modify their arguments in BigInt (are no-ops),
2946 # but do so in BigFloat:
2948 $x->bfloor(); # return integer less or equal than $x
2949 $x->bceil(); # return integer greater or equal than $x
2951 # The following do not modify their arguments:
2953 # greatest common divisor (no OO style)
2954 my $gcd = Math::BigInt::bgcd(@values);
2955 # lowest common multiplicator (no OO style)
2956 my $lcm = Math::BigInt::blcm(@values);
2958 $x->length(); # return number of digits in number
2959 ($xl,$f) = $x->length(); # length of number and length of fraction part,
2960 # latter is always 0 digits long for BigInts
2962 $x->exponent(); # return exponent as BigInt
2963 $x->mantissa(); # return (signed) mantissa as BigInt
2964 $x->parts(); # return (mantissa,exponent) as BigInt
2965 $x->copy(); # make a true copy of $x (unlike $y = $x;)
2966 $x->as_int(); # return as BigInt (in BigInt: same as copy())
2967 $x->numify(); # return as scalar (might overflow!)
2969 # conversation to string (do not modify their argument)
2970 $x->bstr(); # normalized string (e.g. '3')
2971 $x->bsstr(); # norm. string in scientific notation (e.g. '3E0')
2972 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
2973 $x->as_bin(); # as signed binary string with prefixed 0b
2974 $x->as_oct(); # as signed octal string with prefixed 0
2977 # precision and accuracy (see section about rounding for more)
2978 $x->precision(); # return P of $x (or global, if P of $x undef)
2979 $x->precision($n); # set P of $x to $n
2980 $x->accuracy(); # return A of $x (or global, if A of $x undef)
2981 $x->accuracy($n); # set A $x to $n
2984 Math::BigInt->precision(); # get/set global P for all BigInt objects
2985 Math::BigInt->accuracy(); # get/set global A for all BigInt objects
2986 Math::BigInt->round_mode(); # get/set global round mode, one of
2987 # 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'
2988 Math::BigInt->config(); # return hash containing configuration
2992 All operators (including basic math operations) are overloaded if you
2993 declare your big integers as
2995 $i = new Math::BigInt '123_456_789_123_456_789';
2997 Operations with overloaded operators preserve the arguments which is
2998 exactly what you expect.
3004 Input values to these routines may be any string, that looks like a number
3005 and results in an integer, including hexadecimal and binary numbers.
3007 Scalars holding numbers may also be passed, but note that non-integer numbers
3008 may already have lost precision due to the conversation to float. Quote
3009 your input if you want BigInt to see all the digits:
3011 $x = Math::BigInt->new(12345678890123456789); # bad
3012 $x = Math::BigInt->new('12345678901234567890'); # good
3014 You can include one underscore between any two digits.
3016 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
3017 Non-integer values result in NaN.
3019 Hexadecimal (prefixed with "0x") and binary numbers (prefixed with "0b")
3020 are accepted, too. Please note that octal numbers are not recognized
3021 by new(), so the following will print "123":
3023 perl -MMath::BigInt -le 'print Math::BigInt->new("0123")'
3025 To convert an octal number, use from_oct();
3027 perl -MMath::BigInt -le 'print Math::BigInt->from_oct("0123")'
3029 Currently, Math::BigInt::new() defaults to 0, while Math::BigInt::new('')
3030 results in 'NaN'. This might change in the future, so use always the following
3031 explicit forms to get a zero or NaN:
3033 $zero = Math::BigInt->bzero();
3034 $nan = Math::BigInt->bnan();
3036 C<bnorm()> on a BigInt object is now effectively a no-op, since the numbers
3037 are always stored in normalized form. If passed a string, creates a BigInt
3038 object from the input.
3042 Output values are BigInt objects (normalized), except for the methods which
3043 return a string (see L<SYNOPSIS>).
3045 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
3046 C<is_nan()>, etc.) return true or false, while others (C<bcmp()>, C<bacmp()>)
3047 return either undef (if NaN is involved), <0, 0 or >0 and are suited for sort.
3053 Each of the methods below (except config(), accuracy() and precision())
3054 accepts three additional parameters. These arguments C<$A>, C<$P> and C<$R>
3055 are C<accuracy>, C<precision> and C<round_mode>. Please see the section about
3056 L<ACCURACY and PRECISION> for more information.
3062 print Dumper ( Math::BigInt->config() );
3063 print Math::BigInt->config()->{lib},"\n";
3065 Returns a hash containing the configuration, e.g. the version number, lib
3066 loaded etc. The following hash keys are currently filled in with the
3067 appropriate information.
3071 ============================================================
3072 lib Name of the low-level math library
3074 lib_version Version of low-level math library (see 'lib')
3076 class The class name of config() you just called
3078 upgrade To which class math operations might be upgraded
3080 downgrade To which class math operations might be downgraded
3082 precision Global precision
3084 accuracy Global accuracy
3086 round_mode Global round mode
3088 version version number of the class you used
3090 div_scale Fallback accuracy for div
3092 trap_nan If true, traps creation of NaN via croak()
3094 trap_inf If true, traps creation of +inf/-inf via croak()
3097 The following values can be set by passing C<config()> a reference to a hash:
3100 upgrade downgrade precision accuracy round_mode div_scale
3104 $new_cfg = Math::BigInt->config( { trap_inf => 1, precision => 5 } );
3108 $x->accuracy(5); # local for $x
3109 CLASS->accuracy(5); # global for all members of CLASS
3110 # Note: This also applies to new()!
3112 $A = $x->accuracy(); # read out accuracy that affects $x
3113 $A = CLASS->accuracy(); # read out global accuracy
3115 Set or get the global or local accuracy, aka how many significant digits the
3116 results have. If you set a global accuracy, then this also applies to new()!
3118 Warning! The accuracy I<sticks>, e.g. once you created a number under the
3119 influence of C<< CLASS->accuracy($A) >>, all results from math operations with
3120 that number will also be rounded.
3122 In most cases, you should probably round the results explicitly using one of
3123 L<round()>, L<bround()> or L<bfround()> or by passing the desired accuracy
3124 to the math operation as additional parameter:
3126 my $x = Math::BigInt->new(30000);
3127 my $y = Math::BigInt->new(7);
3128 print scalar $x->copy()->bdiv($y, 2); # print 4300
3129 print scalar $x->copy()->bdiv($y)->bround(2); # print 4300
3131 Please see the section about L<ACCURACY AND PRECISION> for further details.
3133 Value must be greater than zero. Pass an undef value to disable it:
3135 $x->accuracy(undef);
3136 Math::BigInt->accuracy(undef);
3138 Returns the current accuracy. For C<$x->accuracy()> it will return either the
3139 local accuracy, or if not defined, the global. This means the return value
3140 represents the accuracy that will be in effect for $x:
3142 $y = Math::BigInt->new(1234567); # unrounded
3143 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
3144 $x = Math::BigInt->new(123456); # $x will be automatically rounded!
3145 print "$x $y\n"; # '123500 1234567'
3146 print $x->accuracy(),"\n"; # will be 4
3147 print $y->accuracy(),"\n"; # also 4, since global is 4
3148 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
3149 print $x->accuracy(),"\n"; # still 4
3150 print $y->accuracy(),"\n"; # 5, since global is 5
3152 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
3153 globals separated from Math::BigInt, but it is possible to subclass
3154 Math::BigInt and make the globals of the subclass aliases to the ones from
3159 $x->precision(-2); # local for $x, round at the second digit right of the dot
3160 $x->precision(2); # ditto, round at the second digit left of the dot
3162 CLASS->precision(5); # Global for all members of CLASS
3163 # This also applies to new()!
3164 CLASS->precision(-5); # ditto
3166 $P = CLASS->precision(); # read out global precision
3167 $P = $x->precision(); # read out precision that affects $x
3169 Note: You probably want to use L<accuracy()> instead. With L<accuracy> you
3170 set the number of digits each result should have, with L<precision> you
3171 set the place where to round!
3173 C<precision()> sets or gets the global or local precision, aka at which digit
3174 before or after the dot to round all results. A set global precision also
3175 applies to all newly created numbers!
3177 In Math::BigInt, passing a negative number precision has no effect since no
3178 numbers have digits after the dot. In L<Math::BigFloat>, it will round all
3179 results to P digits after the dot.
3181 Please see the section about L<ACCURACY AND PRECISION> for further details.
3183 Pass an undef value to disable it:
3185 $x->precision(undef);
3186 Math::BigInt->precision(undef);
3188 Returns the current precision. For C<$x->precision()> it will return either the
3189 local precision of $x, or if not defined, the global. This means the return
3190 value represents the prevision that will be in effect for $x:
3192 $y = Math::BigInt->new(1234567); # unrounded
3193 print Math::BigInt->precision(4),"\n"; # set 4, print 4
3194 $x = Math::BigInt->new(123456); # will be automatically rounded
3195 print $x; # print "120000"!
3197 Note: Works also for subclasses like L<Math::BigFloat>. Each class has its
3198 own globals separated from Math::BigInt, but it is possible to subclass
3199 Math::BigInt and make the globals of the subclass aliases to the ones from
3206 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
3207 2, but others work, too.
3209 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
3213 $x = Math::BigInt->new(10);
3214 $x->brsft(1); # same as $x >> 1: 5
3215 $x = Math::BigInt->new(1234);
3216 $x->brsft(2,10); # result 12
3218 There is one exception, and that is base 2 with negative $x:
3221 $x = Math::BigInt->new(-5);
3224 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
3229 $x = Math::BigInt->new($str,$A,$P,$R);
3231 Creates a new BigInt object from a scalar or another BigInt object. The
3232 input is accepted as decimal, hex (with leading '0x') or binary (with leading
3235 See L<Input> for more info on accepted input formats.
3239 $x = Math::BigIn->from_oct("0775"); # input is octal
3243 $x = Math::BigIn->from_hex("0xcafe"); # input is hexadecimal
3247 $x = Math::BigIn->from_oct("0x10011"); # input is binary
3251 $x = Math::BigInt->bnan();
3253 Creates a new BigInt object representing NaN (Not A Number).
3254 If used on an object, it will set it to NaN:
3260 $x = Math::BigInt->bzero();
3262 Creates a new BigInt object representing zero.
3263 If used on an object, it will set it to zero:
3269 $x = Math::BigInt->binf($sign);
3271 Creates a new BigInt object representing infinity. The optional argument is
3272 either '-' or '+', indicating whether you want infinity or minus infinity.
3273 If used on an object, it will set it to infinity:
3280 $x = Math::BigInt->binf($sign);
3282 Creates a new BigInt object representing one. The optional argument is
3283 either '-' or '+', indicating whether you want one or minus one.
3284 If used on an object, it will set it to one:
3289 =head2 is_one()/is_zero()/is_nan()/is_inf()
3292 $x->is_zero(); # true if arg is +0
3293 $x->is_nan(); # true if arg is NaN
3294 $x->is_one(); # true if arg is +1
3295 $x->is_one('-'); # true if arg is -1
3296 $x->is_inf(); # true if +inf
3297 $x->is_inf('-'); # true if -inf (sign is default '+')
3299 These methods all test the BigInt for being one specific value and return
3300 true or false depending on the input. These are faster than doing something
3305 =head2 is_pos()/is_neg()/is_positive()/is_negative()
3307 $x->is_pos(); # true if > 0
3308 $x->is_neg(); # true if < 0
3310 The methods return true if the argument is positive or negative, respectively.
3311 C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
3312 C<-inf> is negative. A C<zero> is neither positive nor negative.
3314 These methods are only testing the sign, and not the value.
3316 C<is_positive()> and C<is_negative()> are aliases to C<is_pos()> and
3317 C<is_neg()>, respectively. C<is_positive()> and C<is_negative()> were
3318 introduced in v1.36, while C<is_pos()> and C<is_neg()> were only introduced
3321 =head2 is_odd()/is_even()/is_int()
3323 $x->is_odd(); # true if odd, false for even
3324 $x->is_even(); # true if even, false for odd
3325 $x->is_int(); # true if $x is an integer
3327 The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
3328 C<-inf> are not integers and are neither odd nor even.
3330 In BigInt, all numbers except C<NaN>, C<+inf> and C<-inf> are integers.
3336 Compares $x with $y and takes the sign into account.
3337 Returns -1, 0, 1 or undef.
3343 Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
3349 Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
3351 If you want $x to have a certain sign, use one of the following methods:
3354 $x->babs()->bneg(); # '-'
3356 $x->binf(); # '+inf'
3357 $x->binf('-'); # '-inf'
3361 $x->digit($n); # return the nth digit, counting from right
3363 If C<$n> is negative, returns the digit counting from left.
3369 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
3370 and '-inf', respectively. Does nothing for NaN or zero.
3376 Set the number to it's absolute value, e.g. change the sign from '-' to '+'
3377 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
3382 $x->bnorm(); # normalize (no-op)
3388 Two's complement (bit wise not). This is equivalent to
3396 $x->binc(); # increment x by 1
3400 $x->bdec(); # decrement x by 1
3404 $x->badd($y); # addition (add $y to $x)
3408 $x->bsub($y); # subtraction (subtract $y from $x)
3412 $x->bmul($y); # multiplication (multiply $x by $y)
3416 $x->bdiv($y); # divide, set $x to quotient
3417 # return (quo,rem) or quo if scalar
3421 $x->bmod($y); # modulus (x % y)
3425 num->bmodinv($mod); # modular inverse
3427 Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
3428 returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
3429 C<bgcd($num, $mod)==1>.
3433 $num->bmodpow($exp,$mod); # modular exponentation
3434 # ($num**$exp % $mod)
3436 Returns the value of C<$num> taken to the power C<$exp> in the modulus
3437 C<$mod> using binary exponentation. C<bmodpow> is far superior to
3442 because it is much faster - it reduces internal variables into
3443 the modulus whenever possible, so it operates on smaller numbers.
3445 C<bmodpow> also supports negative exponents.
3447 bmodpow($num, -1, $mod)
3449 is exactly equivalent to
3455 $x->bpow($y); # power of arguments (x ** y)
3459 $x->blog($base, $accuracy); # logarithm of x to the base $base
3461 If C<$base> is not defined, Euler's number (e) is used:
3463 print $x->blog(undef, 100); # log(x) to 100 digits
3467 $x->bexp($accuracy); # calculate e ** X
3469 Calculates the expression C<e ** $x> where C<e> is Euler's number.
3471 This method was added in v1.82 of Math::BigInt (April 2007).
3477 $x->bnok($y); # x over y (binomial coefficient n over k)
3479 Calculates the binomial coefficient n over k, also called the "choose"
3480 function. The result is equivalent to:
3486 This method was added in v1.84 of Math::BigInt (April 2007).
3490 $x->blsft($y); # left shift in base 2
3491 $x->blsft($y,$n); # left shift, in base $n (like 10)
3495 $x->brsft($y); # right shift in base 2
3496 $x->brsft($y,$n); # right shift, in base $n (like 10)
3500 $x->band($y); # bitwise and
3504 $x->bior($y); # bitwise inclusive or
3508 $x->bxor($y); # bitwise exclusive or
3512 $x->bnot(); # bitwise not (two's complement)
3516 $x->bsqrt(); # calculate square-root
3520 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3524 $x->round($A,$P,$round_mode);
3526 Round $x to accuracy C<$A> or precision C<$P> using the round mode
3531 $x->bround($N); # accuracy: preserve $N digits
3535 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3541 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3542 does change $x in BigFloat.
3548 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3549 does change $x in BigFloat.
3553 bgcd(@values); # greatest common divisor (no OO style)
3557 blcm(@values); # lowest common multiplicator (no OO style)
3562 ($xl,$fl) = $x->length();
3564 Returns the number of digits in the decimal representation of the number.
3565 In list context, returns the length of the integer and fraction part. For
3566 BigInt's, the length of the fraction part will always be 0.
3572 Return the exponent of $x as BigInt.
3578 Return the signed mantissa of $x as BigInt.
3582 $x->parts(); # return (mantissa,exponent) as BigInt
3586 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3588 =head2 as_int()/as_number()
3592 Returns $x as a BigInt (truncated towards zero). In BigInt this is the same as
3595 C<as_number()> is an alias to this method. C<as_number> was introduced in
3596 v1.22, while C<as_int()> was only introduced in v1.68.
3602 Returns a normalized string representation of C<$x>.
3606 $x->bsstr(); # normalized string in scientific notation
3610 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3614 $x->as_bin(); # as signed binary string with prefixed 0b
3618 $x->as_oct(); # as signed octal string with prefixed 0
3624 This returns a normal Perl scalar from $x. It is used automatically
3625 whenever a scalar is needed, for instance in array index operations.
3627 This loses precision, to avoid this use L<as_int()> instead.
3631 $x->modify('bpowd');
3633 This method returns 0 if the object can be modified with the given
3634 peration, or 1 if not.
3636 This is used for instance by L<Math::BigInt::Constant>.
3638 =head2 upgrade()/downgrade()
3640 Set/get the class for downgrade/upgrade operations. Thuis is used
3641 for instance by L<bignum>. The defaults are '', thus the following
3642 operation will create a BigInt, not a BigFloat:
3644 my $i = Math::BigInt->new(123);
3645 my $f = Math::BigFloat->new('123.1');
3647 print $i + $f,"\n"; # print 246
3651 Set/get the number of digits for the default precision in divide
3656 Set/get the current round mode.
3658 =head1 ACCURACY and PRECISION
3660 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3661 accuracy and precision based rounding, both automatically after every
3662 operation, as well as manually.
3664 This section describes the accuracy/precision handling in Math::Big* as it
3665 used to be and as it is now, complete with an explanation of all terms and
3668 Not yet implemented things (but with correct description) are marked with '!',
3669 things that need to be answered are marked with '?'.
3671 In the next paragraph follows a short description of terms used here (because
3672 these may differ from terms used by others people or documentation).
3674 During the rest of this document, the shortcuts A (for accuracy), P (for
3675 precision), F (fallback) and R (rounding mode) will be used.
3679 A fixed number of digits before (positive) or after (negative)
3680 the decimal point. For example, 123.45 has a precision of -2. 0 means an
3681 integer like 123 (or 120). A precision of 2 means two digits to the left
3682 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3683 numbers with zeros before the decimal point may have different precisions,
3684 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3685 was). It could also have p < 0, when the digits after the decimal point
3688 The string output (of floating point numbers) will be padded with zeros:
3690 Initial value P A Result String
3691 ------------------------------------------------------------
3692 1234.01 -3 1000 1000
3695 1234.001 1 1234 1234.0
3697 1234.01 2 1234.01 1234.01
3698 1234.01 5 1234.01 1234.01000
3700 For BigInts, no padding occurs.
3704 Number of significant digits. Leading zeros are not counted. A
3705 number may have an accuracy greater than the non-zero digits
3706 when there are zeros in it or trailing zeros. For example, 123.456 has
3707 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3709 The string output (of floating point numbers) will be padded with zeros:
3711 Initial value P A Result String
3712 ------------------------------------------------------------
3714 1234.01 6 1234.01 1234.01
3715 1234.1 8 1234.1 1234.1000
3717 For BigInts, no padding occurs.
3721 When both A and P are undefined, this is used as a fallback accuracy when
3724 =head2 Rounding mode R
3726 When rounding a number, different 'styles' or 'kinds'
3727 of rounding are possible. (Note that random rounding, as in
3728 Math::Round, is not implemented.)
3734 truncation invariably removes all digits following the
3735 rounding place, replacing them with zeros. Thus, 987.65 rounded
3736 to tens (P=1) becomes 980, and rounded to the fourth sigdig
3737 becomes 987.6 (A=4). 123.456 rounded to the second place after the
3738 decimal point (P=-2) becomes 123.46.
3740 All other implemented styles of rounding attempt to round to the
3741 "nearest digit." If the digit D immediately to the right of the
3742 rounding place (skipping the decimal point) is greater than 5, the
3743 number is incremented at the rounding place (possibly causing a
3744 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3745 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3746 truncated at the rounding place: e.g. when rounding to units, 0.4
3747 rounds to 0, and -19.4 rounds to -19.
3749 However the results of other styles of rounding differ if the
3750 digit immediately to the right of the rounding place (skipping the
3751 decimal point) is 5 and if there are no digits, or no digits other
3752 than 0, after that 5. In such cases:
3756 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3757 if it is not already. E.g., when rounding to the first sigdig, 0.45
3758 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3762 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3763 it is not already. E.g., when rounding to the first sigdig, 0.45
3764 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3768 round to plus infinity, i.e. always round up. E.g., when
3769 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3770 and 0.4501 also becomes 0.5.
3774 round to minus infinity, i.e. always round down. E.g., when
3775 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3776 but 0.4501 becomes 0.5.
3780 round to zero, i.e. positive numbers down, negative ones up.
3781 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3782 becomes -0.5, but 0.4501 becomes 0.5.
3786 round up if the digit immediately to the right of the rounding place
3787 is 5 or greater, otherwise round down. E.g., 0.15 becomes 0.2 and
3792 The handling of A & P in MBI/MBF (the old core code shipped with Perl
3793 versions <= 5.7.2) is like this:
3799 * ffround($p) is able to round to $p number of digits after the decimal
3801 * otherwise P is unused
3803 =item Accuracy (significant digits)
3805 * fround($a) rounds to $a significant digits
3806 * only fdiv() and fsqrt() take A as (optional) paramater
3807 + other operations simply create the same number (fneg etc), or more (fmul)
3809 + rounding/truncating is only done when explicitly calling one of fround
3810 or ffround, and never for BigInt (not implemented)
3811 * fsqrt() simply hands its accuracy argument over to fdiv.
3812 * the documentation and the comment in the code indicate two different ways
3813 on how fdiv() determines the maximum number of digits it should calculate,
3814 and the actual code does yet another thing
3816 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3818 result has at most max(scale, length(dividend), length(divisor)) digits
3820 scale = max(scale, length(dividend)-1,length(divisor)-1);
3821 scale += length(divisor) - length(dividend);
3822 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3823 Actually, the 'difference' added to the scale is calculated from the
3824 number of "significant digits" in dividend and divisor, which is derived
3825 by looking at the length of the mantissa. Which is wrong, since it includes
3826 the + sign (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops
3827 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3828 assumption that 124 has 3 significant digits, while 120/7 will get you
3829 '17', not '17.1' since 120 is thought to have 2 significant digits.
3830 The rounding after the division then uses the remainder and $y to determine
3831 wether it must round up or down.
3832 ? I have no idea which is the right way. That's why I used a slightly more
3833 ? simple scheme and tweaked the few failing testcases to match it.
3837 This is how it works now:
3841 =item Setting/Accessing
3843 * You can set the A global via C<< Math::BigInt->accuracy() >> or
3844 C<< Math::BigFloat->accuracy() >> or whatever class you are using.
3845 * You can also set P globally by using C<< Math::SomeClass->precision() >>
3847 * Globals are classwide, and not inherited by subclasses.
3848 * to undefine A, use C<< Math::SomeCLass->accuracy(undef); >>
3849 * to undefine P, use C<< Math::SomeClass->precision(undef); >>
3850 * Setting C<< Math::SomeClass->accuracy() >> clears automatically
3851 C<< Math::SomeClass->precision() >>, and vice versa.
3852 * To be valid, A must be > 0, P can have any value.
3853 * If P is negative, this means round to the P'th place to the right of the
3854 decimal point; positive values mean to the left of the decimal point.
3855 P of 0 means round to integer.
3856 * to find out the current global A, use C<< Math::SomeClass->accuracy() >>
3857 * to find out the current global P, use C<< Math::SomeClass->precision() >>
3858 * use C<< $x->accuracy() >> respective C<< $x->precision() >> for the local
3859 setting of C<< $x >>.
3860 * Please note that C<< $x->accuracy() >> respective C<< $x->precision() >>
3861 return eventually defined global A or P, when C<< $x >>'s A or P is not
3864 =item Creating numbers
3866 * When you create a number, you can give it's desired A or P via:
3867 $x = Math::BigInt->new($number,$A,$P);
3868 * Only one of A or P can be defined, otherwise the result is NaN
3869 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
3870 globals (if set) will be used. Thus changing the global defaults later on
3871 will not change the A or P of previously created numbers (i.e., A and P of
3872 $x will be what was in effect when $x was created)
3873 * If given undef for A and P, B<no> rounding will occur, and the globals will
3874 B<not> be used. This is used by subclasses to create numbers without
3875 suffering rounding in the parent. Thus a subclass is able to have it's own
3876 globals enforced upon creation of a number by using
3877 C<< $x = Math::BigInt->new($number,undef,undef) >>:
3879 use Math::BigInt::SomeSubclass;
3882 Math::BigInt->accuracy(2);
3883 Math::BigInt::SomeSubClass->accuracy(3);
3884 $x = Math::BigInt::SomeSubClass->new(1234);
3886 $x is now 1230, and not 1200. A subclass might choose to implement
3887 this otherwise, e.g. falling back to the parent's A and P.
3891 * If A or P are enabled/defined, they are used to round the result of each
3892 operation according to the rules below
3893 * Negative P is ignored in Math::BigInt, since BigInts never have digits
3894 after the decimal point
3895 * Math::BigFloat uses Math::BigInt internally, but setting A or P inside
3896 Math::BigInt as globals does not tamper with the parts of a BigFloat.
3897 A flag is used to mark all Math::BigFloat numbers as 'never round'.
3901 * It only makes sense that a number has only one of A or P at a time.
3902 If you set either A or P on one object, or globally, the other one will
3903 be automatically cleared.
3904 * If two objects are involved in an operation, and one of them has A in
3905 effect, and the other P, this results in an error (NaN).
3906 * A takes precedence over P (Hint: A comes before P).
3907 If neither of them is defined, nothing is used, i.e. the result will have
3908 as many digits as it can (with an exception for fdiv/fsqrt) and will not
3910 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
3911 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
3912 If either the dividend's or the divisor's mantissa has more digits than
3913 the value of F, the higher value will be used instead of F.
3914 This is to limit the digits (A) of the result (just consider what would
3915 happen with unlimited A and P in the case of 1/3 :-)
3916 * fdiv will calculate (at least) 4 more digits than required (determined by
3917 A, P or F), and, if F is not used, round the result
3918 (this will still fail in the case of a result like 0.12345000000001 with A
3919 or P of 5, but this can not be helped - or can it?)
3920 * Thus you can have the math done by on Math::Big* class in two modi:
3921 + never round (this is the default):
3922 This is done by setting A and P to undef. No math operation
3923 will round the result, with fdiv() and fsqrt() as exceptions to guard
3924 against overflows. You must explicitly call bround(), bfround() or
3925 round() (the latter with parameters).
3926 Note: Once you have rounded a number, the settings will 'stick' on it
3927 and 'infect' all other numbers engaged in math operations with it, since
3928 local settings have the highest precedence. So, to get SaferRound[tm],
3929 use a copy() before rounding like this:
3931 $x = Math::BigFloat->new(12.34);
3932 $y = Math::BigFloat->new(98.76);
3933 $z = $x * $y; # 1218.6984
3934 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
3935 $z = $x * $y; # still 1218.6984, without
3936 # copy would have been 1210!
3938 + round after each op:
3939 After each single operation (except for testing like is_zero()), the
3940 method round() is called and the result is rounded appropriately. By
3941 setting proper values for A and P, you can have all-the-same-A or
3942 all-the-same-P modes. For example, Math::Currency might set A to undef,
3943 and P to -2, globally.
3945 ?Maybe an extra option that forbids local A & P settings would be in order,
3946 ?so that intermediate rounding does not 'poison' further math?
3948 =item Overriding globals
3950 * you will be able to give A, P and R as an argument to all the calculation
3951 routines; the second parameter is A, the third one is P, and the fourth is
3952 R (shift right by one for binary operations like badd). P is used only if
3953 the first parameter (A) is undefined. These three parameters override the
3954 globals in the order detailed as follows, i.e. the first defined value
3956 (local: per object, global: global default, parameter: argument to sub)
3959 + local A (if defined on both of the operands: smaller one is taken)
3960 + local P (if defined on both of the operands: bigger one is taken)
3964 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
3965 arguments (A and P) instead of one
3967 =item Local settings
3969 * You can set A or P locally by using C<< $x->accuracy() >> or
3970 C<< $x->precision() >>
3971 and thus force different A and P for different objects/numbers.
3972 * Setting A or P this way immediately rounds $x to the new value.
3973 * C<< $x->accuracy() >> clears C<< $x->precision() >>, and vice versa.
3977 * the rounding routines will use the respective global or local settings.
3978 fround()/bround() is for accuracy rounding, while ffround()/bfround()
3980 * the two rounding functions take as the second parameter one of the
3981 following rounding modes (R):
3982 'even', 'odd', '+inf', '-inf', 'zero', 'trunc', 'common'
3983 * you can set/get the global R by using C<< Math::SomeClass->round_mode() >>
3984 or by setting C<< $Math::SomeClass::round_mode >>
3985 * after each operation, C<< $result->round() >> is called, and the result may
3986 eventually be rounded (that is, if A or P were set either locally,
3987 globally or as parameter to the operation)
3988 * to manually round a number, call C<< $x->round($A,$P,$round_mode); >>
3989 this will round the number by using the appropriate rounding function
3990 and then normalize it.
3991 * rounding modifies the local settings of the number:
3993 $x = Math::BigFloat->new(123.456);
3997 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
3998 will be 4 from now on.
4000 =item Default values
4009 * The defaults are set up so that the new code gives the same results as
4010 the old code (except in a few cases on fdiv):
4011 + Both A and P are undefined and thus will not be used for rounding
4012 after each operation.
4013 + round() is thus a no-op, unless given extra parameters A and P
4017 =head1 Infinity and Not a Number
4019 While BigInt has extensive handling of inf and NaN, certain quirks remain.
4025 These perl routines currently (as of Perl v.5.8.6) cannot handle passed
4028 te@linux:~> perl -wle 'print 2 ** 3333'
4030 te@linux:~> perl -wle 'print 2 ** 3333 == 2 ** 3333'
4032 te@linux:~> perl -wle 'print oct(2 ** 3333)'
4034 te@linux:~> perl -wle 'print hex(2 ** 3333)'
4035 Illegal hexadecimal digit 'i' ignored at -e line 1.
4038 The same problems occur if you pass them Math::BigInt->binf() objects. Since
4039 overloading these routines is not possible, this cannot be fixed from BigInt.
4041 =item ==, !=, <, >, <=, >= with NaNs
4043 BigInt's bcmp() routine currently returns undef to signal that a NaN was
4044 involved in a comparison. However, the overload code turns that into
4045 either 1 or '' and thus operations like C<< NaN != NaN >> might return
4050 C<< log(-inf) >> is highly weird. Since log(-x)=pi*i+log(x), then
4051 log(-inf)=pi*i+inf. However, since the imaginary part is finite, the real
4052 infinity "overshadows" it, so the number might as well just be infinity.
4053 However, the result is a complex number, and since BigInt/BigFloat can only
4054 have real numbers as results, the result is NaN.
4056 =item exp(), cos(), sin(), atan2()
4058 These all might have problems handling infinity right.
4064 The actual numbers are stored as unsigned big integers (with seperate sign).
4066 You should neither care about nor depend on the internal representation; it
4067 might change without notice. Use B<ONLY> method calls like C<< $x->sign(); >>
4068 instead relying on the internal representation.
4072 Math with the numbers is done (by default) by a module called
4073 C<Math::BigInt::Calc>. This is equivalent to saying:
4075 use Math::BigInt lib => 'Calc';
4077 You can change this by using:
4079 use Math::BigInt lib => 'BitVect';
4081 The following would first try to find Math::BigInt::Foo, then
4082 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
4084 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
4086 Since Math::BigInt::GMP is in almost all cases faster than Calc (especially in
4087 math involving really big numbers, where it is B<much> faster), and there is
4088 no penalty if Math::BigInt::GMP is not installed, it is a good idea to always
4091 use Math::BigInt lib => 'GMP';
4093 Different low-level libraries use different formats to store the
4094 numbers. You should B<NOT> depend on the number having a specific format
4097 See the respective math library module documentation for further details.
4101 The sign is either '+', '-', 'NaN', '+inf' or '-inf'.
4103 A sign of 'NaN' is used to represent the result when input arguments are not
4104 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
4105 minus infinity. You will get '+inf' when dividing a positive number by 0, and
4106 '-inf' when dividing any negative number by 0.
4108 =head2 mantissa(), exponent() and parts()
4110 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
4113 $m = $x->mantissa();
4114 $e = $x->exponent();
4115 $y = $m * ( 10 ** $e );
4116 print "ok\n" if $x == $y;
4118 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
4119 in one go. Both the returned mantissa and exponent have a sign.
4121 Currently, for BigInts C<$e> is always 0, except +inf and -inf, where it is
4122 C<+inf>; and for NaN, where it is C<NaN>; and for C<$x == 0>, where it is C<1>
4123 (to be compatible with Math::BigFloat's internal representation of a zero as
4126 C<$m> is currently just a copy of the original number. The relation between
4127 C<$e> and C<$m> will stay always the same, though their real values might
4134 sub bint { Math::BigInt->new(shift); }
4136 $x = Math::BigInt->bstr("1234") # string "1234"
4137 $x = "$x"; # same as bstr()
4138 $x = Math::BigInt->bneg("1234"); # BigInt "-1234"
4139 $x = Math::BigInt->babs("-12345"); # BigInt "12345"
4140 $x = Math::BigInt->bnorm("-0.00"); # BigInt "0"
4141 $x = bint(1) + bint(2); # BigInt "3"
4142 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
4143 $x = bint(1); # BigInt "1"
4144 $x = $x + 5 / 2; # BigInt "3"
4145 $x = $x ** 3; # BigInt "27"
4146 $x *= 2; # BigInt "54"
4147 $x = Math::BigInt->new(0); # BigInt "0"
4149 $x = Math::BigInt->badd(4,5) # BigInt "9"
4150 print $x->bsstr(); # 9e+0
4152 Examples for rounding:
4157 $x = Math::BigFloat->new(123.4567);
4158 $y = Math::BigFloat->new(123.456789);
4159 Math::BigFloat->accuracy(4); # no more A than 4
4161 ok ($x->copy()->fround(),123.4); # even rounding
4162 print $x->copy()->fround(),"\n"; # 123.4
4163 Math::BigFloat->round_mode('odd'); # round to odd
4164 print $x->copy()->fround(),"\n"; # 123.5
4165 Math::BigFloat->accuracy(5); # no more A than 5
4166 Math::BigFloat->round_mode('odd'); # round to odd
4167 print $x->copy()->fround(),"\n"; # 123.46
4168 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
4169 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
4171 Math::BigFloat->accuracy(undef); # A not important now
4172 Math::BigFloat->precision(2); # P important
4173 print $x->copy()->bnorm(),"\n"; # 123.46
4174 print $x->copy()->fround(),"\n"; # 123.46
4176 Examples for converting:
4178 my $x = Math::BigInt->new('0b1'.'01' x 123);
4179 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
4181 =head1 Autocreating constants
4183 After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
4184 and binary constants in the given scope are converted to C<Math::BigInt>.
4185 This conversion happens at compile time.
4189 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
4191 prints the integer value of C<2**100>. Note that without conversion of
4192 constants the expression 2**100 will be calculated as perl scalar.
4194 Please note that strings and floating point constants are not affected,
4197 use Math::BigInt qw/:constant/;
4199 $x = 1234567890123456789012345678901234567890
4200 + 123456789123456789;
4201 $y = '1234567890123456789012345678901234567890'
4202 + '123456789123456789';
4204 do not work. You need an explicit Math::BigInt->new() around one of the
4205 operands. You should also quote large constants to protect loss of precision:
4209 $x = Math::BigInt->new('1234567889123456789123456789123456789');
4211 Without the quotes Perl would convert the large number to a floating point
4212 constant at compile time and then hand the result to BigInt, which results in
4213 an truncated result or a NaN.
4215 This also applies to integers that look like floating point constants:
4217 use Math::BigInt ':constant';
4219 print ref(123e2),"\n";
4220 print ref(123.2e2),"\n";
4222 will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
4223 to get this to work.
4227 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
4228 must be made in the second case. For long numbers, the copy can eat up to 20%
4229 of the work (in the case of addition/subtraction, less for
4230 multiplication/division). If $y is very small compared to $x, the form
4231 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
4232 more time then the actual addition.
4234 With a technique called copy-on-write, the cost of copying with overload could
4235 be minimized or even completely avoided. A test implementation of COW did show
4236 performance gains for overloaded math, but introduced a performance loss due
4237 to a constant overhead for all other operations. So Math::BigInt does currently
4240 The rewritten version of this module (vs. v0.01) is slower on certain
4241 operations, like C<new()>, C<bstr()> and C<numify()>. The reason are that it
4242 does now more work and handles much more cases. The time spent in these
4243 operations is usually gained in the other math operations so that code on
4244 the average should get (much) faster. If they don't, please contact the author.
4246 Some operations may be slower for small numbers, but are significantly faster
4247 for big numbers. Other operations are now constant (O(1), like C<bneg()>,
4248 C<babs()> etc), instead of O(N) and thus nearly always take much less time.
4249 These optimizations were done on purpose.
4251 If you find the Calc module to slow, try to install any of the replacement
4252 modules and see if they help you.
4254 =head2 Alternative math libraries
4256 You can use an alternative library to drive Math::BigInt via:
4258 use Math::BigInt lib => 'Module';
4260 See L<MATH LIBRARY> for more information.
4262 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
4266 =head1 Subclassing Math::BigInt
4268 The basic design of Math::BigInt allows simple subclasses with very little
4269 work, as long as a few simple rules are followed:
4275 The public API must remain consistent, i.e. if a sub-class is overloading
4276 addition, the sub-class must use the same name, in this case badd(). The
4277 reason for this is that Math::BigInt is optimized to call the object methods
4282 The private object hash keys like C<$x->{sign}> may not be changed, but
4283 additional keys can be added, like C<$x->{_custom}>.
4287 Accessor functions are available for all existing object hash keys and should
4288 be used instead of directly accessing the internal hash keys. The reason for
4289 this is that Math::BigInt itself has a pluggable interface which permits it
4290 to support different storage methods.
4294 More complex sub-classes may have to replicate more of the logic internal of
4295 Math::BigInt if they need to change more basic behaviors. A subclass that
4296 needs to merely change the output only needs to overload C<bstr()>.
4298 All other object methods and overloaded functions can be directly inherited
4299 from the parent class.
4301 At the very minimum, any subclass will need to provide it's own C<new()> and can
4302 store additional hash keys in the object. There are also some package globals
4303 that must be defined, e.g.:
4307 $precision = -2; # round to 2 decimal places
4308 $round_mode = 'even';
4311 Additionally, you might want to provide the following two globals to allow
4312 auto-upgrading and auto-downgrading to work correctly:
4317 This allows Math::BigInt to correctly retrieve package globals from the
4318 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
4319 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
4325 in your subclass to automatically inherit the overloading from the parent. If
4326 you like, you can change part of the overloading, look at Math::String for an
4331 When used like this:
4333 use Math::BigInt upgrade => 'Foo::Bar';
4335 certain operations will 'upgrade' their calculation and thus the result to
4336 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
4338 use Math::BigInt upgrade => 'Math::BigFloat';
4340 As a shortcut, you can use the module C<bignum>:
4344 Also good for oneliners:
4346 perl -Mbignum -le 'print 2 ** 255'
4348 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
4349 as well es preserve accuracy (as in sqrt(3)).
4351 Beware: This feature is not fully implemented yet.
4355 The following methods upgrade themselves unconditionally; that is if upgrade
4356 is in effect, they will always hand up their work:
4370 Beware: This list is not complete.
4372 All other methods upgrade themselves only when one (or all) of their
4373 arguments are of the class mentioned in $upgrade (This might change in later
4374 versions to a more sophisticated scheme):
4380 =item broot() does not work
4382 The broot() function in BigInt may only work for small values. This will be
4383 fixed in a later version.
4385 =item Out of Memory!
4387 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
4388 C<eval()> in your code will crash with "Out of memory". This is probably an
4389 overload/exporter bug. You can workaround by not having C<eval()>
4390 and ':constant' at the same time or upgrade your Perl to a newer version.
4392 =item Fails to load Calc on Perl prior 5.6.0
4394 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
4395 will fall back to eval { require ... } when loading the math lib on Perls
4396 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
4397 filesystems using a different seperator.
4403 Some things might not work as you expect them. Below is documented what is
4404 known to be troublesome:
4408 =item bstr(), bsstr() and 'cmp'
4410 Both C<bstr()> and C<bsstr()> as well as automated stringify via overload now
4411 drop the leading '+'. The old code would return '+3', the new returns '3'.
4412 This is to be consistent with Perl and to make C<cmp> (especially with
4413 overloading) to work as you expect. It also solves problems with C<Test.pm>,
4414 because it's C<ok()> uses 'eq' internally.
4416 Mark Biggar said, when asked about to drop the '+' altogether, or make only
4419 I agree (with the first alternative), don't add the '+' on positive
4420 numbers. It's not as important anymore with the new internal
4421 form for numbers. It made doing things like abs and neg easier,
4422 but those have to be done differently now anyway.
4424 So, the following examples will now work all as expected:
4427 BEGIN { plan tests => 1 }
4430 my $x = new Math::BigInt 3*3;
4431 my $y = new Math::BigInt 3*3;
4434 print "$x eq 9" if $x eq $y;
4435 print "$x eq 9" if $x eq '9';
4436 print "$x eq 9" if $x eq 3*3;
4438 Additionally, the following still works:
4440 print "$x == 9" if $x == $y;
4441 print "$x == 9" if $x == 9;
4442 print "$x == 9" if $x == 3*3;
4444 There is now a C<bsstr()> method to get the string in scientific notation aka
4445 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
4446 for comparison, but Perl will represent some numbers as 100 and others
4447 as 1e+308. If in doubt, convert both arguments to Math::BigInt before
4448 comparing them as strings:
4451 BEGIN { plan tests => 3 }
4454 $x = Math::BigInt->new('1e56'); $y = 1e56;
4455 ok ($x,$y); # will fail
4456 ok ($x->bsstr(),$y); # okay
4457 $y = Math::BigInt->new($y);
4460 Alternatively, simple use C<< <=> >> for comparisons, this will get it
4461 always right. There is not yet a way to get a number automatically represented
4462 as a string that matches exactly the way Perl represents it.
4464 See also the section about L<Infinity and Not a Number> for problems in
4469 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
4472 $x = Math::BigInt->new(123);
4473 $y = int($x); # BigInt 123
4474 $x = Math::BigFloat->new(123.45);
4475 $y = int($x); # BigInt 123
4477 In all Perl versions you can use C<as_number()> or C<as_int> for the same
4480 $x = Math::BigFloat->new(123.45);
4481 $y = $x->as_number(); # BigInt 123
4482 $y = $x->as_int(); # ditto
4484 This also works for other subclasses, like Math::String.
4486 If you want a real Perl scalar, use C<numify()>:
4488 $y = $x->numify(); # 123 as scalar
4490 This is seldom necessary, though, because this is done automatically, like
4491 when you access an array:
4493 $z = $array[$x]; # does work automatically
4497 The following will probably not do what you expect:
4499 $c = Math::BigInt->new(123);
4500 print $c->length(),"\n"; # prints 30
4502 It prints both the number of digits in the number and in the fraction part
4503 since print calls C<length()> in list context. Use something like:
4505 print scalar $c->length(),"\n"; # prints 3
4509 The following will probably not do what you expect:
4511 print $c->bdiv(10000),"\n";
4513 It prints both quotient and remainder since print calls C<bdiv()> in list
4514 context. Also, C<bdiv()> will modify $c, so be careful. You probably want
4517 print $c / 10000,"\n";
4518 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
4522 The quotient is always the greatest integer less than or equal to the
4523 real-valued quotient of the two operands, and the remainder (when it is
4524 nonzero) always has the same sign as the second operand; so, for
4534 As a consequence, the behavior of the operator % agrees with the
4535 behavior of Perl's built-in % operator (as documented in the perlop
4536 manpage), and the equation
4538 $x == ($x / $y) * $y + ($x % $y)
4540 holds true for any $x and $y, which justifies calling the two return
4541 values of bdiv() the quotient and remainder. The only exception to this rule
4542 are when $y == 0 and $x is negative, then the remainder will also be
4543 negative. See below under "infinity handling" for the reasoning behind this.
4545 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
4546 not change BigInt's way to do things. This is because under 'use integer' Perl
4547 will do what the underlying C thinks is right and this is different for each
4548 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
4549 the author to implement it ;)
4551 =item infinity handling
4553 Here are some examples that explain the reasons why certain results occur while
4556 The following table shows the result of the division and the remainder, so that
4557 the equation above holds true. Some "ordinary" cases are strewn in to show more
4558 clearly the reasoning:
4560 A / B = C, R so that C * B + R = A
4561 =========================================================
4562 5 / 8 = 0, 5 0 * 8 + 5 = 5
4563 0 / 8 = 0, 0 0 * 8 + 0 = 0
4564 0 / inf = 0, 0 0 * inf + 0 = 0
4565 0 /-inf = 0, 0 0 * -inf + 0 = 0
4566 5 / inf = 0, 5 0 * inf + 5 = 5
4567 5 /-inf = 0, 5 0 * -inf + 5 = 5
4568 -5/ inf = 0, -5 0 * inf + -5 = -5
4569 -5/-inf = 0, -5 0 * -inf + -5 = -5
4570 inf/ 5 = inf, 0 inf * 5 + 0 = inf
4571 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
4572 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
4573 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
4574 5/ 5 = 1, 0 1 * 5 + 0 = 5
4575 -5/ -5 = 1, 0 1 * -5 + 0 = -5
4576 inf/ inf = 1, 0 1 * inf + 0 = inf
4577 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
4578 inf/-inf = -1, 0 -1 * -inf + 0 = inf
4579 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
4580 8/ 0 = inf, 8 inf * 0 + 8 = 8
4581 inf/ 0 = inf, inf inf * 0 + inf = inf
4584 These cases below violate the "remainder has the sign of the second of the two
4585 arguments", since they wouldn't match up otherwise.
4587 A / B = C, R so that C * B + R = A
4588 ========================================================
4589 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
4590 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
4592 =item Modifying and =
4596 $x = Math::BigFloat->new(5);
4599 It will not do what you think, e.g. making a copy of $x. Instead it just makes
4600 a second reference to the B<same> object and stores it in $y. Thus anything
4601 that modifies $x (except overloaded operators) will modify $y, and vice versa.
4602 Or in other words, C<=> is only safe if you modify your BigInts only via
4603 overloaded math. As soon as you use a method call it breaks:
4606 print "$x, $y\n"; # prints '10, 10'
4608 If you want a true copy of $x, use:
4612 You can also chain the calls like this, this will make first a copy and then
4615 $y = $x->copy()->bmul(2);
4617 See also the documentation for overload.pm regarding C<=>.
4621 C<bpow()> (and the rounding functions) now modifies the first argument and
4622 returns it, unlike the old code which left it alone and only returned the
4623 result. This is to be consistent with C<badd()> etc. The first three will
4624 modify $x, the last one won't:
4626 print bpow($x,$i),"\n"; # modify $x
4627 print $x->bpow($i),"\n"; # ditto
4628 print $x **= $i,"\n"; # the same
4629 print $x ** $i,"\n"; # leave $x alone
4631 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4633 =item Overloading -$x
4643 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4644 needs to preserve $x since it does not know that it later will get overwritten.
4645 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4647 =item Mixing different object types
4649 In Perl you will get a floating point value if you do one of the following:
4655 With overloaded math, only the first two variants will result in a BigFloat:
4660 $mbf = Math::BigFloat->new(5);
4661 $mbi2 = Math::BigInteger->new(5);
4662 $mbi = Math::BigInteger->new(2);
4664 # what actually gets called:
4665 $float = $mbf + $mbi; # $mbf->badd()
4666 $float = $mbf / $mbi; # $mbf->bdiv()
4667 $integer = $mbi + $mbf; # $mbi->badd()
4668 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4669 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4671 This is because math with overloaded operators follows the first (dominating)
4672 operand, and the operation of that is called and returns thus the result. So,
4673 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4674 the result should be a Math::BigFloat or the second operant is one.
4676 To get a Math::BigFloat you either need to call the operation manually,
4677 make sure the operands are already of the proper type or casted to that type
4678 via Math::BigFloat->new():
4680 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4682 Beware of simple "casting" the entire expression, this would only convert
4683 the already computed result:
4685 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4687 Beware also of the order of more complicated expressions like:
4689 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4690 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4692 If in doubt, break the expression into simpler terms, or cast all operands
4693 to the desired resulting type.
4695 Scalar values are a bit different, since:
4700 will both result in the proper type due to the way the overloaded math works.
4702 This section also applies to other overloaded math packages, like Math::String.
4704 One solution to you problem might be autoupgrading|upgrading. See the
4705 pragmas L<bignum>, L<bigint> and L<bigrat> for an easy way to do this.
4709 C<bsqrt()> works only good if the result is a big integer, e.g. the square
4710 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4711 mode. The reason is that the result is always truncated to an integer.
4713 If you want a better approximation of the square root, then use:
4715 $x = Math::BigFloat->new(12);
4716 Math::BigFloat->precision(0);
4717 Math::BigFloat->round_mode('even');
4718 print $x->copy->bsqrt(),"\n"; # 4
4720 Math::BigFloat->precision(2);
4721 print $x->bsqrt(),"\n"; # 3.46
4722 print $x->bsqrt(3),"\n"; # 3.464
4726 For negative numbers in base see also L<brsft|brsft>.
4732 This program is free software; you may redistribute it and/or modify it under
4733 the same terms as Perl itself.
4737 L<Math::BigFloat>, L<Math::BigRat> and L<Math::Big> as well as
4738 L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4740 The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest
4741 because they solve the autoupgrading/downgrading issue, at least partly.
4744 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4745 more documentation including a full version history, testcases, empty
4746 subclass files and benchmarks.
4750 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4751 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001 - 2006
4752 and still at it in 2007.
4754 Many people contributed in one or more ways to the final beast, see the file
4755 CREDITS for an (incomplete) list. If you miss your name, please drop me a