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 e ** $x (Euler's number to the power of X), truncated to
1272 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1273 return $x if $x->modify('bexp');
1275 # inf, -inf, NaN, <0 => NaN
1276 return $x->bnan() if $x->{sign} eq 'NaN';
1277 return $x->bone() if $x->is_zero();
1278 return $x if $x->{sign} eq '+inf';
1279 return $x->bzero() if $x->{sign} eq '-inf';
1283 # run through Math::BigFloat unless told otherwise
1284 local $upgrade = 'Math::BigFloat' unless defined $upgrade;
1285 # calculate result, truncate it to integer
1286 $u = $upgrade->bexp($upgrade->new($x),@r);
1289 if (!defined $upgrade)
1292 # modify $x in place
1293 $x->{value} = $u->{value};
1301 # (BINT or num_str, BINT or num_str) return BINT
1302 # does not modify arguments, but returns new object
1303 # Lowest Common Multiplicator
1305 my $y = shift; my ($x);
1312 $x = $class->new($y);
1317 my $y = shift; $y = $self->new($y) if !ref ($y);
1325 # (BINT or num_str, BINT or num_str) return BINT
1326 # does not modify arguments, but returns new object
1327 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1330 $y = $class->new($y) if !ref($y);
1332 my $x = $y->copy()->babs(); # keep arguments
1333 return $x->bnan() if $x->{sign} !~ /^[+-]$/; # x NaN?
1337 $y = shift; $y = $self->new($y) if !ref($y);
1338 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1339 $x->{value} = $CALC->_gcd($x->{value},$y->{value});
1340 last if $CALC->_is_one($x->{value});
1347 # (num_str or BINT) return BINT
1348 # represent ~x as twos-complement number
1349 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1350 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1352 return $x if $x->modify('bnot');
1353 $x->binc()->bneg(); # binc already does round
1356 ##############################################################################
1357 # is_foo test routines
1358 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1362 # return true if arg (BINT or num_str) is zero (array '+', '0')
1363 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1365 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1366 $CALC->_is_zero($x->{value});
1371 # return true if arg (BINT or num_str) is NaN
1372 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1374 $x->{sign} eq $nan ? 1 : 0;
1379 # return true if arg (BINT or num_str) is +-inf
1380 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1384 $sign = '[+-]inf' if $sign eq ''; # +- doesn't matter, only that's inf
1385 $sign = "[$1]inf" if $sign =~ /^([+-])(inf)?$/; # extract '+' or '-'
1386 return $x->{sign} =~ /^$sign$/ ? 1 : 0;
1388 $x->{sign} =~ /^[+-]inf$/ ? 1 : 0; # only +-inf is infinity
1393 # return true if arg (BINT or num_str) is +1, or -1 if sign is given
1394 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1396 $sign = '+' if !defined $sign || $sign ne '-';
1398 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1399 $CALC->_is_one($x->{value});
1404 # return true when arg (BINT or num_str) is odd, false for even
1405 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1407 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1408 $CALC->_is_odd($x->{value});
1413 # return true when arg (BINT or num_str) is even, false for odd
1414 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1416 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1417 $CALC->_is_even($x->{value});
1422 # return true when arg (BINT or num_str) is positive (>= 0)
1423 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1425 return 1 if $x->{sign} eq '+inf'; # +inf is positive
1427 # 0+ is neither positive nor negative
1428 ($x->{sign} eq '+' && !$x->is_zero()) ? 1 : 0;
1433 # return true when arg (BINT or num_str) is negative (< 0)
1434 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1436 $x->{sign} =~ /^-/ ? 1 : 0; # -inf is negative, but NaN is not
1441 # return true when arg (BINT or num_str) is an integer
1442 # always true for BigInt, but different for BigFloats
1443 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1445 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1448 ###############################################################################
1452 # multiply two numbers -- stolen from Knuth Vol 2 pg 233
1453 # (BINT or num_str, BINT or num_str) return BINT
1456 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1457 # objectify is costly, so avoid it
1458 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1460 ($self,$x,$y,@r) = objectify(2,@_);
1463 return $x if $x->modify('bmul');
1465 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1468 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1470 return $x->bnan() if $x->is_zero() || $y->is_zero();
1471 # result will always be +-inf:
1472 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1473 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1474 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1475 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1476 return $x->binf('-');
1479 return $upgrade->bmul($x,$upgrade->new($y),@r)
1480 if defined $upgrade && !$y->isa($self);
1482 $r[3] = $y; # no push here
1484 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1486 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1487 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1494 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1495 my ($self,$x,$y) = @_;
1497 # NaN if x == NaN or y == NaN or x==y==0
1498 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1499 if (($x->is_nan() || $y->is_nan()) ||
1500 ($x->is_zero() && $y->is_zero()));
1502 # +-inf / +-inf == NaN, reminder also NaN
1503 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1505 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1507 # x / +-inf => 0, remainder x (works even if x == 0)
1508 if ($y->{sign} =~ /^[+-]inf$/)
1510 my $t = $x->copy(); # bzero clobbers up $x
1511 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1514 # 5 / 0 => +inf, -6 / 0 => -inf
1515 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1516 # exception: -8 / 0 has remainder -8, not 8
1517 # exception: -inf / 0 has remainder -inf, not inf
1520 # +-inf / 0 => special case for -inf
1521 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1522 if (!$x->is_zero() && !$x->is_inf())
1524 my $t = $x->copy(); # binf clobbers up $x
1526 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1530 # last case: +-inf / ordinary number
1532 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1534 return wantarray ? ($x,$self->bzero()) : $x;
1539 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1540 # (BINT,BINT) (quo,rem) or BINT (only rem)
1543 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1544 # objectify is costly, so avoid it
1545 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1547 ($self,$x,$y,@r) = objectify(2,@_);
1550 return $x if $x->modify('bdiv');
1552 return $self->_div_inf($x,$y)
1553 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1555 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1556 if defined $upgrade;
1558 $r[3] = $y; # no push!
1560 # calc new sign and in case $y == +/- 1, return $x
1561 my $xsign = $x->{sign}; # keep
1562 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1566 my $rem = $self->bzero();
1567 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1568 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1569 $rem->{_a} = $x->{_a};
1570 $rem->{_p} = $x->{_p};
1572 if (! $CALC->_is_zero($rem->{value}))
1574 $rem->{sign} = $y->{sign};
1575 $rem = $y->copy()->bsub($rem) if $xsign ne $y->{sign}; # one of them '-'
1579 $rem->{sign} = '+'; # dont leave -0
1585 $x->{value} = $CALC->_div($x->{value},$y->{value});
1586 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1591 ###############################################################################
1596 # modulus (or remainder)
1597 # (BINT or num_str, BINT or num_str) return BINT
1600 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1601 # objectify is costly, so avoid it
1602 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1604 ($self,$x,$y,@r) = objectify(2,@_);
1607 return $x if $x->modify('bmod');
1608 $r[3] = $y; # no push!
1609 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1611 my ($d,$r) = $self->_div_inf($x,$y);
1612 $x->{sign} = $r->{sign};
1613 $x->{value} = $r->{value};
1614 return $x->round(@r);
1617 # calc new sign and in case $y == +/- 1, return $x
1618 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1619 if (!$CALC->_is_zero($x->{value}))
1621 $x->{value} = $CALC->_sub($y->{value},$x->{value},1) # $y-$x
1622 if ($x->{sign} ne $y->{sign});
1623 $x->{sign} = $y->{sign};
1627 $x->{sign} = '+'; # dont leave -0
1634 # Modular inverse. given a number which is (hopefully) relatively
1635 # prime to the modulus, calculate its inverse using Euclid's
1636 # alogrithm. If the number is not relatively prime to the modulus
1637 # (i.e. their gcd is not one) then NaN is returned.
1640 my ($self,$x,$y,@r) = (undef,@_);
1641 # objectify is costly, so avoid it
1642 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1644 ($self,$x,$y,@r) = objectify(2,@_);
1647 return $x if $x->modify('bmodinv');
1650 if ($y->{sign} ne '+' # -, NaN, +inf, -inf
1651 || $x->is_zero() # or num == 0
1652 || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1655 # put least residue into $x if $x was negative, and thus make it positive
1656 $x->bmod($y) if $x->{sign} eq '-';
1659 ($x->{value},$sign) = $CALC->_modinv($x->{value},$y->{value});
1660 return $x->bnan() if !defined $x->{value}; # in case no GCD found
1661 return $x if !defined $sign; # already real result
1662 $x->{sign} = $sign; # flip/flop see below
1663 $x->bmod($y); # calc real result
1669 # takes a very large number to a very large exponent in a given very
1670 # large modulus, quickly, thanks to binary exponentation. supports
1671 # negative exponents.
1672 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1674 return $num if $num->modify('bmodpow');
1676 # check modulus for valid values
1677 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1678 || $mod->is_zero());
1680 # check exponent for valid values
1681 if ($exp->{sign} =~ /\w/)
1683 # i.e., if it's NaN, +inf, or -inf...
1684 return $num->bnan();
1687 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
1689 # check num for valid values (also NaN if there was no inverse but $exp < 0)
1690 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1692 # $mod is positive, sign on $exp is ignored, result also positive
1693 $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
1697 ###############################################################################
1701 # (BINT or num_str, BINT or num_str) return BINT
1702 # compute factorial number from $x, modify $x in place
1703 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1705 return $x if $x->modify('bfac') || $x->{sign} eq '+inf'; # inf => inf
1706 return $x->bnan() if $x->{sign} ne '+'; # NaN, <0 etc => NaN
1708 $x->{value} = $CALC->_fac($x->{value});
1714 # (BINT or num_str, BINT or num_str) return BINT
1715 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1716 # modifies first argument
1719 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1720 # objectify is costly, so avoid it
1721 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1723 ($self,$x,$y,@r) = objectify(2,@_);
1726 return $x if $x->modify('bpow');
1728 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1731 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1733 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1739 if ($x->{sign} =~ /^[+-]inf/)
1742 return $x->bnan() if $y->is_zero();
1743 # -inf ** -1 => 1/inf => 0
1744 return $x->bzero() if $y->is_one('-') && $x->is_negative();
1747 return $x if $x->{sign} eq '+inf';
1749 # -inf ** Y => -inf if Y is odd
1750 return $x if $y->is_odd();
1756 return $x if $x->is_one();
1759 return $x if $x->is_zero() && $y->{sign} =~ /^[+]/;
1762 return $x->binf() if $x->is_zero();
1765 return $x->bnan() if $x->is_one('-') && $y->{sign} =~ /^[-]/;
1768 return $x->bzero() if $x->{sign} eq '-' && $y->{sign} =~ /^[-]/;
1771 return $x->bnan() if $x->{sign} eq '-';
1774 return $x->binf() if $y->{sign} =~ /^[+]/;
1779 return $upgrade->bpow($upgrade->new($x),$y,@r)
1780 if defined $upgrade && (!$y->isa($self) || $y->{sign} eq '-');
1782 $r[3] = $y; # no push!
1784 # cases 0 ** Y, X ** 0, X ** 1, 1 ** Y are handled by Calc or Emu
1787 $new_sign = $y->is_odd() ? '-' : '+' if ($x->{sign} ne '+');
1789 # 0 ** -7 => ( 1 / (0 ** 7)) => 1 / 0 => +inf
1791 if $y->{sign} eq '-' && $x->{sign} eq '+' && $CALC->_is_zero($x->{value});
1792 # 1 ** -y => 1 / (1 ** |y|)
1793 # so do test for negative $y after above's clause
1794 return $x->bnan() if $y->{sign} eq '-' && !$CALC->_is_one($x->{value});
1796 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1797 $x->{sign} = $new_sign;
1798 $x->{sign} = '+' if $CALC->_is_zero($y->{value});
1804 # (BINT or num_str, BINT or num_str) return BINT
1805 # compute x << y, base n, y >= 0
1808 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1809 # objectify is costly, so avoid it
1810 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1812 ($self,$x,$y,$n,@r) = objectify(2,@_);
1815 return $x if $x->modify('blsft');
1816 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1817 return $x->round(@r) if $y->is_zero();
1819 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1821 $x->{value} = $CALC->_lsft($x->{value},$y->{value},$n);
1827 # (BINT or num_str, BINT or num_str) return BINT
1828 # compute x >> y, base n, y >= 0
1831 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1832 # objectify is costly, so avoid it
1833 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1835 ($self,$x,$y,$n,@r) = objectify(2,@_);
1838 return $x if $x->modify('brsft');
1839 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1840 return $x->round(@r) if $y->is_zero();
1841 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
1843 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1845 # this only works for negative numbers when shifting in base 2
1846 if (($x->{sign} eq '-') && ($n == 2))
1848 return $x->round(@r) if $x->is_one('-'); # -1 => -1
1851 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1852 # but perhaps there is a better emulation for two's complement shift...
1853 # if $y != 1, we must simulate it by doing:
1854 # convert to bin, flip all bits, shift, and be done
1855 $x->binc(); # -3 => -2
1856 my $bin = $x->as_bin();
1857 $bin =~ s/^-0b//; # strip '-0b' prefix
1858 $bin =~ tr/10/01/; # flip bits
1860 if ($y >= CORE::length($bin))
1862 $bin = '0'; # shifting to far right creates -1
1863 # 0, because later increment makes
1864 # that 1, attached '-' makes it '-1'
1865 # because -1 >> x == -1 !
1869 $bin =~ s/.{$y}$//; # cut off at the right side
1870 $bin = '1' . $bin; # extend left side by one dummy '1'
1871 $bin =~ tr/10/01/; # flip bits back
1873 my $res = $self->new('0b'.$bin); # add prefix and convert back
1874 $res->binc(); # remember to increment
1875 $x->{value} = $res->{value}; # take over value
1876 return $x->round(@r); # we are done now, magic, isn't?
1878 # x < 0, n == 2, y == 1
1879 $x->bdec(); # n == 2, but $y == 1: this fixes it
1882 $x->{value} = $CALC->_rsft($x->{value},$y->{value},$n);
1888 #(BINT or num_str, BINT or num_str) return BINT
1892 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1893 # objectify is costly, so avoid it
1894 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1896 ($self,$x,$y,@r) = objectify(2,@_);
1899 return $x if $x->modify('band');
1901 $r[3] = $y; # no push!
1903 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1905 my $sx = $x->{sign} eq '+' ? 1 : -1;
1906 my $sy = $y->{sign} eq '+' ? 1 : -1;
1908 if ($sx == 1 && $sy == 1)
1910 $x->{value} = $CALC->_and($x->{value},$y->{value});
1911 return $x->round(@r);
1914 if ($CAN{signed_and})
1916 $x->{value} = $CALC->_signed_and($x->{value},$y->{value},$sx,$sy);
1917 return $x->round(@r);
1921 __emu_band($self,$x,$y,$sx,$sy,@r);
1926 #(BINT or num_str, BINT or num_str) return BINT
1930 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1931 # objectify is costly, so avoid it
1932 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1934 ($self,$x,$y,@r) = objectify(2,@_);
1937 return $x if $x->modify('bior');
1938 $r[3] = $y; # no push!
1940 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1942 my $sx = $x->{sign} eq '+' ? 1 : -1;
1943 my $sy = $y->{sign} eq '+' ? 1 : -1;
1945 # the sign of X follows the sign of X, e.g. sign of Y irrelevant for bior()
1947 # don't use lib for negative values
1948 if ($sx == 1 && $sy == 1)
1950 $x->{value} = $CALC->_or($x->{value},$y->{value});
1951 return $x->round(@r);
1954 # if lib can do negative values, let it handle this
1955 if ($CAN{signed_or})
1957 $x->{value} = $CALC->_signed_or($x->{value},$y->{value},$sx,$sy);
1958 return $x->round(@r);
1962 __emu_bior($self,$x,$y,$sx,$sy,@r);
1967 #(BINT or num_str, BINT or num_str) return BINT
1971 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1972 # objectify is costly, so avoid it
1973 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1975 ($self,$x,$y,@r) = objectify(2,@_);
1978 return $x if $x->modify('bxor');
1979 $r[3] = $y; # no push!
1981 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1983 my $sx = $x->{sign} eq '+' ? 1 : -1;
1984 my $sy = $y->{sign} eq '+' ? 1 : -1;
1986 # don't use lib for negative values
1987 if ($sx == 1 && $sy == 1)
1989 $x->{value} = $CALC->_xor($x->{value},$y->{value});
1990 return $x->round(@r);
1993 # if lib can do negative values, let it handle this
1994 if ($CAN{signed_xor})
1996 $x->{value} = $CALC->_signed_xor($x->{value},$y->{value},$sx,$sy);
1997 return $x->round(@r);
2001 __emu_bxor($self,$x,$y,$sx,$sy,@r);
2006 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2008 my $e = $CALC->_len($x->{value});
2009 wantarray ? ($e,0) : $e;
2014 # return the nth decimal digit, negative values count backward, 0 is right
2015 my ($self,$x,$n) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2017 $n = $n->numify() if ref($n);
2018 $CALC->_digit($x->{value},$n||0);
2023 # return the amount of trailing zeros in $x (as scalar)
2025 $x = $class->new($x) unless ref $x;
2027 return 0 if $x->{sign} !~ /^[+-]$/; # NaN, inf, -inf etc
2029 $CALC->_zeros($x->{value}); # must handle odd values, 0 etc
2034 # calculate square root of $x
2035 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2037 return $x if $x->modify('bsqrt');
2039 return $x->bnan() if $x->{sign} !~ /^\+/; # -x or -inf or NaN => NaN
2040 return $x if $x->{sign} eq '+inf'; # sqrt(+inf) == inf
2042 return $upgrade->bsqrt($x,@r) if defined $upgrade;
2044 $x->{value} = $CALC->_sqrt($x->{value});
2050 # calculate $y'th root of $x
2053 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2055 $y = $self->new(2) unless defined $y;
2057 # objectify is costly, so avoid it
2058 if ((!ref($x)) || (ref($x) ne ref($y)))
2060 ($self,$x,$y,@r) = objectify(2,$self || $class,@_);
2063 return $x if $x->modify('broot');
2065 # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
2066 return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
2067 $y->{sign} !~ /^\+$/;
2069 return $x->round(@r)
2070 if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
2072 return $upgrade->new($x)->broot($upgrade->new($y),@r) if defined $upgrade;
2074 $x->{value} = $CALC->_root($x->{value},$y->{value});
2080 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
2081 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2083 if ($x->{sign} !~ /^[+-]$/)
2085 my $s = $x->{sign}; $s =~ s/^[+-]//; # NaN, -inf,+inf => NaN or inf
2086 return $self->new($s);
2088 return $self->bone() if $x->is_zero();
2090 # 12300 => 2 trailing zeros => exponent is 2
2091 $self->new( $CALC->_zeros($x->{value}) );
2096 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
2097 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2099 if ($x->{sign} !~ /^[+-]$/)
2101 # for NaN, +inf, -inf: keep the sign
2102 return $self->new($x->{sign});
2104 my $m = $x->copy(); delete $m->{_p}; delete $m->{_a};
2106 # that's a bit inefficient:
2107 my $zeros = $CALC->_zeros($m->{value});
2108 $m->brsft($zeros,10) if $zeros != 0;
2114 # return a copy of both the exponent and the mantissa
2115 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2117 ($x->mantissa(),$x->exponent());
2120 ##############################################################################
2121 # rounding functions
2125 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
2126 # $n == 0 || $n == 1 => round to integer
2127 my $x = shift; my $self = ref($x) || $x; $x = $self->new($x) unless ref $x;
2129 my ($scale,$mode) = $x->_scale_p(@_);
2131 return $x if !defined $scale || $x->modify('bfround'); # no-op
2133 # no-op for BigInts if $n <= 0
2134 $x->bround( $x->length()-$scale, $mode) if $scale > 0;
2136 delete $x->{_a}; # delete to save memory
2137 $x->{_p} = $scale; # store new _p
2141 sub _scan_for_nonzero
2143 # internal, used by bround() to scan for non-zeros after a '5'
2144 my ($x,$pad,$xs,$len) = @_;
2146 return 0 if $len == 1; # "5" is trailed by invisible zeros
2147 my $follow = $pad - 1;
2148 return 0 if $follow > $len || $follow < 1;
2150 # use the string form to check whether only '0's follow or not
2151 substr ($xs,-$follow) =~ /[^0]/ ? 1 : 0;
2156 # Exists to make life easier for switch between MBF and MBI (should we
2157 # autoload fxxx() like MBF does for bxxx()?)
2158 my $x = shift; $x = $class->new($x) unless ref $x;
2164 # accuracy: +$n preserve $n digits from left,
2165 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
2167 # and overwrite the rest with 0's, return normalized number
2168 # do not return $x->bnorm(), but $x
2170 my $x = shift; $x = $class->new($x) unless ref $x;
2171 my ($scale,$mode) = $x->_scale_a(@_);
2172 return $x if !defined $scale || $x->modify('bround'); # no-op
2174 if ($x->is_zero() || $scale == 0)
2176 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2179 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2181 # we have fewer digits than we want to scale to
2182 my $len = $x->length();
2183 # convert $scale to a scalar in case it is an object (put's a limit on the
2184 # number length, but this would already limited by memory constraints), makes
2186 $scale = $scale->numify() if ref ($scale);
2188 # scale < 0, but > -len (not >=!)
2189 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2191 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2195 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2196 my ($pad,$digit_round,$digit_after);
2197 $pad = $len - $scale;
2198 $pad = abs($scale-1) if $scale < 0;
2200 # do not use digit(), it is very costly for binary => decimal
2201 # getting the entire string is also costly, but we need to do it only once
2202 my $xs = $CALC->_str($x->{value});
2205 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2206 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2207 $digit_round = '0'; $digit_round = substr($xs,$pl,1) if $pad <= $len;
2208 $pl++; $pl ++ if $pad >= $len;
2209 $digit_after = '0'; $digit_after = substr($xs,$pl,1) if $pad > 0;
2211 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2212 # closer at the remaining digits of the original $x, remember decision
2213 my $round_up = 1; # default round up
2215 ($mode eq 'trunc') || # trunc by round down
2216 ($digit_after =~ /[01234]/) || # round down anyway,
2218 ($digit_after eq '5') && # not 5000...0000
2219 ($x->_scan_for_nonzero($pad,$xs,$len) == 0) &&
2221 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2222 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2223 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2224 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2225 ($mode eq 'zero') # round down if zero, sign adjusted below
2227 my $put_back = 0; # not yet modified
2229 if (($pad > 0) && ($pad <= $len))
2231 substr($xs,-$pad,$pad) = '0' x $pad; # replace with '00...'
2232 $put_back = 1; # need to put back
2236 $x->bzero(); # round to '0'
2239 if ($round_up) # what gave test above?
2241 $put_back = 1; # need to put back
2242 $pad = $len, $xs = '0' x $pad if $scale < 0; # tlr: whack 0.51=>1.0
2244 # we modify directly the string variant instead of creating a number and
2245 # adding it, since that is faster (we already have the string)
2246 my $c = 0; $pad ++; # for $pad == $len case
2247 while ($pad <= $len)
2249 $c = substr($xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2250 substr($xs,-$pad,1) = $c; $pad++;
2251 last if $c != 0; # no overflow => early out
2253 $xs = '1'.$xs if $c == 0;
2256 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back, if needed
2258 $x->{_a} = $scale if $scale >= 0;
2261 $x->{_a} = $len+$scale;
2262 $x->{_a} = 0 if $scale < -$len;
2269 # return integer less or equal then number; no-op since it's already integer
2270 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2277 # return integer greater or equal then number; no-op since it's already int
2278 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2285 # An object might be asked to return itself as bigint on certain overloaded
2286 # operations. This does exactly this, so that sub classes can simple inherit
2287 # it or override with their own integer conversion routine.
2293 # return as hex string, with prefixed 0x
2294 my $x = shift; $x = $class->new($x) if !ref($x);
2296 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2299 $s = $x->{sign} if $x->{sign} eq '-';
2300 $s . $CALC->_as_hex($x->{value});
2305 # return as binary string, with prefixed 0b
2306 my $x = shift; $x = $class->new($x) if !ref($x);
2308 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2310 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2311 return $s . $CALC->_as_bin($x->{value});
2316 # return as octal string, with prefixed 0
2317 my $x = shift; $x = $class->new($x) if !ref($x);
2319 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2321 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2322 return $s . $CALC->_as_oct($x->{value});
2325 ##############################################################################
2326 # private stuff (internal use only)
2330 # check for strings, if yes, return objects instead
2332 # the first argument is number of args objectify() should look at it will
2333 # return $count+1 elements, the first will be a classname. This is because
2334 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2335 # useless objects being created and thrown away. So we cannot simple loop
2336 # over @_. If the given count is 0, all arguments will be used.
2338 # If the second arg is a ref, use it as class.
2339 # If not, try to use it as classname, unless undef, then use $class
2340 # (aka Math::BigInt). The latter shouldn't happen,though.
2343 # $x->badd(1); => ref x, scalar y
2344 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2345 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2346 # Math::BigInt::badd(1,2); => scalar x, scalar y
2347 # In the last case we check number of arguments to turn it silently into
2348 # $class,1,2. (We can not take '1' as class ;o)
2349 # badd($class,1) is not supported (it should, eventually, try to add undef)
2350 # currently it tries 'Math::BigInt' + 1, which will not work.
2352 # some shortcut for the common cases
2354 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2356 my $count = abs(shift || 0);
2358 my (@a,$k,$d); # resulting array, temp, and downgrade
2361 # okay, got object as first
2366 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2368 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2372 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2373 if (defined ${"$a[0]::downgrade"})
2375 $d = ${"$a[0]::downgrade"};
2376 ${"$a[0]::downgrade"} = undef;
2379 my $up = ${"$a[0]::upgrade"};
2380 # print STDERR "# Now in objectify, my class is today $a[0], count = $count\n";
2388 $k = $a[0]->new($k);
2390 elsif (!defined $up && ref($k) ne $a[0])
2392 # foreign object, try to convert to integer
2393 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2406 $k = $a[0]->new($k);
2408 elsif (!defined $up && ref($k) ne $a[0])
2410 # foreign object, try to convert to integer
2411 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2415 push @a,@_; # return other params, too
2419 require Carp; Carp::croak ("$class objectify needs list context");
2421 ${"$a[0]::downgrade"} = $d;
2425 sub _register_callback
2427 my ($class,$callback) = @_;
2429 if (ref($callback) ne 'CODE')
2432 Carp::croak ("$callback is not a coderef");
2434 $CALLBACKS{$class} = $callback;
2441 $IMPORT++; # remember we did import()
2442 my @a; my $l = scalar @_;
2443 my $warn_or_die = 0; # 0 - no warn, 1 - warn, 2 - die
2444 for ( my $i = 0; $i < $l ; $i++ )
2446 if ($_[$i] eq ':constant')
2448 # this causes overlord er load to step in
2450 integer => sub { $self->new(shift) },
2451 binary => sub { $self->new(shift) };
2453 elsif ($_[$i] eq 'upgrade')
2455 # this causes upgrading
2456 $upgrade = $_[$i+1]; # or undef to disable
2459 elsif ($_[$i] =~ /^(lib|try|only)\z/)
2461 # this causes a different low lib to take care...
2462 $CALC = $_[$i+1] || '';
2463 # lib => 1 (warn on fallback), try => 0 (no warn), only => 2 (die on fallback)
2464 $warn_or_die = 1 if $_[$i] eq 'lib';
2465 $warn_or_die = 2 if $_[$i] eq 'only';
2473 # any non :constant stuff is handled by our parent, Exporter
2478 $self->SUPER::import(@a); # need it for subclasses
2479 $self->export_to_level(1,$self,@a); # need it for MBF
2482 # try to load core math lib
2483 my @c = split /\s*,\s*/,$CALC;
2486 $_ =~ tr/a-zA-Z0-9://cd; # limit to sane characters
2488 push @c, \'FastCalc', \'Calc' # if all fail, try these
2489 if $warn_or_die < 2; # but not for "only"
2490 $CALC = ''; # signal error
2493 # fallback libraries are "marked" as \'string', extract string if nec.
2494 my $lib = $l; $lib = $$l if ref($l);
2496 next if ($lib || '') eq '';
2497 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2501 # Perl < 5.6.0 dies with "out of memory!" when eval("") and ':constant' is
2502 # used in the same script, or eval("") inside import().
2503 my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
2504 my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
2506 $file = File::Spec->catfile (@parts, $file);
2507 eval { require "$file"; $lib->import( @c ); }
2511 eval "use $lib qw/@c/;";
2516 # loaded it ok, see if the api_version() is high enough
2517 if ($lib->can('api_version') && $lib->api_version() >= 1.0)
2520 # api_version matches, check if it really provides anything we need
2524 add mul div sub dec inc
2525 acmp len digit is_one is_zero is_even is_odd
2527 zeros new copy check
2528 from_hex from_oct from_bin as_hex as_bin as_oct
2529 rsft lsft xor and or
2530 mod sqrt root fac pow modinv modpow log_int gcd
2533 if (!$lib->can("_$method"))
2535 if (($WARN{$lib}||0) < 2)
2538 Carp::carp ("$lib is missing method '_$method'");
2539 $WARN{$lib} = 1; # still warn about the lib
2548 if ($warn_or_die > 0 && ref($l))
2551 my $msg = "Math::BigInt: couldn't load specified math lib(s), fallback to $lib";
2552 Carp::carp ($msg) if $warn_or_die == 1;
2553 Carp::croak ($msg) if $warn_or_die == 2;
2555 last; # found a usable one, break
2559 if (($WARN{$lib}||0) < 2)
2561 my $ver = eval "\$$lib\::VERSION" || 'unknown';
2563 Carp::carp ("Cannot load outdated $lib v$ver, please upgrade");
2564 $WARN{$lib} = 2; # never warn again
2572 if ($warn_or_die == 2)
2574 Carp::croak ("Couldn't load specified math lib(s) and fallback disallowed");
2578 Carp::croak ("Couldn't load any math lib(s), not even fallback to Calc.pm");
2583 foreach my $class (keys %CALLBACKS)
2585 &{$CALLBACKS{$class}}($CALC);
2588 # Fill $CAN with the results of $CALC->can(...) for emulating lower math lib
2592 for my $method (qw/ signed_and signed_or signed_xor /)
2594 $CAN{$method} = $CALC->can("_$method") ? 1 : 0;
2602 # create a bigint from a hexadecimal string
2603 my ($self, $hs) = @_;
2605 my $rc = $self->__from_hex($hs);
2607 return $self->bnan() unless defined $rc;
2614 # create a bigint from a hexadecimal string
2615 my ($self, $bs) = @_;
2617 my $rc = $self->__from_bin($bs);
2619 return $self->bnan() unless defined $rc;
2626 # create a bigint from a hexadecimal string
2627 my ($self, $os) = @_;
2629 my $x = $self->bzero();
2632 $os =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2633 $os =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2635 return $x->bnan() if $os !~ /^[\-\+]?0[0-9]+$/;
2637 my $sign = '+'; $sign = '-' if $os =~ /^-/;
2639 $os =~ s/^[+-]//; # strip sign
2640 $x->{value} = $CALC->_from_oct($os);
2641 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2648 # convert a (ref to) big hex string to BigInt, return undef for error
2651 my $x = Math::BigInt->bzero();
2654 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2655 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2657 return $x->bnan() if $hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2659 my $sign = '+'; $sign = '-' if $hs =~ /^-/;
2661 $hs =~ s/^[+-]//; # strip sign
2662 $x->{value} = $CALC->_from_hex($hs);
2663 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2670 # convert a (ref to) big binary string to BigInt, return undef for error
2673 my $x = Math::BigInt->bzero();
2676 $bs =~ s/([01])_([01])/$1$2/g;
2677 $bs =~ s/([01])_([01])/$1$2/g;
2678 return $x->bnan() if $bs !~ /^[+-]?0b[01]+$/;
2680 my $sign = '+'; $sign = '-' if $bs =~ /^\-/;
2681 $bs =~ s/^[+-]//; # strip sign
2683 $x->{value} = $CALC->_from_bin($bs);
2684 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2690 # input: num_str; output: undef for invalid or
2691 # (\$mantissa_sign,\$mantissa_value,\$mantissa_fraction,\$exp_sign,\$exp_value)
2692 # Internal, take apart a string and return the pieces.
2693 # Strip leading/trailing whitespace, leading zeros, underscore and reject
2697 # strip white space at front, also extranous leading zeros
2698 $x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2699 $x =~ s/^\s+//; # but this will
2700 $x =~ s/\s+$//g; # strip white space at end
2702 # shortcut, if nothing to split, return early
2703 if ($x =~ /^[+-]?[0-9]+\z/)
2705 $x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2706 return (\$sign, \$x, \'', \'', \0);
2709 # invalid starting char?
2710 return if $x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2712 return __from_hex($x) if $x =~ /^[\-\+]?0x/; # hex string
2713 return __from_bin($x) if $x =~ /^[\-\+]?0b/; # binary string
2715 # strip underscores between digits
2716 $x =~ s/([0-9])_([0-9])/$1$2/g;
2717 $x =~ s/([0-9])_([0-9])/$1$2/g; # do twice for 1_2_3
2719 # some possible inputs:
2720 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2721 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 # 0e999
2723 my ($m,$e,$last) = split /[Ee]/,$x;
2724 return if defined $last; # last defined => 1e2E3 or others
2725 $e = '0' if !defined $e || $e eq "";
2727 # sign,value for exponent,mantint,mantfrac
2728 my ($es,$ev,$mis,$miv,$mfv);
2730 if ($e =~ /^([+-]?)0*([0-9]+)$/) # strip leading zeros
2734 return if $m eq '.' || $m eq '';
2735 my ($mi,$mf,$lastf) = split /\./,$m;
2736 return if defined $lastf; # lastf defined => 1.2.3 or others
2737 $mi = '0' if !defined $mi;
2738 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2739 $mf = '0' if !defined $mf || $mf eq '';
2740 if ($mi =~ /^([+-]?)0*([0-9]+)$/) # strip leading zeros
2742 $mis = $1||'+'; $miv = $2;
2743 return unless ($mf =~ /^([0-9]*?)0*$/); # strip trailing zeros
2745 # handle the 0e999 case here
2746 $ev = 0 if $miv eq '0' && $mfv eq '';
2747 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2750 return; # NaN, not a number
2753 ##############################################################################
2754 # internal calculation routines (others are in Math::BigInt::Calc etc)
2758 # (BINT or num_str, BINT or num_str) return BINT
2759 # does modify first argument
2763 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2764 my $method = ref($x) . '::bgcd';
2766 $x * $ty / &$method($x,$ty);
2769 ###############################################################################
2770 # this method returns 0 if the object can be modified, or 1 if not.
2771 # We use a fast constant sub() here, to avoid costly calls. Subclasses
2772 # may override it with special code (f.i. Math::BigInt::Constant does so)
2774 sub modify () { 0; }
2783 Math::BigInt - Arbitrary size integer/float math package
2789 # or make it faster: install (optional) Math::BigInt::GMP
2790 # and always use (it will fall back to pure Perl if the
2791 # GMP library is not installed):
2793 # will warn if Math::BigInt::GMP cannot be found
2794 use Math::BigInt lib => 'GMP';
2796 # to supress the warning use this:
2797 # use Math::BigInt try => 'GMP';
2799 my $str = '1234567890';
2800 my @values = (64,74,18);
2801 my $n = 1; my $sign = '-';
2804 $x = Math::BigInt->new($str); # defaults to 0
2805 $y = $x->copy(); # make a true copy
2806 $nan = Math::BigInt->bnan(); # create a NotANumber
2807 $zero = Math::BigInt->bzero(); # create a +0
2808 $inf = Math::BigInt->binf(); # create a +inf
2809 $inf = Math::BigInt->binf('-'); # create a -inf
2810 $one = Math::BigInt->bone(); # create a +1
2811 $one = Math::BigInt->bone('-'); # create a -1
2813 $h = Math::BigInt->new('0x123'); # from hexadecimal
2814 $b = Math::BigInt->new('0b101'); # from binary
2815 $o = Math::BigInt->from_oct('0101'); # from octal
2817 # Testing (don't modify their arguments)
2818 # (return true if the condition is met, otherwise false)
2820 $x->is_zero(); # if $x is +0
2821 $x->is_nan(); # if $x is NaN
2822 $x->is_one(); # if $x is +1
2823 $x->is_one('-'); # if $x is -1
2824 $x->is_odd(); # if $x is odd
2825 $x->is_even(); # if $x is even
2826 $x->is_pos(); # if $x >= 0
2827 $x->is_neg(); # if $x < 0
2828 $x->is_inf($sign); # if $x is +inf, or -inf (sign is default '+')
2829 $x->is_int(); # if $x is an integer (not a float)
2831 # comparing and digit/sign extraction
2832 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2833 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2834 $x->sign(); # return the sign, either +,- or NaN
2835 $x->digit($n); # return the nth digit, counting from right
2836 $x->digit(-$n); # return the nth digit, counting from left
2838 # The following all modify their first argument. If you want to preserve
2839 # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
2840 # necessary when mixing $a = $b assignments with non-overloaded math.
2842 $x->bzero(); # set $x to 0
2843 $x->bnan(); # set $x to NaN
2844 $x->bone(); # set $x to +1
2845 $x->bone('-'); # set $x to -1
2846 $x->binf(); # set $x to inf
2847 $x->binf('-'); # set $x to -inf
2849 $x->bneg(); # negation
2850 $x->babs(); # absolute value
2851 $x->bnorm(); # normalize (no-op in BigInt)
2852 $x->bnot(); # two's complement (bit wise not)
2853 $x->binc(); # increment $x by 1
2854 $x->bdec(); # decrement $x by 1
2856 $x->badd($y); # addition (add $y to $x)
2857 $x->bsub($y); # subtraction (subtract $y from $x)
2858 $x->bmul($y); # multiplication (multiply $x by $y)
2859 $x->bdiv($y); # divide, set $x to quotient
2860 # return (quo,rem) or quo if scalar
2862 $x->bmod($y); # modulus (x % y)
2863 $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
2864 $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
2866 $x->bpow($y); # power of arguments (x ** y)
2867 $x->blsft($y); # left shift in base 2
2868 $x->brsft($y); # right shift in base 2
2869 # returns (quo,rem) or quo if in scalar context
2870 $x->blsft($y,$n); # left shift by $y places in base $n
2871 $x->brsft($y,$n); # right shift by $y places in base $n
2872 # returns (quo,rem) or quo if in scalar context
2874 $x->band($y); # bitwise and
2875 $x->bior($y); # bitwise inclusive or
2876 $x->bxor($y); # bitwise exclusive or
2877 $x->bnot(); # bitwise not (two's complement)
2879 $x->bsqrt(); # calculate square-root
2880 $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
2881 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2883 $x->blog(); # logarithm of $x to base e (Euler's number)
2884 $x->blog($base); # logarithm of $x to base $base (f.i. 2)
2885 $x->bexp(); # calculate e ** $x where e is Euler's number
2887 $x->round($A,$P,$mode); # round to accuracy or precision using mode $mode
2888 $x->bround($n); # accuracy: preserve $n digits
2889 $x->bfround($n); # round to $nth digit, no-op for BigInts
2891 # The following do not modify their arguments in BigInt (are no-ops),
2892 # but do so in BigFloat:
2894 $x->bfloor(); # return integer less or equal than $x
2895 $x->bceil(); # return integer greater or equal than $x
2897 # The following do not modify their arguments:
2899 # greatest common divisor (no OO style)
2900 my $gcd = Math::BigInt::bgcd(@values);
2901 # lowest common multiplicator (no OO style)
2902 my $lcm = Math::BigInt::blcm(@values);
2904 $x->length(); # return number of digits in number
2905 ($xl,$f) = $x->length(); # length of number and length of fraction part,
2906 # latter is always 0 digits long for BigInts
2908 $x->exponent(); # return exponent as BigInt
2909 $x->mantissa(); # return (signed) mantissa as BigInt
2910 $x->parts(); # return (mantissa,exponent) as BigInt
2911 $x->copy(); # make a true copy of $x (unlike $y = $x;)
2912 $x->as_int(); # return as BigInt (in BigInt: same as copy())
2913 $x->numify(); # return as scalar (might overflow!)
2915 # conversation to string (do not modify their argument)
2916 $x->bstr(); # normalized string (e.g. '3')
2917 $x->bsstr(); # norm. string in scientific notation (e.g. '3E0')
2918 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
2919 $x->as_bin(); # as signed binary string with prefixed 0b
2920 $x->as_oct(); # as signed octal string with prefixed 0
2923 # precision and accuracy (see section about rounding for more)
2924 $x->precision(); # return P of $x (or global, if P of $x undef)
2925 $x->precision($n); # set P of $x to $n
2926 $x->accuracy(); # return A of $x (or global, if A of $x undef)
2927 $x->accuracy($n); # set A $x to $n
2930 Math::BigInt->precision(); # get/set global P for all BigInt objects
2931 Math::BigInt->accuracy(); # get/set global A for all BigInt objects
2932 Math::BigInt->round_mode(); # get/set global round mode, one of
2933 # 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'
2934 Math::BigInt->config(); # return hash containing configuration
2938 All operators (including basic math operations) are overloaded if you
2939 declare your big integers as
2941 $i = new Math::BigInt '123_456_789_123_456_789';
2943 Operations with overloaded operators preserve the arguments which is
2944 exactly what you expect.
2950 Input values to these routines may be any string, that looks like a number
2951 and results in an integer, including hexadecimal and binary numbers.
2953 Scalars holding numbers may also be passed, but note that non-integer numbers
2954 may already have lost precision due to the conversation to float. Quote
2955 your input if you want BigInt to see all the digits:
2957 $x = Math::BigInt->new(12345678890123456789); # bad
2958 $x = Math::BigInt->new('12345678901234567890'); # good
2960 You can include one underscore between any two digits.
2962 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
2963 Non-integer values result in NaN.
2965 Hexadecimal (prefixed with "0x") and binary numbers (prefixed with "0b")
2966 are accepted, too. Please note that octal numbers are not recognized
2967 by new(), so the following will print "123":
2969 perl -MMath::BigInt -le 'print Math::BigInt->new("0123")'
2971 To convert an octal number, use from_oct();
2973 perl -MMath::BigInt -le 'print Math::BigInt->from_oct("0123")'
2975 Currently, Math::BigInt::new() defaults to 0, while Math::BigInt::new('')
2976 results in 'NaN'. This might change in the future, so use always the following
2977 explicit forms to get a zero or NaN:
2979 $zero = Math::BigInt->bzero();
2980 $nan = Math::BigInt->bnan();
2982 C<bnorm()> on a BigInt object is now effectively a no-op, since the numbers
2983 are always stored in normalized form. If passed a string, creates a BigInt
2984 object from the input.
2988 Output values are BigInt objects (normalized), except for the methods which
2989 return a string (see L<SYNOPSIS>).
2991 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
2992 C<is_nan()>, etc.) return true or false, while others (C<bcmp()>, C<bacmp()>)
2993 return either undef (if NaN is involved), <0, 0 or >0 and are suited for sort.
2999 Each of the methods below (except config(), accuracy() and precision())
3000 accepts three additional parameters. These arguments C<$A>, C<$P> and C<$R>
3001 are C<accuracy>, C<precision> and C<round_mode>. Please see the section about
3002 L<ACCURACY and PRECISION> for more information.
3008 print Dumper ( Math::BigInt->config() );
3009 print Math::BigInt->config()->{lib},"\n";
3011 Returns a hash containing the configuration, e.g. the version number, lib
3012 loaded etc. The following hash keys are currently filled in with the
3013 appropriate information.
3017 ============================================================
3018 lib Name of the low-level math library
3020 lib_version Version of low-level math library (see 'lib')
3022 class The class name of config() you just called
3024 upgrade To which class math operations might be upgraded
3026 downgrade To which class math operations might be downgraded
3028 precision Global precision
3030 accuracy Global accuracy
3032 round_mode Global round mode
3034 version version number of the class you used
3036 div_scale Fallback accuracy for div
3038 trap_nan If true, traps creation of NaN via croak()
3040 trap_inf If true, traps creation of +inf/-inf via croak()
3043 The following values can be set by passing C<config()> a reference to a hash:
3046 upgrade downgrade precision accuracy round_mode div_scale
3050 $new_cfg = Math::BigInt->config( { trap_inf => 1, precision => 5 } );
3054 $x->accuracy(5); # local for $x
3055 CLASS->accuracy(5); # global for all members of CLASS
3056 # Note: This also applies to new()!
3058 $A = $x->accuracy(); # read out accuracy that affects $x
3059 $A = CLASS->accuracy(); # read out global accuracy
3061 Set or get the global or local accuracy, aka how many significant digits the
3062 results have. If you set a global accuracy, then this also applies to new()!
3064 Warning! The accuracy I<sticks>, e.g. once you created a number under the
3065 influence of C<< CLASS->accuracy($A) >>, all results from math operations with
3066 that number will also be rounded.
3068 In most cases, you should probably round the results explicitly using one of
3069 L<round()>, L<bround()> or L<bfround()> or by passing the desired accuracy
3070 to the math operation as additional parameter:
3072 my $x = Math::BigInt->new(30000);
3073 my $y = Math::BigInt->new(7);
3074 print scalar $x->copy()->bdiv($y, 2); # print 4300
3075 print scalar $x->copy()->bdiv($y)->bround(2); # print 4300
3077 Please see the section about L<ACCURACY AND PRECISION> for further details.
3079 Value must be greater than zero. Pass an undef value to disable it:
3081 $x->accuracy(undef);
3082 Math::BigInt->accuracy(undef);
3084 Returns the current accuracy. For C<$x->accuracy()> it will return either the
3085 local accuracy, or if not defined, the global. This means the return value
3086 represents the accuracy that will be in effect for $x:
3088 $y = Math::BigInt->new(1234567); # unrounded
3089 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
3090 $x = Math::BigInt->new(123456); # $x will be automatically rounded!
3091 print "$x $y\n"; # '123500 1234567'
3092 print $x->accuracy(),"\n"; # will be 4
3093 print $y->accuracy(),"\n"; # also 4, since global is 4
3094 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
3095 print $x->accuracy(),"\n"; # still 4
3096 print $y->accuracy(),"\n"; # 5, since global is 5
3098 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
3099 globals separated from Math::BigInt, but it is possible to subclass
3100 Math::BigInt and make the globals of the subclass aliases to the ones from
3105 $x->precision(-2); # local for $x, round at the second digit right of the dot
3106 $x->precision(2); # ditto, round at the second digit left of the dot
3108 CLASS->precision(5); # Global for all members of CLASS
3109 # This also applies to new()!
3110 CLASS->precision(-5); # ditto
3112 $P = CLASS->precision(); # read out global precision
3113 $P = $x->precision(); # read out precision that affects $x
3115 Note: You probably want to use L<accuracy()> instead. With L<accuracy> you
3116 set the number of digits each result should have, with L<precision> you
3117 set the place where to round!
3119 C<precision()> sets or gets the global or local precision, aka at which digit
3120 before or after the dot to round all results. A set global precision also
3121 applies to all newly created numbers!
3123 In Math::BigInt, passing a negative number precision has no effect since no
3124 numbers have digits after the dot. In L<Math::BigFloat>, it will round all
3125 results to P digits after the dot.
3127 Please see the section about L<ACCURACY AND PRECISION> for further details.
3129 Pass an undef value to disable it:
3131 $x->precision(undef);
3132 Math::BigInt->precision(undef);
3134 Returns the current precision. For C<$x->precision()> it will return either the
3135 local precision of $x, or if not defined, the global. This means the return
3136 value represents the prevision that will be in effect for $x:
3138 $y = Math::BigInt->new(1234567); # unrounded
3139 print Math::BigInt->precision(4),"\n"; # set 4, print 4
3140 $x = Math::BigInt->new(123456); # will be automatically rounded
3141 print $x; # print "120000"!
3143 Note: Works also for subclasses like L<Math::BigFloat>. Each class has its
3144 own globals separated from Math::BigInt, but it is possible to subclass
3145 Math::BigInt and make the globals of the subclass aliases to the ones from
3152 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
3153 2, but others work, too.
3155 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
3159 $x = Math::BigInt->new(10);
3160 $x->brsft(1); # same as $x >> 1: 5
3161 $x = Math::BigInt->new(1234);
3162 $x->brsft(2,10); # result 12
3164 There is one exception, and that is base 2 with negative $x:
3167 $x = Math::BigInt->new(-5);
3170 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
3175 $x = Math::BigInt->new($str,$A,$P,$R);
3177 Creates a new BigInt object from a scalar or another BigInt object. The
3178 input is accepted as decimal, hex (with leading '0x') or binary (with leading
3181 See L<Input> for more info on accepted input formats.
3185 $x = Math::BigIn->from_oct("0775"); # input is octal
3189 $x = Math::BigIn->from_hex("0xcafe"); # input is hexadecimal
3193 $x = Math::BigIn->from_oct("0x10011"); # input is binary
3197 $x = Math::BigInt->bnan();
3199 Creates a new BigInt object representing NaN (Not A Number).
3200 If used on an object, it will set it to NaN:
3206 $x = Math::BigInt->bzero();
3208 Creates a new BigInt object representing zero.
3209 If used on an object, it will set it to zero:
3215 $x = Math::BigInt->binf($sign);
3217 Creates a new BigInt object representing infinity. The optional argument is
3218 either '-' or '+', indicating whether you want infinity or minus infinity.
3219 If used on an object, it will set it to infinity:
3226 $x = Math::BigInt->binf($sign);
3228 Creates a new BigInt object representing one. The optional argument is
3229 either '-' or '+', indicating whether you want one or minus one.
3230 If used on an object, it will set it to one:
3235 =head2 is_one()/is_zero()/is_nan()/is_inf()
3238 $x->is_zero(); # true if arg is +0
3239 $x->is_nan(); # true if arg is NaN
3240 $x->is_one(); # true if arg is +1
3241 $x->is_one('-'); # true if arg is -1
3242 $x->is_inf(); # true if +inf
3243 $x->is_inf('-'); # true if -inf (sign is default '+')
3245 These methods all test the BigInt for being one specific value and return
3246 true or false depending on the input. These are faster than doing something
3251 =head2 is_pos()/is_neg()/is_positive()/is_negative()
3253 $x->is_pos(); # true if > 0
3254 $x->is_neg(); # true if < 0
3256 The methods return true if the argument is positive or negative, respectively.
3257 C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
3258 C<-inf> is negative. A C<zero> is neither positive nor negative.
3260 These methods are only testing the sign, and not the value.
3262 C<is_positive()> and C<is_negative()> are aliases to C<is_pos()> and
3263 C<is_neg()>, respectively. C<is_positive()> and C<is_negative()> were
3264 introduced in v1.36, while C<is_pos()> and C<is_neg()> were only introduced
3267 =head2 is_odd()/is_even()/is_int()
3269 $x->is_odd(); # true if odd, false for even
3270 $x->is_even(); # true if even, false for odd
3271 $x->is_int(); # true if $x is an integer
3273 The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
3274 C<-inf> are not integers and are neither odd nor even.
3276 In BigInt, all numbers except C<NaN>, C<+inf> and C<-inf> are integers.
3282 Compares $x with $y and takes the sign into account.
3283 Returns -1, 0, 1 or undef.
3289 Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
3295 Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
3297 If you want $x to have a certain sign, use one of the following methods:
3300 $x->babs()->bneg(); # '-'
3302 $x->binf(); # '+inf'
3303 $x->binf('-'); # '-inf'
3307 $x->digit($n); # return the nth digit, counting from right
3309 If C<$n> is negative, returns the digit counting from left.
3315 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
3316 and '-inf', respectively. Does nothing for NaN or zero.
3322 Set the number to it's absolute value, e.g. change the sign from '-' to '+'
3323 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
3328 $x->bnorm(); # normalize (no-op)
3334 Two's complement (bit wise not). This is equivalent to
3342 $x->binc(); # increment x by 1
3346 $x->bdec(); # decrement x by 1
3350 $x->badd($y); # addition (add $y to $x)
3354 $x->bsub($y); # subtraction (subtract $y from $x)
3358 $x->bmul($y); # multiplication (multiply $x by $y)
3362 $x->bdiv($y); # divide, set $x to quotient
3363 # return (quo,rem) or quo if scalar
3367 $x->bmod($y); # modulus (x % y)
3371 num->bmodinv($mod); # modular inverse
3373 Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
3374 returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
3375 C<bgcd($num, $mod)==1>.
3379 $num->bmodpow($exp,$mod); # modular exponentation
3380 # ($num**$exp % $mod)
3382 Returns the value of C<$num> taken to the power C<$exp> in the modulus
3383 C<$mod> using binary exponentation. C<bmodpow> is far superior to
3388 because it is much faster - it reduces internal variables into
3389 the modulus whenever possible, so it operates on smaller numbers.
3391 C<bmodpow> also supports negative exponents.
3393 bmodpow($num, -1, $mod)
3395 is exactly equivalent to
3401 $x->bpow($y); # power of arguments (x ** y)
3405 $x->blog($base, $accuracy); # logarithm of x to the base $base
3407 If C<$base> is not defined, Euler's number (e) is used:
3409 print $x->blog(undef, 100); # log(x) to 100 digits
3413 $x->bexp($accuracy); # calculate e ** X
3415 Calculates the expression C<e ** $x> where C<e> is Euler's number.
3417 This method was added in v1.82 of Math::BigInt (April 2007).
3423 $x->blsft($y); # left shift in base 2
3424 $x->blsft($y,$n); # left shift, in base $n (like 10)
3428 $x->brsft($y); # right shift in base 2
3429 $x->brsft($y,$n); # right shift, in base $n (like 10)
3433 $x->band($y); # bitwise and
3437 $x->bior($y); # bitwise inclusive or
3441 $x->bxor($y); # bitwise exclusive or
3445 $x->bnot(); # bitwise not (two's complement)
3449 $x->bsqrt(); # calculate square-root
3453 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3457 $x->round($A,$P,$round_mode);
3459 Round $x to accuracy C<$A> or precision C<$P> using the round mode
3464 $x->bround($N); # accuracy: preserve $N digits
3468 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3474 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3475 does change $x in BigFloat.
3481 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3482 does change $x in BigFloat.
3486 bgcd(@values); # greatest common divisor (no OO style)
3490 blcm(@values); # lowest common multiplicator (no OO style)
3495 ($xl,$fl) = $x->length();
3497 Returns the number of digits in the decimal representation of the number.
3498 In list context, returns the length of the integer and fraction part. For
3499 BigInt's, the length of the fraction part will always be 0.
3505 Return the exponent of $x as BigInt.
3511 Return the signed mantissa of $x as BigInt.
3515 $x->parts(); # return (mantissa,exponent) as BigInt
3519 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3521 =head2 as_int()/as_number()
3525 Returns $x as a BigInt (truncated towards zero). In BigInt this is the same as
3528 C<as_number()> is an alias to this method. C<as_number> was introduced in
3529 v1.22, while C<as_int()> was only introduced in v1.68.
3535 Returns a normalized string representation of C<$x>.
3539 $x->bsstr(); # normalized string in scientific notation
3543 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3547 $x->as_bin(); # as signed binary string with prefixed 0b
3551 $x->as_oct(); # as signed octal string with prefixed 0
3557 This returns a normal Perl scalar from $x. It is used automatically
3558 whenever a scalar is needed, for instance in array index operations.
3560 This loses precision, to avoid this use L<as_int()> instead.
3564 $x->modify('bpowd');
3566 This method returns 0 if the object can be modified with the given
3567 peration, or 1 if not.
3569 This is used for instance by L<Math::BigInt::Constant>.
3571 =head2 upgrade()/downgrade()
3573 Set/get the class for downgrade/upgrade operations. Thuis is used
3574 for instance by L<bignum>. The defaults are '', thus the following
3575 operation will create a BigInt, not a BigFloat:
3577 my $i = Math::BigInt->new(123);
3578 my $f = Math::BigFloat->new('123.1');
3580 print $i + $f,"\n"; # print 246
3584 Set/get the number of digits for the default precision in divide
3589 Set/get the current round mode.
3591 =head1 ACCURACY and PRECISION
3593 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3594 accuracy and precision based rounding, both automatically after every
3595 operation, as well as manually.
3597 This section describes the accuracy/precision handling in Math::Big* as it
3598 used to be and as it is now, complete with an explanation of all terms and
3601 Not yet implemented things (but with correct description) are marked with '!',
3602 things that need to be answered are marked with '?'.
3604 In the next paragraph follows a short description of terms used here (because
3605 these may differ from terms used by others people or documentation).
3607 During the rest of this document, the shortcuts A (for accuracy), P (for
3608 precision), F (fallback) and R (rounding mode) will be used.
3612 A fixed number of digits before (positive) or after (negative)
3613 the decimal point. For example, 123.45 has a precision of -2. 0 means an
3614 integer like 123 (or 120). A precision of 2 means two digits to the left
3615 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3616 numbers with zeros before the decimal point may have different precisions,
3617 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3618 was). It could also have p < 0, when the digits after the decimal point
3621 The string output (of floating point numbers) will be padded with zeros:
3623 Initial value P A Result String
3624 ------------------------------------------------------------
3625 1234.01 -3 1000 1000
3628 1234.001 1 1234 1234.0
3630 1234.01 2 1234.01 1234.01
3631 1234.01 5 1234.01 1234.01000
3633 For BigInts, no padding occurs.
3637 Number of significant digits. Leading zeros are not counted. A
3638 number may have an accuracy greater than the non-zero digits
3639 when there are zeros in it or trailing zeros. For example, 123.456 has
3640 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3642 The string output (of floating point numbers) will be padded with zeros:
3644 Initial value P A Result String
3645 ------------------------------------------------------------
3647 1234.01 6 1234.01 1234.01
3648 1234.1 8 1234.1 1234.1000
3650 For BigInts, no padding occurs.
3654 When both A and P are undefined, this is used as a fallback accuracy when
3657 =head2 Rounding mode R
3659 When rounding a number, different 'styles' or 'kinds'
3660 of rounding are possible. (Note that random rounding, as in
3661 Math::Round, is not implemented.)
3667 truncation invariably removes all digits following the
3668 rounding place, replacing them with zeros. Thus, 987.65 rounded
3669 to tens (P=1) becomes 980, and rounded to the fourth sigdig
3670 becomes 987.6 (A=4). 123.456 rounded to the second place after the
3671 decimal point (P=-2) becomes 123.46.
3673 All other implemented styles of rounding attempt to round to the
3674 "nearest digit." If the digit D immediately to the right of the
3675 rounding place (skipping the decimal point) is greater than 5, the
3676 number is incremented at the rounding place (possibly causing a
3677 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3678 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3679 truncated at the rounding place: e.g. when rounding to units, 0.4
3680 rounds to 0, and -19.4 rounds to -19.
3682 However the results of other styles of rounding differ if the
3683 digit immediately to the right of the rounding place (skipping the
3684 decimal point) is 5 and if there are no digits, or no digits other
3685 than 0, after that 5. In such cases:
3689 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3690 if it is not already. E.g., when rounding to the first sigdig, 0.45
3691 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3695 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3696 it is not already. E.g., when rounding to the first sigdig, 0.45
3697 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3701 round to plus infinity, i.e. always round up. E.g., when
3702 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3703 and 0.4501 also becomes 0.5.
3707 round to minus infinity, i.e. always round down. E.g., when
3708 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3709 but 0.4501 becomes 0.5.
3713 round to zero, i.e. positive numbers down, negative ones up.
3714 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3715 becomes -0.5, but 0.4501 becomes 0.5.
3719 round up if the digit immediately to the right of the rounding place
3720 is 5 or greater, otherwise round down. E.g., 0.15 becomes 0.2 and
3725 The handling of A & P in MBI/MBF (the old core code shipped with Perl
3726 versions <= 5.7.2) is like this:
3732 * ffround($p) is able to round to $p number of digits after the decimal
3734 * otherwise P is unused
3736 =item Accuracy (significant digits)
3738 * fround($a) rounds to $a significant digits
3739 * only fdiv() and fsqrt() take A as (optional) paramater
3740 + other operations simply create the same number (fneg etc), or more (fmul)
3742 + rounding/truncating is only done when explicitly calling one of fround
3743 or ffround, and never for BigInt (not implemented)
3744 * fsqrt() simply hands its accuracy argument over to fdiv.
3745 * the documentation and the comment in the code indicate two different ways
3746 on how fdiv() determines the maximum number of digits it should calculate,
3747 and the actual code does yet another thing
3749 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3751 result has at most max(scale, length(dividend), length(divisor)) digits
3753 scale = max(scale, length(dividend)-1,length(divisor)-1);
3754 scale += length(divisor) - length(dividend);
3755 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3756 Actually, the 'difference' added to the scale is calculated from the
3757 number of "significant digits" in dividend and divisor, which is derived
3758 by looking at the length of the mantissa. Which is wrong, since it includes
3759 the + sign (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops
3760 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3761 assumption that 124 has 3 significant digits, while 120/7 will get you
3762 '17', not '17.1' since 120 is thought to have 2 significant digits.
3763 The rounding after the division then uses the remainder and $y to determine
3764 wether it must round up or down.
3765 ? I have no idea which is the right way. That's why I used a slightly more
3766 ? simple scheme and tweaked the few failing testcases to match it.
3770 This is how it works now:
3774 =item Setting/Accessing
3776 * You can set the A global via C<< Math::BigInt->accuracy() >> or
3777 C<< Math::BigFloat->accuracy() >> or whatever class you are using.
3778 * You can also set P globally by using C<< Math::SomeClass->precision() >>
3780 * Globals are classwide, and not inherited by subclasses.
3781 * to undefine A, use C<< Math::SomeCLass->accuracy(undef); >>
3782 * to undefine P, use C<< Math::SomeClass->precision(undef); >>
3783 * Setting C<< Math::SomeClass->accuracy() >> clears automatically
3784 C<< Math::SomeClass->precision() >>, and vice versa.
3785 * To be valid, A must be > 0, P can have any value.
3786 * If P is negative, this means round to the P'th place to the right of the
3787 decimal point; positive values mean to the left of the decimal point.
3788 P of 0 means round to integer.
3789 * to find out the current global A, use C<< Math::SomeClass->accuracy() >>
3790 * to find out the current global P, use C<< Math::SomeClass->precision() >>
3791 * use C<< $x->accuracy() >> respective C<< $x->precision() >> for the local
3792 setting of C<< $x >>.
3793 * Please note that C<< $x->accuracy() >> respective C<< $x->precision() >>
3794 return eventually defined global A or P, when C<< $x >>'s A or P is not
3797 =item Creating numbers
3799 * When you create a number, you can give it's desired A or P via:
3800 $x = Math::BigInt->new($number,$A,$P);
3801 * Only one of A or P can be defined, otherwise the result is NaN
3802 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
3803 globals (if set) will be used. Thus changing the global defaults later on
3804 will not change the A or P of previously created numbers (i.e., A and P of
3805 $x will be what was in effect when $x was created)
3806 * If given undef for A and P, B<no> rounding will occur, and the globals will
3807 B<not> be used. This is used by subclasses to create numbers without
3808 suffering rounding in the parent. Thus a subclass is able to have it's own
3809 globals enforced upon creation of a number by using
3810 C<< $x = Math::BigInt->new($number,undef,undef) >>:
3812 use Math::BigInt::SomeSubclass;
3815 Math::BigInt->accuracy(2);
3816 Math::BigInt::SomeSubClass->accuracy(3);
3817 $x = Math::BigInt::SomeSubClass->new(1234);
3819 $x is now 1230, and not 1200. A subclass might choose to implement
3820 this otherwise, e.g. falling back to the parent's A and P.
3824 * If A or P are enabled/defined, they are used to round the result of each
3825 operation according to the rules below
3826 * Negative P is ignored in Math::BigInt, since BigInts never have digits
3827 after the decimal point
3828 * Math::BigFloat uses Math::BigInt internally, but setting A or P inside
3829 Math::BigInt as globals does not tamper with the parts of a BigFloat.
3830 A flag is used to mark all Math::BigFloat numbers as 'never round'.
3834 * It only makes sense that a number has only one of A or P at a time.
3835 If you set either A or P on one object, or globally, the other one will
3836 be automatically cleared.
3837 * If two objects are involved in an operation, and one of them has A in
3838 effect, and the other P, this results in an error (NaN).
3839 * A takes precedence over P (Hint: A comes before P).
3840 If neither of them is defined, nothing is used, i.e. the result will have
3841 as many digits as it can (with an exception for fdiv/fsqrt) and will not
3843 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
3844 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
3845 If either the dividend's or the divisor's mantissa has more digits than
3846 the value of F, the higher value will be used instead of F.
3847 This is to limit the digits (A) of the result (just consider what would
3848 happen with unlimited A and P in the case of 1/3 :-)
3849 * fdiv will calculate (at least) 4 more digits than required (determined by
3850 A, P or F), and, if F is not used, round the result
3851 (this will still fail in the case of a result like 0.12345000000001 with A
3852 or P of 5, but this can not be helped - or can it?)
3853 * Thus you can have the math done by on Math::Big* class in two modi:
3854 + never round (this is the default):
3855 This is done by setting A and P to undef. No math operation
3856 will round the result, with fdiv() and fsqrt() as exceptions to guard
3857 against overflows. You must explicitly call bround(), bfround() or
3858 round() (the latter with parameters).
3859 Note: Once you have rounded a number, the settings will 'stick' on it
3860 and 'infect' all other numbers engaged in math operations with it, since
3861 local settings have the highest precedence. So, to get SaferRound[tm],
3862 use a copy() before rounding like this:
3864 $x = Math::BigFloat->new(12.34);
3865 $y = Math::BigFloat->new(98.76);
3866 $z = $x * $y; # 1218.6984
3867 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
3868 $z = $x * $y; # still 1218.6984, without
3869 # copy would have been 1210!
3871 + round after each op:
3872 After each single operation (except for testing like is_zero()), the
3873 method round() is called and the result is rounded appropriately. By
3874 setting proper values for A and P, you can have all-the-same-A or
3875 all-the-same-P modes. For example, Math::Currency might set A to undef,
3876 and P to -2, globally.
3878 ?Maybe an extra option that forbids local A & P settings would be in order,
3879 ?so that intermediate rounding does not 'poison' further math?
3881 =item Overriding globals
3883 * you will be able to give A, P and R as an argument to all the calculation
3884 routines; the second parameter is A, the third one is P, and the fourth is
3885 R (shift right by one for binary operations like badd). P is used only if
3886 the first parameter (A) is undefined. These three parameters override the
3887 globals in the order detailed as follows, i.e. the first defined value
3889 (local: per object, global: global default, parameter: argument to sub)
3892 + local A (if defined on both of the operands: smaller one is taken)
3893 + local P (if defined on both of the operands: bigger one is taken)
3897 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
3898 arguments (A and P) instead of one
3900 =item Local settings
3902 * You can set A or P locally by using C<< $x->accuracy() >> or
3903 C<< $x->precision() >>
3904 and thus force different A and P for different objects/numbers.
3905 * Setting A or P this way immediately rounds $x to the new value.
3906 * C<< $x->accuracy() >> clears C<< $x->precision() >>, and vice versa.
3910 * the rounding routines will use the respective global or local settings.
3911 fround()/bround() is for accuracy rounding, while ffround()/bfround()
3913 * the two rounding functions take as the second parameter one of the
3914 following rounding modes (R):
3915 'even', 'odd', '+inf', '-inf', 'zero', 'trunc', 'common'
3916 * you can set/get the global R by using C<< Math::SomeClass->round_mode() >>
3917 or by setting C<< $Math::SomeClass::round_mode >>
3918 * after each operation, C<< $result->round() >> is called, and the result may
3919 eventually be rounded (that is, if A or P were set either locally,
3920 globally or as parameter to the operation)
3921 * to manually round a number, call C<< $x->round($A,$P,$round_mode); >>
3922 this will round the number by using the appropriate rounding function
3923 and then normalize it.
3924 * rounding modifies the local settings of the number:
3926 $x = Math::BigFloat->new(123.456);
3930 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
3931 will be 4 from now on.
3933 =item Default values
3942 * The defaults are set up so that the new code gives the same results as
3943 the old code (except in a few cases on fdiv):
3944 + Both A and P are undefined and thus will not be used for rounding
3945 after each operation.
3946 + round() is thus a no-op, unless given extra parameters A and P
3950 =head1 Infinity and Not a Number
3952 While BigInt has extensive handling of inf and NaN, certain quirks remain.
3958 These perl routines currently (as of Perl v.5.8.6) cannot handle passed
3961 te@linux:~> perl -wle 'print 2 ** 3333'
3963 te@linux:~> perl -wle 'print 2 ** 3333 == 2 ** 3333'
3965 te@linux:~> perl -wle 'print oct(2 ** 3333)'
3967 te@linux:~> perl -wle 'print hex(2 ** 3333)'
3968 Illegal hexadecimal digit 'i' ignored at -e line 1.
3971 The same problems occur if you pass them Math::BigInt->binf() objects. Since
3972 overloading these routines is not possible, this cannot be fixed from BigInt.
3974 =item ==, !=, <, >, <=, >= with NaNs
3976 BigInt's bcmp() routine currently returns undef to signal that a NaN was
3977 involved in a comparison. However, the overload code turns that into
3978 either 1 or '' and thus operations like C<< NaN != NaN >> might return
3983 C<< log(-inf) >> is highly weird. Since log(-x)=pi*i+log(x), then
3984 log(-inf)=pi*i+inf. However, since the imaginary part is finite, the real
3985 infinity "overshadows" it, so the number might as well just be infinity.
3986 However, the result is a complex number, and since BigInt/BigFloat can only
3987 have real numbers as results, the result is NaN.
3989 =item exp(), cos(), sin(), atan2()
3991 These all might have problems handling infinity right.
3997 The actual numbers are stored as unsigned big integers (with seperate sign).
3999 You should neither care about nor depend on the internal representation; it
4000 might change without notice. Use B<ONLY> method calls like C<< $x->sign(); >>
4001 instead relying on the internal representation.
4005 Math with the numbers is done (by default) by a module called
4006 C<Math::BigInt::Calc>. This is equivalent to saying:
4008 use Math::BigInt lib => 'Calc';
4010 You can change this by using:
4012 use Math::BigInt lib => 'BitVect';
4014 The following would first try to find Math::BigInt::Foo, then
4015 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
4017 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
4019 Since Math::BigInt::GMP is in almost all cases faster than Calc (especially in
4020 math involving really big numbers, where it is B<much> faster), and there is
4021 no penalty if Math::BigInt::GMP is not installed, it is a good idea to always
4024 use Math::BigInt lib => 'GMP';
4026 Different low-level libraries use different formats to store the
4027 numbers. You should B<NOT> depend on the number having a specific format
4030 See the respective math library module documentation for further details.
4034 The sign is either '+', '-', 'NaN', '+inf' or '-inf'.
4036 A sign of 'NaN' is used to represent the result when input arguments are not
4037 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
4038 minus infinity. You will get '+inf' when dividing a positive number by 0, and
4039 '-inf' when dividing any negative number by 0.
4041 =head2 mantissa(), exponent() and parts()
4043 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
4046 $m = $x->mantissa();
4047 $e = $x->exponent();
4048 $y = $m * ( 10 ** $e );
4049 print "ok\n" if $x == $y;
4051 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
4052 in one go. Both the returned mantissa and exponent have a sign.
4054 Currently, for BigInts C<$e> is always 0, except +inf and -inf, where it is
4055 C<+inf>; and for NaN, where it is C<NaN>; and for C<$x == 0>, where it is C<1>
4056 (to be compatible with Math::BigFloat's internal representation of a zero as
4059 C<$m> is currently just a copy of the original number. The relation between
4060 C<$e> and C<$m> will stay always the same, though their real values might
4067 sub bint { Math::BigInt->new(shift); }
4069 $x = Math::BigInt->bstr("1234") # string "1234"
4070 $x = "$x"; # same as bstr()
4071 $x = Math::BigInt->bneg("1234"); # BigInt "-1234"
4072 $x = Math::BigInt->babs("-12345"); # BigInt "12345"
4073 $x = Math::BigInt->bnorm("-0.00"); # BigInt "0"
4074 $x = bint(1) + bint(2); # BigInt "3"
4075 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
4076 $x = bint(1); # BigInt "1"
4077 $x = $x + 5 / 2; # BigInt "3"
4078 $x = $x ** 3; # BigInt "27"
4079 $x *= 2; # BigInt "54"
4080 $x = Math::BigInt->new(0); # BigInt "0"
4082 $x = Math::BigInt->badd(4,5) # BigInt "9"
4083 print $x->bsstr(); # 9e+0
4085 Examples for rounding:
4090 $x = Math::BigFloat->new(123.4567);
4091 $y = Math::BigFloat->new(123.456789);
4092 Math::BigFloat->accuracy(4); # no more A than 4
4094 ok ($x->copy()->fround(),123.4); # even rounding
4095 print $x->copy()->fround(),"\n"; # 123.4
4096 Math::BigFloat->round_mode('odd'); # round to odd
4097 print $x->copy()->fround(),"\n"; # 123.5
4098 Math::BigFloat->accuracy(5); # no more A than 5
4099 Math::BigFloat->round_mode('odd'); # round to odd
4100 print $x->copy()->fround(),"\n"; # 123.46
4101 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
4102 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
4104 Math::BigFloat->accuracy(undef); # A not important now
4105 Math::BigFloat->precision(2); # P important
4106 print $x->copy()->bnorm(),"\n"; # 123.46
4107 print $x->copy()->fround(),"\n"; # 123.46
4109 Examples for converting:
4111 my $x = Math::BigInt->new('0b1'.'01' x 123);
4112 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
4114 =head1 Autocreating constants
4116 After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
4117 and binary constants in the given scope are converted to C<Math::BigInt>.
4118 This conversion happens at compile time.
4122 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
4124 prints the integer value of C<2**100>. Note that without conversion of
4125 constants the expression 2**100 will be calculated as perl scalar.
4127 Please note that strings and floating point constants are not affected,
4130 use Math::BigInt qw/:constant/;
4132 $x = 1234567890123456789012345678901234567890
4133 + 123456789123456789;
4134 $y = '1234567890123456789012345678901234567890'
4135 + '123456789123456789';
4137 do not work. You need an explicit Math::BigInt->new() around one of the
4138 operands. You should also quote large constants to protect loss of precision:
4142 $x = Math::BigInt->new('1234567889123456789123456789123456789');
4144 Without the quotes Perl would convert the large number to a floating point
4145 constant at compile time and then hand the result to BigInt, which results in
4146 an truncated result or a NaN.
4148 This also applies to integers that look like floating point constants:
4150 use Math::BigInt ':constant';
4152 print ref(123e2),"\n";
4153 print ref(123.2e2),"\n";
4155 will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
4156 to get this to work.
4160 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
4161 must be made in the second case. For long numbers, the copy can eat up to 20%
4162 of the work (in the case of addition/subtraction, less for
4163 multiplication/division). If $y is very small compared to $x, the form
4164 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
4165 more time then the actual addition.
4167 With a technique called copy-on-write, the cost of copying with overload could
4168 be minimized or even completely avoided. A test implementation of COW did show
4169 performance gains for overloaded math, but introduced a performance loss due
4170 to a constant overhead for all other operations. So Math::BigInt does currently
4173 The rewritten version of this module (vs. v0.01) is slower on certain
4174 operations, like C<new()>, C<bstr()> and C<numify()>. The reason are that it
4175 does now more work and handles much more cases. The time spent in these
4176 operations is usually gained in the other math operations so that code on
4177 the average should get (much) faster. If they don't, please contact the author.
4179 Some operations may be slower for small numbers, but are significantly faster
4180 for big numbers. Other operations are now constant (O(1), like C<bneg()>,
4181 C<babs()> etc), instead of O(N) and thus nearly always take much less time.
4182 These optimizations were done on purpose.
4184 If you find the Calc module to slow, try to install any of the replacement
4185 modules and see if they help you.
4187 =head2 Alternative math libraries
4189 You can use an alternative library to drive Math::BigInt via:
4191 use Math::BigInt lib => 'Module';
4193 See L<MATH LIBRARY> for more information.
4195 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
4199 =head1 Subclassing Math::BigInt
4201 The basic design of Math::BigInt allows simple subclasses with very little
4202 work, as long as a few simple rules are followed:
4208 The public API must remain consistent, i.e. if a sub-class is overloading
4209 addition, the sub-class must use the same name, in this case badd(). The
4210 reason for this is that Math::BigInt is optimized to call the object methods
4215 The private object hash keys like C<$x->{sign}> may not be changed, but
4216 additional keys can be added, like C<$x->{_custom}>.
4220 Accessor functions are available for all existing object hash keys and should
4221 be used instead of directly accessing the internal hash keys. The reason for
4222 this is that Math::BigInt itself has a pluggable interface which permits it
4223 to support different storage methods.
4227 More complex sub-classes may have to replicate more of the logic internal of
4228 Math::BigInt if they need to change more basic behaviors. A subclass that
4229 needs to merely change the output only needs to overload C<bstr()>.
4231 All other object methods and overloaded functions can be directly inherited
4232 from the parent class.
4234 At the very minimum, any subclass will need to provide it's own C<new()> and can
4235 store additional hash keys in the object. There are also some package globals
4236 that must be defined, e.g.:
4240 $precision = -2; # round to 2 decimal places
4241 $round_mode = 'even';
4244 Additionally, you might want to provide the following two globals to allow
4245 auto-upgrading and auto-downgrading to work correctly:
4250 This allows Math::BigInt to correctly retrieve package globals from the
4251 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
4252 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
4258 in your subclass to automatically inherit the overloading from the parent. If
4259 you like, you can change part of the overloading, look at Math::String for an
4264 When used like this:
4266 use Math::BigInt upgrade => 'Foo::Bar';
4268 certain operations will 'upgrade' their calculation and thus the result to
4269 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
4271 use Math::BigInt upgrade => 'Math::BigFloat';
4273 As a shortcut, you can use the module C<bignum>:
4277 Also good for oneliners:
4279 perl -Mbignum -le 'print 2 ** 255'
4281 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
4282 as well es preserve accuracy (as in sqrt(3)).
4284 Beware: This feature is not fully implemented yet.
4288 The following methods upgrade themselves unconditionally; that is if upgrade
4289 is in effect, they will always hand up their work:
4303 Beware: This list is not complete.
4305 All other methods upgrade themselves only when one (or all) of their
4306 arguments are of the class mentioned in $upgrade (This might change in later
4307 versions to a more sophisticated scheme):
4313 =item broot() does not work
4315 The broot() function in BigInt may only work for small values. This will be
4316 fixed in a later version.
4318 =item Out of Memory!
4320 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
4321 C<eval()> in your code will crash with "Out of memory". This is probably an
4322 overload/exporter bug. You can workaround by not having C<eval()>
4323 and ':constant' at the same time or upgrade your Perl to a newer version.
4325 =item Fails to load Calc on Perl prior 5.6.0
4327 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
4328 will fall back to eval { require ... } when loading the math lib on Perls
4329 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
4330 filesystems using a different seperator.
4336 Some things might not work as you expect them. Below is documented what is
4337 known to be troublesome:
4341 =item bstr(), bsstr() and 'cmp'
4343 Both C<bstr()> and C<bsstr()> as well as automated stringify via overload now
4344 drop the leading '+'. The old code would return '+3', the new returns '3'.
4345 This is to be consistent with Perl and to make C<cmp> (especially with
4346 overloading) to work as you expect. It also solves problems with C<Test.pm>,
4347 because it's C<ok()> uses 'eq' internally.
4349 Mark Biggar said, when asked about to drop the '+' altogether, or make only
4352 I agree (with the first alternative), don't add the '+' on positive
4353 numbers. It's not as important anymore with the new internal
4354 form for numbers. It made doing things like abs and neg easier,
4355 but those have to be done differently now anyway.
4357 So, the following examples will now work all as expected:
4360 BEGIN { plan tests => 1 }
4363 my $x = new Math::BigInt 3*3;
4364 my $y = new Math::BigInt 3*3;
4367 print "$x eq 9" if $x eq $y;
4368 print "$x eq 9" if $x eq '9';
4369 print "$x eq 9" if $x eq 3*3;
4371 Additionally, the following still works:
4373 print "$x == 9" if $x == $y;
4374 print "$x == 9" if $x == 9;
4375 print "$x == 9" if $x == 3*3;
4377 There is now a C<bsstr()> method to get the string in scientific notation aka
4378 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
4379 for comparison, but Perl will represent some numbers as 100 and others
4380 as 1e+308. If in doubt, convert both arguments to Math::BigInt before
4381 comparing them as strings:
4384 BEGIN { plan tests => 3 }
4387 $x = Math::BigInt->new('1e56'); $y = 1e56;
4388 ok ($x,$y); # will fail
4389 ok ($x->bsstr(),$y); # okay
4390 $y = Math::BigInt->new($y);
4393 Alternatively, simple use C<< <=> >> for comparisons, this will get it
4394 always right. There is not yet a way to get a number automatically represented
4395 as a string that matches exactly the way Perl represents it.
4397 See also the section about L<Infinity and Not a Number> for problems in
4402 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
4405 $x = Math::BigInt->new(123);
4406 $y = int($x); # BigInt 123
4407 $x = Math::BigFloat->new(123.45);
4408 $y = int($x); # BigInt 123
4410 In all Perl versions you can use C<as_number()> or C<as_int> for the same
4413 $x = Math::BigFloat->new(123.45);
4414 $y = $x->as_number(); # BigInt 123
4415 $y = $x->as_int(); # ditto
4417 This also works for other subclasses, like Math::String.
4419 If you want a real Perl scalar, use C<numify()>:
4421 $y = $x->numify(); # 123 as scalar
4423 This is seldom necessary, though, because this is done automatically, like
4424 when you access an array:
4426 $z = $array[$x]; # does work automatically
4430 The following will probably not do what you expect:
4432 $c = Math::BigInt->new(123);
4433 print $c->length(),"\n"; # prints 30
4435 It prints both the number of digits in the number and in the fraction part
4436 since print calls C<length()> in list context. Use something like:
4438 print scalar $c->length(),"\n"; # prints 3
4442 The following will probably not do what you expect:
4444 print $c->bdiv(10000),"\n";
4446 It prints both quotient and remainder since print calls C<bdiv()> in list
4447 context. Also, C<bdiv()> will modify $c, so be careful. You probably want
4450 print $c / 10000,"\n";
4451 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
4455 The quotient is always the greatest integer less than or equal to the
4456 real-valued quotient of the two operands, and the remainder (when it is
4457 nonzero) always has the same sign as the second operand; so, for
4467 As a consequence, the behavior of the operator % agrees with the
4468 behavior of Perl's built-in % operator (as documented in the perlop
4469 manpage), and the equation
4471 $x == ($x / $y) * $y + ($x % $y)
4473 holds true for any $x and $y, which justifies calling the two return
4474 values of bdiv() the quotient and remainder. The only exception to this rule
4475 are when $y == 0 and $x is negative, then the remainder will also be
4476 negative. See below under "infinity handling" for the reasoning behind this.
4478 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
4479 not change BigInt's way to do things. This is because under 'use integer' Perl
4480 will do what the underlying C thinks is right and this is different for each
4481 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
4482 the author to implement it ;)
4484 =item infinity handling
4486 Here are some examples that explain the reasons why certain results occur while
4489 The following table shows the result of the division and the remainder, so that
4490 the equation above holds true. Some "ordinary" cases are strewn in to show more
4491 clearly the reasoning:
4493 A / B = C, R so that C * B + R = A
4494 =========================================================
4495 5 / 8 = 0, 5 0 * 8 + 5 = 5
4496 0 / 8 = 0, 0 0 * 8 + 0 = 0
4497 0 / inf = 0, 0 0 * inf + 0 = 0
4498 0 /-inf = 0, 0 0 * -inf + 0 = 0
4499 5 / inf = 0, 5 0 * inf + 5 = 5
4500 5 /-inf = 0, 5 0 * -inf + 5 = 5
4501 -5/ inf = 0, -5 0 * inf + -5 = -5
4502 -5/-inf = 0, -5 0 * -inf + -5 = -5
4503 inf/ 5 = inf, 0 inf * 5 + 0 = inf
4504 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
4505 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
4506 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
4507 5/ 5 = 1, 0 1 * 5 + 0 = 5
4508 -5/ -5 = 1, 0 1 * -5 + 0 = -5
4509 inf/ inf = 1, 0 1 * inf + 0 = inf
4510 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
4511 inf/-inf = -1, 0 -1 * -inf + 0 = inf
4512 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
4513 8/ 0 = inf, 8 inf * 0 + 8 = 8
4514 inf/ 0 = inf, inf inf * 0 + inf = inf
4517 These cases below violate the "remainder has the sign of the second of the two
4518 arguments", since they wouldn't match up otherwise.
4520 A / B = C, R so that C * B + R = A
4521 ========================================================
4522 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
4523 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
4525 =item Modifying and =
4529 $x = Math::BigFloat->new(5);
4532 It will not do what you think, e.g. making a copy of $x. Instead it just makes
4533 a second reference to the B<same> object and stores it in $y. Thus anything
4534 that modifies $x (except overloaded operators) will modify $y, and vice versa.
4535 Or in other words, C<=> is only safe if you modify your BigInts only via
4536 overloaded math. As soon as you use a method call it breaks:
4539 print "$x, $y\n"; # prints '10, 10'
4541 If you want a true copy of $x, use:
4545 You can also chain the calls like this, this will make first a copy and then
4548 $y = $x->copy()->bmul(2);
4550 See also the documentation for overload.pm regarding C<=>.
4554 C<bpow()> (and the rounding functions) now modifies the first argument and
4555 returns it, unlike the old code which left it alone and only returned the
4556 result. This is to be consistent with C<badd()> etc. The first three will
4557 modify $x, the last one won't:
4559 print bpow($x,$i),"\n"; # modify $x
4560 print $x->bpow($i),"\n"; # ditto
4561 print $x **= $i,"\n"; # the same
4562 print $x ** $i,"\n"; # leave $x alone
4564 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4566 =item Overloading -$x
4576 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4577 needs to preserve $x since it does not know that it later will get overwritten.
4578 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4580 =item Mixing different object types
4582 In Perl you will get a floating point value if you do one of the following:
4588 With overloaded math, only the first two variants will result in a BigFloat:
4593 $mbf = Math::BigFloat->new(5);
4594 $mbi2 = Math::BigInteger->new(5);
4595 $mbi = Math::BigInteger->new(2);
4597 # what actually gets called:
4598 $float = $mbf + $mbi; # $mbf->badd()
4599 $float = $mbf / $mbi; # $mbf->bdiv()
4600 $integer = $mbi + $mbf; # $mbi->badd()
4601 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4602 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4604 This is because math with overloaded operators follows the first (dominating)
4605 operand, and the operation of that is called and returns thus the result. So,
4606 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4607 the result should be a Math::BigFloat or the second operant is one.
4609 To get a Math::BigFloat you either need to call the operation manually,
4610 make sure the operands are already of the proper type or casted to that type
4611 via Math::BigFloat->new():
4613 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4615 Beware of simple "casting" the entire expression, this would only convert
4616 the already computed result:
4618 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4620 Beware also of the order of more complicated expressions like:
4622 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4623 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4625 If in doubt, break the expression into simpler terms, or cast all operands
4626 to the desired resulting type.
4628 Scalar values are a bit different, since:
4633 will both result in the proper type due to the way the overloaded math works.
4635 This section also applies to other overloaded math packages, like Math::String.
4637 One solution to you problem might be autoupgrading|upgrading. See the
4638 pragmas L<bignum>, L<bigint> and L<bigrat> for an easy way to do this.
4642 C<bsqrt()> works only good if the result is a big integer, e.g. the square
4643 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4644 mode. The reason is that the result is always truncated to an integer.
4646 If you want a better approximation of the square root, then use:
4648 $x = Math::BigFloat->new(12);
4649 Math::BigFloat->precision(0);
4650 Math::BigFloat->round_mode('even');
4651 print $x->copy->bsqrt(),"\n"; # 4
4653 Math::BigFloat->precision(2);
4654 print $x->bsqrt(),"\n"; # 3.46
4655 print $x->bsqrt(3),"\n"; # 3.464
4659 For negative numbers in base see also L<brsft|brsft>.
4665 This program is free software; you may redistribute it and/or modify it under
4666 the same terms as Perl itself.
4670 L<Math::BigFloat>, L<Math::BigRat> and L<Math::Big> as well as
4671 L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4673 The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest
4674 because they solve the autoupgrading/downgrading issue, at least partly.
4677 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4678 more documentation including a full version history, testcases, empty
4679 subclass files and benchmarks.
4683 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4684 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001 - 2006
4685 and still at it in 2007.
4687 Many people contributed in one or more ways to the final beast, see the file
4688 CREDITS for an (incomplete) list. If you miss your name, please drop me a