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 'cos' => sub { $_[0]->copy->bcos(); },
85 'sin' => sub { $_[0]->copy->bsin(); },
86 'atan2' => sub { $_[2] ?
87 atan2($_[1],$_[0]->numify()) :
88 atan2($_[0]->numify(),$_[1]) },
90 # are not yet overloadable
91 #'hex' => sub { print "hex"; $_[0]; },
92 #'oct' => sub { print "oct"; $_[0]; },
94 # log(N) is log(N, e), where e is Euler's number
95 'log' => sub { $_[0]->copy()->blog($_[1], undef); },
96 'exp' => sub { $_[0]->copy()->bexp($_[1]); },
97 'int' => sub { $_[0]->copy(); },
98 'neg' => sub { $_[0]->copy()->bneg(); },
99 'abs' => sub { $_[0]->copy()->babs(); },
100 'sqrt' => sub { $_[0]->copy()->bsqrt(); },
101 '~' => sub { $_[0]->copy()->bnot(); },
103 # for subtract it's a bit tricky to not modify b: b-a => -a+b
104 '-' => sub { my $c = $_[0]->copy; $_[2] ?
105 $c->bneg()->badd( $_[1]) :
107 '+' => sub { $_[0]->copy()->badd($_[1]); },
108 '*' => sub { $_[0]->copy()->bmul($_[1]); },
111 $_[2] ? ref($_[0])->new($_[1])->bdiv($_[0]) : $_[0]->copy->bdiv($_[1]);
114 $_[2] ? ref($_[0])->new($_[1])->bmod($_[0]) : $_[0]->copy->bmod($_[1]);
117 $_[2] ? ref($_[0])->new($_[1])->bpow($_[0]) : $_[0]->copy->bpow($_[1]);
120 $_[2] ? ref($_[0])->new($_[1])->blsft($_[0]) : $_[0]->copy->blsft($_[1]);
123 $_[2] ? ref($_[0])->new($_[1])->brsft($_[0]) : $_[0]->copy->brsft($_[1]);
126 $_[2] ? ref($_[0])->new($_[1])->band($_[0]) : $_[0]->copy->band($_[1]);
129 $_[2] ? ref($_[0])->new($_[1])->bior($_[0]) : $_[0]->copy->bior($_[1]);
132 $_[2] ? ref($_[0])->new($_[1])->bxor($_[0]) : $_[0]->copy->bxor($_[1]);
135 # can modify arg of ++ and --, so avoid a copy() for speed, but don't
136 # use $_[0]->bone(), it would modify $_[0] to be 1!
137 '++' => sub { $_[0]->binc() },
138 '--' => sub { $_[0]->bdec() },
140 # if overloaded, O(1) instead of O(N) and twice as fast for small numbers
142 # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/
143 # v5.6.1 dumps on this: return !$_[0]->is_zero() || undef; :-(
145 $t = 1 if !$_[0]->is_zero();
149 # the original qw() does not work with the TIESCALAR below, why?
150 # Order of arguments unsignificant
151 '""' => sub { $_[0]->bstr(); },
152 '0+' => sub { $_[0]->numify(); }
155 ##############################################################################
156 # global constants, flags and accessory
158 # These vars are public, but their direct usage is not recommended, use the
159 # accessor methods instead
161 $round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'
166 $upgrade = undef; # default is no upgrade
167 $downgrade = undef; # default is no downgrade
169 # These are internally, and not to be used from the outside at all
171 $_trap_nan = 0; # are NaNs ok? set w/ config()
172 $_trap_inf = 0; # are infs ok? set w/ config()
173 my $nan = 'NaN'; # constants for easier life
175 my $CALC = 'Math::BigInt::FastCalc'; # module to do the low level math
176 # default is FastCalc.pm
177 my $IMPORT = 0; # was import() called yet?
178 # used to make require work
179 my %WARN; # warn only once for low-level libs
180 my %CAN; # cache for $CALC->can(...)
181 my %CALLBACKS; # callbacks to notify on lib loads
182 my $EMU_LIB = 'Math/BigInt/CalcEmu.pm'; # emulate low-level math
184 ##############################################################################
185 # the old code had $rnd_mode, so we need to support it, too
188 sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; }
189 sub FETCH { return $round_mode; }
190 sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
194 # tie to enable $rnd_mode to work transparently
195 tie $rnd_mode, 'Math::BigInt';
197 # set up some handy alias names
198 *as_int = \&as_number;
199 *is_pos = \&is_positive;
200 *is_neg = \&is_negative;
203 ##############################################################################
208 # make Class->round_mode() work
210 my $class = ref($self) || $self || __PACKAGE__;
214 if ($m !~ /^(even|odd|\+inf|\-inf|zero|trunc|common)$/)
216 require Carp; Carp::croak ("Unknown round mode '$m'");
218 return ${"${class}::round_mode"} = $m;
220 ${"${class}::round_mode"};
226 # make Class->upgrade() work
228 my $class = ref($self) || $self || __PACKAGE__;
229 # need to set new value?
232 return ${"${class}::upgrade"} = $_[0];
234 ${"${class}::upgrade"};
240 # make Class->downgrade() work
242 my $class = ref($self) || $self || __PACKAGE__;
243 # need to set new value?
246 return ${"${class}::downgrade"} = $_[0];
248 ${"${class}::downgrade"};
254 # make Class->div_scale() work
256 my $class = ref($self) || $self || __PACKAGE__;
261 require Carp; Carp::croak ('div_scale must be greater than zero');
263 ${"${class}::div_scale"} = $_[0];
265 ${"${class}::div_scale"};
270 # $x->accuracy($a); ref($x) $a
271 # $x->accuracy(); ref($x)
272 # Class->accuracy(); class
273 # Class->accuracy($a); class $a
276 my $class = ref($x) || $x || __PACKAGE__;
279 # need to set new value?
283 # convert objects to scalars to avoid deep recursion. If object doesn't
284 # have numify(), then hopefully it will have overloading for int() and
285 # boolean test without wandering into a deep recursion path...
286 $a = $a->numify() if ref($a) && $a->can('numify');
290 # also croak on non-numerical
294 Carp::croak ('Argument to accuracy must be greater than zero');
298 require Carp; Carp::croak ('Argument to accuracy must be an integer');
303 # $object->accuracy() or fallback to global
304 $x->bround($a) if $a; # not for undef, 0
305 $x->{_a} = $a; # set/overwrite, even if not rounded
306 delete $x->{_p}; # clear P
307 $a = ${"${class}::accuracy"} unless defined $a; # proper return value
311 ${"${class}::accuracy"} = $a; # set global A
312 ${"${class}::precision"} = undef; # clear global P
314 return $a; # shortcut
318 # $object->accuracy() or fallback to global
319 $a = $x->{_a} if ref($x);
320 # but don't return global undef, when $x's accuracy is 0!
321 $a = ${"${class}::accuracy"} if !defined $a;
327 # $x->precision($p); ref($x) $p
328 # $x->precision(); ref($x)
329 # Class->precision(); class
330 # Class->precision($p); class $p
333 my $class = ref($x) || $x || __PACKAGE__;
339 # convert objects to scalars to avoid deep recursion. If object doesn't
340 # have numify(), then hopefully it will have overloading for int() and
341 # boolean test without wandering into a deep recursion path...
342 $p = $p->numify() if ref($p) && $p->can('numify');
343 if ((defined $p) && (int($p) != $p))
345 require Carp; Carp::croak ('Argument to precision must be an integer');
349 # $object->precision() or fallback to global
350 $x->bfround($p) if $p; # not for undef, 0
351 $x->{_p} = $p; # set/overwrite, even if not rounded
352 delete $x->{_a}; # clear A
353 $p = ${"${class}::precision"} unless defined $p; # proper return value
357 ${"${class}::precision"} = $p; # set global P
358 ${"${class}::accuracy"} = undef; # clear global A
360 return $p; # shortcut
364 # $object->precision() or fallback to global
365 $p = $x->{_p} if ref($x);
366 # but don't return global undef, when $x's precision is 0!
367 $p = ${"${class}::precision"} if !defined $p;
373 # return (or set) configuration data as hash ref
374 my $class = shift || 'Math::BigInt';
377 if (@_ > 1 || (@_ == 1 && (ref($_[0]) eq 'HASH')))
379 # try to set given options as arguments from hash
382 if (ref($args) ne 'HASH')
386 # these values can be "set"
390 upgrade downgrade precision accuracy round_mode div_scale/
393 $set_args->{$key} = $args->{$key} if exists $args->{$key};
394 delete $args->{$key};
399 Carp::croak ("Illegal key(s) '",
400 join("','",keys %$args),"' passed to $class\->config()");
402 foreach my $key (keys %$set_args)
404 if ($key =~ /^trap_(inf|nan)\z/)
406 ${"${class}::_trap_$1"} = ($set_args->{"trap_$1"} ? 1 : 0);
409 # use a call instead of just setting the $variable to check argument
410 $class->$key($set_args->{$key});
414 # now return actual configuration
418 lib_version => ${"${CALC}::VERSION"},
420 trap_nan => ${"${class}::_trap_nan"},
421 trap_inf => ${"${class}::_trap_inf"},
422 version => ${"${class}::VERSION"},
425 upgrade downgrade precision accuracy round_mode div_scale
428 $cfg->{$key} = ${"${class}::$key"};
430 if (@_ == 1 && (ref($_[0]) ne 'HASH'))
432 # calls of the style config('lib') return just this value
433 return $cfg->{$_[0]};
440 # select accuracy parameter based on precedence,
441 # used by bround() and bfround(), may return undef for scale (means no op)
442 my ($x,$scale,$mode) = @_;
444 $scale = $x->{_a} unless defined $scale;
449 $scale = ${ $class . '::accuracy' } unless defined $scale;
450 $mode = ${ $class . '::round_mode' } unless defined $mode;
457 # select precision parameter based on precedence,
458 # used by bround() and bfround(), may return undef for scale (means no op)
459 my ($x,$scale,$mode) = @_;
461 $scale = $x->{_p} unless defined $scale;
466 $scale = ${ $class . '::precision' } unless defined $scale;
467 $mode = ${ $class . '::round_mode' } unless defined $mode;
472 ##############################################################################
477 # if two arguments, the first one is the class to "swallow" subclasses
481 sign => $_[1]->{sign},
482 value => $CALC->_copy($_[1]->{value}),
485 $self->{_a} = $_[1]->{_a} if defined $_[1]->{_a};
486 $self->{_p} = $_[1]->{_p} if defined $_[1]->{_p};
491 sign => $_[0]->{sign},
492 value => $CALC->_copy($_[0]->{value}),
495 $self->{_a} = $_[0]->{_a} if defined $_[0]->{_a};
496 $self->{_p} = $_[0]->{_p} if defined $_[0]->{_p};
502 # create a new BigInt object from a string or another BigInt object.
503 # see hash keys documented at top
505 # the argument could be an object, so avoid ||, && etc on it, this would
506 # cause costly overloaded code to be called. The only allowed ops are
509 my ($class,$wanted,$a,$p,$r) = @_;
511 # avoid numify-calls by not using || on $wanted!
512 return $class->bzero($a,$p) if !defined $wanted; # default to 0
513 return $class->copy($wanted,$a,$p,$r)
514 if ref($wanted) && $wanted->isa($class); # MBI or subclass
516 $class->import() if $IMPORT == 0; # make require work
518 my $self = bless {}, $class;
520 # shortcut for "normal" numbers
521 if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*\z/))
523 $self->{sign} = $1 || '+';
525 if ($wanted =~ /^[+-]/)
527 # remove sign without touching wanted to make it work with constants
528 my $t = $wanted; $t =~ s/^[+-]//;
529 $self->{value} = $CALC->_new($t);
533 $self->{value} = $CALC->_new($wanted);
536 if ( (defined $a) || (defined $p)
537 || (defined ${"${class}::precision"})
538 || (defined ${"${class}::accuracy"})
541 $self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p);
546 # handle '+inf', '-inf' first
547 if ($wanted =~ /^[+-]?inf\z/)
549 $self->{sign} = $wanted; # set a default sign for bstr()
550 return $self->binf($wanted);
552 # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign
553 my ($mis,$miv,$mfv,$es,$ev) = _split($wanted);
558 require Carp; Carp::croak("$wanted is not a number in $class");
560 $self->{value} = $CALC->_zero();
561 $self->{sign} = $nan;
566 # _from_hex or _from_bin
567 $self->{value} = $mis->{value};
568 $self->{sign} = $mis->{sign};
569 return $self; # throw away $mis
571 # make integer from mantissa by adjusting exp, then convert to bigint
572 $self->{sign} = $$mis; # store sign
573 $self->{value} = $CALC->_zero(); # for all the NaN cases
574 my $e = int("$$es$$ev"); # exponent (avoid recursion)
577 my $diff = $e - CORE::length($$mfv);
578 if ($diff < 0) # Not integer
582 require Carp; Carp::croak("$wanted not an integer in $class");
585 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
586 $self->{sign} = $nan;
590 # adjust fraction and add it to value
591 #print "diff > 0 $$miv\n";
592 $$miv = $$miv . ($$mfv . '0' x $diff);
597 if ($$mfv ne '') # e <= 0
599 # fraction and negative/zero E => NOI
602 require Carp; Carp::croak("$wanted not an integer in $class");
604 #print "NOI 2 \$\$mfv '$$mfv'\n";
605 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
606 $self->{sign} = $nan;
610 # xE-y, and empty mfv
613 if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
617 require Carp; Carp::croak("$wanted not an integer in $class");
620 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
621 $self->{sign} = $nan;
625 $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
626 $self->{value} = $CALC->_new($$miv) if $self->{sign} =~ /^[+-]$/;
627 # if any of the globals is set, use them to round and store them inside $self
628 # do not round for new($x,undef,undef) since that is used by MBF to signal
630 $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p;
636 # create a bigint 'NaN', if given a BigInt, set it to 'NaN'
638 $self = $class if !defined $self;
641 my $c = $self; $self = {}; bless $self, $c;
644 if (${"${class}::_trap_nan"})
647 Carp::croak ("Tried to set $self to NaN in $class\::bnan()");
649 $self->import() if $IMPORT == 0; # make require work
650 return if $self->modify('bnan');
651 if ($self->can('_bnan'))
653 # use subclass to initialize
658 # otherwise do our own thing
659 $self->{value} = $CALC->_zero();
661 $self->{sign} = $nan;
662 delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly
668 # create a bigint '+-inf', if given a BigInt, set it to '+-inf'
669 # the sign is either '+', or if given, used from there
671 my $sign = shift; $sign = '+' if !defined $sign || $sign !~ /^-(inf)?$/;
672 $self = $class if !defined $self;
675 my $c = $self; $self = {}; bless $self, $c;
678 if (${"${class}::_trap_inf"})
681 Carp::croak ("Tried to set $self to +-inf in $class\::binf()");
683 $self->import() if $IMPORT == 0; # make require work
684 return if $self->modify('binf');
685 if ($self->can('_binf'))
687 # use subclass to initialize
692 # otherwise do our own thing
693 $self->{value} = $CALC->_zero();
695 $sign = $sign . 'inf' if $sign !~ /inf$/; # - => -inf
696 $self->{sign} = $sign;
697 ($self->{_a},$self->{_p}) = @_; # take over requested rounding
703 # create a bigint '+0', if given a BigInt, set it to 0
705 $self = __PACKAGE__ if !defined $self;
709 my $c = $self; $self = {}; bless $self, $c;
711 $self->import() if $IMPORT == 0; # make require work
712 return if $self->modify('bzero');
714 if ($self->can('_bzero'))
716 # use subclass to initialize
721 # otherwise do our own thing
722 $self->{value} = $CALC->_zero();
729 # call like: $x->bzero($a,$p,$r,$y);
730 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
735 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
737 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
745 # create a bigint '+1' (or -1 if given sign '-'),
746 # if given a BigInt, set it to +1 or -1, respectively
748 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
749 $self = $class if !defined $self;
753 my $c = $self; $self = {}; bless $self, $c;
755 $self->import() if $IMPORT == 0; # make require work
756 return if $self->modify('bone');
758 if ($self->can('_bone'))
760 # use subclass to initialize
765 # otherwise do our own thing
766 $self->{value} = $CALC->_one();
768 $self->{sign} = $sign;
773 # call like: $x->bone($sign,$a,$p,$r,$y);
774 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
778 # call like: $x->bone($sign,$a,$p,$r);
780 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
782 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
788 ##############################################################################
789 # string conversation
793 # (ref to BFLOAT or num_str ) return num_str
794 # Convert number from internal format to scientific string format.
795 # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
796 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
798 if ($x->{sign} !~ /^[+-]$/)
800 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
803 my ($m,$e) = $x->parts();
804 #$m->bstr() . 'e+' . $e->bstr(); # e can only be positive in BigInt
805 # 'e+' because E can only be positive in BigInt
806 $m->bstr() . 'e+' . $CALC->_str($e->{value});
811 # make a string from bigint object
812 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
814 if ($x->{sign} !~ /^[+-]$/)
816 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
819 my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
820 $es.$CALC->_str($x->{value});
825 # Make a "normal" scalar from a BigInt object
826 my $x = shift; $x = $class->new($x) unless ref $x;
828 return $x->bstr() if $x->{sign} !~ /^[+-]$/;
829 my $num = $CALC->_num($x->{value});
830 return -$num if $x->{sign} eq '-';
834 ##############################################################################
835 # public stuff (usually prefixed with "b")
839 # return the sign of the number: +/-/-inf/+inf/NaN
840 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
845 sub _find_round_parameters
847 # After any operation or when calling round(), the result is rounded by
848 # regarding the A & P from arguments, local parameters, or globals.
850 # !!!!!!! If you change this, remember to change round(), too! !!!!!!!!!!
852 # This procedure finds the round parameters, but it is for speed reasons
853 # duplicated in round. Otherwise, it is tested by the testsuite and used
856 # returns ($self) or ($self,$a,$p,$r) - sets $self to NaN of both A and P
857 # were requested/defined (locally or globally or both)
859 my ($self,$a,$p,$r,@args) = @_;
860 # $a accuracy, if given by caller
861 # $p precision, if given by caller
862 # $r round_mode, if given by caller
863 # @args all 'other' arguments (0 for unary, 1 for binary ops)
865 my $c = ref($self); # find out class of argument(s)
868 # convert to normal scalar for speed and correctness in inner parts
869 $a = $a->can('numify') ? $a->numify() : "$a" if defined $a && ref($a);
870 $p = $p->can('numify') ? $p->numify() : "$p" if defined $p && ref($p);
872 # now pick $a or $p, but only if we have got "arguments"
875 foreach ($self,@args)
877 # take the defined one, or if both defined, the one that is smaller
878 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
883 # even if $a is defined, take $p, to signal error for both defined
884 foreach ($self,@args)
886 # take the defined one, or if both defined, the one that is bigger
888 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
891 # if still none defined, use globals (#2)
892 $a = ${"$c\::accuracy"} unless defined $a;
893 $p = ${"$c\::precision"} unless defined $p;
895 # A == 0 is useless, so undef it to signal no rounding
896 $a = undef if defined $a && $a == 0;
899 return ($self) unless defined $a || defined $p; # early out
901 # set A and set P is an fatal error
902 return ($self->bnan()) if defined $a && defined $p; # error
904 $r = ${"$c\::round_mode"} unless defined $r;
905 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc|common)$/)
907 require Carp; Carp::croak ("Unknown round mode '$r'");
915 # Round $self according to given parameters, or given second argument's
916 # parameters or global defaults
918 # for speed reasons, _find_round_parameters is embeded here:
920 my ($self,$a,$p,$r,@args) = @_;
921 # $a accuracy, if given by caller
922 # $p precision, if given by caller
923 # $r round_mode, if given by caller
924 # @args all 'other' arguments (0 for unary, 1 for binary ops)
926 my $c = ref($self); # find out class of argument(s)
929 # now pick $a or $p, but only if we have got "arguments"
932 foreach ($self,@args)
934 # take the defined one, or if both defined, the one that is smaller
935 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
940 # even if $a is defined, take $p, to signal error for both defined
941 foreach ($self,@args)
943 # take the defined one, or if both defined, the one that is bigger
945 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
948 # if still none defined, use globals (#2)
949 $a = ${"$c\::accuracy"} unless defined $a;
950 $p = ${"$c\::precision"} unless defined $p;
952 # A == 0 is useless, so undef it to signal no rounding
953 $a = undef if defined $a && $a == 0;
956 return $self unless defined $a || defined $p; # early out
958 # set A and set P is an fatal error
959 return $self->bnan() if defined $a && defined $p;
961 $r = ${"$c\::round_mode"} unless defined $r;
962 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc|common)$/)
964 require Carp; Carp::croak ("Unknown round mode '$r'");
967 # now round, by calling either fround or ffround:
970 $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a;
972 else # both can't be undefined due to early out
974 $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p;
976 # bround() or bfround() already callled bnorm() if nec.
982 # (numstr or BINT) return BINT
983 # Normalize number -- no-op here
984 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
990 # (BINT or num_str) return BINT
991 # make number absolute, or return absolute BINT from string
992 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
994 return $x if $x->modify('babs');
995 # post-normalized abs for internal use (does nothing for NaN)
996 $x->{sign} =~ s/^-/+/;
1002 # (BINT or num_str) return BINT
1003 # negate number or make a negated number from string
1004 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1006 return $x if $x->modify('bneg');
1008 # for +0 dont negate (to have always normalized +0). Does nothing for 'NaN'
1009 $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $CALC->_is_zero($x->{value}));
1015 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
1016 # (BINT or num_str, BINT or num_str) return cond_code
1019 my ($self,$x,$y) = (ref($_[0]),@_);
1021 # objectify is costly, so avoid it
1022 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1024 ($self,$x,$y) = objectify(2,@_);
1027 return $upgrade->bcmp($x,$y) if defined $upgrade &&
1028 ((!$x->isa($self)) || (!$y->isa($self)));
1030 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1032 # handle +-inf and NaN
1033 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1034 return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
1035 return +1 if $x->{sign} eq '+inf';
1036 return -1 if $x->{sign} eq '-inf';
1037 return -1 if $y->{sign} eq '+inf';
1040 # check sign for speed first
1041 return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
1042 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
1044 # have same sign, so compare absolute values. Don't make tests for zero here
1045 # because it's actually slower than testin in Calc (especially w/ Pari et al)
1047 # post-normalized compare for internal use (honors signs)
1048 if ($x->{sign} eq '+')
1050 # $x and $y both > 0
1051 return $CALC->_acmp($x->{value},$y->{value});
1055 $CALC->_acmp($y->{value},$x->{value}); # swaped acmp (lib returns 0,1,-1)
1060 # Compares 2 values, ignoring their signs.
1061 # Returns one of undef, <0, =0, >0. (suitable for sort)
1062 # (BINT, BINT) return cond_code
1065 my ($self,$x,$y) = (ref($_[0]),@_);
1066 # objectify is costly, so avoid it
1067 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1069 ($self,$x,$y) = objectify(2,@_);
1072 return $upgrade->bacmp($x,$y) if defined $upgrade &&
1073 ((!$x->isa($self)) || (!$y->isa($self)));
1075 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1077 # handle +-inf and NaN
1078 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1079 return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
1080 return 1 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} !~ /^[+-]inf$/;
1083 $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
1088 # add second arg (BINT or string) to first (BINT) (modifies first)
1089 # return result as BINT
1092 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1093 # objectify is costly, so avoid it
1094 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1096 ($self,$x,$y,@r) = objectify(2,@_);
1099 return $x if $x->modify('badd');
1100 return $upgrade->badd($upgrade->new($x),$upgrade->new($y),@r) if defined $upgrade &&
1101 ((!$x->isa($self)) || (!$y->isa($self)));
1103 $r[3] = $y; # no push!
1104 # inf and NaN handling
1105 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1108 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1110 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1112 # +inf++inf or -inf+-inf => same, rest is NaN
1113 return $x if $x->{sign} eq $y->{sign};
1116 # +-inf + something => +inf
1117 # something +-inf => +-inf
1118 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
1122 my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
1126 $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
1130 my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
1133 $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
1138 # speedup, if equal, set result to 0
1139 $x->{value} = $CALC->_zero();
1144 $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
1152 # (BINT or num_str, BINT or num_str) return BINT
1153 # subtract second arg from first, modify first
1156 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1158 # objectify is costly, so avoid it
1159 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1161 ($self,$x,$y,@r) = objectify(2,@_);
1164 return $x if $x->modify('bsub');
1166 return $upgrade->new($x)->bsub($upgrade->new($y),@r) if defined $upgrade &&
1167 ((!$x->isa($self)) || (!$y->isa($self)));
1169 return $x->round(@r) if $y->is_zero();
1171 # To correctly handle the lone special case $x->bsub($x), we note the sign
1172 # of $x, then flip the sign from $y, and if the sign of $x did change, too,
1173 # then we caught the special case:
1174 my $xsign = $x->{sign};
1175 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
1176 if ($xsign ne $x->{sign})
1178 # special case of $x->bsub($x) results in 0
1179 return $x->bzero(@r) if $xsign =~ /^[+-]$/;
1180 return $x->bnan(); # NaN, -inf, +inf
1182 $x->badd($y,@r); # badd does not leave internal zeros
1183 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
1184 $x; # already rounded by badd() or no round nec.
1189 # increment arg by one
1190 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1191 return $x if $x->modify('binc');
1193 if ($x->{sign} eq '+')
1195 $x->{value} = $CALC->_inc($x->{value});
1196 return $x->round($a,$p,$r);
1198 elsif ($x->{sign} eq '-')
1200 $x->{value} = $CALC->_dec($x->{value});
1201 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1202 return $x->round($a,$p,$r);
1204 # inf, nan handling etc
1205 $x->badd($self->bone(),$a,$p,$r); # badd does round
1210 # decrement arg by one
1211 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1212 return $x if $x->modify('bdec');
1214 if ($x->{sign} eq '-')
1217 $x->{value} = $CALC->_inc($x->{value});
1221 return $x->badd($self->bone('-'),@r) unless $x->{sign} eq '+'; # inf or NaN
1223 if ($CALC->_is_zero($x->{value}))
1226 $x->{value} = $CALC->_one(); $x->{sign} = '-'; # 0 => -1
1231 $x->{value} = $CALC->_dec($x->{value});
1239 # calculate $x = $a ** $base + $b and return $a (e.g. the log() to base
1243 my ($self,$x,$base,@r) = (undef,@_);
1244 # objectify is costly, so avoid it
1245 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1247 ($self,$x,$base,@r) = objectify(1,ref($x),@_);
1250 return $x if $x->modify('blog');
1252 $base = $self->new($base) if defined $base && !ref $base;
1254 # inf, -inf, NaN, <0 => NaN
1256 if $x->{sign} ne '+' || (defined $base && $base->{sign} ne '+');
1258 return $upgrade->blog($upgrade->new($x),$base,@r) if
1261 # fix for bug #24969:
1262 # the default base is e (Euler's number) which is not an integer
1265 require Math::BigFloat;
1266 my $u = Math::BigFloat->blog(Math::BigFloat->new($x))->as_int();
1267 # modify $x in place
1268 $x->{value} = $u->{value};
1269 $x->{sign} = $u->{sign};
1273 my ($rc,$exact) = $CALC->_log_int($x->{value},$base->{value});
1274 return $x->bnan() unless defined $rc; # not possible to take log?
1281 # Calculate n over k (binomial coefficient or "choose" function) as integer.
1283 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1285 # objectify is costly, so avoid it
1286 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1288 ($self,$x,$y,@r) = objectify(2,@_);
1291 return $x if $x->modify('bnok');
1292 return $x->bnan() if $x->{sign} eq 'NaN' || $y->{sign} eq 'NaN';
1293 return $x->binf() if $x->{sign} eq '+inf';
1295 # k > n or k < 0 => 0
1296 my $cmp = $x->bacmp($y);
1297 return $x->bzero() if $cmp < 0 || $y->{sign} =~ /^-/;
1299 return $x->bone(@r) if $cmp == 0;
1301 if ($CALC->can('_nok'))
1303 $x->{value} = $CALC->_nok($x->{value},$y->{value});
1307 # ( 7 ) 7! 7*6*5 * 4*3*2*1 7 * 6 * 5
1308 # ( - ) = --------- = --------------- = ---------
1309 # ( 3 ) 3! (7-3)! 3*2*1 * 4*3*2*1 3 * 2 * 1
1311 # compute n - k + 2 (so we start with 5 in the example above)
1316 my $r = $z->copy(); $z->binc();
1317 my $d = $self->new(2);
1318 while ($z->bacmp($x) <= 0) # f < x ?
1320 $r->bmul($z); $r->bdiv($d);
1321 $z->binc(); $d->binc();
1323 $x->{value} = $r->{value}; $x->{sign} = '+';
1325 else { $x->bone(); }
1332 # Calculate e ** $x (Euler's number to the power of X), truncated to
1334 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1335 return $x if $x->modify('bexp');
1337 # inf, -inf, NaN, <0 => NaN
1338 return $x->bnan() if $x->{sign} eq 'NaN';
1339 return $x->bone() if $x->is_zero();
1340 return $x if $x->{sign} eq '+inf';
1341 return $x->bzero() if $x->{sign} eq '-inf';
1345 # run through Math::BigFloat unless told otherwise
1346 require Math::BigFloat unless defined $upgrade;
1347 local $upgrade = 'Math::BigFloat' unless defined $upgrade;
1348 # calculate result, truncate it to integer
1349 $u = $upgrade->bexp($upgrade->new($x),@r);
1352 if (!defined $upgrade)
1355 # modify $x in place
1356 $x->{value} = $u->{value};
1364 # (BINT or num_str, BINT or num_str) return BINT
1365 # does not modify arguments, but returns new object
1366 # Lowest Common Multiplicator
1368 my $y = shift; my ($x);
1375 $x = $class->new($y);
1380 my $y = shift; $y = $self->new($y) if !ref ($y);
1388 # (BINT or num_str, BINT or num_str) return BINT
1389 # does not modify arguments, but returns new object
1390 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1393 $y = $class->new($y) if !ref($y);
1395 my $x = $y->copy()->babs(); # keep arguments
1396 return $x->bnan() if $x->{sign} !~ /^[+-]$/; # x NaN?
1400 $y = shift; $y = $self->new($y) if !ref($y);
1401 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1402 $x->{value} = $CALC->_gcd($x->{value},$y->{value});
1403 last if $CALC->_is_one($x->{value});
1410 # (num_str or BINT) return BINT
1411 # represent ~x as twos-complement number
1412 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1413 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1415 return $x if $x->modify('bnot');
1416 $x->binc()->bneg(); # binc already does round
1419 ##############################################################################
1420 # is_foo test routines
1421 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1425 # return true if arg (BINT or num_str) is zero (array '+', '0')
1426 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1428 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1429 $CALC->_is_zero($x->{value});
1434 # return true if arg (BINT or num_str) is NaN
1435 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1437 $x->{sign} eq $nan ? 1 : 0;
1442 # return true if arg (BINT or num_str) is +-inf
1443 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1447 $sign = '[+-]inf' if $sign eq ''; # +- doesn't matter, only that's inf
1448 $sign = "[$1]inf" if $sign =~ /^([+-])(inf)?$/; # extract '+' or '-'
1449 return $x->{sign} =~ /^$sign$/ ? 1 : 0;
1451 $x->{sign} =~ /^[+-]inf$/ ? 1 : 0; # only +-inf is infinity
1456 # return true if arg (BINT or num_str) is +1, or -1 if sign is given
1457 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1459 $sign = '+' if !defined $sign || $sign ne '-';
1461 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1462 $CALC->_is_one($x->{value});
1467 # return true when arg (BINT or num_str) is odd, false for even
1468 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1470 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1471 $CALC->_is_odd($x->{value});
1476 # return true when arg (BINT or num_str) is even, false for odd
1477 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1479 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1480 $CALC->_is_even($x->{value});
1485 # return true when arg (BINT or num_str) is positive (>= 0)
1486 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1488 return 1 if $x->{sign} eq '+inf'; # +inf is positive
1490 # 0+ is neither positive nor negative
1491 ($x->{sign} eq '+' && !$x->is_zero()) ? 1 : 0;
1496 # return true when arg (BINT or num_str) is negative (< 0)
1497 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1499 $x->{sign} =~ /^-/ ? 1 : 0; # -inf is negative, but NaN is not
1504 # return true when arg (BINT or num_str) is an integer
1505 # always true for BigInt, but different for BigFloats
1506 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1508 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1511 ###############################################################################
1515 # multiply the first number by the second numbers
1516 # (BINT or num_str, BINT or num_str) return BINT
1519 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1520 # objectify is costly, so avoid it
1521 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1523 ($self,$x,$y,@r) = objectify(2,@_);
1526 return $x if $x->modify('bmul');
1528 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1531 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1533 return $x->bnan() if $x->is_zero() || $y->is_zero();
1534 # result will always be +-inf:
1535 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1536 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1537 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1538 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1539 return $x->binf('-');
1542 return $upgrade->bmul($x,$upgrade->new($y),@r)
1543 if defined $upgrade && !$y->isa($self);
1545 $r[3] = $y; # no push here
1547 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1549 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1550 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1557 # multiply two numbers and then add the third to the result
1558 # (BINT or num_str, BINT or num_str, BINT or num_str) return BINT
1561 my ($self,$x,$y,$z,@r) = (ref($_[0]),@_);
1562 # objectify is costly, so avoid it
1563 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1565 ($self,$x,$y,$z,@r) = objectify(3,@_);
1568 return $x if $x->modify('bmuladd');
1570 return $x->bnan() if ($x->{sign} eq $nan) ||
1571 ($y->{sign} eq $nan) ||
1572 ($z->{sign} eq $nan);
1574 # inf handling of x and y
1575 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1577 return $x->bnan() if $x->is_zero() || $y->is_zero();
1578 # result will always be +-inf:
1579 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1580 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1581 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1582 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1583 return $x->binf('-');
1585 # inf handling x*y and z
1586 if (($z->{sign} =~ /^[+-]inf$/))
1588 # something +-inf => +-inf
1589 $x->{sign} = $z->{sign}, return $x if $z->{sign} =~ /^[+-]inf$/;
1592 return $upgrade->bmuladd($x,$upgrade->new($y),$upgrade->new($z),@r)
1593 if defined $upgrade && (!$y->isa($self) || !$z->isa($self) || !$x->isa($self));
1595 # TODO: what it $y and $z have A or P set?
1596 $r[3] = $z; # no push here
1598 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1600 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1601 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1603 my ($sx, $sz) = ( $x->{sign}, $z->{sign} ); # get signs
1607 $x->{value} = $CALC->_add($x->{value},$z->{value}); # same sign, abs add
1611 my $a = $CALC->_acmp ($z->{value},$x->{value}); # absolute compare
1614 $x->{value} = $CALC->_sub($z->{value},$x->{value},1); # abs sub w/ swap
1619 # speedup, if equal, set result to 0
1620 $x->{value} = $CALC->_zero();
1625 $x->{value} = $CALC->_sub($x->{value}, $z->{value}); # abs sub
1633 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1634 my ($self,$x,$y) = @_;
1636 # NaN if x == NaN or y == NaN or x==y==0
1637 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1638 if (($x->is_nan() || $y->is_nan()) ||
1639 ($x->is_zero() && $y->is_zero()));
1641 # +-inf / +-inf == NaN, reminder also NaN
1642 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1644 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1646 # x / +-inf => 0, remainder x (works even if x == 0)
1647 if ($y->{sign} =~ /^[+-]inf$/)
1649 my $t = $x->copy(); # bzero clobbers up $x
1650 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1653 # 5 / 0 => +inf, -6 / 0 => -inf
1654 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1655 # exception: -8 / 0 has remainder -8, not 8
1656 # exception: -inf / 0 has remainder -inf, not inf
1659 # +-inf / 0 => special case for -inf
1660 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1661 if (!$x->is_zero() && !$x->is_inf())
1663 my $t = $x->copy(); # binf clobbers up $x
1665 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1669 # last case: +-inf / ordinary number
1671 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1673 return wantarray ? ($x,$self->bzero()) : $x;
1678 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1679 # (BINT,BINT) (quo,rem) or BINT (only rem)
1682 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1683 # objectify is costly, so avoid it
1684 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1686 ($self,$x,$y,@r) = objectify(2,@_);
1689 return $x if $x->modify('bdiv');
1691 return $self->_div_inf($x,$y)
1692 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1694 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1695 if defined $upgrade;
1697 $r[3] = $y; # no push!
1699 # calc new sign and in case $y == +/- 1, return $x
1700 my $xsign = $x->{sign}; # keep
1701 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1705 my $rem = $self->bzero();
1706 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1707 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1708 $rem->{_a} = $x->{_a};
1709 $rem->{_p} = $x->{_p};
1711 if (! $CALC->_is_zero($rem->{value}))
1713 $rem->{sign} = $y->{sign};
1714 $rem = $y->copy()->bsub($rem) if $xsign ne $y->{sign}; # one of them '-'
1718 $rem->{sign} = '+'; # dont leave -0
1724 $x->{value} = $CALC->_div($x->{value},$y->{value});
1725 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1730 ###############################################################################
1735 # modulus (or remainder)
1736 # (BINT or num_str, BINT or num_str) return BINT
1739 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1740 # objectify is costly, so avoid it
1741 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1743 ($self,$x,$y,@r) = objectify(2,@_);
1746 return $x if $x->modify('bmod');
1747 $r[3] = $y; # no push!
1748 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1750 my ($d,$r) = $self->_div_inf($x,$y);
1751 $x->{sign} = $r->{sign};
1752 $x->{value} = $r->{value};
1753 return $x->round(@r);
1756 # calc new sign and in case $y == +/- 1, return $x
1757 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1758 if (!$CALC->_is_zero($x->{value}))
1760 $x->{value} = $CALC->_sub($y->{value},$x->{value},1) # $y-$x
1761 if ($x->{sign} ne $y->{sign});
1762 $x->{sign} = $y->{sign};
1766 $x->{sign} = '+'; # dont leave -0
1773 # Modular inverse. given a number which is (hopefully) relatively
1774 # prime to the modulus, calculate its inverse using Euclid's
1775 # alogrithm. If the number is not relatively prime to the modulus
1776 # (i.e. their gcd is not one) then NaN is returned.
1779 my ($self,$x,$y,@r) = (undef,@_);
1780 # objectify is costly, so avoid it
1781 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1783 ($self,$x,$y,@r) = objectify(2,@_);
1786 return $x if $x->modify('bmodinv');
1789 if ($y->{sign} ne '+' # -, NaN, +inf, -inf
1790 || $x->is_zero() # or num == 0
1791 || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1794 # put least residue into $x if $x was negative, and thus make it positive
1795 $x->bmod($y) if $x->{sign} eq '-';
1798 ($x->{value},$sign) = $CALC->_modinv($x->{value},$y->{value});
1799 return $x->bnan() if !defined $x->{value}; # in case no GCD found
1800 return $x if !defined $sign; # already real result
1801 $x->{sign} = $sign; # flip/flop see below
1802 $x->bmod($y); # calc real result
1808 # takes a very large number to a very large exponent in a given very
1809 # large modulus, quickly, thanks to binary exponentation. Supports
1810 # negative exponents.
1811 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1813 return $num if $num->modify('bmodpow');
1815 # check modulus for valid values
1816 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1817 || $mod->is_zero());
1819 # check exponent for valid values
1820 if ($exp->{sign} =~ /\w/)
1822 # i.e., if it's NaN, +inf, or -inf...
1823 return $num->bnan();
1826 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
1828 # check num for valid values (also NaN if there was no inverse but $exp < 0)
1829 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1831 # $mod is positive, sign on $exp is ignored, result also positive
1832 $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
1836 ###############################################################################
1840 # (BINT or num_str, BINT or num_str) return BINT
1841 # compute factorial number from $x, modify $x in place
1842 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1844 return $x if $x->modify('bfac') || $x->{sign} eq '+inf'; # inf => inf
1845 return $x->bnan() if $x->{sign} ne '+'; # NaN, <0 etc => NaN
1847 $x->{value} = $CALC->_fac($x->{value});
1853 # (BINT or num_str, BINT or num_str) return BINT
1854 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1855 # modifies first argument
1858 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1859 # objectify is costly, so avoid it
1860 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1862 ($self,$x,$y,@r) = objectify(2,@_);
1865 return $x if $x->modify('bpow');
1867 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1870 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1872 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1878 if ($x->{sign} =~ /^[+-]inf/)
1881 return $x->bnan() if $y->is_zero();
1882 # -inf ** -1 => 1/inf => 0
1883 return $x->bzero() if $y->is_one('-') && $x->is_negative();
1886 return $x if $x->{sign} eq '+inf';
1888 # -inf ** Y => -inf if Y is odd
1889 return $x if $y->is_odd();
1895 return $x if $x->is_one();
1898 return $x if $x->is_zero() && $y->{sign} =~ /^[+]/;
1901 return $x->binf() if $x->is_zero();
1904 return $x->bnan() if $x->is_one('-') && $y->{sign} =~ /^[-]/;
1907 return $x->bzero() if $x->{sign} eq '-' && $y->{sign} =~ /^[-]/;
1910 return $x->bnan() if $x->{sign} eq '-';
1913 return $x->binf() if $y->{sign} =~ /^[+]/;
1918 return $upgrade->bpow($upgrade->new($x),$y,@r)
1919 if defined $upgrade && (!$y->isa($self) || $y->{sign} eq '-');
1921 $r[3] = $y; # no push!
1923 # cases 0 ** Y, X ** 0, X ** 1, 1 ** Y are handled by Calc or Emu
1926 $new_sign = $y->is_odd() ? '-' : '+' if ($x->{sign} ne '+');
1928 # 0 ** -7 => ( 1 / (0 ** 7)) => 1 / 0 => +inf
1930 if $y->{sign} eq '-' && $x->{sign} eq '+' && $CALC->_is_zero($x->{value});
1931 # 1 ** -y => 1 / (1 ** |y|)
1932 # so do test for negative $y after above's clause
1933 return $x->bnan() if $y->{sign} eq '-' && !$CALC->_is_one($x->{value});
1935 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1936 $x->{sign} = $new_sign;
1937 $x->{sign} = '+' if $CALC->_is_zero($y->{value});
1943 # (BINT or num_str, BINT or num_str) return BINT
1944 # compute x << y, base n, y >= 0
1947 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1948 # objectify is costly, so avoid it
1949 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1951 ($self,$x,$y,$n,@r) = objectify(2,@_);
1954 return $x if $x->modify('blsft');
1955 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1956 return $x->round(@r) if $y->is_zero();
1958 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1960 $x->{value} = $CALC->_lsft($x->{value},$y->{value},$n);
1966 # (BINT or num_str, BINT or num_str) return BINT
1967 # compute x >> y, base n, y >= 0
1970 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1971 # objectify is costly, so avoid it
1972 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1974 ($self,$x,$y,$n,@r) = objectify(2,@_);
1977 return $x if $x->modify('brsft');
1978 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1979 return $x->round(@r) if $y->is_zero();
1980 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
1982 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1984 # this only works for negative numbers when shifting in base 2
1985 if (($x->{sign} eq '-') && ($n == 2))
1987 return $x->round(@r) if $x->is_one('-'); # -1 => -1
1990 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1991 # but perhaps there is a better emulation for two's complement shift...
1992 # if $y != 1, we must simulate it by doing:
1993 # convert to bin, flip all bits, shift, and be done
1994 $x->binc(); # -3 => -2
1995 my $bin = $x->as_bin();
1996 $bin =~ s/^-0b//; # strip '-0b' prefix
1997 $bin =~ tr/10/01/; # flip bits
1999 if ($y >= CORE::length($bin))
2001 $bin = '0'; # shifting to far right creates -1
2002 # 0, because later increment makes
2003 # that 1, attached '-' makes it '-1'
2004 # because -1 >> x == -1 !
2008 $bin =~ s/.{$y}$//; # cut off at the right side
2009 $bin = '1' . $bin; # extend left side by one dummy '1'
2010 $bin =~ tr/10/01/; # flip bits back
2012 my $res = $self->new('0b'.$bin); # add prefix and convert back
2013 $res->binc(); # remember to increment
2014 $x->{value} = $res->{value}; # take over value
2015 return $x->round(@r); # we are done now, magic, isn't?
2017 # x < 0, n == 2, y == 1
2018 $x->bdec(); # n == 2, but $y == 1: this fixes it
2021 $x->{value} = $CALC->_rsft($x->{value},$y->{value},$n);
2027 #(BINT or num_str, BINT or num_str) return BINT
2031 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2032 # objectify is costly, so avoid it
2033 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2035 ($self,$x,$y,@r) = objectify(2,@_);
2038 return $x if $x->modify('band');
2040 $r[3] = $y; # no push!
2042 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
2044 my $sx = $x->{sign} eq '+' ? 1 : -1;
2045 my $sy = $y->{sign} eq '+' ? 1 : -1;
2047 if ($sx == 1 && $sy == 1)
2049 $x->{value} = $CALC->_and($x->{value},$y->{value});
2050 return $x->round(@r);
2053 if ($CAN{signed_and})
2055 $x->{value} = $CALC->_signed_and($x->{value},$y->{value},$sx,$sy);
2056 return $x->round(@r);
2060 __emu_band($self,$x,$y,$sx,$sy,@r);
2065 #(BINT or num_str, BINT or num_str) return BINT
2069 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2070 # objectify is costly, so avoid it
2071 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2073 ($self,$x,$y,@r) = objectify(2,@_);
2076 return $x if $x->modify('bior');
2077 $r[3] = $y; # no push!
2079 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
2081 my $sx = $x->{sign} eq '+' ? 1 : -1;
2082 my $sy = $y->{sign} eq '+' ? 1 : -1;
2084 # the sign of X follows the sign of X, e.g. sign of Y irrelevant for bior()
2086 # don't use lib for negative values
2087 if ($sx == 1 && $sy == 1)
2089 $x->{value} = $CALC->_or($x->{value},$y->{value});
2090 return $x->round(@r);
2093 # if lib can do negative values, let it handle this
2094 if ($CAN{signed_or})
2096 $x->{value} = $CALC->_signed_or($x->{value},$y->{value},$sx,$sy);
2097 return $x->round(@r);
2101 __emu_bior($self,$x,$y,$sx,$sy,@r);
2106 #(BINT or num_str, BINT or num_str) return BINT
2110 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2111 # objectify is costly, so avoid it
2112 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2114 ($self,$x,$y,@r) = objectify(2,@_);
2117 return $x if $x->modify('bxor');
2118 $r[3] = $y; # no push!
2120 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
2122 my $sx = $x->{sign} eq '+' ? 1 : -1;
2123 my $sy = $y->{sign} eq '+' ? 1 : -1;
2125 # don't use lib for negative values
2126 if ($sx == 1 && $sy == 1)
2128 $x->{value} = $CALC->_xor($x->{value},$y->{value});
2129 return $x->round(@r);
2132 # if lib can do negative values, let it handle this
2133 if ($CAN{signed_xor})
2135 $x->{value} = $CALC->_signed_xor($x->{value},$y->{value},$sx,$sy);
2136 return $x->round(@r);
2140 __emu_bxor($self,$x,$y,$sx,$sy,@r);
2145 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2147 my $e = $CALC->_len($x->{value});
2148 wantarray ? ($e,0) : $e;
2153 # return the nth decimal digit, negative values count backward, 0 is right
2154 my ($self,$x,$n) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2156 $n = $n->numify() if ref($n);
2157 $CALC->_digit($x->{value},$n||0);
2162 # return the amount of trailing zeros in $x (as scalar)
2164 $x = $class->new($x) unless ref $x;
2166 return 0 if $x->{sign} !~ /^[+-]$/; # NaN, inf, -inf etc
2168 $CALC->_zeros($x->{value}); # must handle odd values, 0 etc
2173 # calculate square root of $x
2174 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2176 return $x if $x->modify('bsqrt');
2178 return $x->bnan() if $x->{sign} !~ /^\+/; # -x or -inf or NaN => NaN
2179 return $x if $x->{sign} eq '+inf'; # sqrt(+inf) == inf
2181 return $upgrade->bsqrt($x,@r) if defined $upgrade;
2183 $x->{value} = $CALC->_sqrt($x->{value});
2189 # calculate $y'th root of $x
2192 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2194 $y = $self->new(2) unless defined $y;
2196 # objectify is costly, so avoid it
2197 if ((!ref($x)) || (ref($x) ne ref($y)))
2199 ($self,$x,$y,@r) = objectify(2,$self || $class,@_);
2202 return $x if $x->modify('broot');
2204 # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
2205 return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
2206 $y->{sign} !~ /^\+$/;
2208 return $x->round(@r)
2209 if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
2211 return $upgrade->new($x)->broot($upgrade->new($y),@r) if defined $upgrade;
2213 $x->{value} = $CALC->_root($x->{value},$y->{value});
2219 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
2220 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2222 if ($x->{sign} !~ /^[+-]$/)
2224 my $s = $x->{sign}; $s =~ s/^[+-]//; # NaN, -inf,+inf => NaN or inf
2225 return $self->new($s);
2227 return $self->bone() if $x->is_zero();
2229 # 12300 => 2 trailing zeros => exponent is 2
2230 $self->new( $CALC->_zeros($x->{value}) );
2235 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
2236 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2238 if ($x->{sign} !~ /^[+-]$/)
2240 # for NaN, +inf, -inf: keep the sign
2241 return $self->new($x->{sign});
2243 my $m = $x->copy(); delete $m->{_p}; delete $m->{_a};
2245 # that's a bit inefficient:
2246 my $zeros = $CALC->_zeros($m->{value});
2247 $m->brsft($zeros,10) if $zeros != 0;
2253 # return a copy of both the exponent and the mantissa
2254 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2256 ($x->mantissa(),$x->exponent());
2259 ##############################################################################
2260 # rounding functions
2264 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
2265 # $n == 0 || $n == 1 => round to integer
2266 my $x = shift; my $self = ref($x) || $x; $x = $self->new($x) unless ref $x;
2268 my ($scale,$mode) = $x->_scale_p(@_);
2270 return $x if !defined $scale || $x->modify('bfround'); # no-op
2272 # no-op for BigInts if $n <= 0
2273 $x->bround( $x->length()-$scale, $mode) if $scale > 0;
2275 delete $x->{_a}; # delete to save memory
2276 $x->{_p} = $scale; # store new _p
2280 sub _scan_for_nonzero
2282 # internal, used by bround() to scan for non-zeros after a '5'
2283 my ($x,$pad,$xs,$len) = @_;
2285 return 0 if $len == 1; # "5" is trailed by invisible zeros
2286 my $follow = $pad - 1;
2287 return 0 if $follow > $len || $follow < 1;
2289 # use the string form to check whether only '0's follow or not
2290 substr ($xs,-$follow) =~ /[^0]/ ? 1 : 0;
2295 # Exists to make life easier for switch between MBF and MBI (should we
2296 # autoload fxxx() like MBF does for bxxx()?)
2297 my $x = shift; $x = $class->new($x) unless ref $x;
2303 # accuracy: +$n preserve $n digits from left,
2304 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
2306 # and overwrite the rest with 0's, return normalized number
2307 # do not return $x->bnorm(), but $x
2309 my $x = shift; $x = $class->new($x) unless ref $x;
2310 my ($scale,$mode) = $x->_scale_a(@_);
2311 return $x if !defined $scale || $x->modify('bround'); # no-op
2313 if ($x->is_zero() || $scale == 0)
2315 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2318 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2320 # we have fewer digits than we want to scale to
2321 my $len = $x->length();
2322 # convert $scale to a scalar in case it is an object (put's a limit on the
2323 # number length, but this would already limited by memory constraints), makes
2325 $scale = $scale->numify() if ref ($scale);
2327 # scale < 0, but > -len (not >=!)
2328 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2330 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2334 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2335 my ($pad,$digit_round,$digit_after);
2336 $pad = $len - $scale;
2337 $pad = abs($scale-1) if $scale < 0;
2339 # do not use digit(), it is very costly for binary => decimal
2340 # getting the entire string is also costly, but we need to do it only once
2341 my $xs = $CALC->_str($x->{value});
2344 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2345 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2346 $digit_round = '0'; $digit_round = substr($xs,$pl,1) if $pad <= $len;
2347 $pl++; $pl ++ if $pad >= $len;
2348 $digit_after = '0'; $digit_after = substr($xs,$pl,1) if $pad > 0;
2350 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2351 # closer at the remaining digits of the original $x, remember decision
2352 my $round_up = 1; # default round up
2354 ($mode eq 'trunc') || # trunc by round down
2355 ($digit_after =~ /[01234]/) || # round down anyway,
2357 ($digit_after eq '5') && # not 5000...0000
2358 ($x->_scan_for_nonzero($pad,$xs,$len) == 0) &&
2360 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2361 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2362 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2363 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2364 ($mode eq 'zero') # round down if zero, sign adjusted below
2366 my $put_back = 0; # not yet modified
2368 if (($pad > 0) && ($pad <= $len))
2370 substr($xs,-$pad,$pad) = '0' x $pad; # replace with '00...'
2371 $put_back = 1; # need to put back
2375 $x->bzero(); # round to '0'
2378 if ($round_up) # what gave test above?
2380 $put_back = 1; # need to put back
2381 $pad = $len, $xs = '0' x $pad if $scale < 0; # tlr: whack 0.51=>1.0
2383 # we modify directly the string variant instead of creating a number and
2384 # adding it, since that is faster (we already have the string)
2385 my $c = 0; $pad ++; # for $pad == $len case
2386 while ($pad <= $len)
2388 $c = substr($xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2389 substr($xs,-$pad,1) = $c; $pad++;
2390 last if $c != 0; # no overflow => early out
2392 $xs = '1'.$xs if $c == 0;
2395 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back, if needed
2397 $x->{_a} = $scale if $scale >= 0;
2400 $x->{_a} = $len+$scale;
2401 $x->{_a} = 0 if $scale < -$len;
2408 # return integer less or equal then number; no-op since it's already integer
2409 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2416 # return integer greater or equal then number; no-op since it's already int
2417 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2424 # An object might be asked to return itself as bigint on certain overloaded
2425 # operations. This does exactly this, so that sub classes can simple inherit
2426 # it or override with their own integer conversion routine.
2432 # return as hex string, with prefixed 0x
2433 my $x = shift; $x = $class->new($x) if !ref($x);
2435 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2438 $s = $x->{sign} if $x->{sign} eq '-';
2439 $s . $CALC->_as_hex($x->{value});
2444 # return as binary string, with prefixed 0b
2445 my $x = shift; $x = $class->new($x) if !ref($x);
2447 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2449 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2450 return $s . $CALC->_as_bin($x->{value});
2455 # return as octal string, with prefixed 0
2456 my $x = shift; $x = $class->new($x) if !ref($x);
2458 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2460 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2461 return $s . $CALC->_as_oct($x->{value});
2464 ##############################################################################
2465 # private stuff (internal use only)
2469 # check for strings, if yes, return objects instead
2471 # the first argument is number of args objectify() should look at it will
2472 # return $count+1 elements, the first will be a classname. This is because
2473 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2474 # useless objects being created and thrown away. So we cannot simple loop
2475 # over @_. If the given count is 0, all arguments will be used.
2477 # If the second arg is a ref, use it as class.
2478 # If not, try to use it as classname, unless undef, then use $class
2479 # (aka Math::BigInt). The latter shouldn't happen,though.
2482 # $x->badd(1); => ref x, scalar y
2483 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2484 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2485 # Math::BigInt::badd(1,2); => scalar x, scalar y
2486 # In the last case we check number of arguments to turn it silently into
2487 # $class,1,2. (We can not take '1' as class ;o)
2488 # badd($class,1) is not supported (it should, eventually, try to add undef)
2489 # currently it tries 'Math::BigInt' + 1, which will not work.
2491 # some shortcut for the common cases
2493 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2495 my $count = abs(shift || 0);
2497 my (@a,$k,$d); # resulting array, temp, and downgrade
2500 # okay, got object as first
2505 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2507 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2511 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2512 if (defined ${"$a[0]::downgrade"})
2514 $d = ${"$a[0]::downgrade"};
2515 ${"$a[0]::downgrade"} = undef;
2518 my $up = ${"$a[0]::upgrade"};
2519 # print STDERR "# Now in objectify, my class is today $a[0], count = $count\n";
2527 $k = $a[0]->new($k);
2529 elsif (!defined $up && ref($k) ne $a[0])
2531 # foreign object, try to convert to integer
2532 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2545 $k = $a[0]->new($k);
2547 elsif (!defined $up && ref($k) ne $a[0])
2549 # foreign object, try to convert to integer
2550 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2554 push @a,@_; # return other params, too
2558 require Carp; Carp::croak ("$class objectify needs list context");
2560 ${"$a[0]::downgrade"} = $d;
2564 sub _register_callback
2566 my ($class,$callback) = @_;
2568 if (ref($callback) ne 'CODE')
2571 Carp::croak ("$callback is not a coderef");
2573 $CALLBACKS{$class} = $callback;
2580 $IMPORT++; # remember we did import()
2581 my @a; my $l = scalar @_;
2582 my $warn_or_die = 0; # 0 - no warn, 1 - warn, 2 - die
2583 for ( my $i = 0; $i < $l ; $i++ )
2585 if ($_[$i] eq ':constant')
2587 # this causes overlord er load to step in
2589 integer => sub { $self->new(shift) },
2590 binary => sub { $self->new(shift) };
2592 elsif ($_[$i] eq 'upgrade')
2594 # this causes upgrading
2595 $upgrade = $_[$i+1]; # or undef to disable
2598 elsif ($_[$i] =~ /^(lib|try|only)\z/)
2600 # this causes a different low lib to take care...
2601 $CALC = $_[$i+1] || '';
2602 # lib => 1 (warn on fallback), try => 0 (no warn), only => 2 (die on fallback)
2603 $warn_or_die = 1 if $_[$i] eq 'lib';
2604 $warn_or_die = 2 if $_[$i] eq 'only';
2612 # any non :constant stuff is handled by our parent, Exporter
2617 $self->SUPER::import(@a); # need it for subclasses
2618 $self->export_to_level(1,$self,@a); # need it for MBF
2621 # try to load core math lib
2622 my @c = split /\s*,\s*/,$CALC;
2625 $_ =~ tr/a-zA-Z0-9://cd; # limit to sane characters
2627 push @c, \'FastCalc', \'Calc' # if all fail, try these
2628 if $warn_or_die < 2; # but not for "only"
2629 $CALC = ''; # signal error
2632 # fallback libraries are "marked" as \'string', extract string if nec.
2633 my $lib = $l; $lib = $$l if ref($l);
2635 next if ($lib || '') eq '';
2636 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2640 # Perl < 5.6.0 dies with "out of memory!" when eval("") and ':constant' is
2641 # used in the same script, or eval("") inside import().
2642 my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
2643 my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
2645 $file = File::Spec->catfile (@parts, $file);
2646 eval { require "$file"; $lib->import( @c ); }
2650 eval "use $lib qw/@c/;";
2655 # loaded it ok, see if the api_version() is high enough
2656 if ($lib->can('api_version') && $lib->api_version() >= 1.0)
2659 # api_version matches, check if it really provides anything we need
2663 add mul div sub dec inc
2664 acmp len digit is_one is_zero is_even is_odd
2666 zeros new copy check
2667 from_hex from_oct from_bin as_hex as_bin as_oct
2668 rsft lsft xor and or
2669 mod sqrt root fac pow modinv modpow log_int gcd
2672 if (!$lib->can("_$method"))
2674 if (($WARN{$lib}||0) < 2)
2677 Carp::carp ("$lib is missing method '_$method'");
2678 $WARN{$lib} = 1; # still warn about the lib
2687 if ($warn_or_die > 0 && ref($l))
2690 my $msg = "Math::BigInt: couldn't load specified math lib(s), fallback to $lib";
2691 Carp::carp ($msg) if $warn_or_die == 1;
2692 Carp::croak ($msg) if $warn_or_die == 2;
2694 last; # found a usable one, break
2698 if (($WARN{$lib}||0) < 2)
2700 my $ver = eval "\$$lib\::VERSION" || 'unknown';
2702 Carp::carp ("Cannot load outdated $lib v$ver, please upgrade");
2703 $WARN{$lib} = 2; # never warn again
2711 if ($warn_or_die == 2)
2713 Carp::croak ("Couldn't load specified math lib(s) and fallback disallowed");
2717 Carp::croak ("Couldn't load any math lib(s), not even fallback to Calc.pm");
2722 foreach my $class (keys %CALLBACKS)
2724 &{$CALLBACKS{$class}}($CALC);
2727 # Fill $CAN with the results of $CALC->can(...) for emulating lower math lib
2731 for my $method (qw/ signed_and signed_or signed_xor /)
2733 $CAN{$method} = $CALC->can("_$method") ? 1 : 0;
2741 # create a bigint from a hexadecimal string
2742 my ($self, $hs) = @_;
2744 my $rc = $self->__from_hex($hs);
2746 return $self->bnan() unless defined $rc;
2753 # create a bigint from a hexadecimal string
2754 my ($self, $bs) = @_;
2756 my $rc = $self->__from_bin($bs);
2758 return $self->bnan() unless defined $rc;
2765 # create a bigint from a hexadecimal string
2766 my ($self, $os) = @_;
2768 my $x = $self->bzero();
2771 $os =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2772 $os =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2774 return $x->bnan() if $os !~ /^[\-\+]?0[0-9]+$/;
2776 my $sign = '+'; $sign = '-' if $os =~ /^-/;
2778 $os =~ s/^[+-]//; # strip sign
2779 $x->{value} = $CALC->_from_oct($os);
2780 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2787 # convert a (ref to) big hex string to BigInt, return undef for error
2790 my $x = Math::BigInt->bzero();
2793 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2794 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2796 return $x->bnan() if $hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2798 my $sign = '+'; $sign = '-' if $hs =~ /^-/;
2800 $hs =~ s/^[+-]//; # strip sign
2801 $x->{value} = $CALC->_from_hex($hs);
2802 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2809 # convert a (ref to) big binary string to BigInt, return undef for error
2812 my $x = Math::BigInt->bzero();
2815 $bs =~ s/([01])_([01])/$1$2/g;
2816 $bs =~ s/([01])_([01])/$1$2/g;
2817 return $x->bnan() if $bs !~ /^[+-]?0b[01]+$/;
2819 my $sign = '+'; $sign = '-' if $bs =~ /^\-/;
2820 $bs =~ s/^[+-]//; # strip sign
2822 $x->{value} = $CALC->_from_bin($bs);
2823 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2829 # input: num_str; output: undef for invalid or
2830 # (\$mantissa_sign,\$mantissa_value,\$mantissa_fraction,\$exp_sign,\$exp_value)
2831 # Internal, take apart a string and return the pieces.
2832 # Strip leading/trailing whitespace, leading zeros, underscore and reject
2836 # strip white space at front, also extranous leading zeros
2837 $x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2838 $x =~ s/^\s+//; # but this will
2839 $x =~ s/\s+$//g; # strip white space at end
2841 # shortcut, if nothing to split, return early
2842 if ($x =~ /^[+-]?[0-9]+\z/)
2844 $x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2845 return (\$sign, \$x, \'', \'', \0);
2848 # invalid starting char?
2849 return if $x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2851 return __from_hex($x) if $x =~ /^[\-\+]?0x/; # hex string
2852 return __from_bin($x) if $x =~ /^[\-\+]?0b/; # binary string
2854 # strip underscores between digits
2855 $x =~ s/([0-9])_([0-9])/$1$2/g;
2856 $x =~ s/([0-9])_([0-9])/$1$2/g; # do twice for 1_2_3
2858 # some possible inputs:
2859 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2860 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 # 0e999
2862 my ($m,$e,$last) = split /[Ee]/,$x;
2863 return if defined $last; # last defined => 1e2E3 or others
2864 $e = '0' if !defined $e || $e eq "";
2866 # sign,value for exponent,mantint,mantfrac
2867 my ($es,$ev,$mis,$miv,$mfv);
2869 if ($e =~ /^([+-]?)0*([0-9]+)$/) # strip leading zeros
2873 return if $m eq '.' || $m eq '';
2874 my ($mi,$mf,$lastf) = split /\./,$m;
2875 return if defined $lastf; # lastf defined => 1.2.3 or others
2876 $mi = '0' if !defined $mi;
2877 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2878 $mf = '0' if !defined $mf || $mf eq '';
2879 if ($mi =~ /^([+-]?)0*([0-9]+)$/) # strip leading zeros
2881 $mis = $1||'+'; $miv = $2;
2882 return unless ($mf =~ /^([0-9]*?)0*$/); # strip trailing zeros
2884 # handle the 0e999 case here
2885 $ev = 0 if $miv eq '0' && $mfv eq '';
2886 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2889 return; # NaN, not a number
2892 ##############################################################################
2893 # internal calculation routines (others are in Math::BigInt::Calc etc)
2897 # (BINT or num_str, BINT or num_str) return BINT
2898 # does modify first argument
2902 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2903 my $method = ref($x) . '::bgcd';
2905 $x * $ty / &$method($x,$ty);
2908 ###############################################################################
2909 # trigonometric functions
2913 # Calculate PI to N digits. Unless upgrading is in effect, returns the
2914 # result truncated to an integer, that is, always returns '3'.
2918 # called like Math::BigInt::bpi(10);
2919 $n = $self; $self = $class;
2921 $self = ref($self) if ref($self);
2923 return $upgrade->new($n) if defined $upgrade;
2931 # Calculate cosinus(x) to N digits. Unless upgrading is in effect, returns the
2932 # result truncated to an integer.
2933 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2935 return $x if $x->modify('bcos');
2937 return $x->bnan() if $x->{sign} !~ /^[+-]\z/; # -inf +inf or NaN => NaN
2939 return $upgrade->new($x)->bcos(@r) if defined $upgrade;
2941 # calculate the result and truncate it to integer
2942 my $t = Math::BigFloat->new($x)->bcos(@r)->as_int();
2944 $x->bone() if $t->is_one();
2945 $x->bzero() if $t->is_zero();
2951 # Calculate sinus(x) to N digits. Unless upgrading is in effect, returns the
2952 # result truncated to an integer.
2953 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2955 return $x if $x->modify('bsin');
2957 return $x->bnan() if $x->{sign} !~ /^[+-]\z/; # -inf +inf or NaN => NaN
2959 return $upgrade->new($x)->bsin(@r) if defined $upgrade;
2961 # calculate the result and truncate it to integer
2962 my $t = Math::BigFloat->new($x)->bsin(@r)->as_int();
2964 $x->bone() if $t->is_one();
2965 $x->bzero() if $t->is_zero();
2971 # Calculate arcus tangens of x to N digits. Unless upgrading is in effect, returns the
2972 # result truncated to an integer.
2973 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2975 return $x if $x->modify('batan');
2977 return $x->bnan() if $x->{sign} !~ /^[+-]\z/; # -inf +inf or NaN => NaN
2979 return $upgrade->new($x)->batan(@r) if defined $upgrade;
2981 # calculate the result and truncate it to integer
2982 my $t = Math::BigFloat->new($x)->batan(@r);
2984 $x->{value} = $CALC->_new( $x->as_int()->bstr() );
2988 ###############################################################################
2989 # this method returns 0 if the object can be modified, or 1 if not.
2990 # We use a fast constant sub() here, to avoid costly calls. Subclasses
2991 # may override it with special code (f.i. Math::BigInt::Constant does so)
2993 sub modify () { 0; }
3002 Math::BigInt - Arbitrary size integer/float math package
3008 # or make it faster: install (optional) Math::BigInt::GMP
3009 # and always use (it will fall back to pure Perl if the
3010 # GMP library is not installed):
3012 # will warn if Math::BigInt::GMP cannot be found
3013 use Math::BigInt lib => 'GMP';
3015 # to supress the warning use this:
3016 # use Math::BigInt try => 'GMP';
3018 my $str = '1234567890';
3019 my @values = (64,74,18);
3020 my $n = 1; my $sign = '-';
3023 my $x = Math::BigInt->new($str); # defaults to 0
3024 my $y = $x->copy(); # make a true copy
3025 my $nan = Math::BigInt->bnan(); # create a NotANumber
3026 my $zero = Math::BigInt->bzero(); # create a +0
3027 my $inf = Math::BigInt->binf(); # create a +inf
3028 my $inf = Math::BigInt->binf('-'); # create a -inf
3029 my $one = Math::BigInt->bone(); # create a +1
3030 my $mone = Math::BigInt->bone('-'); # create a -1
3032 my $pi = Math::BigInt->bpi(); # returns '3'
3033 # see Math::BigFloat::bpi()
3035 $h = Math::BigInt->new('0x123'); # from hexadecimal
3036 $b = Math::BigInt->new('0b101'); # from binary
3037 $o = Math::BigInt->from_oct('0101'); # from octal
3039 # Testing (don't modify their arguments)
3040 # (return true if the condition is met, otherwise false)
3042 $x->is_zero(); # if $x is +0
3043 $x->is_nan(); # if $x is NaN
3044 $x->is_one(); # if $x is +1
3045 $x->is_one('-'); # if $x is -1
3046 $x->is_odd(); # if $x is odd
3047 $x->is_even(); # if $x is even
3048 $x->is_pos(); # if $x >= 0
3049 $x->is_neg(); # if $x < 0
3050 $x->is_inf($sign); # if $x is +inf, or -inf (sign is default '+')
3051 $x->is_int(); # if $x is an integer (not a float)
3053 # comparing and digit/sign extraction
3054 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
3055 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
3056 $x->sign(); # return the sign, either +,- or NaN
3057 $x->digit($n); # return the nth digit, counting from right
3058 $x->digit(-$n); # return the nth digit, counting from left
3060 # The following all modify their first argument. If you want to preserve
3061 # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
3062 # necessary when mixing $a = $b assignments with non-overloaded math.
3064 $x->bzero(); # set $x to 0
3065 $x->bnan(); # set $x to NaN
3066 $x->bone(); # set $x to +1
3067 $x->bone('-'); # set $x to -1
3068 $x->binf(); # set $x to inf
3069 $x->binf('-'); # set $x to -inf
3071 $x->bneg(); # negation
3072 $x->babs(); # absolute value
3073 $x->bnorm(); # normalize (no-op in BigInt)
3074 $x->bnot(); # two's complement (bit wise not)
3075 $x->binc(); # increment $x by 1
3076 $x->bdec(); # decrement $x by 1
3078 $x->badd($y); # addition (add $y to $x)
3079 $x->bsub($y); # subtraction (subtract $y from $x)
3080 $x->bmul($y); # multiplication (multiply $x by $y)
3081 $x->bdiv($y); # divide, set $x to quotient
3082 # return (quo,rem) or quo if scalar
3084 $x->bmuladd($y,$z); # $x = $x * $y + $z
3086 $x->bmod($y); # modulus (x % y)
3087 $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
3088 $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
3090 $x->bpow($y); # power of arguments (x ** y)
3091 $x->blsft($y); # left shift in base 2
3092 $x->brsft($y); # right shift in base 2
3093 # returns (quo,rem) or quo if in scalar context
3094 $x->blsft($y,$n); # left shift by $y places in base $n
3095 $x->brsft($y,$n); # right shift by $y places in base $n
3096 # returns (quo,rem) or quo if in scalar context
3098 $x->band($y); # bitwise and
3099 $x->bior($y); # bitwise inclusive or
3100 $x->bxor($y); # bitwise exclusive or
3101 $x->bnot(); # bitwise not (two's complement)
3103 $x->bsqrt(); # calculate square-root
3104 $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
3105 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3107 $x->bnok($y); # x over y (binomial coefficient n over k)
3109 $x->blog(); # logarithm of $x to base e (Euler's number)
3110 $x->blog($base); # logarithm of $x to base $base (f.i. 2)
3111 $x->bexp(); # calculate e ** $x where e is Euler's number
3113 $x->round($A,$P,$mode); # round to accuracy or precision using mode $mode
3114 $x->bround($n); # accuracy: preserve $n digits
3115 $x->bfround($n); # round to $nth digit, no-op for BigInts
3117 # The following do not modify their arguments in BigInt (are no-ops),
3118 # but do so in BigFloat:
3120 $x->bfloor(); # return integer less or equal than $x
3121 $x->bceil(); # return integer greater or equal than $x
3123 # The following do not modify their arguments:
3125 # greatest common divisor (no OO style)
3126 my $gcd = Math::BigInt::bgcd(@values);
3127 # lowest common multiplicator (no OO style)
3128 my $lcm = Math::BigInt::blcm(@values);
3130 $x->length(); # return number of digits in number
3131 ($xl,$f) = $x->length(); # length of number and length of fraction part,
3132 # latter is always 0 digits long for BigInts
3134 $x->exponent(); # return exponent as BigInt
3135 $x->mantissa(); # return (signed) mantissa as BigInt
3136 $x->parts(); # return (mantissa,exponent) as BigInt
3137 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3138 $x->as_int(); # return as BigInt (in BigInt: same as copy())
3139 $x->numify(); # return as scalar (might overflow!)
3141 # conversation to string (do not modify their argument)
3142 $x->bstr(); # normalized string (e.g. '3')
3143 $x->bsstr(); # norm. string in scientific notation (e.g. '3E0')
3144 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3145 $x->as_bin(); # as signed binary string with prefixed 0b
3146 $x->as_oct(); # as signed octal string with prefixed 0
3149 # precision and accuracy (see section about rounding for more)
3150 $x->precision(); # return P of $x (or global, if P of $x undef)
3151 $x->precision($n); # set P of $x to $n
3152 $x->accuracy(); # return A of $x (or global, if A of $x undef)
3153 $x->accuracy($n); # set A $x to $n
3156 Math::BigInt->precision(); # get/set global P for all BigInt objects
3157 Math::BigInt->accuracy(); # get/set global A for all BigInt objects
3158 Math::BigInt->round_mode(); # get/set global round mode, one of
3159 # 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'
3160 Math::BigInt->config(); # return hash containing configuration
3164 All operators (including basic math operations) are overloaded if you
3165 declare your big integers as
3167 $i = new Math::BigInt '123_456_789_123_456_789';
3169 Operations with overloaded operators preserve the arguments which is
3170 exactly what you expect.
3176 Input values to these routines may be any string, that looks like a number
3177 and results in an integer, including hexadecimal and binary numbers.
3179 Scalars holding numbers may also be passed, but note that non-integer numbers
3180 may already have lost precision due to the conversation to float. Quote
3181 your input if you want BigInt to see all the digits:
3183 $x = Math::BigInt->new(12345678890123456789); # bad
3184 $x = Math::BigInt->new('12345678901234567890'); # good
3186 You can include one underscore between any two digits.
3188 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
3189 Non-integer values result in NaN.
3191 Hexadecimal (prefixed with "0x") and binary numbers (prefixed with "0b")
3192 are accepted, too. Please note that octal numbers are not recognized
3193 by new(), so the following will print "123":
3195 perl -MMath::BigInt -le 'print Math::BigInt->new("0123")'
3197 To convert an octal number, use from_oct();
3199 perl -MMath::BigInt -le 'print Math::BigInt->from_oct("0123")'
3201 Currently, Math::BigInt::new() defaults to 0, while Math::BigInt::new('')
3202 results in 'NaN'. This might change in the future, so use always the following
3203 explicit forms to get a zero or NaN:
3205 $zero = Math::BigInt->bzero();
3206 $nan = Math::BigInt->bnan();
3208 C<bnorm()> on a BigInt object is now effectively a no-op, since the numbers
3209 are always stored in normalized form. If passed a string, creates a BigInt
3210 object from the input.
3214 Output values are BigInt objects (normalized), except for the methods which
3215 return a string (see L<SYNOPSIS>).
3217 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
3218 C<is_nan()>, etc.) return true or false, while others (C<bcmp()>, C<bacmp()>)
3219 return either undef (if NaN is involved), <0, 0 or >0 and are suited for sort.
3225 Each of the methods below (except config(), accuracy() and precision())
3226 accepts three additional parameters. These arguments C<$A>, C<$P> and C<$R>
3227 are C<accuracy>, C<precision> and C<round_mode>. Please see the section about
3228 L<ACCURACY and PRECISION> for more information.
3234 print Dumper ( Math::BigInt->config() );
3235 print Math::BigInt->config()->{lib},"\n";
3237 Returns a hash containing the configuration, e.g. the version number, lib
3238 loaded etc. The following hash keys are currently filled in with the
3239 appropriate information.
3243 ============================================================
3244 lib Name of the low-level math library
3246 lib_version Version of low-level math library (see 'lib')
3248 class The class name of config() you just called
3250 upgrade To which class math operations might be upgraded
3252 downgrade To which class math operations might be downgraded
3254 precision Global precision
3256 accuracy Global accuracy
3258 round_mode Global round mode
3260 version version number of the class you used
3262 div_scale Fallback accuracy for div
3264 trap_nan If true, traps creation of NaN via croak()
3266 trap_inf If true, traps creation of +inf/-inf via croak()
3269 The following values can be set by passing C<config()> a reference to a hash:
3272 upgrade downgrade precision accuracy round_mode div_scale
3276 $new_cfg = Math::BigInt->config( { trap_inf => 1, precision => 5 } );
3280 $x->accuracy(5); # local for $x
3281 CLASS->accuracy(5); # global for all members of CLASS
3282 # Note: This also applies to new()!
3284 $A = $x->accuracy(); # read out accuracy that affects $x
3285 $A = CLASS->accuracy(); # read out global accuracy
3287 Set or get the global or local accuracy, aka how many significant digits the
3288 results have. If you set a global accuracy, then this also applies to new()!
3290 Warning! The accuracy I<sticks>, e.g. once you created a number under the
3291 influence of C<< CLASS->accuracy($A) >>, all results from math operations with
3292 that number will also be rounded.
3294 In most cases, you should probably round the results explicitly using one of
3295 L<round()>, L<bround()> or L<bfround()> or by passing the desired accuracy
3296 to the math operation as additional parameter:
3298 my $x = Math::BigInt->new(30000);
3299 my $y = Math::BigInt->new(7);
3300 print scalar $x->copy()->bdiv($y, 2); # print 4300
3301 print scalar $x->copy()->bdiv($y)->bround(2); # print 4300
3303 Please see the section about L<ACCURACY AND PRECISION> for further details.
3305 Value must be greater than zero. Pass an undef value to disable it:
3307 $x->accuracy(undef);
3308 Math::BigInt->accuracy(undef);
3310 Returns the current accuracy. For C<$x->accuracy()> it will return either the
3311 local accuracy, or if not defined, the global. This means the return value
3312 represents the accuracy that will be in effect for $x:
3314 $y = Math::BigInt->new(1234567); # unrounded
3315 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
3316 $x = Math::BigInt->new(123456); # $x will be automatically rounded!
3317 print "$x $y\n"; # '123500 1234567'
3318 print $x->accuracy(),"\n"; # will be 4
3319 print $y->accuracy(),"\n"; # also 4, since global is 4
3320 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
3321 print $x->accuracy(),"\n"; # still 4
3322 print $y->accuracy(),"\n"; # 5, since global is 5
3324 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
3325 globals separated from Math::BigInt, but it is possible to subclass
3326 Math::BigInt and make the globals of the subclass aliases to the ones from
3331 $x->precision(-2); # local for $x, round at the second digit right of the dot
3332 $x->precision(2); # ditto, round at the second digit left of the dot
3334 CLASS->precision(5); # Global for all members of CLASS
3335 # This also applies to new()!
3336 CLASS->precision(-5); # ditto
3338 $P = CLASS->precision(); # read out global precision
3339 $P = $x->precision(); # read out precision that affects $x
3341 Note: You probably want to use L<accuracy()> instead. With L<accuracy> you
3342 set the number of digits each result should have, with L<precision> you
3343 set the place where to round!
3345 C<precision()> sets or gets the global or local precision, aka at which digit
3346 before or after the dot to round all results. A set global precision also
3347 applies to all newly created numbers!
3349 In Math::BigInt, passing a negative number precision has no effect since no
3350 numbers have digits after the dot. In L<Math::BigFloat>, it will round all
3351 results to P digits after the dot.
3353 Please see the section about L<ACCURACY AND PRECISION> for further details.
3355 Pass an undef value to disable it:
3357 $x->precision(undef);
3358 Math::BigInt->precision(undef);
3360 Returns the current precision. For C<$x->precision()> it will return either the
3361 local precision of $x, or if not defined, the global. This means the return
3362 value represents the prevision that will be in effect for $x:
3364 $y = Math::BigInt->new(1234567); # unrounded
3365 print Math::BigInt->precision(4),"\n"; # set 4, print 4
3366 $x = Math::BigInt->new(123456); # will be automatically rounded
3367 print $x; # print "120000"!
3369 Note: Works also for subclasses like L<Math::BigFloat>. Each class has its
3370 own globals separated from Math::BigInt, but it is possible to subclass
3371 Math::BigInt and make the globals of the subclass aliases to the ones from
3378 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
3379 2, but others work, too.
3381 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
3385 $x = Math::BigInt->new(10);
3386 $x->brsft(1); # same as $x >> 1: 5
3387 $x = Math::BigInt->new(1234);
3388 $x->brsft(2,10); # result 12
3390 There is one exception, and that is base 2 with negative $x:
3393 $x = Math::BigInt->new(-5);
3396 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
3401 $x = Math::BigInt->new($str,$A,$P,$R);
3403 Creates a new BigInt object from a scalar or another BigInt object. The
3404 input is accepted as decimal, hex (with leading '0x') or binary (with leading
3407 See L<Input> for more info on accepted input formats.
3411 $x = Math::BigIn->from_oct("0775"); # input is octal
3415 $x = Math::BigIn->from_hex("0xcafe"); # input is hexadecimal
3419 $x = Math::BigIn->from_oct("0x10011"); # input is binary
3423 $x = Math::BigInt->bnan();
3425 Creates a new BigInt object representing NaN (Not A Number).
3426 If used on an object, it will set it to NaN:
3432 $x = Math::BigInt->bzero();
3434 Creates a new BigInt object representing zero.
3435 If used on an object, it will set it to zero:
3441 $x = Math::BigInt->binf($sign);
3443 Creates a new BigInt object representing infinity. The optional argument is
3444 either '-' or '+', indicating whether you want infinity or minus infinity.
3445 If used on an object, it will set it to infinity:
3452 $x = Math::BigInt->binf($sign);
3454 Creates a new BigInt object representing one. The optional argument is
3455 either '-' or '+', indicating whether you want one or minus one.
3456 If used on an object, it will set it to one:
3461 =head2 is_one()/is_zero()/is_nan()/is_inf()
3464 $x->is_zero(); # true if arg is +0
3465 $x->is_nan(); # true if arg is NaN
3466 $x->is_one(); # true if arg is +1
3467 $x->is_one('-'); # true if arg is -1
3468 $x->is_inf(); # true if +inf
3469 $x->is_inf('-'); # true if -inf (sign is default '+')
3471 These methods all test the BigInt for being one specific value and return
3472 true or false depending on the input. These are faster than doing something
3477 =head2 is_pos()/is_neg()/is_positive()/is_negative()
3479 $x->is_pos(); # true if > 0
3480 $x->is_neg(); # true if < 0
3482 The methods return true if the argument is positive or negative, respectively.
3483 C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
3484 C<-inf> is negative. A C<zero> is neither positive nor negative.
3486 These methods are only testing the sign, and not the value.
3488 C<is_positive()> and C<is_negative()> are aliases to C<is_pos()> and
3489 C<is_neg()>, respectively. C<is_positive()> and C<is_negative()> were
3490 introduced in v1.36, while C<is_pos()> and C<is_neg()> were only introduced
3493 =head2 is_odd()/is_even()/is_int()
3495 $x->is_odd(); # true if odd, false for even
3496 $x->is_even(); # true if even, false for odd
3497 $x->is_int(); # true if $x is an integer
3499 The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
3500 C<-inf> are not integers and are neither odd nor even.
3502 In BigInt, all numbers except C<NaN>, C<+inf> and C<-inf> are integers.
3508 Compares $x with $y and takes the sign into account.
3509 Returns -1, 0, 1 or undef.
3515 Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
3521 Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
3523 If you want $x to have a certain sign, use one of the following methods:
3526 $x->babs()->bneg(); # '-'
3528 $x->binf(); # '+inf'
3529 $x->binf('-'); # '-inf'
3533 $x->digit($n); # return the nth digit, counting from right
3535 If C<$n> is negative, returns the digit counting from left.
3541 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
3542 and '-inf', respectively. Does nothing for NaN or zero.
3548 Set the number to its absolute value, e.g. change the sign from '-' to '+'
3549 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
3554 $x->bnorm(); # normalize (no-op)
3560 Two's complement (bitwise not). This is equivalent to
3568 $x->binc(); # increment x by 1
3572 $x->bdec(); # decrement x by 1
3576 $x->badd($y); # addition (add $y to $x)
3580 $x->bsub($y); # subtraction (subtract $y from $x)
3584 $x->bmul($y); # multiplication (multiply $x by $y)
3590 Multiply $x by $y, and then add $z to the result,
3592 This method was added in v1.87 of Math::BigInt (June 2007).
3596 $x->bdiv($y); # divide, set $x to quotient
3597 # return (quo,rem) or quo if scalar
3601 $x->bmod($y); # modulus (x % y)
3605 num->bmodinv($mod); # modular inverse
3607 Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
3608 returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
3609 C<bgcd($num, $mod)==1>.
3613 $num->bmodpow($exp,$mod); # modular exponentation
3614 # ($num**$exp % $mod)
3616 Returns the value of C<$num> taken to the power C<$exp> in the modulus
3617 C<$mod> using binary exponentation. C<bmodpow> is far superior to
3622 because it is much faster - it reduces internal variables into
3623 the modulus whenever possible, so it operates on smaller numbers.
3625 C<bmodpow> also supports negative exponents.
3627 bmodpow($num, -1, $mod)
3629 is exactly equivalent to
3635 $x->bpow($y); # power of arguments (x ** y)
3639 $x->blog($base, $accuracy); # logarithm of x to the base $base
3641 If C<$base> is not defined, Euler's number (e) is used:
3643 print $x->blog(undef, 100); # log(x) to 100 digits
3647 $x->bexp($accuracy); # calculate e ** X
3649 Calculates the expression C<e ** $x> where C<e> is Euler's number.
3651 This method was added in v1.82 of Math::BigInt (April 2007).
3657 $x->bnok($y); # x over y (binomial coefficient n over k)
3659 Calculates the binomial coefficient n over k, also called the "choose"
3660 function. The result is equivalent to:
3666 This method was added in v1.84 of Math::BigInt (April 2007).
3670 print Math::BigInt->bpi(100), "\n"; # 3
3672 Returns PI truncated to an integer, with the argument being ignored. that
3673 is it always returns C<3>.
3675 If upgrading is in effect, returns PI to N digits (including the "3"
3679 use Math::BigInt upgrade => Math::BigFloat;
3680 print Math::BigInt->bpi(3), "\n"; # 3.14
3681 print Math::BigInt->bpi(100), "\n"; # 3.1415....
3683 This method was added in v1.87 of Math::BigInt (June 2007).
3687 my $x = Math::BigFloat->new(1);
3688 print $x->bcos(100), "\n";
3690 Calculate the cosinus of $x, modifying $x in place.
3692 This method was added in v1.87 of Math::BigInt (June 2007).
3696 my $x = Math::BigFloat->new(1);
3697 print $x->bsin(100), "\n";
3699 Calculate the sinus of $x, modifying $x in place.
3701 This method was added in v1.87 of Math::BigInt (June 2007).
3705 my $x = Math::BigFloat->new(0.5);
3706 print $x->batan(100), "\n";
3708 Calculate the arcus tanges of $x, modifying $x in place.
3710 This method was added in v1.87 of Math::BigInt (June 2007).
3714 $x->blsft($y); # left shift in base 2
3715 $x->blsft($y,$n); # left shift, in base $n (like 10)
3719 $x->brsft($y); # right shift in base 2
3720 $x->brsft($y,$n); # right shift, in base $n (like 10)
3724 $x->band($y); # bitwise and
3728 $x->bior($y); # bitwise inclusive or
3732 $x->bxor($y); # bitwise exclusive or
3736 $x->bnot(); # bitwise not (two's complement)
3740 $x->bsqrt(); # calculate square-root
3744 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3748 $x->round($A,$P,$round_mode);
3750 Round $x to accuracy C<$A> or precision C<$P> using the round mode
3755 $x->bround($N); # accuracy: preserve $N digits
3759 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3765 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3766 does change $x in BigFloat.
3772 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3773 does change $x in BigFloat.
3777 bgcd(@values); # greatest common divisor (no OO style)
3781 blcm(@values); # lowest common multiplicator (no OO style)
3786 ($xl,$fl) = $x->length();
3788 Returns the number of digits in the decimal representation of the number.
3789 In list context, returns the length of the integer and fraction part. For
3790 BigInt's, the length of the fraction part will always be 0.
3796 Return the exponent of $x as BigInt.
3802 Return the signed mantissa of $x as BigInt.
3806 $x->parts(); # return (mantissa,exponent) as BigInt
3810 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3812 =head2 as_int()/as_number()
3816 Returns $x as a BigInt (truncated towards zero). In BigInt this is the same as
3819 C<as_number()> is an alias to this method. C<as_number> was introduced in
3820 v1.22, while C<as_int()> was only introduced in v1.68.
3826 Returns a normalized string representation of C<$x>.
3830 $x->bsstr(); # normalized string in scientific notation
3834 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3838 $x->as_bin(); # as signed binary string with prefixed 0b
3842 $x->as_oct(); # as signed octal string with prefixed 0
3848 This returns a normal Perl scalar from $x. It is used automatically
3849 whenever a scalar is needed, for instance in array index operations.
3851 This loses precision, to avoid this use L<as_int()> instead.
3855 $x->modify('bpowd');
3857 This method returns 0 if the object can be modified with the given
3858 peration, or 1 if not.
3860 This is used for instance by L<Math::BigInt::Constant>.
3862 =head2 upgrade()/downgrade()
3864 Set/get the class for downgrade/upgrade operations. Thuis is used
3865 for instance by L<bignum>. The defaults are '', thus the following
3866 operation will create a BigInt, not a BigFloat:
3868 my $i = Math::BigInt->new(123);
3869 my $f = Math::BigFloat->new('123.1');
3871 print $i + $f,"\n"; # print 246
3875 Set/get the number of digits for the default precision in divide
3880 Set/get the current round mode.
3882 =head1 ACCURACY and PRECISION
3884 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3885 accuracy and precision based rounding, both automatically after every
3886 operation, as well as manually.
3888 This section describes the accuracy/precision handling in Math::Big* as it
3889 used to be and as it is now, complete with an explanation of all terms and
3892 Not yet implemented things (but with correct description) are marked with '!',
3893 things that need to be answered are marked with '?'.
3895 In the next paragraph follows a short description of terms used here (because
3896 these may differ from terms used by others people or documentation).
3898 During the rest of this document, the shortcuts A (for accuracy), P (for
3899 precision), F (fallback) and R (rounding mode) will be used.
3903 A fixed number of digits before (positive) or after (negative)
3904 the decimal point. For example, 123.45 has a precision of -2. 0 means an
3905 integer like 123 (or 120). A precision of 2 means two digits to the left
3906 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3907 numbers with zeros before the decimal point may have different precisions,
3908 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3909 was). It could also have p < 0, when the digits after the decimal point
3912 The string output (of floating point numbers) will be padded with zeros:
3914 Initial value P A Result String
3915 ------------------------------------------------------------
3916 1234.01 -3 1000 1000
3919 1234.001 1 1234 1234.0
3921 1234.01 2 1234.01 1234.01
3922 1234.01 5 1234.01 1234.01000
3924 For BigInts, no padding occurs.
3928 Number of significant digits. Leading zeros are not counted. A
3929 number may have an accuracy greater than the non-zero digits
3930 when there are zeros in it or trailing zeros. For example, 123.456 has
3931 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3933 The string output (of floating point numbers) will be padded with zeros:
3935 Initial value P A Result String
3936 ------------------------------------------------------------
3938 1234.01 6 1234.01 1234.01
3939 1234.1 8 1234.1 1234.1000
3941 For BigInts, no padding occurs.
3945 When both A and P are undefined, this is used as a fallback accuracy when
3948 =head2 Rounding mode R
3950 When rounding a number, different 'styles' or 'kinds'
3951 of rounding are possible. (Note that random rounding, as in
3952 Math::Round, is not implemented.)
3958 truncation invariably removes all digits following the
3959 rounding place, replacing them with zeros. Thus, 987.65 rounded
3960 to tens (P=1) becomes 980, and rounded to the fourth sigdig
3961 becomes 987.6 (A=4). 123.456 rounded to the second place after the
3962 decimal point (P=-2) becomes 123.46.
3964 All other implemented styles of rounding attempt to round to the
3965 "nearest digit." If the digit D immediately to the right of the
3966 rounding place (skipping the decimal point) is greater than 5, the
3967 number is incremented at the rounding place (possibly causing a
3968 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3969 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3970 truncated at the rounding place: e.g. when rounding to units, 0.4
3971 rounds to 0, and -19.4 rounds to -19.
3973 However the results of other styles of rounding differ if the
3974 digit immediately to the right of the rounding place (skipping the
3975 decimal point) is 5 and if there are no digits, or no digits other
3976 than 0, after that 5. In such cases:
3980 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3981 if it is not already. E.g., when rounding to the first sigdig, 0.45
3982 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3986 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3987 it is not already. E.g., when rounding to the first sigdig, 0.45
3988 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3992 round to plus infinity, i.e. always round up. E.g., when
3993 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3994 and 0.4501 also becomes 0.5.
3998 round to minus infinity, i.e. always round down. E.g., when
3999 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
4000 but 0.4501 becomes 0.5.
4004 round to zero, i.e. positive numbers down, negative ones up.
4005 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
4006 becomes -0.5, but 0.4501 becomes 0.5.
4010 round up if the digit immediately to the right of the rounding place
4011 is 5 or greater, otherwise round down. E.g., 0.15 becomes 0.2 and
4016 The handling of A & P in MBI/MBF (the old core code shipped with Perl
4017 versions <= 5.7.2) is like this:
4023 * ffround($p) is able to round to $p number of digits after the decimal
4025 * otherwise P is unused
4027 =item Accuracy (significant digits)
4029 * fround($a) rounds to $a significant digits
4030 * only fdiv() and fsqrt() take A as (optional) paramater
4031 + other operations simply create the same number (fneg etc), or more (fmul)
4033 + rounding/truncating is only done when explicitly calling one of fround
4034 or ffround, and never for BigInt (not implemented)
4035 * fsqrt() simply hands its accuracy argument over to fdiv.
4036 * the documentation and the comment in the code indicate two different ways
4037 on how fdiv() determines the maximum number of digits it should calculate,
4038 and the actual code does yet another thing
4040 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
4042 result has at most max(scale, length(dividend), length(divisor)) digits
4044 scale = max(scale, length(dividend)-1,length(divisor)-1);
4045 scale += length(divisor) - length(dividend);
4046 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
4047 Actually, the 'difference' added to the scale is calculated from the
4048 number of "significant digits" in dividend and divisor, which is derived
4049 by looking at the length of the mantissa. Which is wrong, since it includes
4050 the + sign (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops
4051 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
4052 assumption that 124 has 3 significant digits, while 120/7 will get you
4053 '17', not '17.1' since 120 is thought to have 2 significant digits.
4054 The rounding after the division then uses the remainder and $y to determine
4055 wether it must round up or down.
4056 ? I have no idea which is the right way. That's why I used a slightly more
4057 ? simple scheme and tweaked the few failing testcases to match it.
4061 This is how it works now:
4065 =item Setting/Accessing
4067 * You can set the A global via C<< Math::BigInt->accuracy() >> or
4068 C<< Math::BigFloat->accuracy() >> or whatever class you are using.
4069 * You can also set P globally by using C<< Math::SomeClass->precision() >>
4071 * Globals are classwide, and not inherited by subclasses.
4072 * to undefine A, use C<< Math::SomeCLass->accuracy(undef); >>
4073 * to undefine P, use C<< Math::SomeClass->precision(undef); >>
4074 * Setting C<< Math::SomeClass->accuracy() >> clears automatically
4075 C<< Math::SomeClass->precision() >>, and vice versa.
4076 * To be valid, A must be > 0, P can have any value.
4077 * If P is negative, this means round to the P'th place to the right of the
4078 decimal point; positive values mean to the left of the decimal point.
4079 P of 0 means round to integer.
4080 * to find out the current global A, use C<< Math::SomeClass->accuracy() >>
4081 * to find out the current global P, use C<< Math::SomeClass->precision() >>
4082 * use C<< $x->accuracy() >> respective C<< $x->precision() >> for the local
4083 setting of C<< $x >>.
4084 * Please note that C<< $x->accuracy() >> respective C<< $x->precision() >>
4085 return eventually defined global A or P, when C<< $x >>'s A or P is not
4088 =item Creating numbers
4090 * When you create a number, you can give the desired A or P via:
4091 $x = Math::BigInt->new($number,$A,$P);
4092 * Only one of A or P can be defined, otherwise the result is NaN
4093 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
4094 globals (if set) will be used. Thus changing the global defaults later on
4095 will not change the A or P of previously created numbers (i.e., A and P of
4096 $x will be what was in effect when $x was created)
4097 * If given undef for A and P, B<no> rounding will occur, and the globals will
4098 B<not> be used. This is used by subclasses to create numbers without
4099 suffering rounding in the parent. Thus a subclass is able to have its own
4100 globals enforced upon creation of a number by using
4101 C<< $x = Math::BigInt->new($number,undef,undef) >>:
4103 use Math::BigInt::SomeSubclass;
4106 Math::BigInt->accuracy(2);
4107 Math::BigInt::SomeSubClass->accuracy(3);
4108 $x = Math::BigInt::SomeSubClass->new(1234);
4110 $x is now 1230, and not 1200. A subclass might choose to implement
4111 this otherwise, e.g. falling back to the parent's A and P.
4115 * If A or P are enabled/defined, they are used to round the result of each
4116 operation according to the rules below
4117 * Negative P is ignored in Math::BigInt, since BigInts never have digits
4118 after the decimal point
4119 * Math::BigFloat uses Math::BigInt internally, but setting A or P inside
4120 Math::BigInt as globals does not tamper with the parts of a BigFloat.
4121 A flag is used to mark all Math::BigFloat numbers as 'never round'.
4125 * It only makes sense that a number has only one of A or P at a time.
4126 If you set either A or P on one object, or globally, the other one will
4127 be automatically cleared.
4128 * If two objects are involved in an operation, and one of them has A in
4129 effect, and the other P, this results in an error (NaN).
4130 * A takes precedence over P (Hint: A comes before P).
4131 If neither of them is defined, nothing is used, i.e. the result will have
4132 as many digits as it can (with an exception for fdiv/fsqrt) and will not
4134 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
4135 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
4136 If either the dividend's or the divisor's mantissa has more digits than
4137 the value of F, the higher value will be used instead of F.
4138 This is to limit the digits (A) of the result (just consider what would
4139 happen with unlimited A and P in the case of 1/3 :-)
4140 * fdiv will calculate (at least) 4 more digits than required (determined by
4141 A, P or F), and, if F is not used, round the result
4142 (this will still fail in the case of a result like 0.12345000000001 with A
4143 or P of 5, but this can not be helped - or can it?)
4144 * Thus you can have the math done by on Math::Big* class in two modi:
4145 + never round (this is the default):
4146 This is done by setting A and P to undef. No math operation
4147 will round the result, with fdiv() and fsqrt() as exceptions to guard
4148 against overflows. You must explicitly call bround(), bfround() or
4149 round() (the latter with parameters).
4150 Note: Once you have rounded a number, the settings will 'stick' on it
4151 and 'infect' all other numbers engaged in math operations with it, since
4152 local settings have the highest precedence. So, to get SaferRound[tm],
4153 use a copy() before rounding like this:
4155 $x = Math::BigFloat->new(12.34);
4156 $y = Math::BigFloat->new(98.76);
4157 $z = $x * $y; # 1218.6984
4158 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
4159 $z = $x * $y; # still 1218.6984, without
4160 # copy would have been 1210!
4162 + round after each op:
4163 After each single operation (except for testing like is_zero()), the
4164 method round() is called and the result is rounded appropriately. By
4165 setting proper values for A and P, you can have all-the-same-A or
4166 all-the-same-P modes. For example, Math::Currency might set A to undef,
4167 and P to -2, globally.
4169 ?Maybe an extra option that forbids local A & P settings would be in order,
4170 ?so that intermediate rounding does not 'poison' further math?
4172 =item Overriding globals
4174 * you will be able to give A, P and R as an argument to all the calculation
4175 routines; the second parameter is A, the third one is P, and the fourth is
4176 R (shift right by one for binary operations like badd). P is used only if
4177 the first parameter (A) is undefined. These three parameters override the
4178 globals in the order detailed as follows, i.e. the first defined value
4180 (local: per object, global: global default, parameter: argument to sub)
4183 + local A (if defined on both of the operands: smaller one is taken)
4184 + local P (if defined on both of the operands: bigger one is taken)
4188 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
4189 arguments (A and P) instead of one
4191 =item Local settings
4193 * You can set A or P locally by using C<< $x->accuracy() >> or
4194 C<< $x->precision() >>
4195 and thus force different A and P for different objects/numbers.
4196 * Setting A or P this way immediately rounds $x to the new value.
4197 * C<< $x->accuracy() >> clears C<< $x->precision() >>, and vice versa.
4201 * the rounding routines will use the respective global or local settings.
4202 fround()/bround() is for accuracy rounding, while ffround()/bfround()
4204 * the two rounding functions take as the second parameter one of the
4205 following rounding modes (R):
4206 'even', 'odd', '+inf', '-inf', 'zero', 'trunc', 'common'
4207 * you can set/get the global R by using C<< Math::SomeClass->round_mode() >>
4208 or by setting C<< $Math::SomeClass::round_mode >>
4209 * after each operation, C<< $result->round() >> is called, and the result may
4210 eventually be rounded (that is, if A or P were set either locally,
4211 globally or as parameter to the operation)
4212 * to manually round a number, call C<< $x->round($A,$P,$round_mode); >>
4213 this will round the number by using the appropriate rounding function
4214 and then normalize it.
4215 * rounding modifies the local settings of the number:
4217 $x = Math::BigFloat->new(123.456);
4221 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
4222 will be 4 from now on.
4224 =item Default values
4233 * The defaults are set up so that the new code gives the same results as
4234 the old code (except in a few cases on fdiv):
4235 + Both A and P are undefined and thus will not be used for rounding
4236 after each operation.
4237 + round() is thus a no-op, unless given extra parameters A and P
4241 =head1 Infinity and Not a Number
4243 While BigInt has extensive handling of inf and NaN, certain quirks remain.
4249 These perl routines currently (as of Perl v.5.8.6) cannot handle passed
4252 te@linux:~> perl -wle 'print 2 ** 3333'
4254 te@linux:~> perl -wle 'print 2 ** 3333 == 2 ** 3333'
4256 te@linux:~> perl -wle 'print oct(2 ** 3333)'
4258 te@linux:~> perl -wle 'print hex(2 ** 3333)'
4259 Illegal hexadecimal digit 'i' ignored at -e line 1.
4262 The same problems occur if you pass them Math::BigInt->binf() objects. Since
4263 overloading these routines is not possible, this cannot be fixed from BigInt.
4265 =item ==, !=, <, >, <=, >= with NaNs
4267 BigInt's bcmp() routine currently returns undef to signal that a NaN was
4268 involved in a comparison. However, the overload code turns that into
4269 either 1 or '' and thus operations like C<< NaN != NaN >> might return
4274 C<< log(-inf) >> is highly weird. Since log(-x)=pi*i+log(x), then
4275 log(-inf)=pi*i+inf. However, since the imaginary part is finite, the real
4276 infinity "overshadows" it, so the number might as well just be infinity.
4277 However, the result is a complex number, and since BigInt/BigFloat can only
4278 have real numbers as results, the result is NaN.
4280 =item exp(), cos(), sin(), atan2()
4282 These all might have problems handling infinity right.
4288 The actual numbers are stored as unsigned big integers (with seperate sign).
4290 You should neither care about nor depend on the internal representation; it
4291 might change without notice. Use B<ONLY> method calls like C<< $x->sign(); >>
4292 instead relying on the internal representation.
4296 Math with the numbers is done (by default) by a module called
4297 C<Math::BigInt::Calc>. This is equivalent to saying:
4299 use Math::BigInt lib => 'Calc';
4301 You can change this by using:
4303 use Math::BigInt lib => 'BitVect';
4305 The following would first try to find Math::BigInt::Foo, then
4306 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
4308 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
4310 Since Math::BigInt::GMP is in almost all cases faster than Calc (especially in
4311 math involving really big numbers, where it is B<much> faster), and there is
4312 no penalty if Math::BigInt::GMP is not installed, it is a good idea to always
4315 use Math::BigInt lib => 'GMP';
4317 Different low-level libraries use different formats to store the
4318 numbers. You should B<NOT> depend on the number having a specific format
4321 See the respective math library module documentation for further details.
4325 The sign is either '+', '-', 'NaN', '+inf' or '-inf'.
4327 A sign of 'NaN' is used to represent the result when input arguments are not
4328 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
4329 minus infinity. You will get '+inf' when dividing a positive number by 0, and
4330 '-inf' when dividing any negative number by 0.
4332 =head2 mantissa(), exponent() and parts()
4334 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
4337 $m = $x->mantissa();
4338 $e = $x->exponent();
4339 $y = $m * ( 10 ** $e );
4340 print "ok\n" if $x == $y;
4342 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
4343 in one go. Both the returned mantissa and exponent have a sign.
4345 Currently, for BigInts C<$e> is always 0, except +inf and -inf, where it is
4346 C<+inf>; and for NaN, where it is C<NaN>; and for C<$x == 0>, where it is C<1>
4347 (to be compatible with Math::BigFloat's internal representation of a zero as
4350 C<$m> is currently just a copy of the original number. The relation between
4351 C<$e> and C<$m> will stay always the same, though their real values might
4358 sub bint { Math::BigInt->new(shift); }
4360 $x = Math::BigInt->bstr("1234") # string "1234"
4361 $x = "$x"; # same as bstr()
4362 $x = Math::BigInt->bneg("1234"); # BigInt "-1234"
4363 $x = Math::BigInt->babs("-12345"); # BigInt "12345"
4364 $x = Math::BigInt->bnorm("-0.00"); # BigInt "0"
4365 $x = bint(1) + bint(2); # BigInt "3"
4366 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
4367 $x = bint(1); # BigInt "1"
4368 $x = $x + 5 / 2; # BigInt "3"
4369 $x = $x ** 3; # BigInt "27"
4370 $x *= 2; # BigInt "54"
4371 $x = Math::BigInt->new(0); # BigInt "0"
4373 $x = Math::BigInt->badd(4,5) # BigInt "9"
4374 print $x->bsstr(); # 9e+0
4376 Examples for rounding:
4381 $x = Math::BigFloat->new(123.4567);
4382 $y = Math::BigFloat->new(123.456789);
4383 Math::BigFloat->accuracy(4); # no more A than 4
4385 ok ($x->copy()->fround(),123.4); # even rounding
4386 print $x->copy()->fround(),"\n"; # 123.4
4387 Math::BigFloat->round_mode('odd'); # round to odd
4388 print $x->copy()->fround(),"\n"; # 123.5
4389 Math::BigFloat->accuracy(5); # no more A than 5
4390 Math::BigFloat->round_mode('odd'); # round to odd
4391 print $x->copy()->fround(),"\n"; # 123.46
4392 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
4393 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
4395 Math::BigFloat->accuracy(undef); # A not important now
4396 Math::BigFloat->precision(2); # P important
4397 print $x->copy()->bnorm(),"\n"; # 123.46
4398 print $x->copy()->fround(),"\n"; # 123.46
4400 Examples for converting:
4402 my $x = Math::BigInt->new('0b1'.'01' x 123);
4403 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
4405 =head1 Autocreating constants
4407 After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
4408 and binary constants in the given scope are converted to C<Math::BigInt>.
4409 This conversion happens at compile time.
4413 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
4415 prints the integer value of C<2**100>. Note that without conversion of
4416 constants the expression 2**100 will be calculated as perl scalar.
4418 Please note that strings and floating point constants are not affected,
4421 use Math::BigInt qw/:constant/;
4423 $x = 1234567890123456789012345678901234567890
4424 + 123456789123456789;
4425 $y = '1234567890123456789012345678901234567890'
4426 + '123456789123456789';
4428 do not work. You need an explicit Math::BigInt->new() around one of the
4429 operands. You should also quote large constants to protect loss of precision:
4433 $x = Math::BigInt->new('1234567889123456789123456789123456789');
4435 Without the quotes Perl would convert the large number to a floating point
4436 constant at compile time and then hand the result to BigInt, which results in
4437 an truncated result or a NaN.
4439 This also applies to integers that look like floating point constants:
4441 use Math::BigInt ':constant';
4443 print ref(123e2),"\n";
4444 print ref(123.2e2),"\n";
4446 will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
4447 to get this to work.
4451 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
4452 must be made in the second case. For long numbers, the copy can eat up to 20%
4453 of the work (in the case of addition/subtraction, less for
4454 multiplication/division). If $y is very small compared to $x, the form
4455 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
4456 more time then the actual addition.
4458 With a technique called copy-on-write, the cost of copying with overload could
4459 be minimized or even completely avoided. A test implementation of COW did show
4460 performance gains for overloaded math, but introduced a performance loss due
4461 to a constant overhead for all other operations. So Math::BigInt does currently
4464 The rewritten version of this module (vs. v0.01) is slower on certain
4465 operations, like C<new()>, C<bstr()> and C<numify()>. The reason are that it
4466 does now more work and handles much more cases. The time spent in these
4467 operations is usually gained in the other math operations so that code on
4468 the average should get (much) faster. If they don't, please contact the author.
4470 Some operations may be slower for small numbers, but are significantly faster
4471 for big numbers. Other operations are now constant (O(1), like C<bneg()>,
4472 C<babs()> etc), instead of O(N) and thus nearly always take much less time.
4473 These optimizations were done on purpose.
4475 If you find the Calc module to slow, try to install any of the replacement
4476 modules and see if they help you.
4478 =head2 Alternative math libraries
4480 You can use an alternative library to drive Math::BigInt via:
4482 use Math::BigInt lib => 'Module';
4484 See L<MATH LIBRARY> for more information.
4486 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
4490 =head1 Subclassing Math::BigInt
4492 The basic design of Math::BigInt allows simple subclasses with very little
4493 work, as long as a few simple rules are followed:
4499 The public API must remain consistent, i.e. if a sub-class is overloading
4500 addition, the sub-class must use the same name, in this case badd(). The
4501 reason for this is that Math::BigInt is optimized to call the object methods
4506 The private object hash keys like C<$x->{sign}> may not be changed, but
4507 additional keys can be added, like C<$x->{_custom}>.
4511 Accessor functions are available for all existing object hash keys and should
4512 be used instead of directly accessing the internal hash keys. The reason for
4513 this is that Math::BigInt itself has a pluggable interface which permits it
4514 to support different storage methods.
4518 More complex sub-classes may have to replicate more of the logic internal of
4519 Math::BigInt if they need to change more basic behaviors. A subclass that
4520 needs to merely change the output only needs to overload C<bstr()>.
4522 All other object methods and overloaded functions can be directly inherited
4523 from the parent class.
4525 At the very minimum, any subclass will need to provide its own C<new()> and can
4526 store additional hash keys in the object. There are also some package globals
4527 that must be defined, e.g.:
4531 $precision = -2; # round to 2 decimal places
4532 $round_mode = 'even';
4535 Additionally, you might want to provide the following two globals to allow
4536 auto-upgrading and auto-downgrading to work correctly:
4541 This allows Math::BigInt to correctly retrieve package globals from the
4542 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
4543 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
4549 in your subclass to automatically inherit the overloading from the parent. If
4550 you like, you can change part of the overloading, look at Math::String for an
4555 When used like this:
4557 use Math::BigInt upgrade => 'Foo::Bar';
4559 certain operations will 'upgrade' their calculation and thus the result to
4560 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
4562 use Math::BigInt upgrade => 'Math::BigFloat';
4564 As a shortcut, you can use the module C<bignum>:
4568 Also good for oneliners:
4570 perl -Mbignum -le 'print 2 ** 255'
4572 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
4573 as well es preserve accuracy (as in sqrt(3)).
4575 Beware: This feature is not fully implemented yet.
4579 The following methods upgrade themselves unconditionally; that is if upgrade
4580 is in effect, they will always hand up their work:
4594 Beware: This list is not complete.
4596 All other methods upgrade themselves only when one (or all) of their
4597 arguments are of the class mentioned in $upgrade (This might change in later
4598 versions to a more sophisticated scheme):
4602 C<Math::BigInt> exports nothing by default, but can export the following methods:
4611 =item broot() does not work
4613 The broot() function in BigInt may only work for small values. This will be
4614 fixed in a later version.
4616 =item Out of Memory!
4618 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
4619 C<eval()> in your code will crash with "Out of memory". This is probably an
4620 overload/exporter bug. You can workaround by not having C<eval()>
4621 and ':constant' at the same time or upgrade your Perl to a newer version.
4623 =item Fails to load Calc on Perl prior 5.6.0
4625 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
4626 will fall back to eval { require ... } when loading the math lib on Perls
4627 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
4628 filesystems using a different seperator.
4634 Some things might not work as you expect them. Below is documented what is
4635 known to be troublesome:
4639 =item bstr(), bsstr() and 'cmp'
4641 Both C<bstr()> and C<bsstr()> as well as automated stringify via overload now
4642 drop the leading '+'. The old code would return '+3', the new returns '3'.
4643 This is to be consistent with Perl and to make C<cmp> (especially with
4644 overloading) to work as you expect. It also solves problems with C<Test.pm>,
4645 because its C<ok()> uses 'eq' internally.
4647 Mark Biggar said, when asked about to drop the '+' altogether, or make only
4650 I agree (with the first alternative), don't add the '+' on positive
4651 numbers. It's not as important anymore with the new internal
4652 form for numbers. It made doing things like abs and neg easier,
4653 but those have to be done differently now anyway.
4655 So, the following examples will now work all as expected:
4658 BEGIN { plan tests => 1 }
4661 my $x = new Math::BigInt 3*3;
4662 my $y = new Math::BigInt 3*3;
4665 print "$x eq 9" if $x eq $y;
4666 print "$x eq 9" if $x eq '9';
4667 print "$x eq 9" if $x eq 3*3;
4669 Additionally, the following still works:
4671 print "$x == 9" if $x == $y;
4672 print "$x == 9" if $x == 9;
4673 print "$x == 9" if $x == 3*3;
4675 There is now a C<bsstr()> method to get the string in scientific notation aka
4676 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
4677 for comparison, but Perl will represent some numbers as 100 and others
4678 as 1e+308. If in doubt, convert both arguments to Math::BigInt before
4679 comparing them as strings:
4682 BEGIN { plan tests => 3 }
4685 $x = Math::BigInt->new('1e56'); $y = 1e56;
4686 ok ($x,$y); # will fail
4687 ok ($x->bsstr(),$y); # okay
4688 $y = Math::BigInt->new($y);
4691 Alternatively, simple use C<< <=> >> for comparisons, this will get it
4692 always right. There is not yet a way to get a number automatically represented
4693 as a string that matches exactly the way Perl represents it.
4695 See also the section about L<Infinity and Not a Number> for problems in
4700 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
4703 $x = Math::BigInt->new(123);
4704 $y = int($x); # BigInt 123
4705 $x = Math::BigFloat->new(123.45);
4706 $y = int($x); # BigInt 123
4708 In all Perl versions you can use C<as_number()> or C<as_int> for the same
4711 $x = Math::BigFloat->new(123.45);
4712 $y = $x->as_number(); # BigInt 123
4713 $y = $x->as_int(); # ditto
4715 This also works for other subclasses, like Math::String.
4717 If you want a real Perl scalar, use C<numify()>:
4719 $y = $x->numify(); # 123 as scalar
4721 This is seldom necessary, though, because this is done automatically, like
4722 when you access an array:
4724 $z = $array[$x]; # does work automatically
4728 The following will probably not do what you expect:
4730 $c = Math::BigInt->new(123);
4731 print $c->length(),"\n"; # prints 30
4733 It prints both the number of digits in the number and in the fraction part
4734 since print calls C<length()> in list context. Use something like:
4736 print scalar $c->length(),"\n"; # prints 3
4740 The following will probably not do what you expect:
4742 print $c->bdiv(10000),"\n";
4744 It prints both quotient and remainder since print calls C<bdiv()> in list
4745 context. Also, C<bdiv()> will modify $c, so be careful. You probably want
4748 print $c / 10000,"\n";
4749 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
4753 The quotient is always the greatest integer less than or equal to the
4754 real-valued quotient of the two operands, and the remainder (when it is
4755 nonzero) always has the same sign as the second operand; so, for
4765 As a consequence, the behavior of the operator % agrees with the
4766 behavior of Perl's built-in % operator (as documented in the perlop
4767 manpage), and the equation
4769 $x == ($x / $y) * $y + ($x % $y)
4771 holds true for any $x and $y, which justifies calling the two return
4772 values of bdiv() the quotient and remainder. The only exception to this rule
4773 are when $y == 0 and $x is negative, then the remainder will also be
4774 negative. See below under "infinity handling" for the reasoning behind this.
4776 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
4777 not change BigInt's way to do things. This is because under 'use integer' Perl
4778 will do what the underlying C thinks is right and this is different for each
4779 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
4780 the author to implement it ;)
4782 =item infinity handling
4784 Here are some examples that explain the reasons why certain results occur while
4787 The following table shows the result of the division and the remainder, so that
4788 the equation above holds true. Some "ordinary" cases are strewn in to show more
4789 clearly the reasoning:
4791 A / B = C, R so that C * B + R = A
4792 =========================================================
4793 5 / 8 = 0, 5 0 * 8 + 5 = 5
4794 0 / 8 = 0, 0 0 * 8 + 0 = 0
4795 0 / inf = 0, 0 0 * inf + 0 = 0
4796 0 /-inf = 0, 0 0 * -inf + 0 = 0
4797 5 / inf = 0, 5 0 * inf + 5 = 5
4798 5 /-inf = 0, 5 0 * -inf + 5 = 5
4799 -5/ inf = 0, -5 0 * inf + -5 = -5
4800 -5/-inf = 0, -5 0 * -inf + -5 = -5
4801 inf/ 5 = inf, 0 inf * 5 + 0 = inf
4802 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
4803 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
4804 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
4805 5/ 5 = 1, 0 1 * 5 + 0 = 5
4806 -5/ -5 = 1, 0 1 * -5 + 0 = -5
4807 inf/ inf = 1, 0 1 * inf + 0 = inf
4808 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
4809 inf/-inf = -1, 0 -1 * -inf + 0 = inf
4810 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
4811 8/ 0 = inf, 8 inf * 0 + 8 = 8
4812 inf/ 0 = inf, inf inf * 0 + inf = inf
4815 These cases below violate the "remainder has the sign of the second of the two
4816 arguments", since they wouldn't match up otherwise.
4818 A / B = C, R so that C * B + R = A
4819 ========================================================
4820 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
4821 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
4823 =item Modifying and =
4827 $x = Math::BigFloat->new(5);
4830 It will not do what you think, e.g. making a copy of $x. Instead it just makes
4831 a second reference to the B<same> object and stores it in $y. Thus anything
4832 that modifies $x (except overloaded operators) will modify $y, and vice versa.
4833 Or in other words, C<=> is only safe if you modify your BigInts only via
4834 overloaded math. As soon as you use a method call it breaks:
4837 print "$x, $y\n"; # prints '10, 10'
4839 If you want a true copy of $x, use:
4843 You can also chain the calls like this, this will make first a copy and then
4846 $y = $x->copy()->bmul(2);
4848 See also the documentation for overload.pm regarding C<=>.
4852 C<bpow()> (and the rounding functions) now modifies the first argument and
4853 returns it, unlike the old code which left it alone and only returned the
4854 result. This is to be consistent with C<badd()> etc. The first three will
4855 modify $x, the last one won't:
4857 print bpow($x,$i),"\n"; # modify $x
4858 print $x->bpow($i),"\n"; # ditto
4859 print $x **= $i,"\n"; # the same
4860 print $x ** $i,"\n"; # leave $x alone
4862 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4864 =item Overloading -$x
4874 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4875 needs to preserve $x since it does not know that it later will get overwritten.
4876 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4878 =item Mixing different object types
4880 In Perl you will get a floating point value if you do one of the following:
4886 With overloaded math, only the first two variants will result in a BigFloat:
4891 $mbf = Math::BigFloat->new(5);
4892 $mbi2 = Math::BigInteger->new(5);
4893 $mbi = Math::BigInteger->new(2);
4895 # what actually gets called:
4896 $float = $mbf + $mbi; # $mbf->badd()
4897 $float = $mbf / $mbi; # $mbf->bdiv()
4898 $integer = $mbi + $mbf; # $mbi->badd()
4899 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4900 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4902 This is because math with overloaded operators follows the first (dominating)
4903 operand, and the operation of that is called and returns thus the result. So,
4904 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4905 the result should be a Math::BigFloat or the second operant is one.
4907 To get a Math::BigFloat you either need to call the operation manually,
4908 make sure the operands are already of the proper type or casted to that type
4909 via Math::BigFloat->new():
4911 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4913 Beware of simple "casting" the entire expression, this would only convert
4914 the already computed result:
4916 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4918 Beware also of the order of more complicated expressions like:
4920 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4921 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4923 If in doubt, break the expression into simpler terms, or cast all operands
4924 to the desired resulting type.
4926 Scalar values are a bit different, since:
4931 will both result in the proper type due to the way the overloaded math works.
4933 This section also applies to other overloaded math packages, like Math::String.
4935 One solution to you problem might be autoupgrading|upgrading. See the
4936 pragmas L<bignum>, L<bigint> and L<bigrat> for an easy way to do this.
4940 C<bsqrt()> works only good if the result is a big integer, e.g. the square
4941 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4942 mode. The reason is that the result is always truncated to an integer.
4944 If you want a better approximation of the square root, then use:
4946 $x = Math::BigFloat->new(12);
4947 Math::BigFloat->precision(0);
4948 Math::BigFloat->round_mode('even');
4949 print $x->copy->bsqrt(),"\n"; # 4
4951 Math::BigFloat->precision(2);
4952 print $x->bsqrt(),"\n"; # 3.46
4953 print $x->bsqrt(3),"\n"; # 3.464
4957 For negative numbers in base see also L<brsft|brsft>.
4963 This program is free software; you may redistribute it and/or modify it under
4964 the same terms as Perl itself.
4968 L<Math::BigFloat>, L<Math::BigRat> and L<Math::Big> as well as
4969 L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4971 The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest
4972 because they solve the autoupgrading/downgrading issue, at least partly.
4975 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4976 more documentation including a full version history, testcases, empty
4977 subclass files and benchmarks.
4981 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4982 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001 - 2006
4983 and still at it in 2007.
4985 Many people contributed in one or more ways to the final beast, see the file
4986 CREDITS for an (incomplete) list. If you miss your name, please drop me a