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 { $_[2] ?
87 ref($_[0])->new($_[1])->batan2($_[0]) :
88 $_[0]->copy()->batan2($_[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');
299 Carp::croak ('Argument to accuracy must be an integer');
304 # $object->accuracy() or fallback to global
305 $x->bround($a) if $a; # not for undef, 0
306 $x->{_a} = $a; # set/overwrite, even if not rounded
307 delete $x->{_p}; # clear P
308 $a = ${"${class}::accuracy"} unless defined $a; # proper return value
312 ${"${class}::accuracy"} = $a; # set global A
313 ${"${class}::precision"} = undef; # clear global P
315 return $a; # shortcut
319 # $object->accuracy() or fallback to global
320 $a = $x->{_a} if ref($x);
321 # but don't return global undef, when $x's accuracy is 0!
322 $a = ${"${class}::accuracy"} if !defined $a;
328 # $x->precision($p); ref($x) $p
329 # $x->precision(); ref($x)
330 # Class->precision(); class
331 # Class->precision($p); class $p
334 my $class = ref($x) || $x || __PACKAGE__;
340 # convert objects to scalars to avoid deep recursion. If object doesn't
341 # have numify(), then hopefully it will have overloading for int() and
342 # boolean test without wandering into a deep recursion path...
343 $p = $p->numify() if ref($p) && $p->can('numify');
344 if ((defined $p) && (int($p) != $p))
346 require Carp; Carp::croak ('Argument to precision must be an integer');
350 # $object->precision() or fallback to global
351 $x->bfround($p) if $p; # not for undef, 0
352 $x->{_p} = $p; # set/overwrite, even if not rounded
353 delete $x->{_a}; # clear A
354 $p = ${"${class}::precision"} unless defined $p; # proper return value
358 ${"${class}::precision"} = $p; # set global P
359 ${"${class}::accuracy"} = undef; # clear global A
361 return $p; # shortcut
365 # $object->precision() or fallback to global
366 $p = $x->{_p} if ref($x);
367 # but don't return global undef, when $x's precision is 0!
368 $p = ${"${class}::precision"} if !defined $p;
374 # return (or set) configuration data as hash ref
375 my $class = shift || 'Math::BigInt';
378 if (@_ > 1 || (@_ == 1 && (ref($_[0]) eq 'HASH')))
380 # try to set given options as arguments from hash
383 if (ref($args) ne 'HASH')
387 # these values can be "set"
391 upgrade downgrade precision accuracy round_mode div_scale/
394 $set_args->{$key} = $args->{$key} if exists $args->{$key};
395 delete $args->{$key};
400 Carp::croak ("Illegal key(s) '",
401 join("','",keys %$args),"' passed to $class\->config()");
403 foreach my $key (keys %$set_args)
405 if ($key =~ /^trap_(inf|nan)\z/)
407 ${"${class}::_trap_$1"} = ($set_args->{"trap_$1"} ? 1 : 0);
410 # use a call instead of just setting the $variable to check argument
411 $class->$key($set_args->{$key});
415 # now return actual configuration
419 lib_version => ${"${CALC}::VERSION"},
421 trap_nan => ${"${class}::_trap_nan"},
422 trap_inf => ${"${class}::_trap_inf"},
423 version => ${"${class}::VERSION"},
426 upgrade downgrade precision accuracy round_mode div_scale
429 $cfg->{$key} = ${"${class}::$key"};
431 if (@_ == 1 && (ref($_[0]) ne 'HASH'))
433 # calls of the style config('lib') return just this value
434 return $cfg->{$_[0]};
441 # select accuracy parameter based on precedence,
442 # used by bround() and bfround(), may return undef for scale (means no op)
443 my ($x,$scale,$mode) = @_;
445 $scale = $x->{_a} unless defined $scale;
450 $scale = ${ $class . '::accuracy' } unless defined $scale;
451 $mode = ${ $class . '::round_mode' } unless defined $mode;
455 $scale = $scale->can('numify') ? $scale->numify() : "$scale" if ref($scale);
456 $scale = int($scale);
464 # select precision parameter based on precedence,
465 # used by bround() and bfround(), may return undef for scale (means no op)
466 my ($x,$scale,$mode) = @_;
468 $scale = $x->{_p} unless defined $scale;
473 $scale = ${ $class . '::precision' } unless defined $scale;
474 $mode = ${ $class . '::round_mode' } unless defined $mode;
478 $scale = $scale->can('numify') ? $scale->numify() : "$scale" if ref($scale);
479 $scale = int($scale);
485 ##############################################################################
490 # if two arguments, the first one is the class to "swallow" subclasses
494 sign => $_[1]->{sign},
495 value => $CALC->_copy($_[1]->{value}),
498 $self->{_a} = $_[1]->{_a} if defined $_[1]->{_a};
499 $self->{_p} = $_[1]->{_p} if defined $_[1]->{_p};
504 sign => $_[0]->{sign},
505 value => $CALC->_copy($_[0]->{value}),
508 $self->{_a} = $_[0]->{_a} if defined $_[0]->{_a};
509 $self->{_p} = $_[0]->{_p} if defined $_[0]->{_p};
515 # create a new BigInt object from a string or another BigInt object.
516 # see hash keys documented at top
518 # the argument could be an object, so avoid ||, && etc on it, this would
519 # cause costly overloaded code to be called. The only allowed ops are
522 my ($class,$wanted,$a,$p,$r) = @_;
524 # avoid numify-calls by not using || on $wanted!
525 return $class->bzero($a,$p) if !defined $wanted; # default to 0
526 return $class->copy($wanted,$a,$p,$r)
527 if ref($wanted) && $wanted->isa($class); # MBI or subclass
529 $class->import() if $IMPORT == 0; # make require work
531 my $self = bless {}, $class;
533 # shortcut for "normal" numbers
534 if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*\z/))
536 $self->{sign} = $1 || '+';
538 if ($wanted =~ /^[+-]/)
540 # remove sign without touching wanted to make it work with constants
541 my $t = $wanted; $t =~ s/^[+-]//;
542 $self->{value} = $CALC->_new($t);
546 $self->{value} = $CALC->_new($wanted);
549 if ( (defined $a) || (defined $p)
550 || (defined ${"${class}::precision"})
551 || (defined ${"${class}::accuracy"})
554 $self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p);
559 # handle '+inf', '-inf' first
560 if ($wanted =~ /^[+-]?inf\z/)
562 $self->{sign} = $wanted; # set a default sign for bstr()
563 return $self->binf($wanted);
565 # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign
566 my ($mis,$miv,$mfv,$es,$ev) = _split($wanted);
571 require Carp; Carp::croak("$wanted is not a number in $class");
573 $self->{value} = $CALC->_zero();
574 $self->{sign} = $nan;
579 # _from_hex or _from_bin
580 $self->{value} = $mis->{value};
581 $self->{sign} = $mis->{sign};
582 return $self; # throw away $mis
584 # make integer from mantissa by adjusting exp, then convert to bigint
585 $self->{sign} = $$mis; # store sign
586 $self->{value} = $CALC->_zero(); # for all the NaN cases
587 my $e = int("$$es$$ev"); # exponent (avoid recursion)
590 my $diff = $e - CORE::length($$mfv);
591 if ($diff < 0) # Not integer
595 require Carp; Carp::croak("$wanted not an integer in $class");
598 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
599 $self->{sign} = $nan;
603 # adjust fraction and add it to value
604 #print "diff > 0 $$miv\n";
605 $$miv = $$miv . ($$mfv . '0' x $diff);
610 if ($$mfv ne '') # e <= 0
612 # fraction and negative/zero E => NOI
615 require Carp; Carp::croak("$wanted not an integer in $class");
617 #print "NOI 2 \$\$mfv '$$mfv'\n";
618 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
619 $self->{sign} = $nan;
623 # xE-y, and empty mfv
626 if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
630 require Carp; Carp::croak("$wanted not an integer in $class");
633 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
634 $self->{sign} = $nan;
638 $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
639 $self->{value} = $CALC->_new($$miv) if $self->{sign} =~ /^[+-]$/;
640 # if any of the globals is set, use them to round and store them inside $self
641 # do not round for new($x,undef,undef) since that is used by MBF to signal
643 $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p;
649 # create a bigint 'NaN', if given a BigInt, set it to 'NaN'
651 $self = $class if !defined $self;
654 my $c = $self; $self = {}; bless $self, $c;
657 if (${"${class}::_trap_nan"})
660 Carp::croak ("Tried to set $self to NaN in $class\::bnan()");
662 $self->import() if $IMPORT == 0; # make require work
663 return if $self->modify('bnan');
664 if ($self->can('_bnan'))
666 # use subclass to initialize
671 # otherwise do our own thing
672 $self->{value} = $CALC->_zero();
674 $self->{sign} = $nan;
675 delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly
681 # create a bigint '+-inf', if given a BigInt, set it to '+-inf'
682 # the sign is either '+', or if given, used from there
684 my $sign = shift; $sign = '+' if !defined $sign || $sign !~ /^-(inf)?$/;
685 $self = $class if !defined $self;
688 my $c = $self; $self = {}; bless $self, $c;
691 if (${"${class}::_trap_inf"})
694 Carp::croak ("Tried to set $self to +-inf in $class\::binf()");
696 $self->import() if $IMPORT == 0; # make require work
697 return if $self->modify('binf');
698 if ($self->can('_binf'))
700 # use subclass to initialize
705 # otherwise do our own thing
706 $self->{value} = $CALC->_zero();
708 $sign = $sign . 'inf' if $sign !~ /inf$/; # - => -inf
709 $self->{sign} = $sign;
710 ($self->{_a},$self->{_p}) = @_; # take over requested rounding
716 # create a bigint '+0', if given a BigInt, set it to 0
718 $self = __PACKAGE__ if !defined $self;
722 my $c = $self; $self = {}; bless $self, $c;
724 $self->import() if $IMPORT == 0; # make require work
725 return if $self->modify('bzero');
727 if ($self->can('_bzero'))
729 # use subclass to initialize
734 # otherwise do our own thing
735 $self->{value} = $CALC->_zero();
742 # call like: $x->bzero($a,$p,$r,$y);
743 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
748 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
750 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
758 # create a bigint '+1' (or -1 if given sign '-'),
759 # if given a BigInt, set it to +1 or -1, respectively
761 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
762 $self = $class if !defined $self;
766 my $c = $self; $self = {}; bless $self, $c;
768 $self->import() if $IMPORT == 0; # make require work
769 return if $self->modify('bone');
771 if ($self->can('_bone'))
773 # use subclass to initialize
778 # otherwise do our own thing
779 $self->{value} = $CALC->_one();
781 $self->{sign} = $sign;
786 # call like: $x->bone($sign,$a,$p,$r,$y);
787 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
791 # call like: $x->bone($sign,$a,$p,$r);
793 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
795 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
801 ##############################################################################
802 # string conversation
806 # (ref to BFLOAT or num_str ) return num_str
807 # Convert number from internal format to scientific string format.
808 # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
809 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
811 if ($x->{sign} !~ /^[+-]$/)
813 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
816 my ($m,$e) = $x->parts();
817 #$m->bstr() . 'e+' . $e->bstr(); # e can only be positive in BigInt
818 # 'e+' because E can only be positive in BigInt
819 $m->bstr() . 'e+' . $CALC->_str($e->{value});
824 # make a string from bigint object
825 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
827 if ($x->{sign} !~ /^[+-]$/)
829 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
832 my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
833 $es.$CALC->_str($x->{value});
838 # Make a "normal" scalar from a BigInt object
839 my $x = shift; $x = $class->new($x) unless ref $x;
841 return $x->bstr() if $x->{sign} !~ /^[+-]$/;
842 my $num = $CALC->_num($x->{value});
843 return -$num if $x->{sign} eq '-';
847 ##############################################################################
848 # public stuff (usually prefixed with "b")
852 # return the sign of the number: +/-/-inf/+inf/NaN
853 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
858 sub _find_round_parameters
860 # After any operation or when calling round(), the result is rounded by
861 # regarding the A & P from arguments, local parameters, or globals.
863 # !!!!!!! If you change this, remember to change round(), too! !!!!!!!!!!
865 # This procedure finds the round parameters, but it is for speed reasons
866 # duplicated in round. Otherwise, it is tested by the testsuite and used
869 # returns ($self) or ($self,$a,$p,$r) - sets $self to NaN of both A and P
870 # were requested/defined (locally or globally or both)
872 my ($self,$a,$p,$r,@args) = @_;
873 # $a accuracy, if given by caller
874 # $p precision, if given by caller
875 # $r round_mode, if given by caller
876 # @args all 'other' arguments (0 for unary, 1 for binary ops)
878 my $c = ref($self); # find out class of argument(s)
881 # convert to normal scalar for speed and correctness in inner parts
882 $a = $a->can('numify') ? $a->numify() : "$a" if defined $a && ref($a);
883 $p = $p->can('numify') ? $p->numify() : "$p" if defined $p && ref($p);
885 # now pick $a or $p, but only if we have got "arguments"
888 foreach ($self,@args)
890 # take the defined one, or if both defined, the one that is smaller
891 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
896 # even if $a is defined, take $p, to signal error for both defined
897 foreach ($self,@args)
899 # take the defined one, or if both defined, the one that is bigger
901 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
904 # if still none defined, use globals (#2)
905 $a = ${"$c\::accuracy"} unless defined $a;
906 $p = ${"$c\::precision"} unless defined $p;
908 # A == 0 is useless, so undef it to signal no rounding
909 $a = undef if defined $a && $a == 0;
912 return ($self) unless defined $a || defined $p; # early out
914 # set A and set P is an fatal error
915 return ($self->bnan()) if defined $a && defined $p; # error
917 $r = ${"$c\::round_mode"} unless defined $r;
918 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc|common)$/)
920 require Carp; Carp::croak ("Unknown round mode '$r'");
923 $a = int($a) if defined $a;
924 $p = int($p) if defined $p;
931 # Round $self according to given parameters, or given second argument's
932 # parameters or global defaults
934 # for speed reasons, _find_round_parameters is embeded here:
936 my ($self,$a,$p,$r,@args) = @_;
937 # $a accuracy, if given by caller
938 # $p precision, if given by caller
939 # $r round_mode, if given by caller
940 # @args all 'other' arguments (0 for unary, 1 for binary ops)
942 my $c = ref($self); # find out class of argument(s)
945 # now pick $a or $p, but only if we have got "arguments"
948 foreach ($self,@args)
950 # take the defined one, or if both defined, the one that is smaller
951 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
956 # even if $a is defined, take $p, to signal error for both defined
957 foreach ($self,@args)
959 # take the defined one, or if both defined, the one that is bigger
961 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
964 # if still none defined, use globals (#2)
965 $a = ${"$c\::accuracy"} unless defined $a;
966 $p = ${"$c\::precision"} unless defined $p;
968 # A == 0 is useless, so undef it to signal no rounding
969 $a = undef if defined $a && $a == 0;
972 return $self unless defined $a || defined $p; # early out
974 # set A and set P is an fatal error
975 return $self->bnan() if defined $a && defined $p;
977 $r = ${"$c\::round_mode"} unless defined $r;
978 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc|common)$/)
980 require Carp; Carp::croak ("Unknown round mode '$r'");
983 # now round, by calling either fround or ffround:
986 $self->bround(int($a),$r) if !defined $self->{_a} || $self->{_a} >= $a;
988 else # both can't be undefined due to early out
990 $self->bfround(int($p),$r) if !defined $self->{_p} || $self->{_p} <= $p;
992 # bround() or bfround() already callled bnorm() if nec.
998 # (numstr or BINT) return BINT
999 # Normalize number -- no-op here
1000 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1006 # (BINT or num_str) return BINT
1007 # make number absolute, or return absolute BINT from string
1008 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1010 return $x if $x->modify('babs');
1011 # post-normalized abs for internal use (does nothing for NaN)
1012 $x->{sign} =~ s/^-/+/;
1018 # (BINT or num_str) return BINT
1019 # negate number or make a negated number from string
1020 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1022 return $x if $x->modify('bneg');
1024 # for +0 dont negate (to have always normalized +0). Does nothing for 'NaN'
1025 $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $CALC->_is_zero($x->{value}));
1031 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
1032 # (BINT or num_str, BINT or num_str) return cond_code
1035 my ($self,$x,$y) = (ref($_[0]),@_);
1037 # objectify is costly, so avoid it
1038 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1040 ($self,$x,$y) = objectify(2,@_);
1043 return $upgrade->bcmp($x,$y) if defined $upgrade &&
1044 ((!$x->isa($self)) || (!$y->isa($self)));
1046 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1048 # handle +-inf and NaN
1049 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1050 return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
1051 return +1 if $x->{sign} eq '+inf';
1052 return -1 if $x->{sign} eq '-inf';
1053 return -1 if $y->{sign} eq '+inf';
1056 # check sign for speed first
1057 return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
1058 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
1060 # have same sign, so compare absolute values. Don't make tests for zero here
1061 # because it's actually slower than testin in Calc (especially w/ Pari et al)
1063 # post-normalized compare for internal use (honors signs)
1064 if ($x->{sign} eq '+')
1066 # $x and $y both > 0
1067 return $CALC->_acmp($x->{value},$y->{value});
1071 $CALC->_acmp($y->{value},$x->{value}); # swaped acmp (lib returns 0,1,-1)
1076 # Compares 2 values, ignoring their signs.
1077 # Returns one of undef, <0, =0, >0. (suitable for sort)
1078 # (BINT, BINT) return cond_code
1081 my ($self,$x,$y) = (ref($_[0]),@_);
1082 # objectify is costly, so avoid it
1083 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1085 ($self,$x,$y) = objectify(2,@_);
1088 return $upgrade->bacmp($x,$y) if defined $upgrade &&
1089 ((!$x->isa($self)) || (!$y->isa($self)));
1091 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1093 # handle +-inf and NaN
1094 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1095 return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
1096 return 1 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} !~ /^[+-]inf$/;
1099 $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
1104 # add second arg (BINT or string) to first (BINT) (modifies first)
1105 # return result as BINT
1108 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1109 # objectify is costly, so avoid it
1110 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1112 ($self,$x,$y,@r) = objectify(2,@_);
1115 return $x if $x->modify('badd');
1116 return $upgrade->badd($upgrade->new($x),$upgrade->new($y),@r) if defined $upgrade &&
1117 ((!$x->isa($self)) || (!$y->isa($self)));
1119 $r[3] = $y; # no push!
1120 # inf and NaN handling
1121 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1124 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1126 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1128 # +inf++inf or -inf+-inf => same, rest is NaN
1129 return $x if $x->{sign} eq $y->{sign};
1132 # +-inf + something => +inf
1133 # something +-inf => +-inf
1134 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
1138 my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
1142 $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
1146 my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
1149 $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
1154 # speedup, if equal, set result to 0
1155 $x->{value} = $CALC->_zero();
1160 $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
1168 # (BINT or num_str, BINT or num_str) return BINT
1169 # subtract second arg from first, modify first
1172 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1174 # objectify is costly, so avoid it
1175 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1177 ($self,$x,$y,@r) = objectify(2,@_);
1180 return $x if $x->modify('bsub');
1182 return $upgrade->new($x)->bsub($upgrade->new($y),@r) if defined $upgrade &&
1183 ((!$x->isa($self)) || (!$y->isa($self)));
1185 return $x->round(@r) if $y->is_zero();
1187 # To correctly handle the lone special case $x->bsub($x), we note the sign
1188 # of $x, then flip the sign from $y, and if the sign of $x did change, too,
1189 # then we caught the special case:
1190 my $xsign = $x->{sign};
1191 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
1192 if ($xsign ne $x->{sign})
1194 # special case of $x->bsub($x) results in 0
1195 return $x->bzero(@r) if $xsign =~ /^[+-]$/;
1196 return $x->bnan(); # NaN, -inf, +inf
1198 $x->badd($y,@r); # badd does not leave internal zeros
1199 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
1200 $x; # already rounded by badd() or no round nec.
1205 # increment arg by one
1206 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1207 return $x if $x->modify('binc');
1209 if ($x->{sign} eq '+')
1211 $x->{value} = $CALC->_inc($x->{value});
1212 return $x->round($a,$p,$r);
1214 elsif ($x->{sign} eq '-')
1216 $x->{value} = $CALC->_dec($x->{value});
1217 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1218 return $x->round($a,$p,$r);
1220 # inf, nan handling etc
1221 $x->badd($self->bone(),$a,$p,$r); # badd does round
1226 # decrement arg by one
1227 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1228 return $x if $x->modify('bdec');
1230 if ($x->{sign} eq '-')
1233 $x->{value} = $CALC->_inc($x->{value});
1237 return $x->badd($self->bone('-'),@r) unless $x->{sign} eq '+'; # inf or NaN
1239 if ($CALC->_is_zero($x->{value}))
1242 $x->{value} = $CALC->_one(); $x->{sign} = '-'; # 0 => -1
1247 $x->{value} = $CALC->_dec($x->{value});
1255 # calculate $x = $a ** $base + $b and return $a (e.g. the log() to base
1259 my ($self,$x,$base,@r) = (undef,@_);
1260 # objectify is costly, so avoid it
1261 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1263 ($self,$x,$base,@r) = objectify(1,ref($x),@_);
1266 return $x if $x->modify('blog');
1268 $base = $self->new($base) if defined $base && !ref $base;
1270 # inf, -inf, NaN, <0 => NaN
1272 if $x->{sign} ne '+' || (defined $base && $base->{sign} ne '+');
1274 return $upgrade->blog($upgrade->new($x),$base,@r) if
1277 # fix for bug #24969:
1278 # the default base is e (Euler's number) which is not an integer
1281 require Math::BigFloat;
1282 my $u = Math::BigFloat->blog(Math::BigFloat->new($x))->as_int();
1283 # modify $x in place
1284 $x->{value} = $u->{value};
1285 $x->{sign} = $u->{sign};
1289 my ($rc,$exact) = $CALC->_log_int($x->{value},$base->{value});
1290 return $x->bnan() unless defined $rc; # not possible to take log?
1297 # Calculate n over k (binomial coefficient or "choose" function) as integer.
1299 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1301 # objectify is costly, so avoid it
1302 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1304 ($self,$x,$y,@r) = objectify(2,@_);
1307 return $x if $x->modify('bnok');
1308 return $x->bnan() if $x->{sign} eq 'NaN' || $y->{sign} eq 'NaN';
1309 return $x->binf() if $x->{sign} eq '+inf';
1311 # k > n or k < 0 => 0
1312 my $cmp = $x->bacmp($y);
1313 return $x->bzero() if $cmp < 0 || $y->{sign} =~ /^-/;
1315 return $x->bone(@r) if $cmp == 0;
1317 if ($CALC->can('_nok'))
1319 $x->{value} = $CALC->_nok($x->{value},$y->{value});
1323 # ( 7 ) 7! 7*6*5 * 4*3*2*1 7 * 6 * 5
1324 # ( - ) = --------- = --------------- = ---------
1325 # ( 3 ) 3! (7-3)! 3*2*1 * 4*3*2*1 3 * 2 * 1
1327 # compute n - k + 2 (so we start with 5 in the example above)
1332 my $r = $z->copy(); $z->binc();
1333 my $d = $self->new(2);
1334 while ($z->bacmp($x) <= 0) # f < x ?
1336 $r->bmul($z); $r->bdiv($d);
1337 $z->binc(); $d->binc();
1339 $x->{value} = $r->{value}; $x->{sign} = '+';
1341 else { $x->bone(); }
1348 # Calculate e ** $x (Euler's number to the power of X), truncated to
1350 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1351 return $x if $x->modify('bexp');
1353 # inf, -inf, NaN, <0 => NaN
1354 return $x->bnan() if $x->{sign} eq 'NaN';
1355 return $x->bone() if $x->is_zero();
1356 return $x if $x->{sign} eq '+inf';
1357 return $x->bzero() if $x->{sign} eq '-inf';
1361 # run through Math::BigFloat unless told otherwise
1362 require Math::BigFloat unless defined $upgrade;
1363 local $upgrade = 'Math::BigFloat' unless defined $upgrade;
1364 # calculate result, truncate it to integer
1365 $u = $upgrade->bexp($upgrade->new($x),@r);
1368 if (!defined $upgrade)
1371 # modify $x in place
1372 $x->{value} = $u->{value};
1380 # (BINT or num_str, BINT or num_str) return BINT
1381 # does not modify arguments, but returns new object
1382 # Lowest Common Multiplicator
1384 my $y = shift; my ($x);
1391 $x = $class->new($y);
1396 my $y = shift; $y = $self->new($y) if !ref ($y);
1404 # (BINT or num_str, BINT or num_str) return BINT
1405 # does not modify arguments, but returns new object
1406 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1409 $y = $class->new($y) if !ref($y);
1411 my $x = $y->copy()->babs(); # keep arguments
1412 return $x->bnan() if $x->{sign} !~ /^[+-]$/; # x NaN?
1416 $y = shift; $y = $self->new($y) if !ref($y);
1417 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1418 $x->{value} = $CALC->_gcd($x->{value},$y->{value});
1419 last if $CALC->_is_one($x->{value});
1426 # (num_str or BINT) return BINT
1427 # represent ~x as twos-complement number
1428 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1429 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1431 return $x if $x->modify('bnot');
1432 $x->binc()->bneg(); # binc already does round
1435 ##############################################################################
1436 # is_foo test routines
1437 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1441 # return true if arg (BINT or num_str) is zero (array '+', '0')
1442 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1444 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1445 $CALC->_is_zero($x->{value});
1450 # return true if arg (BINT or num_str) is NaN
1451 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1453 $x->{sign} eq $nan ? 1 : 0;
1458 # return true if arg (BINT or num_str) is +-inf
1459 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1463 $sign = '[+-]inf' if $sign eq ''; # +- doesn't matter, only that's inf
1464 $sign = "[$1]inf" if $sign =~ /^([+-])(inf)?$/; # extract '+' or '-'
1465 return $x->{sign} =~ /^$sign$/ ? 1 : 0;
1467 $x->{sign} =~ /^[+-]inf$/ ? 1 : 0; # only +-inf is infinity
1472 # return true if arg (BINT or num_str) is +1, or -1 if sign is given
1473 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1475 $sign = '+' if !defined $sign || $sign ne '-';
1477 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1478 $CALC->_is_one($x->{value});
1483 # return true when arg (BINT or num_str) is odd, false for even
1484 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1486 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1487 $CALC->_is_odd($x->{value});
1492 # return true when arg (BINT or num_str) is even, false for odd
1493 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1495 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1496 $CALC->_is_even($x->{value});
1501 # return true when arg (BINT or num_str) is positive (>= 0)
1502 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1504 return 1 if $x->{sign} eq '+inf'; # +inf is positive
1506 # 0+ is neither positive nor negative
1507 ($x->{sign} eq '+' && !$x->is_zero()) ? 1 : 0;
1512 # return true when arg (BINT or num_str) is negative (< 0)
1513 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1515 $x->{sign} =~ /^-/ ? 1 : 0; # -inf is negative, but NaN is not
1520 # return true when arg (BINT or num_str) is an integer
1521 # always true for BigInt, but different for BigFloats
1522 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1524 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1527 ###############################################################################
1531 # multiply the first number by the second number
1532 # (BINT or num_str, BINT or num_str) return BINT
1535 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1536 # objectify is costly, so avoid it
1537 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1539 ($self,$x,$y,@r) = objectify(2,@_);
1542 return $x if $x->modify('bmul');
1544 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1547 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1549 return $x->bnan() if $x->is_zero() || $y->is_zero();
1550 # result will always be +-inf:
1551 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1552 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1553 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1554 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1555 return $x->binf('-');
1558 return $upgrade->bmul($x,$upgrade->new($y),@r)
1559 if defined $upgrade && !$y->isa($self);
1561 $r[3] = $y; # no push here
1563 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1565 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1566 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1573 # multiply two numbers and then add the third to the result
1574 # (BINT or num_str, BINT or num_str, BINT or num_str) return BINT
1577 my ($self,$x,$y,$z,@r) = (ref($_[0]),@_);
1578 # objectify is costly, so avoid it
1579 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1581 ($self,$x,$y,$z,@r) = objectify(3,@_);
1584 return $x if $x->modify('bmuladd');
1586 return $x->bnan() if ($x->{sign} eq $nan) ||
1587 ($y->{sign} eq $nan) ||
1588 ($z->{sign} eq $nan);
1590 # inf handling of x and y
1591 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1593 return $x->bnan() if $x->is_zero() || $y->is_zero();
1594 # result will always be +-inf:
1595 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1596 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1597 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1598 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1599 return $x->binf('-');
1601 # inf handling x*y and z
1602 if (($z->{sign} =~ /^[+-]inf$/))
1604 # something +-inf => +-inf
1605 $x->{sign} = $z->{sign}, return $x if $z->{sign} =~ /^[+-]inf$/;
1608 return $upgrade->bmuladd($x,$upgrade->new($y),$upgrade->new($z),@r)
1609 if defined $upgrade && (!$y->isa($self) || !$z->isa($self) || !$x->isa($self));
1611 # TODO: what if $y and $z have A or P set?
1612 $r[3] = $z; # no push here
1614 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1616 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1617 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1619 my ($sx, $sz) = ( $x->{sign}, $z->{sign} ); # get signs
1623 $x->{value} = $CALC->_add($x->{value},$z->{value}); # same sign, abs add
1627 my $a = $CALC->_acmp ($z->{value},$x->{value}); # absolute compare
1630 $x->{value} = $CALC->_sub($z->{value},$x->{value},1); # abs sub w/ swap
1635 # speedup, if equal, set result to 0
1636 $x->{value} = $CALC->_zero();
1641 $x->{value} = $CALC->_sub($x->{value}, $z->{value}); # abs sub
1649 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1650 my ($self,$x,$y) = @_;
1652 # NaN if x == NaN or y == NaN or x==y==0
1653 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1654 if (($x->is_nan() || $y->is_nan()) ||
1655 ($x->is_zero() && $y->is_zero()));
1657 # +-inf / +-inf == NaN, reminder also NaN
1658 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1660 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1662 # x / +-inf => 0, remainder x (works even if x == 0)
1663 if ($y->{sign} =~ /^[+-]inf$/)
1665 my $t = $x->copy(); # bzero clobbers up $x
1666 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1669 # 5 / 0 => +inf, -6 / 0 => -inf
1670 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1671 # exception: -8 / 0 has remainder -8, not 8
1672 # exception: -inf / 0 has remainder -inf, not inf
1675 # +-inf / 0 => special case for -inf
1676 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1677 if (!$x->is_zero() && !$x->is_inf())
1679 my $t = $x->copy(); # binf clobbers up $x
1681 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1685 # last case: +-inf / ordinary number
1687 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1689 return wantarray ? ($x,$self->bzero()) : $x;
1694 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1695 # (BINT,BINT) (quo,rem) or BINT (only rem)
1698 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1699 # objectify is costly, so avoid it
1700 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1702 ($self,$x,$y,@r) = objectify(2,@_);
1705 return $x if $x->modify('bdiv');
1707 return $self->_div_inf($x,$y)
1708 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1710 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1711 if defined $upgrade;
1713 $r[3] = $y; # no push!
1715 # calc new sign and in case $y == +/- 1, return $x
1716 my $xsign = $x->{sign}; # keep
1717 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1721 my $rem = $self->bzero();
1722 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1723 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1724 $rem->{_a} = $x->{_a};
1725 $rem->{_p} = $x->{_p};
1727 if (! $CALC->_is_zero($rem->{value}))
1729 $rem->{sign} = $y->{sign};
1730 $rem = $y->copy()->bsub($rem) if $xsign ne $y->{sign}; # one of them '-'
1734 $rem->{sign} = '+'; # dont leave -0
1740 $x->{value} = $CALC->_div($x->{value},$y->{value});
1741 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1746 ###############################################################################
1751 # modulus (or remainder)
1752 # (BINT or num_str, BINT or num_str) return BINT
1755 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1756 # objectify is costly, so avoid it
1757 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1759 ($self,$x,$y,@r) = objectify(2,@_);
1762 return $x if $x->modify('bmod');
1763 $r[3] = $y; # no push!
1764 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1766 my ($d,$r) = $self->_div_inf($x,$y);
1767 $x->{sign} = $r->{sign};
1768 $x->{value} = $r->{value};
1769 return $x->round(@r);
1772 # calc new sign and in case $y == +/- 1, return $x
1773 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1774 if (!$CALC->_is_zero($x->{value}))
1776 $x->{value} = $CALC->_sub($y->{value},$x->{value},1) # $y-$x
1777 if ($x->{sign} ne $y->{sign});
1778 $x->{sign} = $y->{sign};
1782 $x->{sign} = '+'; # dont leave -0
1789 # Modular inverse. given a number which is (hopefully) relatively
1790 # prime to the modulus, calculate its inverse using Euclid's
1791 # alogrithm. If the number is not relatively prime to the modulus
1792 # (i.e. their gcd is not one) then NaN is returned.
1795 my ($self,$x,$y,@r) = (undef,@_);
1796 # objectify is costly, so avoid it
1797 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1799 ($self,$x,$y,@r) = objectify(2,@_);
1802 return $x if $x->modify('bmodinv');
1805 if ($y->{sign} ne '+' # -, NaN, +inf, -inf
1806 || $x->is_zero() # or num == 0
1807 || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1810 # put least residue into $x if $x was negative, and thus make it positive
1811 $x->bmod($y) if $x->{sign} eq '-';
1814 ($x->{value},$sign) = $CALC->_modinv($x->{value},$y->{value});
1815 return $x->bnan() if !defined $x->{value}; # in case no GCD found
1816 return $x if !defined $sign; # already real result
1817 $x->{sign} = $sign; # flip/flop see below
1818 $x->bmod($y); # calc real result
1824 # takes a very large number to a very large exponent in a given very
1825 # large modulus, quickly, thanks to binary exponentation. Supports
1826 # negative exponents.
1827 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1829 return $num if $num->modify('bmodpow');
1831 # check modulus for valid values
1832 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1833 || $mod->is_zero());
1835 # check exponent for valid values
1836 if ($exp->{sign} =~ /\w/)
1838 # i.e., if it's NaN, +inf, or -inf...
1839 return $num->bnan();
1842 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
1844 # check num for valid values (also NaN if there was no inverse but $exp < 0)
1845 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1847 # $mod is positive, sign on $exp is ignored, result also positive
1848 $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
1852 ###############################################################################
1856 # (BINT or num_str, BINT or num_str) return BINT
1857 # compute factorial number from $x, modify $x in place
1858 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1860 return $x if $x->modify('bfac') || $x->{sign} eq '+inf'; # inf => inf
1861 return $x->bnan() if $x->{sign} ne '+'; # NaN, <0 etc => NaN
1863 $x->{value} = $CALC->_fac($x->{value});
1869 # (BINT or num_str, BINT or num_str) return BINT
1870 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1871 # modifies first argument
1874 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1875 # objectify is costly, so avoid it
1876 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1878 ($self,$x,$y,@r) = objectify(2,@_);
1881 return $x if $x->modify('bpow');
1883 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1886 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1888 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1894 if ($x->{sign} =~ /^[+-]inf/)
1897 return $x->bnan() if $y->is_zero();
1898 # -inf ** -1 => 1/inf => 0
1899 return $x->bzero() if $y->is_one('-') && $x->is_negative();
1902 return $x if $x->{sign} eq '+inf';
1904 # -inf ** Y => -inf if Y is odd
1905 return $x if $y->is_odd();
1911 return $x if $x->is_one();
1914 return $x if $x->is_zero() && $y->{sign} =~ /^[+]/;
1917 return $x->binf() if $x->is_zero();
1920 return $x->bnan() if $x->is_one('-') && $y->{sign} =~ /^[-]/;
1923 return $x->bzero() if $x->{sign} eq '-' && $y->{sign} =~ /^[-]/;
1926 return $x->bnan() if $x->{sign} eq '-';
1929 return $x->binf() if $y->{sign} =~ /^[+]/;
1934 return $upgrade->bpow($upgrade->new($x),$y,@r)
1935 if defined $upgrade && (!$y->isa($self) || $y->{sign} eq '-');
1937 $r[3] = $y; # no push!
1939 # cases 0 ** Y, X ** 0, X ** 1, 1 ** Y are handled by Calc or Emu
1942 $new_sign = $y->is_odd() ? '-' : '+' if ($x->{sign} ne '+');
1944 # 0 ** -7 => ( 1 / (0 ** 7)) => 1 / 0 => +inf
1946 if $y->{sign} eq '-' && $x->{sign} eq '+' && $CALC->_is_zero($x->{value});
1947 # 1 ** -y => 1 / (1 ** |y|)
1948 # so do test for negative $y after above's clause
1949 return $x->bnan() if $y->{sign} eq '-' && !$CALC->_is_one($x->{value});
1951 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1952 $x->{sign} = $new_sign;
1953 $x->{sign} = '+' if $CALC->_is_zero($y->{value});
1959 # (BINT or num_str, BINT or num_str) return BINT
1960 # compute x << y, base n, y >= 0
1963 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1964 # objectify is costly, so avoid it
1965 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1967 ($self,$x,$y,$n,@r) = objectify(2,@_);
1970 return $x if $x->modify('blsft');
1971 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1972 return $x->round(@r) if $y->is_zero();
1974 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1976 $x->{value} = $CALC->_lsft($x->{value},$y->{value},$n);
1982 # (BINT or num_str, BINT or num_str) return BINT
1983 # compute x >> y, base n, y >= 0
1986 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1987 # objectify is costly, so avoid it
1988 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1990 ($self,$x,$y,$n,@r) = objectify(2,@_);
1993 return $x if $x->modify('brsft');
1994 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1995 return $x->round(@r) if $y->is_zero();
1996 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
1998 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
2000 # this only works for negative numbers when shifting in base 2
2001 if (($x->{sign} eq '-') && ($n == 2))
2003 return $x->round(@r) if $x->is_one('-'); # -1 => -1
2006 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
2007 # but perhaps there is a better emulation for two's complement shift...
2008 # if $y != 1, we must simulate it by doing:
2009 # convert to bin, flip all bits, shift, and be done
2010 $x->binc(); # -3 => -2
2011 my $bin = $x->as_bin();
2012 $bin =~ s/^-0b//; # strip '-0b' prefix
2013 $bin =~ tr/10/01/; # flip bits
2015 if ($y >= CORE::length($bin))
2017 $bin = '0'; # shifting to far right creates -1
2018 # 0, because later increment makes
2019 # that 1, attached '-' makes it '-1'
2020 # because -1 >> x == -1 !
2024 $bin =~ s/.{$y}$//; # cut off at the right side
2025 $bin = '1' . $bin; # extend left side by one dummy '1'
2026 $bin =~ tr/10/01/; # flip bits back
2028 my $res = $self->new('0b'.$bin); # add prefix and convert back
2029 $res->binc(); # remember to increment
2030 $x->{value} = $res->{value}; # take over value
2031 return $x->round(@r); # we are done now, magic, isn't?
2033 # x < 0, n == 2, y == 1
2034 $x->bdec(); # n == 2, but $y == 1: this fixes it
2037 $x->{value} = $CALC->_rsft($x->{value},$y->{value},$n);
2043 #(BINT or num_str, BINT or num_str) return BINT
2047 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2048 # objectify is costly, so avoid it
2049 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2051 ($self,$x,$y,@r) = objectify(2,@_);
2054 return $x if $x->modify('band');
2056 $r[3] = $y; # no push!
2058 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
2060 my $sx = $x->{sign} eq '+' ? 1 : -1;
2061 my $sy = $y->{sign} eq '+' ? 1 : -1;
2063 if ($sx == 1 && $sy == 1)
2065 $x->{value} = $CALC->_and($x->{value},$y->{value});
2066 return $x->round(@r);
2069 if ($CAN{signed_and})
2071 $x->{value} = $CALC->_signed_and($x->{value},$y->{value},$sx,$sy);
2072 return $x->round(@r);
2076 __emu_band($self,$x,$y,$sx,$sy,@r);
2081 #(BINT or num_str, BINT or num_str) return BINT
2085 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2086 # objectify is costly, so avoid it
2087 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2089 ($self,$x,$y,@r) = objectify(2,@_);
2092 return $x if $x->modify('bior');
2093 $r[3] = $y; # no push!
2095 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
2097 my $sx = $x->{sign} eq '+' ? 1 : -1;
2098 my $sy = $y->{sign} eq '+' ? 1 : -1;
2100 # the sign of X follows the sign of X, e.g. sign of Y irrelevant for bior()
2102 # don't use lib for negative values
2103 if ($sx == 1 && $sy == 1)
2105 $x->{value} = $CALC->_or($x->{value},$y->{value});
2106 return $x->round(@r);
2109 # if lib can do negative values, let it handle this
2110 if ($CAN{signed_or})
2112 $x->{value} = $CALC->_signed_or($x->{value},$y->{value},$sx,$sy);
2113 return $x->round(@r);
2117 __emu_bior($self,$x,$y,$sx,$sy,@r);
2122 #(BINT or num_str, BINT or num_str) return BINT
2126 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2127 # objectify is costly, so avoid it
2128 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2130 ($self,$x,$y,@r) = objectify(2,@_);
2133 return $x if $x->modify('bxor');
2134 $r[3] = $y; # no push!
2136 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
2138 my $sx = $x->{sign} eq '+' ? 1 : -1;
2139 my $sy = $y->{sign} eq '+' ? 1 : -1;
2141 # don't use lib for negative values
2142 if ($sx == 1 && $sy == 1)
2144 $x->{value} = $CALC->_xor($x->{value},$y->{value});
2145 return $x->round(@r);
2148 # if lib can do negative values, let it handle this
2149 if ($CAN{signed_xor})
2151 $x->{value} = $CALC->_signed_xor($x->{value},$y->{value},$sx,$sy);
2152 return $x->round(@r);
2156 __emu_bxor($self,$x,$y,$sx,$sy,@r);
2161 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2163 my $e = $CALC->_len($x->{value});
2164 wantarray ? ($e,0) : $e;
2169 # return the nth decimal digit, negative values count backward, 0 is right
2170 my ($self,$x,$n) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2172 $n = $n->numify() if ref($n);
2173 $CALC->_digit($x->{value},$n||0);
2178 # return the amount of trailing zeros in $x (as scalar)
2180 $x = $class->new($x) unless ref $x;
2182 return 0 if $x->{sign} !~ /^[+-]$/; # NaN, inf, -inf etc
2184 $CALC->_zeros($x->{value}); # must handle odd values, 0 etc
2189 # calculate square root of $x
2190 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2192 return $x if $x->modify('bsqrt');
2194 return $x->bnan() if $x->{sign} !~ /^\+/; # -x or -inf or NaN => NaN
2195 return $x if $x->{sign} eq '+inf'; # sqrt(+inf) == inf
2197 return $upgrade->bsqrt($x,@r) if defined $upgrade;
2199 $x->{value} = $CALC->_sqrt($x->{value});
2205 # calculate $y'th root of $x
2208 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2210 $y = $self->new(2) unless defined $y;
2212 # objectify is costly, so avoid it
2213 if ((!ref($x)) || (ref($x) ne ref($y)))
2215 ($self,$x,$y,@r) = objectify(2,$self || $class,@_);
2218 return $x if $x->modify('broot');
2220 # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
2221 return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
2222 $y->{sign} !~ /^\+$/;
2224 return $x->round(@r)
2225 if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
2227 return $upgrade->new($x)->broot($upgrade->new($y),@r) if defined $upgrade;
2229 $x->{value} = $CALC->_root($x->{value},$y->{value});
2235 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
2236 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2238 if ($x->{sign} !~ /^[+-]$/)
2240 my $s = $x->{sign}; $s =~ s/^[+-]//; # NaN, -inf,+inf => NaN or inf
2241 return $self->new($s);
2243 return $self->bone() if $x->is_zero();
2245 # 12300 => 2 trailing zeros => exponent is 2
2246 $self->new( $CALC->_zeros($x->{value}) );
2251 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
2252 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2254 if ($x->{sign} !~ /^[+-]$/)
2256 # for NaN, +inf, -inf: keep the sign
2257 return $self->new($x->{sign});
2259 my $m = $x->copy(); delete $m->{_p}; delete $m->{_a};
2261 # that's a bit inefficient:
2262 my $zeros = $CALC->_zeros($m->{value});
2263 $m->brsft($zeros,10) if $zeros != 0;
2269 # return a copy of both the exponent and the mantissa
2270 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2272 ($x->mantissa(),$x->exponent());
2275 ##############################################################################
2276 # rounding functions
2280 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
2281 # $n == 0 || $n == 1 => round to integer
2282 my $x = shift; my $self = ref($x) || $x; $x = $self->new($x) unless ref $x;
2284 my ($scale,$mode) = $x->_scale_p(@_);
2286 return $x if !defined $scale || $x->modify('bfround'); # no-op
2288 # no-op for BigInts if $n <= 0
2289 $x->bround( $x->length()-$scale, $mode) if $scale > 0;
2291 delete $x->{_a}; # delete to save memory
2292 $x->{_p} = $scale; # store new _p
2296 sub _scan_for_nonzero
2298 # internal, used by bround() to scan for non-zeros after a '5'
2299 my ($x,$pad,$xs,$len) = @_;
2301 return 0 if $len == 1; # "5" is trailed by invisible zeros
2302 my $follow = $pad - 1;
2303 return 0 if $follow > $len || $follow < 1;
2305 # use the string form to check whether only '0's follow or not
2306 substr ($xs,-$follow) =~ /[^0]/ ? 1 : 0;
2311 # Exists to make life easier for switch between MBF and MBI (should we
2312 # autoload fxxx() like MBF does for bxxx()?)
2313 my $x = shift; $x = $class->new($x) unless ref $x;
2319 # accuracy: +$n preserve $n digits from left,
2320 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
2322 # and overwrite the rest with 0's, return normalized number
2323 # do not return $x->bnorm(), but $x
2325 my $x = shift; $x = $class->new($x) unless ref $x;
2326 my ($scale,$mode) = $x->_scale_a(@_);
2327 return $x if !defined $scale || $x->modify('bround'); # no-op
2329 if ($x->is_zero() || $scale == 0)
2331 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2334 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2336 # we have fewer digits than we want to scale to
2337 my $len = $x->length();
2338 # convert $scale to a scalar in case it is an object (put's a limit on the
2339 # number length, but this would already limited by memory constraints), makes
2341 $scale = $scale->numify() if ref ($scale);
2343 # scale < 0, but > -len (not >=!)
2344 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2346 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2350 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2351 my ($pad,$digit_round,$digit_after);
2352 $pad = $len - $scale;
2353 $pad = abs($scale-1) if $scale < 0;
2355 # do not use digit(), it is very costly for binary => decimal
2356 # getting the entire string is also costly, but we need to do it only once
2357 my $xs = $CALC->_str($x->{value});
2360 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2361 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2362 $digit_round = '0'; $digit_round = substr($xs,$pl,1) if $pad <= $len;
2363 $pl++; $pl ++ if $pad >= $len;
2364 $digit_after = '0'; $digit_after = substr($xs,$pl,1) if $pad > 0;
2366 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2367 # closer at the remaining digits of the original $x, remember decision
2368 my $round_up = 1; # default round up
2370 ($mode eq 'trunc') || # trunc by round down
2371 ($digit_after =~ /[01234]/) || # round down anyway,
2373 ($digit_after eq '5') && # not 5000...0000
2374 ($x->_scan_for_nonzero($pad,$xs,$len) == 0) &&
2376 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2377 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2378 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2379 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2380 ($mode eq 'zero') # round down if zero, sign adjusted below
2382 my $put_back = 0; # not yet modified
2384 if (($pad > 0) && ($pad <= $len))
2386 substr($xs,-$pad,$pad) = '0' x $pad; # replace with '00...'
2387 $put_back = 1; # need to put back
2391 $x->bzero(); # round to '0'
2394 if ($round_up) # what gave test above?
2396 $put_back = 1; # need to put back
2397 $pad = $len, $xs = '0' x $pad if $scale < 0; # tlr: whack 0.51=>1.0
2399 # we modify directly the string variant instead of creating a number and
2400 # adding it, since that is faster (we already have the string)
2401 my $c = 0; $pad ++; # for $pad == $len case
2402 while ($pad <= $len)
2404 $c = substr($xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2405 substr($xs,-$pad,1) = $c; $pad++;
2406 last if $c != 0; # no overflow => early out
2408 $xs = '1'.$xs if $c == 0;
2411 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back, if needed
2413 $x->{_a} = $scale if $scale >= 0;
2416 $x->{_a} = $len+$scale;
2417 $x->{_a} = 0 if $scale < -$len;
2424 # return integer less or equal then number; no-op since it's already integer
2425 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2432 # return integer greater or equal then number; no-op since it's already int
2433 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2440 # An object might be asked to return itself as bigint on certain overloaded
2441 # operations. This does exactly this, so that sub classes can simple inherit
2442 # it or override with their own integer conversion routine.
2448 # return as hex string, with prefixed 0x
2449 my $x = shift; $x = $class->new($x) if !ref($x);
2451 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2454 $s = $x->{sign} if $x->{sign} eq '-';
2455 $s . $CALC->_as_hex($x->{value});
2460 # return as binary string, with prefixed 0b
2461 my $x = shift; $x = $class->new($x) if !ref($x);
2463 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2465 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2466 return $s . $CALC->_as_bin($x->{value});
2471 # return as octal string, with prefixed 0
2472 my $x = shift; $x = $class->new($x) if !ref($x);
2474 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2476 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2477 return $s . $CALC->_as_oct($x->{value});
2480 ##############################################################################
2481 # private stuff (internal use only)
2485 # check for strings, if yes, return objects instead
2487 # the first argument is number of args objectify() should look at it will
2488 # return $count+1 elements, the first will be a classname. This is because
2489 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2490 # useless objects being created and thrown away. So we cannot simple loop
2491 # over @_. If the given count is 0, all arguments will be used.
2493 # If the second arg is a ref, use it as class.
2494 # If not, try to use it as classname, unless undef, then use $class
2495 # (aka Math::BigInt). The latter shouldn't happen,though.
2498 # $x->badd(1); => ref x, scalar y
2499 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2500 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2501 # Math::BigInt::badd(1,2); => scalar x, scalar y
2502 # In the last case we check number of arguments to turn it silently into
2503 # $class,1,2. (We can not take '1' as class ;o)
2504 # badd($class,1) is not supported (it should, eventually, try to add undef)
2505 # currently it tries 'Math::BigInt' + 1, which will not work.
2507 # some shortcut for the common cases
2509 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2511 my $count = abs(shift || 0);
2513 my (@a,$k,$d); # resulting array, temp, and downgrade
2516 # okay, got object as first
2521 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2523 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2527 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2528 if (defined ${"$a[0]::downgrade"})
2530 $d = ${"$a[0]::downgrade"};
2531 ${"$a[0]::downgrade"} = undef;
2534 my $up = ${"$a[0]::upgrade"};
2535 # print STDERR "# Now in objectify, my class is today $a[0], count = $count\n";
2543 $k = $a[0]->new($k);
2545 elsif (!defined $up && ref($k) ne $a[0])
2547 # foreign object, try to convert to integer
2548 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2561 $k = $a[0]->new($k);
2563 elsif (!defined $up && ref($k) ne $a[0])
2565 # foreign object, try to convert to integer
2566 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2570 push @a,@_; # return other params, too
2574 require Carp; Carp::croak ("$class objectify needs list context");
2576 ${"$a[0]::downgrade"} = $d;
2580 sub _register_callback
2582 my ($class,$callback) = @_;
2584 if (ref($callback) ne 'CODE')
2587 Carp::croak ("$callback is not a coderef");
2589 $CALLBACKS{$class} = $callback;
2596 $IMPORT++; # remember we did import()
2597 my @a; my $l = scalar @_;
2598 my $warn_or_die = 0; # 0 - no warn, 1 - warn, 2 - die
2599 for ( my $i = 0; $i < $l ; $i++ )
2601 if ($_[$i] eq ':constant')
2603 # this causes overlord er load to step in
2605 integer => sub { $self->new(shift) },
2606 binary => sub { $self->new(shift) };
2608 elsif ($_[$i] eq 'upgrade')
2610 # this causes upgrading
2611 $upgrade = $_[$i+1]; # or undef to disable
2614 elsif ($_[$i] =~ /^(lib|try|only)\z/)
2616 # this causes a different low lib to take care...
2617 $CALC = $_[$i+1] || '';
2618 # lib => 1 (warn on fallback), try => 0 (no warn), only => 2 (die on fallback)
2619 $warn_or_die = 1 if $_[$i] eq 'lib';
2620 $warn_or_die = 2 if $_[$i] eq 'only';
2628 # any non :constant stuff is handled by our parent, Exporter
2633 $self->SUPER::import(@a); # need it for subclasses
2634 $self->export_to_level(1,$self,@a); # need it for MBF
2637 # try to load core math lib
2638 my @c = split /\s*,\s*/,$CALC;
2641 $_ =~ tr/a-zA-Z0-9://cd; # limit to sane characters
2643 push @c, \'FastCalc', \'Calc' # if all fail, try these
2644 if $warn_or_die < 2; # but not for "only"
2645 $CALC = ''; # signal error
2648 # fallback libraries are "marked" as \'string', extract string if nec.
2649 my $lib = $l; $lib = $$l if ref($l);
2651 next if ($lib || '') eq '';
2652 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2656 # Perl < 5.6.0 dies with "out of memory!" when eval("") and ':constant' is
2657 # used in the same script, or eval("") inside import().
2658 my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
2659 my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
2661 $file = File::Spec->catfile (@parts, $file);
2662 eval { require "$file"; $lib->import( @c ); }
2666 eval "use $lib qw/@c/;";
2671 # loaded it ok, see if the api_version() is high enough
2672 if ($lib->can('api_version') && $lib->api_version() >= 1.0)
2675 # api_version matches, check if it really provides anything we need
2679 add mul div sub dec inc
2680 acmp len digit is_one is_zero is_even is_odd
2682 zeros new copy check
2683 from_hex from_oct from_bin as_hex as_bin as_oct
2684 rsft lsft xor and or
2685 mod sqrt root fac pow modinv modpow log_int gcd
2688 if (!$lib->can("_$method"))
2690 if (($WARN{$lib}||0) < 2)
2693 Carp::carp ("$lib is missing method '_$method'");
2694 $WARN{$lib} = 1; # still warn about the lib
2703 if ($warn_or_die > 0 && ref($l))
2706 my $msg = "Math::BigInt: couldn't load specified math lib(s), fallback to $lib";
2707 Carp::carp ($msg) if $warn_or_die == 1;
2708 Carp::croak ($msg) if $warn_or_die == 2;
2710 last; # found a usable one, break
2714 if (($WARN{$lib}||0) < 2)
2716 my $ver = eval "\$$lib\::VERSION" || 'unknown';
2718 Carp::carp ("Cannot load outdated $lib v$ver, please upgrade");
2719 $WARN{$lib} = 2; # never warn again
2727 if ($warn_or_die == 2)
2729 Carp::croak ("Couldn't load specified math lib(s) and fallback disallowed");
2733 Carp::croak ("Couldn't load any math lib(s), not even fallback to Calc.pm");
2738 foreach my $class (keys %CALLBACKS)
2740 &{$CALLBACKS{$class}}($CALC);
2743 # Fill $CAN with the results of $CALC->can(...) for emulating lower math lib
2747 for my $method (qw/ signed_and signed_or signed_xor /)
2749 $CAN{$method} = $CALC->can("_$method") ? 1 : 0;
2757 # create a bigint from a hexadecimal string
2758 my ($self, $hs) = @_;
2760 my $rc = __from_hex($hs);
2762 return $self->bnan() unless defined $rc;
2769 # create a bigint from a hexadecimal string
2770 my ($self, $bs) = @_;
2772 my $rc = __from_bin($bs);
2774 return $self->bnan() unless defined $rc;
2781 # create a bigint from a hexadecimal string
2782 my ($self, $os) = @_;
2784 my $x = $self->bzero();
2787 $os =~ s/([0-7])_([0-7])/$1$2/g;
2788 $os =~ s/([0-7])_([0-7])/$1$2/g;
2790 return $x->bnan() if $os !~ /^[\-\+]?0[0-7]+\z/;
2792 my $sign = '+'; $sign = '-' if $os =~ /^-/;
2794 $os =~ s/^[+-]//; # strip sign
2795 $x->{value} = $CALC->_from_oct($os);
2796 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2803 # convert a (ref to) big hex string to BigInt, return undef for error
2806 my $x = Math::BigInt->bzero();
2809 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2810 $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2812 return $x->bnan() if $hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2814 my $sign = '+'; $sign = '-' if $hs =~ /^-/;
2816 $hs =~ s/^[+-]//; # strip sign
2817 $x->{value} = $CALC->_from_hex($hs);
2818 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2825 # convert a (ref to) big binary string to BigInt, return undef for error
2828 my $x = Math::BigInt->bzero();
2831 $bs =~ s/([01])_([01])/$1$2/g;
2832 $bs =~ s/([01])_([01])/$1$2/g;
2833 return $x->bnan() if $bs !~ /^[+-]?0b[01]+$/;
2835 my $sign = '+'; $sign = '-' if $bs =~ /^\-/;
2836 $bs =~ s/^[+-]//; # strip sign
2838 $x->{value} = $CALC->_from_bin($bs);
2839 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2845 # input: num_str; output: undef for invalid or
2846 # (\$mantissa_sign,\$mantissa_value,\$mantissa_fraction,\$exp_sign,\$exp_value)
2847 # Internal, take apart a string and return the pieces.
2848 # Strip leading/trailing whitespace, leading zeros, underscore and reject
2852 # strip white space at front, also extranous leading zeros
2853 $x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2854 $x =~ s/^\s+//; # but this will
2855 $x =~ s/\s+$//g; # strip white space at end
2857 # shortcut, if nothing to split, return early
2858 if ($x =~ /^[+-]?[0-9]+\z/)
2860 $x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2861 return (\$sign, \$x, \'', \'', \0);
2864 # invalid starting char?
2865 return if $x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2867 return __from_hex($x) if $x =~ /^[\-\+]?0x/; # hex string
2868 return __from_bin($x) if $x =~ /^[\-\+]?0b/; # binary string
2870 # strip underscores between digits
2871 $x =~ s/([0-9])_([0-9])/$1$2/g;
2872 $x =~ s/([0-9])_([0-9])/$1$2/g; # do twice for 1_2_3
2874 # some possible inputs:
2875 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2876 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 # 0e999
2878 my ($m,$e,$last) = split /[Ee]/,$x;
2879 return if defined $last; # last defined => 1e2E3 or others
2880 $e = '0' if !defined $e || $e eq "";
2882 # sign,value for exponent,mantint,mantfrac
2883 my ($es,$ev,$mis,$miv,$mfv);
2885 if ($e =~ /^([+-]?)0*([0-9]+)$/) # strip leading zeros
2889 return if $m eq '.' || $m eq '';
2890 my ($mi,$mf,$lastf) = split /\./,$m;
2891 return if defined $lastf; # lastf defined => 1.2.3 or others
2892 $mi = '0' if !defined $mi;
2893 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2894 $mf = '0' if !defined $mf || $mf eq '';
2895 if ($mi =~ /^([+-]?)0*([0-9]+)$/) # strip leading zeros
2897 $mis = $1||'+'; $miv = $2;
2898 return unless ($mf =~ /^([0-9]*?)0*$/); # strip trailing zeros
2900 # handle the 0e999 case here
2901 $ev = 0 if $miv eq '0' && $mfv eq '';
2902 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2905 return; # NaN, not a number
2908 ##############################################################################
2909 # internal calculation routines (others are in Math::BigInt::Calc etc)
2913 # (BINT or num_str, BINT or num_str) return BINT
2914 # does modify first argument
2918 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2919 my $method = ref($x) . '::bgcd';
2921 $x * $ty / &$method($x,$ty);
2924 ###############################################################################
2925 # trigonometric functions
2929 # Calculate PI to N digits. Unless upgrading is in effect, returns the
2930 # result truncated to an integer, that is, always returns '3'.
2934 # called like Math::BigInt::bpi(10);
2935 $n = $self; $self = $class;
2937 $self = ref($self) if ref($self);
2939 return $upgrade->new($n) if defined $upgrade;
2947 # Calculate cosinus(x) to N digits. Unless upgrading is in effect, returns the
2948 # result truncated to an integer.
2949 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2951 return $x if $x->modify('bcos');
2953 return $x->bnan() if $x->{sign} !~ /^[+-]\z/; # -inf +inf or NaN => NaN
2955 return $upgrade->new($x)->bcos(@r) if defined $upgrade;
2957 require Math::BigFloat;
2958 # calculate the result and truncate it to integer
2959 my $t = Math::BigFloat->new($x)->bcos(@r)->as_int();
2961 $x->bone() if $t->is_one();
2962 $x->bzero() if $t->is_zero();
2968 # Calculate sinus(x) to N digits. Unless upgrading is in effect, returns the
2969 # result truncated to an integer.
2970 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2972 return $x if $x->modify('bsin');
2974 return $x->bnan() if $x->{sign} !~ /^[+-]\z/; # -inf +inf or NaN => NaN
2976 return $upgrade->new($x)->bsin(@r) if defined $upgrade;
2978 require Math::BigFloat;
2979 # calculate the result and truncate it to integer
2980 my $t = Math::BigFloat->new($x)->bsin(@r)->as_int();
2982 $x->bone() if $t->is_one();
2983 $x->bzero() if $t->is_zero();
2989 # calculate arcus tangens of ($y/$x)
2992 my ($self,$y,$x,@r) = (ref($_[0]),@_);
2993 # objectify is costly, so avoid it
2994 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2996 ($self,$y,$x,@r) = objectify(2,@_);
2999 return $y if $y->modify('batan2');
3001 return $y->bnan() if ($y->{sign} eq $nan) || ($x->{sign} eq $nan);
3003 return $y->bzero() if $y->is_zero() && $x->{sign} eq '+'; # x >= 0
3006 # +-inf => --PI/2 => +-1
3007 return $y->bone( substr($y->{sign},0,1) ) if $y->{sign} =~ /^[+-]inf$/;
3009 return $upgrade->new($y)->batan2($upgrade->new($x),@r) if defined $upgrade;
3011 require Math::BigFloat;
3012 my $r = Math::BigFloat->new($y)->batan2(Math::BigFloat->new($x),@r)->as_int();
3014 $x->{value} = $r->{value};
3015 $x->{sign} = $r->{sign};
3022 # Calculate arcus tangens of x to N digits. Unless upgrading is in effect, returns the
3023 # result truncated to an integer.
3024 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
3026 return $x if $x->modify('batan');
3028 return $x->bnan() if $x->{sign} !~ /^[+-]\z/; # -inf +inf or NaN => NaN
3030 return $upgrade->new($x)->batan(@r) if defined $upgrade;
3032 # calculate the result and truncate it to integer
3033 my $t = Math::BigFloat->new($x)->batan(@r);
3035 $x->{value} = $CALC->_new( $x->as_int()->bstr() );
3039 ###############################################################################
3040 # this method returns 0 if the object can be modified, or 1 if not.
3041 # We use a fast constant sub() here, to avoid costly calls. Subclasses
3042 # may override it with special code (f.i. Math::BigInt::Constant does so)
3044 sub modify () { 0; }
3053 Math::BigInt - Arbitrary size integer/float math package
3059 # or make it faster: install (optional) Math::BigInt::GMP
3060 # and always use (it will fall back to pure Perl if the
3061 # GMP library is not installed):
3063 # will warn if Math::BigInt::GMP cannot be found
3064 use Math::BigInt lib => 'GMP';
3066 # to supress the warning use this:
3067 # use Math::BigInt try => 'GMP';
3069 my $str = '1234567890';
3070 my @values = (64,74,18);
3071 my $n = 1; my $sign = '-';
3074 my $x = Math::BigInt->new($str); # defaults to 0
3075 my $y = $x->copy(); # make a true copy
3076 my $nan = Math::BigInt->bnan(); # create a NotANumber
3077 my $zero = Math::BigInt->bzero(); # create a +0
3078 my $inf = Math::BigInt->binf(); # create a +inf
3079 my $inf = Math::BigInt->binf('-'); # create a -inf
3080 my $one = Math::BigInt->bone(); # create a +1
3081 my $mone = Math::BigInt->bone('-'); # create a -1
3083 my $pi = Math::BigInt->bpi(); # returns '3'
3084 # see Math::BigFloat::bpi()
3086 $h = Math::BigInt->new('0x123'); # from hexadecimal
3087 $b = Math::BigInt->new('0b101'); # from binary
3088 $o = Math::BigInt->from_oct('0101'); # from octal
3090 # Testing (don't modify their arguments)
3091 # (return true if the condition is met, otherwise false)
3093 $x->is_zero(); # if $x is +0
3094 $x->is_nan(); # if $x is NaN
3095 $x->is_one(); # if $x is +1
3096 $x->is_one('-'); # if $x is -1
3097 $x->is_odd(); # if $x is odd
3098 $x->is_even(); # if $x is even
3099 $x->is_pos(); # if $x >= 0
3100 $x->is_neg(); # if $x < 0
3101 $x->is_inf($sign); # if $x is +inf, or -inf (sign is default '+')
3102 $x->is_int(); # if $x is an integer (not a float)
3104 # comparing and digit/sign extraction
3105 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
3106 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
3107 $x->sign(); # return the sign, either +,- or NaN
3108 $x->digit($n); # return the nth digit, counting from right
3109 $x->digit(-$n); # return the nth digit, counting from left
3111 # The following all modify their first argument. If you want to preserve
3112 # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
3113 # necessary when mixing $a = $b assignments with non-overloaded math.
3115 $x->bzero(); # set $x to 0
3116 $x->bnan(); # set $x to NaN
3117 $x->bone(); # set $x to +1
3118 $x->bone('-'); # set $x to -1
3119 $x->binf(); # set $x to inf
3120 $x->binf('-'); # set $x to -inf
3122 $x->bneg(); # negation
3123 $x->babs(); # absolute value
3124 $x->bnorm(); # normalize (no-op in BigInt)
3125 $x->bnot(); # two's complement (bit wise not)
3126 $x->binc(); # increment $x by 1
3127 $x->bdec(); # decrement $x by 1
3129 $x->badd($y); # addition (add $y to $x)
3130 $x->bsub($y); # subtraction (subtract $y from $x)
3131 $x->bmul($y); # multiplication (multiply $x by $y)
3132 $x->bdiv($y); # divide, set $x to quotient
3133 # return (quo,rem) or quo if scalar
3135 $x->bmuladd($y,$z); # $x = $x * $y + $z
3137 $x->bmod($y); # modulus (x % y)
3138 $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
3139 $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
3141 $x->bpow($y); # power of arguments (x ** y)
3142 $x->blsft($y); # left shift in base 2
3143 $x->brsft($y); # right shift in base 2
3144 # returns (quo,rem) or quo if in scalar context
3145 $x->blsft($y,$n); # left shift by $y places in base $n
3146 $x->brsft($y,$n); # right shift by $y places in base $n
3147 # returns (quo,rem) or quo if in scalar context
3149 $x->band($y); # bitwise and
3150 $x->bior($y); # bitwise inclusive or
3151 $x->bxor($y); # bitwise exclusive or
3152 $x->bnot(); # bitwise not (two's complement)
3154 $x->bsqrt(); # calculate square-root
3155 $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
3156 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3158 $x->bnok($y); # x over y (binomial coefficient n over k)
3160 $x->blog(); # logarithm of $x to base e (Euler's number)
3161 $x->blog($base); # logarithm of $x to base $base (f.i. 2)
3162 $x->bexp(); # calculate e ** $x where e is Euler's number
3164 $x->round($A,$P,$mode); # round to accuracy or precision using mode $mode
3165 $x->bround($n); # accuracy: preserve $n digits
3166 $x->bfround($n); # $n > 0: round $nth digits,
3167 # $n < 0: round to the $nth digit after the
3168 # dot, no-op for BigInts
3170 # The following do not modify their arguments in BigInt (are no-ops),
3171 # but do so in BigFloat:
3173 $x->bfloor(); # return integer less or equal than $x
3174 $x->bceil(); # return integer greater or equal than $x
3176 # The following do not modify their arguments:
3178 # greatest common divisor (no OO style)
3179 my $gcd = Math::BigInt::bgcd(@values);
3180 # lowest common multiplicator (no OO style)
3181 my $lcm = Math::BigInt::blcm(@values);
3183 $x->length(); # return number of digits in number
3184 ($xl,$f) = $x->length(); # length of number and length of fraction part,
3185 # latter is always 0 digits long for BigInts
3187 $x->exponent(); # return exponent as BigInt
3188 $x->mantissa(); # return (signed) mantissa as BigInt
3189 $x->parts(); # return (mantissa,exponent) as BigInt
3190 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3191 $x->as_int(); # return as BigInt (in BigInt: same as copy())
3192 $x->numify(); # return as scalar (might overflow!)
3194 # conversation to string (do not modify their argument)
3195 $x->bstr(); # normalized string (e.g. '3')
3196 $x->bsstr(); # norm. string in scientific notation (e.g. '3E0')
3197 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3198 $x->as_bin(); # as signed binary string with prefixed 0b
3199 $x->as_oct(); # as signed octal string with prefixed 0
3202 # precision and accuracy (see section about rounding for more)
3203 $x->precision(); # return P of $x (or global, if P of $x undef)
3204 $x->precision($n); # set P of $x to $n
3205 $x->accuracy(); # return A of $x (or global, if A of $x undef)
3206 $x->accuracy($n); # set A $x to $n
3209 Math::BigInt->precision(); # get/set global P for all BigInt objects
3210 Math::BigInt->accuracy(); # get/set global A for all BigInt objects
3211 Math::BigInt->round_mode(); # get/set global round mode, one of
3212 # 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'
3213 Math::BigInt->config(); # return hash containing configuration
3217 All operators (including basic math operations) are overloaded if you
3218 declare your big integers as
3220 $i = new Math::BigInt '123_456_789_123_456_789';
3222 Operations with overloaded operators preserve the arguments which is
3223 exactly what you expect.
3229 Input values to these routines may be any string, that looks like a number
3230 and results in an integer, including hexadecimal and binary numbers.
3232 Scalars holding numbers may also be passed, but note that non-integer numbers
3233 may already have lost precision due to the conversation to float. Quote
3234 your input if you want BigInt to see all the digits:
3236 $x = Math::BigInt->new(12345678890123456789); # bad
3237 $x = Math::BigInt->new('12345678901234567890'); # good
3239 You can include one underscore between any two digits.
3241 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
3242 Non-integer values result in NaN.
3244 Hexadecimal (prefixed with "0x") and binary numbers (prefixed with "0b")
3245 are accepted, too. Please note that octal numbers are not recognized
3246 by new(), so the following will print "123":
3248 perl -MMath::BigInt -le 'print Math::BigInt->new("0123")'
3250 To convert an octal number, use from_oct();
3252 perl -MMath::BigInt -le 'print Math::BigInt->from_oct("0123")'
3254 Currently, Math::BigInt::new() defaults to 0, while Math::BigInt::new('')
3255 results in 'NaN'. This might change in the future, so use always the following
3256 explicit forms to get a zero or NaN:
3258 $zero = Math::BigInt->bzero();
3259 $nan = Math::BigInt->bnan();
3261 C<bnorm()> on a BigInt object is now effectively a no-op, since the numbers
3262 are always stored in normalized form. If passed a string, creates a BigInt
3263 object from the input.
3267 Output values are BigInt objects (normalized), except for the methods which
3268 return a string (see L<SYNOPSIS>).
3270 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
3271 C<is_nan()>, etc.) return true or false, while others (C<bcmp()>, C<bacmp()>)
3272 return either undef (if NaN is involved), <0, 0 or >0 and are suited for sort.
3278 Each of the methods below (except config(), accuracy() and precision())
3279 accepts three additional parameters. These arguments C<$A>, C<$P> and C<$R>
3280 are C<accuracy>, C<precision> and C<round_mode>. Please see the section about
3281 L<ACCURACY and PRECISION> for more information.
3287 print Dumper ( Math::BigInt->config() );
3288 print Math::BigInt->config()->{lib},"\n";
3290 Returns a hash containing the configuration, e.g. the version number, lib
3291 loaded etc. The following hash keys are currently filled in with the
3292 appropriate information.
3296 ============================================================
3297 lib Name of the low-level math library
3299 lib_version Version of low-level math library (see 'lib')
3301 class The class name of config() you just called
3303 upgrade To which class math operations might be upgraded
3305 downgrade To which class math operations might be downgraded
3307 precision Global precision
3309 accuracy Global accuracy
3311 round_mode Global round mode
3313 version version number of the class you used
3315 div_scale Fallback accuracy for div
3317 trap_nan If true, traps creation of NaN via croak()
3319 trap_inf If true, traps creation of +inf/-inf via croak()
3322 The following values can be set by passing C<config()> a reference to a hash:
3325 upgrade downgrade precision accuracy round_mode div_scale
3329 $new_cfg = Math::BigInt->config( { trap_inf => 1, precision => 5 } );
3333 $x->accuracy(5); # local for $x
3334 CLASS->accuracy(5); # global for all members of CLASS
3335 # Note: This also applies to new()!
3337 $A = $x->accuracy(); # read out accuracy that affects $x
3338 $A = CLASS->accuracy(); # read out global accuracy
3340 Set or get the global or local accuracy, aka how many significant digits the
3341 results have. If you set a global accuracy, then this also applies to new()!
3343 Warning! The accuracy I<sticks>, e.g. once you created a number under the
3344 influence of C<< CLASS->accuracy($A) >>, all results from math operations with
3345 that number will also be rounded.
3347 In most cases, you should probably round the results explicitly using one of
3348 L<round()>, L<bround()> or L<bfround()> or by passing the desired accuracy
3349 to the math operation as additional parameter:
3351 my $x = Math::BigInt->new(30000);
3352 my $y = Math::BigInt->new(7);
3353 print scalar $x->copy()->bdiv($y, 2); # print 4300
3354 print scalar $x->copy()->bdiv($y)->bround(2); # print 4300
3356 Please see the section about L<ACCURACY AND PRECISION> for further details.
3358 Value must be greater than zero. Pass an undef value to disable it:
3360 $x->accuracy(undef);
3361 Math::BigInt->accuracy(undef);
3363 Returns the current accuracy. For C<$x->accuracy()> it will return either the
3364 local accuracy, or if not defined, the global. This means the return value
3365 represents the accuracy that will be in effect for $x:
3367 $y = Math::BigInt->new(1234567); # unrounded
3368 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
3369 $x = Math::BigInt->new(123456); # $x will be automatically rounded!
3370 print "$x $y\n"; # '123500 1234567'
3371 print $x->accuracy(),"\n"; # will be 4
3372 print $y->accuracy(),"\n"; # also 4, since global is 4
3373 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
3374 print $x->accuracy(),"\n"; # still 4
3375 print $y->accuracy(),"\n"; # 5, since global is 5
3377 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
3378 globals separated from Math::BigInt, but it is possible to subclass
3379 Math::BigInt and make the globals of the subclass aliases to the ones from
3384 $x->precision(-2); # local for $x, round at the second digit right of the dot
3385 $x->precision(2); # ditto, round at the second digit left of the dot
3387 CLASS->precision(5); # Global for all members of CLASS
3388 # This also applies to new()!
3389 CLASS->precision(-5); # ditto
3391 $P = CLASS->precision(); # read out global precision
3392 $P = $x->precision(); # read out precision that affects $x
3394 Note: You probably want to use L<accuracy()> instead. With L<accuracy> you
3395 set the number of digits each result should have, with L<precision> you
3396 set the place where to round!
3398 C<precision()> sets or gets the global or local precision, aka at which digit
3399 before or after the dot to round all results. A set global precision also
3400 applies to all newly created numbers!
3402 In Math::BigInt, passing a negative number precision has no effect since no
3403 numbers have digits after the dot. In L<Math::BigFloat>, it will round all
3404 results to P digits after the dot.
3406 Please see the section about L<ACCURACY AND PRECISION> for further details.
3408 Pass an undef value to disable it:
3410 $x->precision(undef);
3411 Math::BigInt->precision(undef);
3413 Returns the current precision. For C<$x->precision()> it will return either the
3414 local precision of $x, or if not defined, the global. This means the return
3415 value represents the prevision that will be in effect for $x:
3417 $y = Math::BigInt->new(1234567); # unrounded
3418 print Math::BigInt->precision(4),"\n"; # set 4, print 4
3419 $x = Math::BigInt->new(123456); # will be automatically rounded
3420 print $x; # print "120000"!
3422 Note: Works also for subclasses like L<Math::BigFloat>. Each class has its
3423 own globals separated from Math::BigInt, but it is possible to subclass
3424 Math::BigInt and make the globals of the subclass aliases to the ones from
3431 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
3432 2, but others work, too.
3434 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
3438 $x = Math::BigInt->new(10);
3439 $x->brsft(1); # same as $x >> 1: 5
3440 $x = Math::BigInt->new(1234);
3441 $x->brsft(2,10); # result 12
3443 There is one exception, and that is base 2 with negative $x:
3446 $x = Math::BigInt->new(-5);
3449 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
3454 $x = Math::BigInt->new($str,$A,$P,$R);
3456 Creates a new BigInt object from a scalar or another BigInt object. The
3457 input is accepted as decimal, hex (with leading '0x') or binary (with leading
3460 See L<Input> for more info on accepted input formats.
3464 $x = Math::BigInt->from_oct("0775"); # input is octal
3468 $x = Math::BigInt->from_hex("0xcafe"); # input is hexadecimal
3472 $x = Math::BigInt->from_oct("0x10011"); # input is binary
3476 $x = Math::BigInt->bnan();
3478 Creates a new BigInt object representing NaN (Not A Number).
3479 If used on an object, it will set it to NaN:
3485 $x = Math::BigInt->bzero();
3487 Creates a new BigInt object representing zero.
3488 If used on an object, it will set it to zero:
3494 $x = Math::BigInt->binf($sign);
3496 Creates a new BigInt object representing infinity. The optional argument is
3497 either '-' or '+', indicating whether you want infinity or minus infinity.
3498 If used on an object, it will set it to infinity:
3505 $x = Math::BigInt->binf($sign);
3507 Creates a new BigInt object representing one. The optional argument is
3508 either '-' or '+', indicating whether you want one or minus one.
3509 If used on an object, it will set it to one:
3514 =head2 is_one()/is_zero()/is_nan()/is_inf()
3517 $x->is_zero(); # true if arg is +0
3518 $x->is_nan(); # true if arg is NaN
3519 $x->is_one(); # true if arg is +1
3520 $x->is_one('-'); # true if arg is -1
3521 $x->is_inf(); # true if +inf
3522 $x->is_inf('-'); # true if -inf (sign is default '+')
3524 These methods all test the BigInt for being one specific value and return
3525 true or false depending on the input. These are faster than doing something
3530 =head2 is_pos()/is_neg()/is_positive()/is_negative()
3532 $x->is_pos(); # true if > 0
3533 $x->is_neg(); # true if < 0
3535 The methods return true if the argument is positive or negative, respectively.
3536 C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
3537 C<-inf> is negative. A C<zero> is neither positive nor negative.
3539 These methods are only testing the sign, and not the value.
3541 C<is_positive()> and C<is_negative()> are aliases to C<is_pos()> and
3542 C<is_neg()>, respectively. C<is_positive()> and C<is_negative()> were
3543 introduced in v1.36, while C<is_pos()> and C<is_neg()> were only introduced
3546 =head2 is_odd()/is_even()/is_int()
3548 $x->is_odd(); # true if odd, false for even
3549 $x->is_even(); # true if even, false for odd
3550 $x->is_int(); # true if $x is an integer
3552 The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
3553 C<-inf> are not integers and are neither odd nor even.
3555 In BigInt, all numbers except C<NaN>, C<+inf> and C<-inf> are integers.
3561 Compares $x with $y and takes the sign into account.
3562 Returns -1, 0, 1 or undef.
3568 Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
3574 Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
3576 If you want $x to have a certain sign, use one of the following methods:
3579 $x->babs()->bneg(); # '-'
3581 $x->binf(); # '+inf'
3582 $x->binf('-'); # '-inf'
3586 $x->digit($n); # return the nth digit, counting from right
3588 If C<$n> is negative, returns the digit counting from left.
3594 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
3595 and '-inf', respectively. Does nothing for NaN or zero.
3601 Set the number to its absolute value, e.g. change the sign from '-' to '+'
3602 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
3607 $x->bnorm(); # normalize (no-op)
3613 Two's complement (bitwise not). This is equivalent to
3621 $x->binc(); # increment x by 1
3625 $x->bdec(); # decrement x by 1
3629 $x->badd($y); # addition (add $y to $x)
3633 $x->bsub($y); # subtraction (subtract $y from $x)
3637 $x->bmul($y); # multiplication (multiply $x by $y)
3643 Multiply $x by $y, and then add $z to the result,
3645 This method was added in v1.87 of Math::BigInt (June 2007).
3649 $x->bdiv($y); # divide, set $x to quotient
3650 # return (quo,rem) or quo if scalar
3654 $x->bmod($y); # modulus (x % y)
3658 num->bmodinv($mod); # modular inverse
3660 Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
3661 returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
3662 C<bgcd($num, $mod)==1>.
3666 $num->bmodpow($exp,$mod); # modular exponentation
3667 # ($num**$exp % $mod)
3669 Returns the value of C<$num> taken to the power C<$exp> in the modulus
3670 C<$mod> using binary exponentation. C<bmodpow> is far superior to
3675 because it is much faster - it reduces internal variables into
3676 the modulus whenever possible, so it operates on smaller numbers.
3678 C<bmodpow> also supports negative exponents.
3680 bmodpow($num, -1, $mod)
3682 is exactly equivalent to
3688 $x->bpow($y); # power of arguments (x ** y)
3692 $x->blog($base, $accuracy); # logarithm of x to the base $base
3694 If C<$base> is not defined, Euler's number (e) is used:
3696 print $x->blog(undef, 100); # log(x) to 100 digits
3700 $x->bexp($accuracy); # calculate e ** X
3702 Calculates the expression C<e ** $x> where C<e> is Euler's number.
3704 This method was added in v1.82 of Math::BigInt (April 2007).
3710 $x->bnok($y); # x over y (binomial coefficient n over k)
3712 Calculates the binomial coefficient n over k, also called the "choose"
3713 function. The result is equivalent to:
3719 This method was added in v1.84 of Math::BigInt (April 2007).
3723 print Math::BigInt->bpi(100), "\n"; # 3
3725 Returns PI truncated to an integer, with the argument being ignored. This means
3726 under BigInt this always returns C<3>.
3728 If upgrading is in effect, returns PI, rounded to N digits with the
3729 current rounding mode:
3732 use Math::BigInt upgrade => Math::BigFloat;
3733 print Math::BigInt->bpi(3), "\n"; # 3.14
3734 print Math::BigInt->bpi(100), "\n"; # 3.1415....
3736 This method was added in v1.87 of Math::BigInt (June 2007).
3740 my $x = Math::BigInt->new(1);
3741 print $x->bcos(100), "\n";
3743 Calculate the cosinus of $x, modifying $x in place.
3745 In BigInt, unless upgrading is in effect, the result is truncated to an
3748 This method was added in v1.87 of Math::BigInt (June 2007).
3752 my $x = Math::BigInt->new(1);
3753 print $x->bsin(100), "\n";
3755 Calculate the sinus of $x, modifying $x in place.
3757 In BigInt, unless upgrading is in effect, the result is truncated to an
3760 This method was added in v1.87 of Math::BigInt (June 2007).
3764 my $x = Math::BigInt->new(1);
3765 my $y = Math::BigInt->new(1);
3766 print $y->batan2($x), "\n";
3768 Calculate the arcus tangens of C<$y> divided by C<$x>, modifying $y in place.
3770 In BigInt, unless upgrading is in effect, the result is truncated to an
3773 This method was added in v1.87 of Math::BigInt (June 2007).
3777 my $x = Math::BigFloat->new(0.5);
3778 print $x->batan(100), "\n";
3780 Calculate the arcus tangens of $x, modifying $x in place.
3782 In BigInt, unless upgrading is in effect, the result is truncated to an
3785 This method was added in v1.87 of Math::BigInt (June 2007).
3789 $x->blsft($y); # left shift in base 2
3790 $x->blsft($y,$n); # left shift, in base $n (like 10)
3794 $x->brsft($y); # right shift in base 2
3795 $x->brsft($y,$n); # right shift, in base $n (like 10)
3799 $x->band($y); # bitwise and
3803 $x->bior($y); # bitwise inclusive or
3807 $x->bxor($y); # bitwise exclusive or
3811 $x->bnot(); # bitwise not (two's complement)
3815 $x->bsqrt(); # calculate square-root
3821 Calculates the N'th root of C<$x>.
3825 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3829 $x->round($A,$P,$round_mode);
3831 Round $x to accuracy C<$A> or precision C<$P> using the round mode
3836 $x->bround($N); # accuracy: preserve $N digits
3842 If N is > 0, rounds to the Nth digit from the left. If N < 0, rounds to
3843 the Nth digit after the dot. Since BigInts are integers, the case N < 0
3844 is a no-op for them.
3849 ===================================================
3850 123456.123456 3 123500
3851 123456.123456 2 123450
3852 123456.123456 -2 123456.12
3853 123456.123456 -3 123456.123
3859 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3860 does change $x in BigFloat.
3866 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3867 does change $x in BigFloat.
3871 bgcd(@values); # greatest common divisor (no OO style)
3875 blcm(@values); # lowest common multiplicator (no OO style)
3880 ($xl,$fl) = $x->length();
3882 Returns the number of digits in the decimal representation of the number.
3883 In list context, returns the length of the integer and fraction part. For
3884 BigInt's, the length of the fraction part will always be 0.
3890 Return the exponent of $x as BigInt.
3896 Return the signed mantissa of $x as BigInt.
3900 $x->parts(); # return (mantissa,exponent) as BigInt
3904 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3906 =head2 as_int()/as_number()
3910 Returns $x as a BigInt (truncated towards zero). In BigInt this is the same as
3913 C<as_number()> is an alias to this method. C<as_number> was introduced in
3914 v1.22, while C<as_int()> was only introduced in v1.68.
3920 Returns a normalized string representation of C<$x>.
3924 $x->bsstr(); # normalized string in scientific notation
3928 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3932 $x->as_bin(); # as signed binary string with prefixed 0b
3936 $x->as_oct(); # as signed octal string with prefixed 0
3942 This returns a normal Perl scalar from $x. It is used automatically
3943 whenever a scalar is needed, for instance in array index operations.
3945 This loses precision, to avoid this use L<as_int()> instead.
3949 $x->modify('bpowd');
3951 This method returns 0 if the object can be modified with the given
3952 peration, or 1 if not.
3954 This is used for instance by L<Math::BigInt::Constant>.
3956 =head2 upgrade()/downgrade()
3958 Set/get the class for downgrade/upgrade operations. Thuis is used
3959 for instance by L<bignum>. The defaults are '', thus the following
3960 operation will create a BigInt, not a BigFloat:
3962 my $i = Math::BigInt->new(123);
3963 my $f = Math::BigFloat->new('123.1');
3965 print $i + $f,"\n"; # print 246
3969 Set/get the number of digits for the default precision in divide
3974 Set/get the current round mode.
3976 =head1 ACCURACY and PRECISION
3978 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3979 accuracy and precision based rounding, both automatically after every
3980 operation, as well as manually.
3982 This section describes the accuracy/precision handling in Math::Big* as it
3983 used to be and as it is now, complete with an explanation of all terms and
3986 Not yet implemented things (but with correct description) are marked with '!',
3987 things that need to be answered are marked with '?'.
3989 In the next paragraph follows a short description of terms used here (because
3990 these may differ from terms used by others people or documentation).
3992 During the rest of this document, the shortcuts A (for accuracy), P (for
3993 precision), F (fallback) and R (rounding mode) will be used.
3997 A fixed number of digits before (positive) or after (negative)
3998 the decimal point. For example, 123.45 has a precision of -2. 0 means an
3999 integer like 123 (or 120). A precision of 2 means two digits to the left
4000 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
4001 numbers with zeros before the decimal point may have different precisions,
4002 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
4003 was). It could also have p < 0, when the digits after the decimal point
4006 The string output (of floating point numbers) will be padded with zeros:
4008 Initial value P A Result String
4009 ------------------------------------------------------------
4010 1234.01 -3 1000 1000
4013 1234.001 1 1234 1234.0
4015 1234.01 2 1234.01 1234.01
4016 1234.01 5 1234.01 1234.01000
4018 For BigInts, no padding occurs.
4022 Number of significant digits. Leading zeros are not counted. A
4023 number may have an accuracy greater than the non-zero digits
4024 when there are zeros in it or trailing zeros. For example, 123.456 has
4025 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
4027 The string output (of floating point numbers) will be padded with zeros:
4029 Initial value P A Result String
4030 ------------------------------------------------------------
4032 1234.01 6 1234.01 1234.01
4033 1234.1 8 1234.1 1234.1000
4035 For BigInts, no padding occurs.
4039 When both A and P are undefined, this is used as a fallback accuracy when
4042 =head2 Rounding mode R
4044 When rounding a number, different 'styles' or 'kinds'
4045 of rounding are possible. (Note that random rounding, as in
4046 Math::Round, is not implemented.)
4052 truncation invariably removes all digits following the
4053 rounding place, replacing them with zeros. Thus, 987.65 rounded
4054 to tens (P=1) becomes 980, and rounded to the fourth sigdig
4055 becomes 987.6 (A=4). 123.456 rounded to the second place after the
4056 decimal point (P=-2) becomes 123.46.
4058 All other implemented styles of rounding attempt to round to the
4059 "nearest digit." If the digit D immediately to the right of the
4060 rounding place (skipping the decimal point) is greater than 5, the
4061 number is incremented at the rounding place (possibly causing a
4062 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
4063 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
4064 truncated at the rounding place: e.g. when rounding to units, 0.4
4065 rounds to 0, and -19.4 rounds to -19.
4067 However the results of other styles of rounding differ if the
4068 digit immediately to the right of the rounding place (skipping the
4069 decimal point) is 5 and if there are no digits, or no digits other
4070 than 0, after that 5. In such cases:
4074 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
4075 if it is not already. E.g., when rounding to the first sigdig, 0.45
4076 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
4080 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
4081 it is not already. E.g., when rounding to the first sigdig, 0.45
4082 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
4086 round to plus infinity, i.e. always round up. E.g., when
4087 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
4088 and 0.4501 also becomes 0.5.
4092 round to minus infinity, i.e. always round down. E.g., when
4093 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
4094 but 0.4501 becomes 0.5.
4098 round to zero, i.e. positive numbers down, negative ones up.
4099 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
4100 becomes -0.5, but 0.4501 becomes 0.5.
4104 round up if the digit immediately to the right of the rounding place
4105 is 5 or greater, otherwise round down. E.g., 0.15 becomes 0.2 and
4110 The handling of A & P in MBI/MBF (the old core code shipped with Perl
4111 versions <= 5.7.2) is like this:
4117 * ffround($p) is able to round to $p number of digits after the decimal
4119 * otherwise P is unused
4121 =item Accuracy (significant digits)
4123 * fround($a) rounds to $a significant digits
4124 * only fdiv() and fsqrt() take A as (optional) paramater
4125 + other operations simply create the same number (fneg etc), or more (fmul)
4127 + rounding/truncating is only done when explicitly calling one of fround
4128 or ffround, and never for BigInt (not implemented)
4129 * fsqrt() simply hands its accuracy argument over to fdiv.
4130 * the documentation and the comment in the code indicate two different ways
4131 on how fdiv() determines the maximum number of digits it should calculate,
4132 and the actual code does yet another thing
4134 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
4136 result has at most max(scale, length(dividend), length(divisor)) digits
4138 scale = max(scale, length(dividend)-1,length(divisor)-1);
4139 scale += length(divisor) - length(dividend);
4140 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
4141 Actually, the 'difference' added to the scale is calculated from the
4142 number of "significant digits" in dividend and divisor, which is derived
4143 by looking at the length of the mantissa. Which is wrong, since it includes
4144 the + sign (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops
4145 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
4146 assumption that 124 has 3 significant digits, while 120/7 will get you
4147 '17', not '17.1' since 120 is thought to have 2 significant digits.
4148 The rounding after the division then uses the remainder and $y to determine
4149 wether it must round up or down.
4150 ? I have no idea which is the right way. That's why I used a slightly more
4151 ? simple scheme and tweaked the few failing testcases to match it.
4155 This is how it works now:
4159 =item Setting/Accessing
4161 * You can set the A global via C<< Math::BigInt->accuracy() >> or
4162 C<< Math::BigFloat->accuracy() >> or whatever class you are using.
4163 * You can also set P globally by using C<< Math::SomeClass->precision() >>
4165 * Globals are classwide, and not inherited by subclasses.
4166 * to undefine A, use C<< Math::SomeCLass->accuracy(undef); >>
4167 * to undefine P, use C<< Math::SomeClass->precision(undef); >>
4168 * Setting C<< Math::SomeClass->accuracy() >> clears automatically
4169 C<< Math::SomeClass->precision() >>, and vice versa.
4170 * To be valid, A must be > 0, P can have any value.
4171 * If P is negative, this means round to the P'th place to the right of the
4172 decimal point; positive values mean to the left of the decimal point.
4173 P of 0 means round to integer.
4174 * to find out the current global A, use C<< Math::SomeClass->accuracy() >>
4175 * to find out the current global P, use C<< Math::SomeClass->precision() >>
4176 * use C<< $x->accuracy() >> respective C<< $x->precision() >> for the local
4177 setting of C<< $x >>.
4178 * Please note that C<< $x->accuracy() >> respective C<< $x->precision() >>
4179 return eventually defined global A or P, when C<< $x >>'s A or P is not
4182 =item Creating numbers
4184 * When you create a number, you can give the desired A or P via:
4185 $x = Math::BigInt->new($number,$A,$P);
4186 * Only one of A or P can be defined, otherwise the result is NaN
4187 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
4188 globals (if set) will be used. Thus changing the global defaults later on
4189 will not change the A or P of previously created numbers (i.e., A and P of
4190 $x will be what was in effect when $x was created)
4191 * If given undef for A and P, B<no> rounding will occur, and the globals will
4192 B<not> be used. This is used by subclasses to create numbers without
4193 suffering rounding in the parent. Thus a subclass is able to have its own
4194 globals enforced upon creation of a number by using
4195 C<< $x = Math::BigInt->new($number,undef,undef) >>:
4197 use Math::BigInt::SomeSubclass;
4200 Math::BigInt->accuracy(2);
4201 Math::BigInt::SomeSubClass->accuracy(3);
4202 $x = Math::BigInt::SomeSubClass->new(1234);
4204 $x is now 1230, and not 1200. A subclass might choose to implement
4205 this otherwise, e.g. falling back to the parent's A and P.
4209 * If A or P are enabled/defined, they are used to round the result of each
4210 operation according to the rules below
4211 * Negative P is ignored in Math::BigInt, since BigInts never have digits
4212 after the decimal point
4213 * Math::BigFloat uses Math::BigInt internally, but setting A or P inside
4214 Math::BigInt as globals does not tamper with the parts of a BigFloat.
4215 A flag is used to mark all Math::BigFloat numbers as 'never round'.
4219 * It only makes sense that a number has only one of A or P at a time.
4220 If you set either A or P on one object, or globally, the other one will
4221 be automatically cleared.
4222 * If two objects are involved in an operation, and one of them has A in
4223 effect, and the other P, this results in an error (NaN).
4224 * A takes precedence over P (Hint: A comes before P).
4225 If neither of them is defined, nothing is used, i.e. the result will have
4226 as many digits as it can (with an exception for fdiv/fsqrt) and will not
4228 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
4229 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
4230 If either the dividend's or the divisor's mantissa has more digits than
4231 the value of F, the higher value will be used instead of F.
4232 This is to limit the digits (A) of the result (just consider what would
4233 happen with unlimited A and P in the case of 1/3 :-)
4234 * fdiv will calculate (at least) 4 more digits than required (determined by
4235 A, P or F), and, if F is not used, round the result
4236 (this will still fail in the case of a result like 0.12345000000001 with A
4237 or P of 5, but this can not be helped - or can it?)
4238 * Thus you can have the math done by on Math::Big* class in two modi:
4239 + never round (this is the default):
4240 This is done by setting A and P to undef. No math operation
4241 will round the result, with fdiv() and fsqrt() as exceptions to guard
4242 against overflows. You must explicitly call bround(), bfround() or
4243 round() (the latter with parameters).
4244 Note: Once you have rounded a number, the settings will 'stick' on it
4245 and 'infect' all other numbers engaged in math operations with it, since
4246 local settings have the highest precedence. So, to get SaferRound[tm],
4247 use a copy() before rounding like this:
4249 $x = Math::BigFloat->new(12.34);
4250 $y = Math::BigFloat->new(98.76);
4251 $z = $x * $y; # 1218.6984
4252 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
4253 $z = $x * $y; # still 1218.6984, without
4254 # copy would have been 1210!
4256 + round after each op:
4257 After each single operation (except for testing like is_zero()), the
4258 method round() is called and the result is rounded appropriately. By
4259 setting proper values for A and P, you can have all-the-same-A or
4260 all-the-same-P modes. For example, Math::Currency might set A to undef,
4261 and P to -2, globally.
4263 ?Maybe an extra option that forbids local A & P settings would be in order,
4264 ?so that intermediate rounding does not 'poison' further math?
4266 =item Overriding globals
4268 * you will be able to give A, P and R as an argument to all the calculation
4269 routines; the second parameter is A, the third one is P, and the fourth is
4270 R (shift right by one for binary operations like badd). P is used only if
4271 the first parameter (A) is undefined. These three parameters override the
4272 globals in the order detailed as follows, i.e. the first defined value
4274 (local: per object, global: global default, parameter: argument to sub)
4277 + local A (if defined on both of the operands: smaller one is taken)
4278 + local P (if defined on both of the operands: bigger one is taken)
4282 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
4283 arguments (A and P) instead of one
4285 =item Local settings
4287 * You can set A or P locally by using C<< $x->accuracy() >> or
4288 C<< $x->precision() >>
4289 and thus force different A and P for different objects/numbers.
4290 * Setting A or P this way immediately rounds $x to the new value.
4291 * C<< $x->accuracy() >> clears C<< $x->precision() >>, and vice versa.
4295 * the rounding routines will use the respective global or local settings.
4296 fround()/bround() is for accuracy rounding, while ffround()/bfround()
4298 * the two rounding functions take as the second parameter one of the
4299 following rounding modes (R):
4300 'even', 'odd', '+inf', '-inf', 'zero', 'trunc', 'common'
4301 * you can set/get the global R by using C<< Math::SomeClass->round_mode() >>
4302 or by setting C<< $Math::SomeClass::round_mode >>
4303 * after each operation, C<< $result->round() >> is called, and the result may
4304 eventually be rounded (that is, if A or P were set either locally,
4305 globally or as parameter to the operation)
4306 * to manually round a number, call C<< $x->round($A,$P,$round_mode); >>
4307 this will round the number by using the appropriate rounding function
4308 and then normalize it.
4309 * rounding modifies the local settings of the number:
4311 $x = Math::BigFloat->new(123.456);
4315 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
4316 will be 4 from now on.
4318 =item Default values
4327 * The defaults are set up so that the new code gives the same results as
4328 the old code (except in a few cases on fdiv):
4329 + Both A and P are undefined and thus will not be used for rounding
4330 after each operation.
4331 + round() is thus a no-op, unless given extra parameters A and P
4335 =head1 Infinity and Not a Number
4337 While BigInt has extensive handling of inf and NaN, certain quirks remain.
4343 These perl routines currently (as of Perl v.5.8.6) cannot handle passed
4346 te@linux:~> perl -wle 'print 2 ** 3333'
4348 te@linux:~> perl -wle 'print 2 ** 3333 == 2 ** 3333'
4350 te@linux:~> perl -wle 'print oct(2 ** 3333)'
4352 te@linux:~> perl -wle 'print hex(2 ** 3333)'
4353 Illegal hexadecimal digit 'i' ignored at -e line 1.
4356 The same problems occur if you pass them Math::BigInt->binf() objects. Since
4357 overloading these routines is not possible, this cannot be fixed from BigInt.
4359 =item ==, !=, <, >, <=, >= with NaNs
4361 BigInt's bcmp() routine currently returns undef to signal that a NaN was
4362 involved in a comparison. However, the overload code turns that into
4363 either 1 or '' and thus operations like C<< NaN != NaN >> might return
4368 C<< log(-inf) >> is highly weird. Since log(-x)=pi*i+log(x), then
4369 log(-inf)=pi*i+inf. However, since the imaginary part is finite, the real
4370 infinity "overshadows" it, so the number might as well just be infinity.
4371 However, the result is a complex number, and since BigInt/BigFloat can only
4372 have real numbers as results, the result is NaN.
4374 =item exp(), cos(), sin(), atan2()
4376 These all might have problems handling infinity right.
4382 The actual numbers are stored as unsigned big integers (with seperate sign).
4384 You should neither care about nor depend on the internal representation; it
4385 might change without notice. Use B<ONLY> method calls like C<< $x->sign(); >>
4386 instead relying on the internal representation.
4390 Math with the numbers is done (by default) by a module called
4391 C<Math::BigInt::Calc>. This is equivalent to saying:
4393 use Math::BigInt lib => 'Calc';
4395 You can change this by using:
4397 use Math::BigInt lib => 'BitVect';
4399 The following would first try to find Math::BigInt::Foo, then
4400 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
4402 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
4404 Since Math::BigInt::GMP is in almost all cases faster than Calc (especially in
4405 math involving really big numbers, where it is B<much> faster), and there is
4406 no penalty if Math::BigInt::GMP is not installed, it is a good idea to always
4409 use Math::BigInt lib => 'GMP';
4411 Different low-level libraries use different formats to store the
4412 numbers. You should B<NOT> depend on the number having a specific format
4415 See the respective math library module documentation for further details.
4419 The sign is either '+', '-', 'NaN', '+inf' or '-inf'.
4421 A sign of 'NaN' is used to represent the result when input arguments are not
4422 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
4423 minus infinity. You will get '+inf' when dividing a positive number by 0, and
4424 '-inf' when dividing any negative number by 0.
4426 =head2 mantissa(), exponent() and parts()
4428 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
4431 $m = $x->mantissa();
4432 $e = $x->exponent();
4433 $y = $m * ( 10 ** $e );
4434 print "ok\n" if $x == $y;
4436 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
4437 in one go. Both the returned mantissa and exponent have a sign.
4439 Currently, for BigInts C<$e> is always 0, except +inf and -inf, where it is
4440 C<+inf>; and for NaN, where it is C<NaN>; and for C<$x == 0>, where it is C<1>
4441 (to be compatible with Math::BigFloat's internal representation of a zero as
4444 C<$m> is currently just a copy of the original number. The relation between
4445 C<$e> and C<$m> will stay always the same, though their real values might
4452 sub bint { Math::BigInt->new(shift); }
4454 $x = Math::BigInt->bstr("1234") # string "1234"
4455 $x = "$x"; # same as bstr()
4456 $x = Math::BigInt->bneg("1234"); # BigInt "-1234"
4457 $x = Math::BigInt->babs("-12345"); # BigInt "12345"
4458 $x = Math::BigInt->bnorm("-0.00"); # BigInt "0"
4459 $x = bint(1) + bint(2); # BigInt "3"
4460 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
4461 $x = bint(1); # BigInt "1"
4462 $x = $x + 5 / 2; # BigInt "3"
4463 $x = $x ** 3; # BigInt "27"
4464 $x *= 2; # BigInt "54"
4465 $x = Math::BigInt->new(0); # BigInt "0"
4467 $x = Math::BigInt->badd(4,5) # BigInt "9"
4468 print $x->bsstr(); # 9e+0
4470 Examples for rounding:
4475 $x = Math::BigFloat->new(123.4567);
4476 $y = Math::BigFloat->new(123.456789);
4477 Math::BigFloat->accuracy(4); # no more A than 4
4479 ok ($x->copy()->fround(),123.4); # even rounding
4480 print $x->copy()->fround(),"\n"; # 123.4
4481 Math::BigFloat->round_mode('odd'); # round to odd
4482 print $x->copy()->fround(),"\n"; # 123.5
4483 Math::BigFloat->accuracy(5); # no more A than 5
4484 Math::BigFloat->round_mode('odd'); # round to odd
4485 print $x->copy()->fround(),"\n"; # 123.46
4486 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
4487 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
4489 Math::BigFloat->accuracy(undef); # A not important now
4490 Math::BigFloat->precision(2); # P important
4491 print $x->copy()->bnorm(),"\n"; # 123.46
4492 print $x->copy()->fround(),"\n"; # 123.46
4494 Examples for converting:
4496 my $x = Math::BigInt->new('0b1'.'01' x 123);
4497 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
4499 =head1 Autocreating constants
4501 After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
4502 and binary constants in the given scope are converted to C<Math::BigInt>.
4503 This conversion happens at compile time.
4507 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
4509 prints the integer value of C<2**100>. Note that without conversion of
4510 constants the expression 2**100 will be calculated as perl scalar.
4512 Please note that strings and floating point constants are not affected,
4515 use Math::BigInt qw/:constant/;
4517 $x = 1234567890123456789012345678901234567890
4518 + 123456789123456789;
4519 $y = '1234567890123456789012345678901234567890'
4520 + '123456789123456789';
4522 do not work. You need an explicit Math::BigInt->new() around one of the
4523 operands. You should also quote large constants to protect loss of precision:
4527 $x = Math::BigInt->new('1234567889123456789123456789123456789');
4529 Without the quotes Perl would convert the large number to a floating point
4530 constant at compile time and then hand the result to BigInt, which results in
4531 an truncated result or a NaN.
4533 This also applies to integers that look like floating point constants:
4535 use Math::BigInt ':constant';
4537 print ref(123e2),"\n";
4538 print ref(123.2e2),"\n";
4540 will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
4541 to get this to work.
4545 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
4546 must be made in the second case. For long numbers, the copy can eat up to 20%
4547 of the work (in the case of addition/subtraction, less for
4548 multiplication/division). If $y is very small compared to $x, the form
4549 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
4550 more time then the actual addition.
4552 With a technique called copy-on-write, the cost of copying with overload could
4553 be minimized or even completely avoided. A test implementation of COW did show
4554 performance gains for overloaded math, but introduced a performance loss due
4555 to a constant overhead for all other operations. So Math::BigInt does currently
4558 The rewritten version of this module (vs. v0.01) is slower on certain
4559 operations, like C<new()>, C<bstr()> and C<numify()>. The reason are that it
4560 does now more work and handles much more cases. The time spent in these
4561 operations is usually gained in the other math operations so that code on
4562 the average should get (much) faster. If they don't, please contact the author.
4564 Some operations may be slower for small numbers, but are significantly faster
4565 for big numbers. Other operations are now constant (O(1), like C<bneg()>,
4566 C<babs()> etc), instead of O(N) and thus nearly always take much less time.
4567 These optimizations were done on purpose.
4569 If you find the Calc module to slow, try to install any of the replacement
4570 modules and see if they help you.
4572 =head2 Alternative math libraries
4574 You can use an alternative library to drive Math::BigInt via:
4576 use Math::BigInt lib => 'Module';
4578 See L<MATH LIBRARY> for more information.
4580 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
4584 =head1 Subclassing Math::BigInt
4586 The basic design of Math::BigInt allows simple subclasses with very little
4587 work, as long as a few simple rules are followed:
4593 The public API must remain consistent, i.e. if a sub-class is overloading
4594 addition, the sub-class must use the same name, in this case badd(). The
4595 reason for this is that Math::BigInt is optimized to call the object methods
4600 The private object hash keys like C<$x->{sign}> may not be changed, but
4601 additional keys can be added, like C<$x->{_custom}>.
4605 Accessor functions are available for all existing object hash keys and should
4606 be used instead of directly accessing the internal hash keys. The reason for
4607 this is that Math::BigInt itself has a pluggable interface which permits it
4608 to support different storage methods.
4612 More complex sub-classes may have to replicate more of the logic internal of
4613 Math::BigInt if they need to change more basic behaviors. A subclass that
4614 needs to merely change the output only needs to overload C<bstr()>.
4616 All other object methods and overloaded functions can be directly inherited
4617 from the parent class.
4619 At the very minimum, any subclass will need to provide its own C<new()> and can
4620 store additional hash keys in the object. There are also some package globals
4621 that must be defined, e.g.:
4625 $precision = -2; # round to 2 decimal places
4626 $round_mode = 'even';
4629 Additionally, you might want to provide the following two globals to allow
4630 auto-upgrading and auto-downgrading to work correctly:
4635 This allows Math::BigInt to correctly retrieve package globals from the
4636 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
4637 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
4643 in your subclass to automatically inherit the overloading from the parent. If
4644 you like, you can change part of the overloading, look at Math::String for an
4649 When used like this:
4651 use Math::BigInt upgrade => 'Foo::Bar';
4653 certain operations will 'upgrade' their calculation and thus the result to
4654 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
4656 use Math::BigInt upgrade => 'Math::BigFloat';
4658 As a shortcut, you can use the module C<bignum>:
4662 Also good for oneliners:
4664 perl -Mbignum -le 'print 2 ** 255'
4666 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
4667 as well es preserve accuracy (as in sqrt(3)).
4669 Beware: This feature is not fully implemented yet.
4673 The following methods upgrade themselves unconditionally; that is if upgrade
4674 is in effect, they will always hand up their work:
4688 Beware: This list is not complete.
4690 All other methods upgrade themselves only when one (or all) of their
4691 arguments are of the class mentioned in $upgrade (This might change in later
4692 versions to a more sophisticated scheme):
4696 C<Math::BigInt> exports nothing by default, but can export the following methods:
4703 Some things might not work as you expect them. Below is documented what is
4704 known to be troublesome:
4708 =item bstr(), bsstr() and 'cmp'
4710 Both C<bstr()> and C<bsstr()> as well as automated stringify via overload now
4711 drop the leading '+'. The old code would return '+3', the new returns '3'.
4712 This is to be consistent with Perl and to make C<cmp> (especially with
4713 overloading) to work as you expect. It also solves problems with C<Test.pm>,
4714 because its C<ok()> uses 'eq' internally.
4716 Mark Biggar said, when asked about to drop the '+' altogether, or make only
4719 I agree (with the first alternative), don't add the '+' on positive
4720 numbers. It's not as important anymore with the new internal
4721 form for numbers. It made doing things like abs and neg easier,
4722 but those have to be done differently now anyway.
4724 So, the following examples will now work all as expected:
4727 BEGIN { plan tests => 1 }
4730 my $x = new Math::BigInt 3*3;
4731 my $y = new Math::BigInt 3*3;
4734 print "$x eq 9" if $x eq $y;
4735 print "$x eq 9" if $x eq '9';
4736 print "$x eq 9" if $x eq 3*3;
4738 Additionally, the following still works:
4740 print "$x == 9" if $x == $y;
4741 print "$x == 9" if $x == 9;
4742 print "$x == 9" if $x == 3*3;
4744 There is now a C<bsstr()> method to get the string in scientific notation aka
4745 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
4746 for comparison, but Perl will represent some numbers as 100 and others
4747 as 1e+308. If in doubt, convert both arguments to Math::BigInt before
4748 comparing them as strings:
4751 BEGIN { plan tests => 3 }
4754 $x = Math::BigInt->new('1e56'); $y = 1e56;
4755 ok ($x,$y); # will fail
4756 ok ($x->bsstr(),$y); # okay
4757 $y = Math::BigInt->new($y);
4760 Alternatively, simple use C<< <=> >> for comparisons, this will get it
4761 always right. There is not yet a way to get a number automatically represented
4762 as a string that matches exactly the way Perl represents it.
4764 See also the section about L<Infinity and Not a Number> for problems in
4769 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
4772 $x = Math::BigInt->new(123);
4773 $y = int($x); # BigInt 123
4774 $x = Math::BigFloat->new(123.45);
4775 $y = int($x); # BigInt 123
4777 In all Perl versions you can use C<as_number()> or C<as_int> for the same
4780 $x = Math::BigFloat->new(123.45);
4781 $y = $x->as_number(); # BigInt 123
4782 $y = $x->as_int(); # ditto
4784 This also works for other subclasses, like Math::String.
4786 If you want a real Perl scalar, use C<numify()>:
4788 $y = $x->numify(); # 123 as scalar
4790 This is seldom necessary, though, because this is done automatically, like
4791 when you access an array:
4793 $z = $array[$x]; # does work automatically
4797 The following will probably not do what you expect:
4799 $c = Math::BigInt->new(123);
4800 print $c->length(),"\n"; # prints 30
4802 It prints both the number of digits in the number and in the fraction part
4803 since print calls C<length()> in list context. Use something like:
4805 print scalar $c->length(),"\n"; # prints 3
4809 The following will probably not do what you expect:
4811 print $c->bdiv(10000),"\n";
4813 It prints both quotient and remainder since print calls C<bdiv()> in list
4814 context. Also, C<bdiv()> will modify $c, so be careful. You probably want
4817 print $c / 10000,"\n";
4818 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
4822 The quotient is always the greatest integer less than or equal to the
4823 real-valued quotient of the two operands, and the remainder (when it is
4824 nonzero) always has the same sign as the second operand; so, for
4834 As a consequence, the behavior of the operator % agrees with the
4835 behavior of Perl's built-in % operator (as documented in the perlop
4836 manpage), and the equation
4838 $x == ($x / $y) * $y + ($x % $y)
4840 holds true for any $x and $y, which justifies calling the two return
4841 values of bdiv() the quotient and remainder. The only exception to this rule
4842 are when $y == 0 and $x is negative, then the remainder will also be
4843 negative. See below under "infinity handling" for the reasoning behind this.
4845 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
4846 not change BigInt's way to do things. This is because under 'use integer' Perl
4847 will do what the underlying C thinks is right and this is different for each
4848 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
4849 the author to implement it ;)
4851 =item infinity handling
4853 Here are some examples that explain the reasons why certain results occur while
4856 The following table shows the result of the division and the remainder, so that
4857 the equation above holds true. Some "ordinary" cases are strewn in to show more
4858 clearly the reasoning:
4860 A / B = C, R so that C * B + R = A
4861 =========================================================
4862 5 / 8 = 0, 5 0 * 8 + 5 = 5
4863 0 / 8 = 0, 0 0 * 8 + 0 = 0
4864 0 / inf = 0, 0 0 * inf + 0 = 0
4865 0 /-inf = 0, 0 0 * -inf + 0 = 0
4866 5 / inf = 0, 5 0 * inf + 5 = 5
4867 5 /-inf = 0, 5 0 * -inf + 5 = 5
4868 -5/ inf = 0, -5 0 * inf + -5 = -5
4869 -5/-inf = 0, -5 0 * -inf + -5 = -5
4870 inf/ 5 = inf, 0 inf * 5 + 0 = inf
4871 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
4872 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
4873 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
4874 5/ 5 = 1, 0 1 * 5 + 0 = 5
4875 -5/ -5 = 1, 0 1 * -5 + 0 = -5
4876 inf/ inf = 1, 0 1 * inf + 0 = inf
4877 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
4878 inf/-inf = -1, 0 -1 * -inf + 0 = inf
4879 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
4880 8/ 0 = inf, 8 inf * 0 + 8 = 8
4881 inf/ 0 = inf, inf inf * 0 + inf = inf
4884 These cases below violate the "remainder has the sign of the second of the two
4885 arguments", since they wouldn't match up otherwise.
4887 A / B = C, R so that C * B + R = A
4888 ========================================================
4889 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
4890 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
4892 =item Modifying and =
4896 $x = Math::BigFloat->new(5);
4899 It will not do what you think, e.g. making a copy of $x. Instead it just makes
4900 a second reference to the B<same> object and stores it in $y. Thus anything
4901 that modifies $x (except overloaded operators) will modify $y, and vice versa.
4902 Or in other words, C<=> is only safe if you modify your BigInts only via
4903 overloaded math. As soon as you use a method call it breaks:
4906 print "$x, $y\n"; # prints '10, 10'
4908 If you want a true copy of $x, use:
4912 You can also chain the calls like this, this will make first a copy and then
4915 $y = $x->copy()->bmul(2);
4917 See also the documentation for overload.pm regarding C<=>.
4921 C<bpow()> (and the rounding functions) now modifies the first argument and
4922 returns it, unlike the old code which left it alone and only returned the
4923 result. This is to be consistent with C<badd()> etc. The first three will
4924 modify $x, the last one won't:
4926 print bpow($x,$i),"\n"; # modify $x
4927 print $x->bpow($i),"\n"; # ditto
4928 print $x **= $i,"\n"; # the same
4929 print $x ** $i,"\n"; # leave $x alone
4931 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4933 =item Overloading -$x
4943 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4944 needs to preserve $x since it does not know that it later will get overwritten.
4945 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4947 =item Mixing different object types
4949 In Perl you will get a floating point value if you do one of the following:
4955 With overloaded math, only the first two variants will result in a BigFloat:
4960 $mbf = Math::BigFloat->new(5);
4961 $mbi2 = Math::BigInteger->new(5);
4962 $mbi = Math::BigInteger->new(2);
4964 # what actually gets called:
4965 $float = $mbf + $mbi; # $mbf->badd()
4966 $float = $mbf / $mbi; # $mbf->bdiv()
4967 $integer = $mbi + $mbf; # $mbi->badd()
4968 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4969 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4971 This is because math with overloaded operators follows the first (dominating)
4972 operand, and the operation of that is called and returns thus the result. So,
4973 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4974 the result should be a Math::BigFloat or the second operant is one.
4976 To get a Math::BigFloat you either need to call the operation manually,
4977 make sure the operands are already of the proper type or casted to that type
4978 via Math::BigFloat->new():
4980 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4982 Beware of simple "casting" the entire expression, this would only convert
4983 the already computed result:
4985 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4987 Beware also of the order of more complicated expressions like:
4989 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4990 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4992 If in doubt, break the expression into simpler terms, or cast all operands
4993 to the desired resulting type.
4995 Scalar values are a bit different, since:
5000 will both result in the proper type due to the way the overloaded math works.
5002 This section also applies to other overloaded math packages, like Math::String.
5004 One solution to you problem might be autoupgrading|upgrading. See the
5005 pragmas L<bignum>, L<bigint> and L<bigrat> for an easy way to do this.
5009 C<bsqrt()> works only good if the result is a big integer, e.g. the square
5010 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
5011 mode. The reason is that the result is always truncated to an integer.
5013 If you want a better approximation of the square root, then use:
5015 $x = Math::BigFloat->new(12);
5016 Math::BigFloat->precision(0);
5017 Math::BigFloat->round_mode('even');
5018 print $x->copy->bsqrt(),"\n"; # 4
5020 Math::BigFloat->precision(2);
5021 print $x->bsqrt(),"\n"; # 3.46
5022 print $x->bsqrt(3),"\n"; # 3.464
5026 For negative numbers in base see also L<brsft|brsft>.
5032 This program is free software; you may redistribute it and/or modify it under
5033 the same terms as Perl itself.
5037 L<Math::BigFloat>, L<Math::BigRat> and L<Math::Big> as well as
5038 L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
5040 The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest
5041 because they solve the autoupgrading/downgrading issue, at least partly.
5044 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
5045 more documentation including a full version history, testcases, empty
5046 subclass files and benchmarks.
5050 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
5051 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001 - 2006
5052 and still at it in 2007.
5054 Many people contributed in one or more ways to the final beast, see the file
5055 CREDITS for an (incomplete) list. If you miss your name, please drop me a