4 # "Mike had an infinite amount to do and a negative amount of time in which
5 # to do it." - Before and After
8 # The following hash values are used:
9 # value: unsigned int with actual value (as a Math::BigInt::Calc or similiar)
10 # sign : +,-,NaN,+inf,-inf
13 # _f : flags, used by MBF to flag parts of a float as untouchable
15 # Remember not to take shortcuts ala $xs = $x->{value}; $CALC->foo($xs); since
16 # underlying lib might change the reference!
18 my $class = "Math::BigInt";
23 @ISA = qw( Exporter );
24 @EXPORT_OK = qw( objectify bgcd blcm);
25 # _trap_inf and _trap_nan are internal and should never be accessed from the
27 use vars qw/$round_mode $accuracy $precision $div_scale $rnd_mode
28 $upgrade $downgrade $_trap_nan $_trap_inf/;
31 # Inside overload, the first arg is always an object. If the original code had
32 # it reversed (like $x = 2 * $y), then the third paramater is true.
33 # In some cases (like add, $x = $x + 2 is the same as $x = 2 + $x) this makes
34 # no difference, but in some cases it does.
36 # For overloaded ops with only one argument we simple use $_[0]->copy() to
37 # preserve the argument.
39 # Thus inheritance of overload operators becomes possible and transparent for
40 # our subclasses without the need to repeat the entire overload section there.
43 '=' => sub { $_[0]->copy(); },
45 # some shortcuts for speed (assumes that reversed order of arguments is routed
46 # to normal '+' and we thus can always modify first arg. If this is changed,
47 # this breaks and must be adjusted.)
48 '+=' => sub { $_[0]->badd($_[1]); },
49 '-=' => sub { $_[0]->bsub($_[1]); },
50 '*=' => sub { $_[0]->bmul($_[1]); },
51 '/=' => sub { scalar $_[0]->bdiv($_[1]); },
52 '%=' => sub { $_[0]->bmod($_[1]); },
53 '^=' => sub { $_[0]->bxor($_[1]); },
54 '&=' => sub { $_[0]->band($_[1]); },
55 '|=' => sub { $_[0]->bior($_[1]); },
56 '**=' => sub { $_[0]->bpow($_[1]); },
58 # not supported by Perl yet
59 '..' => \&_pointpoint,
61 '<=>' => sub { $_[2] ?
62 ref($_[0])->bcmp($_[1],$_[0]) :
66 "$_[1]" cmp $_[0]->bstr() :
67 $_[0]->bstr() cmp "$_[1]" },
69 # make cos()/sin()/exp() "work" with BigInt's or subclasses
70 'cos' => sub { cos($_[0]->numify()) },
71 'sin' => sub { sin($_[0]->numify()) },
72 'exp' => sub { exp($_[0]->numify()) },
73 'atan2' => sub { atan2($_[0]->numify(),$_[1]) },
75 'log' => sub { $_[0]->copy()->blog($_[1]); },
76 'int' => sub { $_[0]->copy(); },
77 'neg' => sub { $_[0]->copy()->bneg(); },
78 'abs' => sub { $_[0]->copy()->babs(); },
79 'sqrt' => sub { $_[0]->copy()->bsqrt(); },
80 '~' => sub { $_[0]->copy()->bnot(); },
82 # for sub it is a bit tricky to keep b: b-a => -a+b
83 '-' => sub { my $c = $_[0]->copy; $_[2] ?
84 $c->bneg()->badd($_[1]) :
86 '+' => sub { $_[0]->copy()->badd($_[1]); },
87 '*' => sub { $_[0]->copy()->bmul($_[1]); },
90 $_[2] ? ref($_[0])->new($_[1])->bdiv($_[0]) : $_[0]->copy->bdiv($_[1]);
93 $_[2] ? ref($_[0])->new($_[1])->bmod($_[0]) : $_[0]->copy->bmod($_[1]);
96 $_[2] ? ref($_[0])->new($_[1])->bpow($_[0]) : $_[0]->copy->bpow($_[1]);
99 $_[2] ? ref($_[0])->new($_[1])->blsft($_[0]) : $_[0]->copy->blsft($_[1]);
102 $_[2] ? ref($_[0])->new($_[1])->brsft($_[0]) : $_[0]->copy->brsft($_[1]);
105 $_[2] ? ref($_[0])->new($_[1])->band($_[0]) : $_[0]->copy->band($_[1]);
108 $_[2] ? ref($_[0])->new($_[1])->bior($_[0]) : $_[0]->copy->bior($_[1]);
111 $_[2] ? ref($_[0])->new($_[1])->bxor($_[0]) : $_[0]->copy->bxor($_[1]);
114 # can modify arg of ++ and --, so avoid a copy() for speed, but don't
115 # use $_[0]->bone(), it would modify $_[0] to be 1!
116 '++' => sub { $_[0]->binc() },
117 '--' => sub { $_[0]->bdec() },
119 # if overloaded, O(1) instead of O(N) and twice as fast for small numbers
121 # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/
122 # v5.6.1 dumps on this: return !$_[0]->is_zero() || undef; :-(
123 my $t = !$_[0]->is_zero();
128 # the original qw() does not work with the TIESCALAR below, why?
129 # Order of arguments unsignificant
130 '""' => sub { $_[0]->bstr(); },
131 '0+' => sub { $_[0]->numify(); }
134 ##############################################################################
135 # global constants, flags and accessory
137 # these are public, but their usage is not recommended, use the accessor
140 $round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
145 $upgrade = undef; # default is no upgrade
146 $downgrade = undef; # default is no downgrade
148 # these are internally, and not to be used from the outside
150 sub MB_NEVER_ROUND () { 0x0001; }
152 $_trap_nan = 0; # are NaNs ok? set w/ config()
153 $_trap_inf = 0; # are infs ok? set w/ config()
154 my $nan = 'NaN'; # constants for easier life
156 my $CALC = 'Math::BigInt::Calc'; # module to do the low level math
158 my %CAN; # cache for $CALC->can(...)
159 my $IMPORT = 0; # was import() called yet?
160 # used to make require work
162 my $EMU_LIB = 'Math/BigInt/CalcEmu.pm'; # emulate low-level math
163 my $EMU = 'Math::BigInt::CalcEmu'; # emulate low-level math
165 ##############################################################################
166 # the old code had $rnd_mode, so we need to support it, too
169 sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; }
170 sub FETCH { return $round_mode; }
171 sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
175 # tie to enable $rnd_mode to work transparently
176 tie $rnd_mode, 'Math::BigInt';
178 # set up some handy alias names
179 *as_int = \&as_number;
180 *is_pos = \&is_positive;
181 *is_neg = \&is_negative;
184 ##############################################################################
189 # make Class->round_mode() work
191 my $class = ref($self) || $self || __PACKAGE__;
195 if ($m !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
197 require Carp; Carp::croak ("Unknown round mode '$m'");
199 return ${"${class}::round_mode"} = $m;
201 ${"${class}::round_mode"};
207 # make Class->upgrade() work
209 my $class = ref($self) || $self || __PACKAGE__;
210 # need to set new value?
214 return ${"${class}::upgrade"} = $u;
216 ${"${class}::upgrade"};
222 # make Class->downgrade() work
224 my $class = ref($self) || $self || __PACKAGE__;
225 # need to set new value?
229 return ${"${class}::downgrade"} = $u;
231 ${"${class}::downgrade"};
237 # make Class->div_scale() work
239 my $class = ref($self) || $self || __PACKAGE__;
244 require Carp; Carp::croak ('div_scale must be greater than zero');
246 ${"${class}::div_scale"} = shift;
248 ${"${class}::div_scale"};
253 # $x->accuracy($a); ref($x) $a
254 # $x->accuracy(); ref($x)
255 # Class->accuracy(); class
256 # Class->accuracy($a); class $a
259 my $class = ref($x) || $x || __PACKAGE__;
262 # need to set new value?
266 # convert objects to scalars to avoid deep recursion. If object doesn't
267 # have numify(), then hopefully it will have overloading for int() and
268 # boolean test without wandering into a deep recursion path...
269 $a = $a->numify() if ref($a) && $a->can('numify');
273 # also croak on non-numerical
277 Carp::croak ('Argument to accuracy must be greater than zero');
281 require Carp; Carp::croak ('Argument to accuracy must be an integer');
286 # $object->accuracy() or fallback to global
287 $x->bround($a) if $a; # not for undef, 0
288 $x->{_a} = $a; # set/overwrite, even if not rounded
289 $x->{_p} = undef; # clear P
290 $a = ${"${class}::accuracy"} unless defined $a; # proper return value
295 ${"${class}::accuracy"} = $a;
296 ${"${class}::precision"} = undef; # clear P
298 return $a; # shortcut
302 # $object->accuracy() or fallback to global
303 $r = $x->{_a} if ref($x);
304 # but don't return global undef, when $x's accuracy is 0!
305 $r = ${"${class}::accuracy"} if !defined $r;
311 # $x->precision($p); ref($x) $p
312 # $x->precision(); ref($x)
313 # Class->precision(); class
314 # Class->precision($p); class $p
317 my $class = ref($x) || $x || __PACKAGE__;
323 # convert objects to scalars to avoid deep recursion. If object doesn't
324 # have numify(), then hopefully it will have overloading for int() and
325 # boolean test without wandering into a deep recursion path...
326 $p = $p->numify() if ref($p) && $p->can('numify');
327 if ((defined $p) && (int($p) != $p))
329 require Carp; Carp::croak ('Argument to precision must be an integer');
333 # $object->precision() or fallback to global
334 $x->bfround($p) if $p; # not for undef, 0
335 $x->{_p} = $p; # set/overwrite, even if not rounded
336 $x->{_a} = undef; # clear A
337 $p = ${"${class}::precision"} unless defined $p; # proper return value
342 ${"${class}::precision"} = $p;
343 ${"${class}::accuracy"} = undef; # clear A
345 return $p; # shortcut
349 # $object->precision() or fallback to global
350 $r = $x->{_p} if ref($x);
351 # but don't return global undef, when $x's precision is 0!
352 $r = ${"${class}::precision"} if !defined $r;
358 # return (or set) configuration data as hash ref
359 my $class = shift || 'Math::BigInt';
364 # try to set given options as arguments from hash
367 if (ref($args) ne 'HASH')
371 # these values can be "set"
375 upgrade downgrade precision accuracy round_mode div_scale/
378 $set_args->{$key} = $args->{$key} if exists $args->{$key};
379 delete $args->{$key};
384 Carp::croak ("Illegal key(s) '",
385 join("','",keys %$args),"' passed to $class\->config()");
387 foreach my $key (keys %$set_args)
389 if ($key =~ /^trap_(inf|nan)\z/)
391 ${"${class}::_trap_$1"} = ($set_args->{"trap_$1"} ? 1 : 0);
394 # use a call instead of just setting the $variable to check argument
395 $class->$key($set_args->{$key});
399 # now return actual configuration
403 lib_version => ${"${CALC}::VERSION"},
405 trap_nan => ${"${class}::_trap_nan"},
406 trap_inf => ${"${class}::_trap_inf"},
407 version => ${"${class}::VERSION"},
410 upgrade downgrade precision accuracy round_mode div_scale
413 $cfg->{$key} = ${"${class}::$key"};
420 # select accuracy parameter based on precedence,
421 # used by bround() and bfround(), may return undef for scale (means no op)
422 my ($x,$s,$m,$scale,$mode) = @_;
423 $scale = $x->{_a} if !defined $scale;
424 $scale = $s if (!defined $scale);
425 $mode = $m if !defined $mode;
426 return ($scale,$mode);
431 # select precision parameter based on precedence,
432 # used by bround() and bfround(), may return undef for scale (means no op)
433 my ($x,$s,$m,$scale,$mode) = @_;
434 $scale = $x->{_p} if !defined $scale;
435 $scale = $s if (!defined $scale);
436 $mode = $m if !defined $mode;
437 return ($scale,$mode);
440 ##############################################################################
448 # if two arguments, the first one is the class to "swallow" subclasses
456 return unless ref($x); # only for objects
458 my $self = {}; bless $self,$c;
460 foreach my $k (keys %$x)
464 $self->{value} = $CALC->_copy($x->{value}); next;
466 if (!($r = ref($x->{$k})))
468 $self->{$k} = $x->{$k}; next;
472 $self->{$k} = \${$x->{$k}};
474 elsif ($r eq 'ARRAY')
476 $self->{$k} = [ @{$x->{$k}} ];
480 # only one level deep!
481 foreach my $h (keys %{$x->{$k}})
483 $self->{$k}->{$h} = $x->{$k}->{$h};
489 if ($xk->can('copy'))
491 $self->{$k} = $xk->copy();
495 $self->{$k} = $xk->new($xk);
504 # create a new BigInt object from a string or another BigInt object.
505 # see hash keys documented at top
507 # the argument could be an object, so avoid ||, && etc on it, this would
508 # cause costly overloaded code to be called. The only allowed ops are
511 my ($class,$wanted,$a,$p,$r) = @_;
513 # avoid numify-calls by not using || on $wanted!
514 return $class->bzero($a,$p) if !defined $wanted; # default to 0
515 return $class->copy($wanted,$a,$p,$r)
516 if ref($wanted) && $wanted->isa($class); # MBI or subclass
518 $class->import() if $IMPORT == 0; # make require work
520 my $self = bless {}, $class;
522 # shortcut for "normal" numbers
523 if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*\z/))
525 $self->{sign} = $1 || '+';
527 if ($wanted =~ /^[+-]/)
529 # remove sign without touching wanted to make it work with constants
530 my $t = $wanted; $t =~ s/^[+-]//; $ref = \$t;
532 # force to string version (otherwise Pari is unhappy about overflowed
533 # constants, for instance)
534 # not good, BigInt shouldn't need to know about alternative libs:
535 # $ref = \"$$ref" if $CALC eq 'Math::BigInt::Pari';
536 $self->{value} = $CALC->_new($ref);
538 if ( (defined $a) || (defined $p)
539 || (defined ${"${class}::precision"})
540 || (defined ${"${class}::accuracy"})
543 $self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p);
548 # handle '+inf', '-inf' first
549 if ($wanted =~ /^[+-]?inf$/)
551 $self->{value} = $CALC->_zero();
552 $self->{sign} = $wanted; $self->{sign} = '+inf' if $self->{sign} eq 'inf';
555 # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign
556 my ($mis,$miv,$mfv,$es,$ev) = _split(\$wanted);
561 require Carp; Carp::croak("$wanted is not a number in $class");
563 $self->{value} = $CALC->_zero();
564 $self->{sign} = $nan;
569 # _from_hex or _from_bin
570 $self->{value} = $mis->{value};
571 $self->{sign} = $mis->{sign};
572 return $self; # throw away $mis
574 # make integer from mantissa by adjusting exp, then convert to bigint
575 $self->{sign} = $$mis; # store sign
576 $self->{value} = $CALC->_zero(); # for all the NaN cases
577 my $e = int("$$es$$ev"); # exponent (avoid recursion)
580 my $diff = $e - CORE::length($$mfv);
581 if ($diff < 0) # Not integer
585 require Carp; Carp::croak("$wanted not an integer in $class");
588 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
589 $self->{sign} = $nan;
593 # adjust fraction and add it to value
594 #print "diff > 0 $$miv\n";
595 $$miv = $$miv . ($$mfv . '0' x $diff);
600 if ($$mfv ne '') # e <= 0
602 # fraction and negative/zero E => NOI
605 require Carp; Carp::croak("$wanted not an integer in $class");
607 #print "NOI 2 \$\$mfv '$$mfv'\n";
608 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
609 $self->{sign} = $nan;
613 # xE-y, and empty mfv
616 if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
620 require Carp; Carp::croak("$wanted not an integer in $class");
623 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
624 $self->{sign} = $nan;
628 $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
629 $self->{value} = $CALC->_new($miv) if $self->{sign} =~ /^[+-]$/;
630 # if any of the globals is set, use them to round and store them inside $self
631 # do not round for new($x,undef,undef) since that is used by MBF to signal
633 $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p;
639 # create a bigint 'NaN', if given a BigInt, set it to 'NaN'
641 $self = $class if !defined $self;
644 my $c = $self; $self = {}; bless $self, $c;
647 if (${"${class}::_trap_nan"})
650 Carp::croak ("Tried to set $self to NaN in $class\::bnan()");
652 $self->import() if $IMPORT == 0; # make require work
653 return if $self->modify('bnan');
654 if ($self->can('_bnan'))
656 # use subclass to initialize
661 # otherwise do our own thing
662 $self->{value} = $CALC->_zero();
664 $self->{sign} = $nan;
665 delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly
671 # create a bigint '+-inf', if given a BigInt, set it to '+-inf'
672 # the sign is either '+', or if given, used from there
674 my $sign = shift; $sign = '+' if !defined $sign || $sign !~ /^-(inf)?$/;
675 $self = $class if !defined $self;
678 my $c = $self; $self = {}; bless $self, $c;
681 if (${"${class}::_trap_inf"})
684 Carp::croak ("Tried to set $self to +-inf in $class\::binfn()");
686 $self->import() if $IMPORT == 0; # make require work
687 return if $self->modify('binf');
688 if ($self->can('_binf'))
690 # use subclass to initialize
695 # otherwise do our own thing
696 $self->{value} = $CALC->_zero();
698 $sign = $sign . 'inf' if $sign !~ /inf$/; # - => -inf
699 $self->{sign} = $sign;
700 ($self->{_a},$self->{_p}) = @_; # take over requested rounding
706 # create a bigint '+0', if given a BigInt, set it to 0
708 $self = $class if !defined $self;
712 my $c = $self; $self = {}; bless $self, $c;
714 $self->import() if $IMPORT == 0; # make require work
715 return if $self->modify('bzero');
717 if ($self->can('_bzero'))
719 # use subclass to initialize
724 # otherwise do our own thing
725 $self->{value} = $CALC->_zero();
732 # call like: $x->bzero($a,$p,$r,$y);
733 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
738 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
740 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
748 # create a bigint '+1' (or -1 if given sign '-'),
749 # if given a BigInt, set it to +1 or -1, respecively
751 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
752 $self = $class if !defined $self;
756 my $c = $self; $self = {}; bless $self, $c;
758 $self->import() if $IMPORT == 0; # make require work
759 return if $self->modify('bone');
761 if ($self->can('_bone'))
763 # use subclass to initialize
768 # otherwise do our own thing
769 $self->{value} = $CALC->_one();
771 $self->{sign} = $sign;
776 # call like: $x->bone($sign,$a,$p,$r,$y);
777 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
781 # call like: $x->bone($sign,$a,$p,$r);
783 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
785 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
791 ##############################################################################
792 # string conversation
796 # (ref to BFLOAT or num_str ) return num_str
797 # Convert number from internal format to scientific string format.
798 # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
799 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
800 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
802 if ($x->{sign} !~ /^[+-]$/)
804 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
807 my ($m,$e) = $x->parts();
808 #$m->bstr() . 'e+' . $e->bstr(); # e can only be positive in BigInt
809 # 'e+' because E can only be positive in BigInt
810 $m->bstr() . 'e+' . ${$CALC->_str($e->{value})};
815 # make a string from bigint object
816 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
817 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
819 if ($x->{sign} !~ /^[+-]$/)
821 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
824 my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
825 $es.${$CALC->_str($x->{value})};
830 # Make a "normal" scalar from a BigInt object
831 my $x = shift; $x = $class->new($x) unless ref $x;
833 return $x->bstr() if $x->{sign} !~ /^[+-]$/;
834 my $num = $CALC->_num($x->{value});
835 return -$num if $x->{sign} eq '-';
839 ##############################################################################
840 # public stuff (usually prefixed with "b")
844 # return the sign of the number: +/-/-inf/+inf/NaN
845 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
850 sub _find_round_parameters
852 # After any operation or when calling round(), the result is rounded by
853 # regarding the A & P from arguments, local parameters, or globals.
855 # !!!!!!! If you change this, remember to change round(), too! !!!!!!!!!!
857 # This procedure finds the round parameters, but it is for speed reasons
858 # duplicated in round. Otherwise, it is tested by the testsuite and used
861 # returns ($self) or ($self,$a,$p,$r) - sets $self to NaN of both A and P
862 # were requested/defined (locally or globally or both)
864 my ($self,$a,$p,$r,@args) = @_;
865 # $a accuracy, if given by caller
866 # $p precision, if given by caller
867 # $r round_mode, if given by caller
868 # @args all 'other' arguments (0 for unary, 1 for binary ops)
870 # leave bigfloat parts alone
871 return ($self) if exists $self->{_f} && ($self->{_f} & MB_NEVER_ROUND) != 0;
873 my $c = ref($self); # find out class of argument(s)
876 # now pick $a or $p, but only if we have got "arguments"
879 foreach ($self,@args)
881 # take the defined one, or if both defined, the one that is smaller
882 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
887 # even if $a is defined, take $p, to signal error for both defined
888 foreach ($self,@args)
890 # take the defined one, or if both defined, the one that is bigger
892 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
895 # if still none defined, use globals (#2)
896 $a = ${"$c\::accuracy"} unless defined $a;
897 $p = ${"$c\::precision"} unless defined $p;
899 # A == 0 is useless, so undef it to signal no rounding
900 $a = undef if defined $a && $a == 0;
903 return ($self) unless defined $a || defined $p; # early out
905 # set A and set P is an fatal error
906 return ($self->bnan()) if defined $a && defined $p; # error
908 $r = ${"$c\::round_mode"} unless defined $r;
909 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
911 require Carp; Carp::croak ("Unknown round mode '$r'");
919 # Round $self according to given parameters, or given second argument's
920 # parameters or global defaults
922 # for speed reasons, _find_round_parameters is embeded here:
924 my ($self,$a,$p,$r,@args) = @_;
925 # $a accuracy, if given by caller
926 # $p precision, if given by caller
927 # $r round_mode, if given by caller
928 # @args all 'other' arguments (0 for unary, 1 for binary ops)
930 # leave bigfloat parts alone
931 return ($self) if exists $self->{_f} && ($self->{_f} & MB_NEVER_ROUND) != 0;
933 my $c = ref($self); # find out class of argument(s)
936 # now pick $a or $p, but only if we have got "arguments"
939 foreach ($self,@args)
941 # take the defined one, or if both defined, the one that is smaller
942 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
947 # even if $a is defined, take $p, to signal error for both defined
948 foreach ($self,@args)
950 # take the defined one, or if both defined, the one that is bigger
952 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
955 # if still none defined, use globals (#2)
956 $a = ${"$c\::accuracy"} unless defined $a;
957 $p = ${"$c\::precision"} unless defined $p;
959 # A == 0 is useless, so undef it to signal no rounding
960 $a = undef if defined $a && $a == 0;
963 return $self unless defined $a || defined $p; # early out
965 # set A and set P is an fatal error
966 return $self->bnan() if defined $a && defined $p;
968 $r = ${"$c\::round_mode"} unless defined $r;
969 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
971 require Carp; Carp::croak ("Unknown round mode '$r'");
974 # now round, by calling either fround or ffround:
977 $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a;
979 else # both can't be undefined due to early out
981 $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p;
983 $self->bnorm(); # after round, normalize
988 # (numstr or BINT) return BINT
989 # Normalize number -- no-op here
990 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
996 # (BINT or num_str) return BINT
997 # make number absolute, or return absolute BINT from string
998 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1000 return $x if $x->modify('babs');
1001 # post-normalized abs for internal use (does nothing for NaN)
1002 $x->{sign} =~ s/^-/+/;
1008 # (BINT or num_str) return BINT
1009 # negate number or make a negated number from string
1010 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1012 return $x if $x->modify('bneg');
1014 # for +0 dont negate (to have always normalized)
1015 $x->{sign} =~ tr/+-/-+/ if !$x->is_zero(); # does nothing for NaN
1021 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
1022 # (BINT or num_str, BINT or num_str) return cond_code
1025 my ($self,$x,$y) = (ref($_[0]),@_);
1027 # objectify is costly, so avoid it
1028 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1030 ($self,$x,$y) = objectify(2,@_);
1033 return $upgrade->bcmp($x,$y) if defined $upgrade &&
1034 ((!$x->isa($self)) || (!$y->isa($self)));
1036 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1038 # handle +-inf and NaN
1039 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1040 return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
1041 return +1 if $x->{sign} eq '+inf';
1042 return -1 if $x->{sign} eq '-inf';
1043 return -1 if $y->{sign} eq '+inf';
1046 # check sign for speed first
1047 return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
1048 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
1050 # have same sign, so compare absolute values. Don't make tests for zero here
1051 # because it's actually slower than testin in Calc (especially w/ Pari et al)
1053 # post-normalized compare for internal use (honors signs)
1054 if ($x->{sign} eq '+')
1056 # $x and $y both > 0
1057 return $CALC->_acmp($x->{value},$y->{value});
1061 $CALC->_acmp($y->{value},$x->{value}); # swaped acmp (lib returns 0,1,-1)
1066 # Compares 2 values, ignoring their signs.
1067 # Returns one of undef, <0, =0, >0. (suitable for sort)
1068 # (BINT, BINT) return cond_code
1071 my ($self,$x,$y) = (ref($_[0]),@_);
1072 # objectify is costly, so avoid it
1073 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1075 ($self,$x,$y) = objectify(2,@_);
1078 return $upgrade->bacmp($x,$y) if defined $upgrade &&
1079 ((!$x->isa($self)) || (!$y->isa($self)));
1081 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1083 # handle +-inf and NaN
1084 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1085 return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
1086 return +1; # inf is always bigger
1088 $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
1093 # add second arg (BINT or string) to first (BINT) (modifies first)
1094 # return result as BINT
1097 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1098 # objectify is costly, so avoid it
1099 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1101 ($self,$x,$y,@r) = objectify(2,@_);
1104 return $x if $x->modify('badd');
1105 return $upgrade->badd($upgrade->new($x),$upgrade->new($y),@r) if defined $upgrade &&
1106 ((!$x->isa($self)) || (!$y->isa($self)));
1108 $r[3] = $y; # no push!
1109 # inf and NaN handling
1110 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1113 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1115 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1117 # +inf++inf or -inf+-inf => same, rest is NaN
1118 return $x if $x->{sign} eq $y->{sign};
1121 # +-inf + something => +inf
1122 # something +-inf => +-inf
1123 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
1127 my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
1131 $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
1135 my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
1138 $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
1143 # speedup, if equal, set result to 0
1144 $x->{value} = $CALC->_zero();
1149 $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
1152 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1158 # (BINT or num_str, BINT or num_str) return BINT
1159 # subtract second arg from first, modify first
1162 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1163 # objectify is costly, so avoid it
1164 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1166 ($self,$x,$y,@r) = objectify(2,@_);
1169 return $x if $x->modify('bsub');
1171 # upgrade done by badd():
1172 # return $upgrade->badd($x,$y,@r) if defined $upgrade &&
1173 # ((!$x->isa($self)) || (!$y->isa($self)));
1177 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1181 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
1182 $x->badd($y,@r); # badd does not leave internal zeros
1183 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
1184 $x; # already rounded by badd() or no round necc.
1189 # increment arg by one
1190 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1191 return $x if $x->modify('binc');
1193 if ($x->{sign} eq '+')
1195 $x->{value} = $CALC->_inc($x->{value});
1196 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1199 elsif ($x->{sign} eq '-')
1201 $x->{value} = $CALC->_dec($x->{value});
1202 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1203 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1206 # inf, nan handling etc
1207 $x->badd($self->bone(),$a,$p,$r); # badd does round
1212 # decrement arg by one
1213 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1214 return $x if $x->modify('bdec');
1216 if ($x->{sign} eq '-')
1219 $x->{value} = $CALC->_inc($x->{value});
1223 return $x->badd($self->bone('-'),@r) unless $x->{sign} eq '+'; # inf/NaN
1225 if ($CALC->_is_zero($x->{value}))
1228 $x->{value} = $CALC->_one(); $x->{sign} = '-'; # 0 => -1
1233 $x->{value} = $CALC->_dec($x->{value});
1236 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1242 # calculate $x = $a ** $base + $b and return $a (e.g. the log() to base
1246 my ($self,$x,$base,@r) = (ref($_[0]),@_);
1247 # objectify is costly, so avoid it
1248 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1250 ($self,$x,$base,@r) = objectify(2,$class,@_);
1253 # inf, -inf, NaN, <0 => NaN
1255 if $x->{sign} ne '+' || $base->{sign} ne '+';
1257 return $upgrade->blog($upgrade->new($x),$base,@r) if
1258 defined $upgrade && (ref($x) ne $upgrade || ref($base) ne $upgrade);
1262 my ($rc,$exact) = $CALC->_log_int($x->{value},$base->{value});
1263 return $x->bnan() unless defined $rc;
1265 return $x->round(@r);
1269 __emu_blog($self,$x,$base,@r);
1274 # (BINT or num_str, BINT or num_str) return BINT
1275 # does not modify arguments, but returns new object
1276 # Lowest Common Multiplicator
1278 my $y = shift; my ($x);
1285 $x = $class->new($y);
1287 while (@_) { $x = __lcm($x,shift); }
1293 # (BINT or num_str, BINT or num_str) return BINT
1294 # does not modify arguments, but returns new object
1295 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1298 $y = __PACKAGE__->new($y) if !ref($y);
1300 my $x = $y->copy(); # keep arguments
1305 $y = shift; $y = $self->new($y) if !ref($y);
1306 next if $y->is_zero();
1307 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1308 $x->{value} = $CALC->_gcd($x->{value},$y->{value}); last if $x->is_one();
1315 $y = shift; $y = $self->new($y) if !ref($y);
1316 $x = __gcd($x,$y->copy()); last if $x->is_one(); # _gcd handles NaN
1324 # (num_str or BINT) return BINT
1325 # represent ~x as twos-complement number
1326 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1327 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1329 return $x if $x->modify('bnot');
1330 $x->binc()->bneg(); # binc already does round
1333 ##############################################################################
1334 # is_foo test routines
1335 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1339 # return true if arg (BINT or num_str) is zero (array '+', '0')
1340 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1342 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1343 $CALC->_is_zero($x->{value});
1348 # return true if arg (BINT or num_str) is NaN
1349 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1351 $x->{sign} eq $nan ? 1 : 0;
1356 # return true if arg (BINT or num_str) is +-inf
1357 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1361 $sign = '[+-]inf' if $sign eq ''; # +- doesn't matter, only that's inf
1362 $sign = "[$1]inf" if $sign =~ /^([+-])(inf)?$/; # extract '+' or '-'
1363 return $x->{sign} =~ /^$sign$/ ? 1 : 0;
1365 $x->{sign} =~ /^[+-]inf$/ ? 1 : 0; # only +-inf is infinity
1370 # return true if arg (BINT or num_str) is +1, or -1 if sign is given
1371 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1373 $sign = '+' if !defined $sign || $sign ne '-';
1375 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1376 $CALC->_is_one($x->{value});
1381 # return true when arg (BINT or num_str) is odd, false for even
1382 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1384 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1385 $CALC->_is_odd($x->{value});
1390 # return true when arg (BINT or num_str) is even, false for odd
1391 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1393 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1394 $CALC->_is_even($x->{value});
1399 # return true when arg (BINT or num_str) is positive (>= 0)
1400 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1402 $x->{sign} =~ /^\+/ ? 1 : 0; # +inf is also positive, but NaN not
1407 # return true when arg (BINT or num_str) is negative (< 0)
1408 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1410 $x->{sign} =~ /^-/ ? 1 : 0; # -inf is also negative, but NaN not
1415 # return true when arg (BINT or num_str) is an integer
1416 # always true for BigInt, but different for BigFloats
1417 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1419 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1422 ###############################################################################
1426 # multiply two numbers -- stolen from Knuth Vol 2 pg 233
1427 # (BINT or num_str, BINT or num_str) return BINT
1430 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1431 # objectify is costly, so avoid it
1432 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1434 ($self,$x,$y,@r) = objectify(2,@_);
1437 return $x if $x->modify('bmul');
1439 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1442 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1444 return $x->bnan() if $x->is_zero() || $y->is_zero();
1445 # result will always be +-inf:
1446 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1447 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1448 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1449 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1450 return $x->binf('-');
1453 return $upgrade->bmul($x,$y,@r)
1454 if defined $upgrade && $y->isa($upgrade);
1456 $r[3] = $y; # no push here
1458 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1460 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1461 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1463 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1469 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1470 my ($self,$x,$y) = @_;
1472 # NaN if x == NaN or y == NaN or x==y==0
1473 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1474 if (($x->is_nan() || $y->is_nan()) ||
1475 ($x->is_zero() && $y->is_zero()));
1477 # +-inf / +-inf == NaN, reminder also NaN
1478 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1480 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1482 # x / +-inf => 0, remainder x (works even if x == 0)
1483 if ($y->{sign} =~ /^[+-]inf$/)
1485 my $t = $x->copy(); # bzero clobbers up $x
1486 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1489 # 5 / 0 => +inf, -6 / 0 => -inf
1490 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1491 # exception: -8 / 0 has remainder -8, not 8
1492 # exception: -inf / 0 has remainder -inf, not inf
1495 # +-inf / 0 => special case for -inf
1496 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1497 if (!$x->is_zero() && !$x->is_inf())
1499 my $t = $x->copy(); # binf clobbers up $x
1501 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1505 # last case: +-inf / ordinary number
1507 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1509 return wantarray ? ($x,$self->bzero()) : $x;
1514 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1515 # (BINT,BINT) (quo,rem) or BINT (only rem)
1518 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1519 # objectify is costly, so avoid it
1520 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1522 ($self,$x,$y,@r) = objectify(2,@_);
1525 return $x if $x->modify('bdiv');
1527 return $self->_div_inf($x,$y)
1528 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1530 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1531 if defined $upgrade;
1533 $r[3] = $y; # no push!
1535 # calc new sign and in case $y == +/- 1, return $x
1536 my $xsign = $x->{sign}; # keep
1537 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1541 my $rem = $self->bzero();
1542 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1543 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1544 $rem->{_a} = $x->{_a};
1545 $rem->{_p} = $x->{_p};
1546 $x->round(@r) if !exists $x->{_f} || ($x->{_f} & MB_NEVER_ROUND) == 0;
1547 if (! $CALC->_is_zero($rem->{value}))
1549 $rem->{sign} = $y->{sign};
1550 $rem = $y->copy()->bsub($rem) if $xsign ne $y->{sign}; # one of them '-'
1554 $rem->{sign} = '+'; # dont leave -0
1556 $rem->round(@r) if !exists $rem->{_f} || ($rem->{_f} & MB_NEVER_ROUND) == 0;
1560 $x->{value} = $CALC->_div($x->{value},$y->{value});
1561 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1563 $x->round(@r) if !exists $x->{_f} || ($x->{_f} & MB_NEVER_ROUND) == 0;
1567 ###############################################################################
1572 # modulus (or remainder)
1573 # (BINT or num_str, BINT or num_str) return BINT
1576 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1577 # objectify is costly, so avoid it
1578 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1580 ($self,$x,$y,@r) = objectify(2,@_);
1583 return $x if $x->modify('bmod');
1584 $r[3] = $y; # no push!
1585 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1587 my ($d,$r) = $self->_div_inf($x,$y);
1588 $x->{sign} = $r->{sign};
1589 $x->{value} = $r->{value};
1590 return $x->round(@r);
1595 # calc new sign and in case $y == +/- 1, return $x
1596 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1597 if (!$CALC->_is_zero($x->{value}))
1599 my $xsign = $x->{sign};
1600 $x->{sign} = $y->{sign};
1601 if ($xsign ne $y->{sign})
1603 my $t = $CALC->_copy($x->{value}); # copy $x
1604 $x->{value} = $CALC->_sub($y->{value},$t,1); # $y-$x
1609 $x->{sign} = '+'; # dont leave -0
1611 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1614 # disable upgrade temporarily, otherwise endless loop due to bdiv()
1615 local $upgrade = undef;
1616 my ($t,$rem) = $self->bdiv($x->copy(),$y,@r); # slow way (also rounds)
1618 foreach (qw/value sign _a _p/)
1620 $x->{$_} = $rem->{$_};
1627 # Modular inverse. given a number which is (hopefully) relatively
1628 # prime to the modulus, calculate its inverse using Euclid's
1629 # alogrithm. If the number is not relatively prime to the modulus
1630 # (i.e. their gcd is not one) then NaN is returned.
1633 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1634 # objectify is costly, so avoid it
1635 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1637 ($self,$x,$y,@r) = objectify(2,@_);
1640 return $x if $x->modify('bmodinv');
1643 if ($y->{sign} ne '+' # -, NaN, +inf, -inf
1644 || $x->is_zero() # or num == 0
1645 || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1648 # put least residue into $x if $x was negative, and thus make it positive
1649 $x->bmod($y) if $x->{sign} eq '-';
1654 ($x->{value},$sign) = $CALC->_modinv($x->{value},$y->{value});
1655 return $x->bnan() if !defined $x->{value}; # in case no GCD found
1656 return $x if !defined $sign; # already real result
1657 $x->{sign} = $sign; # flip/flop see below
1658 $x->bmod($y); # calc real result
1663 __emu_bmodinv($self,$x,$y,@r);
1668 # takes a very large number to a very large exponent in a given very
1669 # large modulus, quickly, thanks to binary exponentation. supports
1670 # negative exponents.
1671 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1673 return $num if $num->modify('bmodpow');
1675 # check modulus for valid values
1676 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1677 || $mod->is_zero());
1679 # check exponent for valid values
1680 if ($exp->{sign} =~ /\w/)
1682 # i.e., if it's NaN, +inf, or -inf...
1683 return $num->bnan();
1686 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
1688 # check num for valid values (also NaN if there was no inverse but $exp < 0)
1689 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1693 # $mod is positive, sign on $exp is ignored, result also positive
1694 $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
1699 __emu_bmodpow($self,$num,$exp,$mod,@r);
1702 ###############################################################################
1706 # (BINT or num_str, BINT or num_str) return BINT
1707 # compute factorial number from $x, modify $x in place
1708 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1710 return $x if $x->modify('bfac');
1712 return $x if $x->{sign} eq '+inf'; # inf => inf
1713 return $x->bnan() if $x->{sign} ne '+'; # NaN, <0 etc => NaN
1717 $x->{value} = $CALC->_fac($x->{value});
1718 return $x->round(@r);
1722 __emu_bfac($self,$x,@r);
1727 # (BINT or num_str, BINT or num_str) return BINT
1728 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1729 # modifies first argument
1732 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1733 # objectify is costly, so avoid it
1734 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1736 ($self,$x,$y,@r) = objectify(2,@_);
1739 return $x if $x->modify('bpow');
1741 return $upgrade->bpow($upgrade->new($x),$y,@r)
1742 if defined $upgrade && !$y->isa($self);
1744 $r[3] = $y; # no push!
1745 return $x if $x->{sign} =~ /^[+-]inf$/; # -inf/+inf ** x
1746 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1748 # cases 0 ** Y, X ** 0, X ** 1, 1 ** Y are handled by Calc or Emu
1750 if ($x->{sign} eq '-' && $CALC->_is_one($x->{value}))
1752 # if $x == -1 and odd/even y => +1/-1
1753 return $y->is_odd() ? $x->round(@r) : $x->babs()->round(@r);
1754 # my Casio FX-5500L has a bug here: -1 ** 2 is -1, but -1 * -1 is 1;
1756 # 1 ** -y => 1 / (1 ** |y|)
1757 # so do test for negative $y after above's clause
1758 return $x->bnan() if $y->{sign} eq '-' && !$x->is_one();
1762 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1763 $x->{sign} = '+' if $CALC->_is_zero($y->{value});
1764 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1769 __emu_bpow($self,$x,$y,@r);
1774 # (BINT or num_str, BINT or num_str) return BINT
1775 # compute x << y, base n, y >= 0
1778 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1779 # objectify is costly, so avoid it
1780 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1782 ($self,$x,$y,$n,@r) = objectify(2,@_);
1785 return $x if $x->modify('blsft');
1786 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1787 return $x->round(@r) if $y->is_zero();
1789 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1791 my $t; $t = $CALC->_lsft($x->{value},$y->{value},$n) if $CAN{lsft};
1794 $x->{value} = $t; return $x->round(@r);
1797 $x->bmul( $self->bpow($n, $y, @r), @r );
1802 # (BINT or num_str, BINT or num_str) return BINT
1803 # compute x >> y, base n, y >= 0
1806 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1807 # objectify is costly, so avoid it
1808 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1810 ($self,$x,$y,$n,@r) = objectify(2,@_);
1813 return $x if $x->modify('brsft');
1814 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1815 return $x->round(@r) if $y->is_zero();
1816 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
1818 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1820 # this only works for negative numbers when shifting in base 2
1821 if (($x->{sign} eq '-') && ($n == 2))
1823 return $x->round(@r) if $x->is_one('-'); # -1 => -1
1826 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1827 # but perhaps there is a better emulation for two's complement shift...
1828 # if $y != 1, we must simulate it by doing:
1829 # convert to bin, flip all bits, shift, and be done
1830 $x->binc(); # -3 => -2
1831 my $bin = $x->as_bin();
1832 $bin =~ s/^-0b//; # strip '-0b' prefix
1833 $bin =~ tr/10/01/; # flip bits
1835 if (CORE::length($bin) <= $y)
1837 $bin = '0'; # shifting to far right creates -1
1838 # 0, because later increment makes
1839 # that 1, attached '-' makes it '-1'
1840 # because -1 >> x == -1 !
1844 $bin =~ s/.{$y}$//; # cut off at the right side
1845 $bin = '1' . $bin; # extend left side by one dummy '1'
1846 $bin =~ tr/10/01/; # flip bits back
1848 my $res = $self->new('0b'.$bin); # add prefix and convert back
1849 $res->binc(); # remember to increment
1850 $x->{value} = $res->{value}; # take over value
1851 return $x->round(@r); # we are done now, magic, isn't?
1853 # x < 0, n == 2, y == 1
1854 $x->bdec(); # n == 2, but $y == 1: this fixes it
1857 my $t; $t = $CALC->_rsft($x->{value},$y->{value},$n) if $CAN{rsft};
1861 return $x->round(@r);
1864 $x->bdiv($self->bpow($n,$y, @r), @r);
1870 #(BINT or num_str, BINT or num_str) return BINT
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('band');
1883 $r[3] = $y; # no push!
1885 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1887 my $sx = $x->{sign} eq '+' ? 1 : -1;
1888 my $sy = $y->{sign} eq '+' ? 1 : -1;
1890 if ($CAN{and} && $sx == 1 && $sy == 1)
1892 $x->{value} = $CALC->_and($x->{value},$y->{value});
1893 return $x->round(@r);
1896 if ($CAN{signed_and})
1898 $x->{value} = $CALC->_signed_and($x->{value},$y->{value},$sx,$sy);
1899 return $x->round(@r);
1903 __emu_band($self,$x,$y,$sx,$sy,@r);
1908 #(BINT or num_str, BINT or num_str) return BINT
1912 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1913 # objectify is costly, so avoid it
1914 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1916 ($self,$x,$y,@r) = objectify(2,@_);
1919 return $x if $x->modify('bior');
1920 $r[3] = $y; # no push!
1922 local $Math::BigInt::upgrade = undef;
1924 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1926 my $sx = $x->{sign} eq '+' ? 1 : -1;
1927 my $sy = $y->{sign} eq '+' ? 1 : -1;
1929 # the sign of X follows the sign of X, e.g. sign of Y irrelevant for bior()
1931 # don't use lib for negative values
1932 if ($CAN{or} && $sx == 1 && $sy == 1)
1934 $x->{value} = $CALC->_or($x->{value},$y->{value});
1935 return $x->round(@r);
1938 # if lib can do negative values, let it handle this
1939 if ($CAN{signed_or})
1941 $x->{value} = $CALC->_signed_or($x->{value},$y->{value},$sx,$sy);
1942 return $x->round(@r);
1946 __emu_bior($self,$x,$y,$sx,$sy,@r);
1951 #(BINT or num_str, BINT or num_str) return BINT
1955 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1956 # objectify is costly, so avoid it
1957 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1959 ($self,$x,$y,@r) = objectify(2,@_);
1962 return $x if $x->modify('bxor');
1963 $r[3] = $y; # no push!
1965 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1967 my $sx = $x->{sign} eq '+' ? 1 : -1;
1968 my $sy = $y->{sign} eq '+' ? 1 : -1;
1970 # don't use lib for negative values
1971 if ($CAN{xor} && $sx == 1 && $sy == 1)
1973 $x->{value} = $CALC->_xor($x->{value},$y->{value});
1974 return $x->round(@r);
1977 # if lib can do negative values, let it handle this
1978 if ($CAN{signed_xor})
1980 $x->{value} = $CALC->_signed_xor($x->{value},$y->{value},$sx,$sy);
1981 return $x->round(@r);
1985 __emu_bxor($self,$x,$y,$sx,$sy,@r);
1990 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1992 my $e = $CALC->_len($x->{value});
1993 wantarray ? ($e,0) : $e;
1998 # return the nth decimal digit, negative values count backward, 0 is right
1999 my ($self,$x,$n) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2001 $CALC->_digit($x->{value},$n||0);
2006 # return the amount of trailing zeros in $x (as scalar)
2008 $x = $class->new($x) unless ref $x;
2010 return 0 if $x->is_zero() || $x->is_odd() || $x->{sign} !~ /^[+-]$/;
2012 return $CALC->_zeros($x->{value}) if $CAN{zeros};
2014 # if not: since we do not know underlying internal representation:
2015 my $es = "$x"; $es =~ /([0]*)$/;
2016 return 0 if !defined $1; # no zeros
2017 CORE::length("$1"); # as string, not as +0!
2022 # calculate square root of $x
2023 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2025 return $x if $x->modify('bsqrt');
2027 return $x->bnan() if $x->{sign} !~ /^\+/; # -x or -inf or NaN => NaN
2028 return $x if $x->{sign} eq '+inf'; # sqrt(+inf) == inf
2030 return $upgrade->bsqrt($x,@r) if defined $upgrade;
2034 $x->{value} = $CALC->_sqrt($x->{value});
2035 return $x->round(@r);
2039 __emu_bsqrt($self,$x,@r);
2044 # calculate $y'th root of $x
2047 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2049 $y = $self->new(2) unless defined $y;
2051 # objectify is costly, so avoid it
2052 if ((!ref($x)) || (ref($x) ne ref($y)))
2054 ($self,$x,$y,@r) = $self->objectify(2,@_);
2057 return $x if $x->modify('broot');
2059 # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
2060 return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
2061 $y->{sign} !~ /^\+$/;
2063 return $x->round(@r)
2064 if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
2066 return $upgrade->new($x)->broot($upgrade->new($y),@r) if defined $upgrade;
2070 $x->{value} = $CALC->_root($x->{value},$y->{value});
2071 return $x->round(@r);
2075 __emu_broot($self,$x,$y,@r);
2080 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
2081 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2083 if ($x->{sign} !~ /^[+-]$/)
2085 my $s = $x->{sign}; $s =~ s/^[+-]//; # NaN, -inf,+inf => NaN or inf
2086 return $self->new($s);
2088 return $self->bone() if $x->is_zero();
2090 $self->new($x->_trailing_zeros());
2095 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
2096 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2098 if ($x->{sign} !~ /^[+-]$/)
2100 # for NaN, +inf, -inf: keep the sign
2101 return $self->new($x->{sign});
2103 my $m = $x->copy(); delete $m->{_p}; delete $m->{_a};
2104 # that's a bit inefficient:
2105 my $zeros = $m->_trailing_zeros();
2106 $m->brsft($zeros,10) if $zeros != 0;
2112 # return a copy of both the exponent and the mantissa
2113 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
2115 ($x->mantissa(),$x->exponent());
2118 ##############################################################################
2119 # rounding functions
2123 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
2124 # $n == 0 || $n == 1 => round to integer
2125 my $x = shift; $x = $class->new($x) unless ref $x;
2127 my ($scale,$mode) = $x->_scale_p($x->precision(),$x->round_mode(),@_);
2129 return $x if !defined $scale || $x->modify('bfround'); # no-op
2131 # no-op for BigInts if $n <= 0
2132 $x->bround( $x->length()-$scale, $mode) if $scale > 0;
2134 $x->{_a} = undef; # bround sets {_a}
2135 $x->{_p} = $scale; # so correct it
2139 sub _scan_for_nonzero
2141 # internal, used by bround()
2142 my ($x,$pad,$xs) = @_;
2144 my $len = $x->length();
2145 return 0 if $len == 1; # '5' is trailed by invisible zeros
2146 my $follow = $pad - 1;
2147 return 0 if $follow > $len || $follow < 1;
2149 # since we do not know underlying represention of $x, use decimal string
2150 my $r = substr ("$x",-$follow);
2151 $r =~ /[^0]/ ? 1 : 0;
2156 # Exists to make life easier for switch between MBF and MBI (should we
2157 # autoload fxxx() like MBF does for bxxx()?)
2164 # accuracy: +$n preserve $n digits from left,
2165 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
2167 # and overwrite the rest with 0's, return normalized number
2168 # do not return $x->bnorm(), but $x
2170 my $x = shift; $x = $class->new($x) unless ref $x;
2171 my ($scale,$mode) = $x->_scale_a($x->accuracy(),$x->round_mode(),@_);
2172 return $x if !defined $scale; # no-op
2173 return $x if $x->modify('bround');
2175 if ($x->is_zero() || $scale == 0)
2177 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2180 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2182 # we have fewer digits than we want to scale to
2183 my $len = $x->length();
2184 # convert $scale to a scalar in case it is an object (put's a limit on the
2185 # number length, but this would already limited by memory constraints), makes
2187 $scale = $scale->numify() if ref ($scale);
2189 # scale < 0, but > -len (not >=!)
2190 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2192 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2196 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2197 my ($pad,$digit_round,$digit_after);
2198 $pad = $len - $scale;
2199 $pad = abs($scale-1) if $scale < 0;
2201 # do not use digit(), it is costly for binary => decimal
2203 my $xs = $CALC->_str($x->{value});
2206 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2207 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2208 $digit_round = '0'; $digit_round = substr($$xs,$pl,1) if $pad <= $len;
2209 $pl++; $pl ++ if $pad >= $len;
2210 $digit_after = '0'; $digit_after = substr($$xs,$pl,1) if $pad > 0;
2212 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2213 # closer at the remaining digits of the original $x, remember decision
2214 my $round_up = 1; # default round up
2216 ($mode eq 'trunc') || # trunc by round down
2217 ($digit_after =~ /[01234]/) || # round down anyway,
2219 ($digit_after eq '5') && # not 5000...0000
2220 ($x->_scan_for_nonzero($pad,$xs) == 0) &&
2222 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2223 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2224 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2225 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2226 ($mode eq 'zero') # round down if zero, sign adjusted below
2228 my $put_back = 0; # not yet modified
2230 if (($pad > 0) && ($pad <= $len))
2232 substr($$xs,-$pad,$pad) = '0' x $pad;
2237 $x->bzero(); # round to '0'
2240 if ($round_up) # what gave test above?
2243 $pad = $len, $$xs = '0' x $pad if $scale < 0; # tlr: whack 0.51=>1.0
2245 # we modify directly the string variant instead of creating a number and
2246 # adding it, since that is faster (we already have the string)
2247 my $c = 0; $pad ++; # for $pad == $len case
2248 while ($pad <= $len)
2250 $c = substr($$xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2251 substr($$xs,-$pad,1) = $c; $pad++;
2252 last if $c != 0; # no overflow => early out
2254 $$xs = '1'.$$xs if $c == 0;
2257 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back in if needed
2259 $x->{_a} = $scale if $scale >= 0;
2262 $x->{_a} = $len+$scale;
2263 $x->{_a} = 0 if $scale < -$len;
2270 # return integer less or equal then number; no-op since it's already integer
2271 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2278 # return integer greater or equal then number; no-op since it's already int
2279 my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
2286 # An object might be asked to return itself as bigint on certain overloaded
2287 # operations, this does exactly this, so that sub classes can simple inherit
2288 # it or override with their own integer conversion routine.
2294 # return as hex string, with prefixed 0x
2295 my $x = shift; $x = $class->new($x) if !ref($x);
2297 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2300 $s = $x->{sign} if $x->{sign} eq '-';
2303 return $s . ${$CALC->_as_hex($x->{value})};
2307 __emu_as_hex(ref($x),$x,$s);
2312 # return as binary string, with prefixed 0b
2313 my $x = shift; $x = $class->new($x) if !ref($x);
2315 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2317 my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
2320 return $s . ${$CALC->_as_bin($x->{value})};
2324 __emu_as_bin(ref($x),$x,$s);
2328 ##############################################################################
2329 # private stuff (internal use only)
2333 # check for strings, if yes, return objects instead
2335 # the first argument is number of args objectify() should look at it will
2336 # return $count+1 elements, the first will be a classname. This is because
2337 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2338 # useless objects beeing created and thrown away. So we cannot simple loop
2339 # over @_. If the given count is 0, all arguments will be used.
2341 # If the second arg is a ref, use it as class.
2342 # If not, try to use it as classname, unless undef, then use $class
2343 # (aka Math::BigInt). The latter shouldn't happen,though.
2346 # $x->badd(1); => ref x, scalar y
2347 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2348 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2349 # Math::BigInt::badd(1,2); => scalar x, scalar y
2350 # In the last case we check number of arguments to turn it silently into
2351 # $class,1,2. (We can not take '1' as class ;o)
2352 # badd($class,1) is not supported (it should, eventually, try to add undef)
2353 # currently it tries 'Math::BigInt' + 1, which will not work.
2355 # some shortcut for the common cases
2357 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2359 my $count = abs(shift || 0);
2361 my (@a,$k,$d); # resulting array, temp, and downgrade
2364 # okay, got object as first
2369 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2371 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2375 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2376 if (defined ${"$a[0]::downgrade"})
2378 $d = ${"$a[0]::downgrade"};
2379 ${"$a[0]::downgrade"} = undef;
2382 my $up = ${"$a[0]::upgrade"};
2383 #print "Now in objectify, my class is today $a[0], count = $count\n";
2391 $k = $a[0]->new($k);
2393 elsif (!defined $up && ref($k) ne $a[0])
2395 # foreign object, try to convert to integer
2396 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2409 $k = $a[0]->new($k);
2411 elsif (!defined $up && ref($k) ne $a[0])
2413 # foreign object, try to convert to integer
2414 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2418 push @a,@_; # return other params, too
2422 require Carp; Carp::croak ("$class objectify needs list context");
2424 ${"$a[0]::downgrade"} = $d;
2432 $IMPORT++; # remember we did import()
2433 my @a; my $l = scalar @_;
2434 for ( my $i = 0; $i < $l ; $i++ )
2436 if ($_[$i] eq ':constant')
2438 # this causes overlord er load to step in
2440 integer => sub { $self->new(shift) },
2441 binary => sub { $self->new(shift) };
2443 elsif ($_[$i] eq 'upgrade')
2445 # this causes upgrading
2446 $upgrade = $_[$i+1]; # or undef to disable
2449 elsif ($_[$i] =~ /^lib$/i)
2451 # this causes a different low lib to take care...
2452 $CALC = $_[$i+1] || '';
2460 # any non :constant stuff is handled by our parent, Exporter
2461 # even if @_ is empty, to give it a chance
2462 $self->SUPER::import(@a); # need it for subclasses
2463 $self->export_to_level(1,$self,@a); # need it for MBF
2465 # try to load core math lib
2466 my @c = split /\s*,\s*/,$CALC;
2467 push @c,'Calc'; # if all fail, try this
2468 $CALC = ''; # signal error
2469 foreach my $lib (@c)
2471 next if ($lib || '') eq '';
2472 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2476 # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
2477 # used in the same script, or eval inside import().
2478 my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
2479 my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
2481 $file = File::Spec->catfile (@parts, $file);
2482 eval { require "$file"; $lib->import( @c ); }
2486 eval "use $lib qw/@c/;";
2488 $CALC = $lib, last if $@ eq ''; # no error in loading lib?
2493 Carp::croak ("Couldn't load any math lib, not even 'Calc.pm'");
2500 # fill $CAN with the results of $CALC->can(...)
2503 for my $method (qw/gcd mod modinv modpow fac pow lsft rsft
2504 and signed_and or signed_or xor signed_xor
2505 from_hex as_hex from_bin as_bin
2506 zeros sqrt root log_int log
2509 $CAN{$method} = $CALC->can("_$method") ? 1 : 0;
2515 # convert a (ref to) big hex string to BigInt, return undef for error
2518 my $x = Math::BigInt->bzero();
2521 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2522 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2524 return $x->bnan() if $$hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2526 my $sign = '+'; $sign = '-' if ($$hs =~ /^-/);
2528 $$hs =~ s/^[+-]//; # strip sign
2529 if ($CAN{'from_hex'})
2531 $x->{value} = $CALC->_from_hex($hs);
2535 # fallback to pure perl
2536 my $mul = Math::BigInt->bone();
2537 my $x65536 = Math::BigInt->new(65536);
2538 my $len = CORE::length($$hs)-2; # minus 2 for 0x
2539 $len = int($len/4); # 4-digit parts, w/o '0x'
2540 my $val; my $i = -4;
2543 $val = substr($$hs,$i,4);
2544 $val =~ s/^[+-]?0x// if $len == 0; # for last part only because
2545 $val = hex($val); # hex does not like wrong chars
2547 $x += $mul * $val if $val != 0;
2548 $mul *= $x65536 if $len >= 0; # skip last mul
2551 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2557 # convert a (ref to) big binary string to BigInt, return undef for error
2560 my $x = Math::BigInt->bzero();
2562 $$bs =~ s/([01])_([01])/$1$2/g;
2563 $$bs =~ s/([01])_([01])/$1$2/g;
2564 return $x->bnan() if $$bs !~ /^[+-]?0b[01]+$/;
2566 my $sign = '+'; $sign = '-' if ($$bs =~ /^\-/);
2567 $$bs =~ s/^[+-]//; # strip sign
2568 if ($CAN{'from_bin'})
2570 $x->{value} = $CALC->_from_bin($bs);
2574 my $mul = Math::BigInt->bone();
2575 my $x256 = Math::BigInt->new(256);
2576 my $len = CORE::length($$bs)-2; # minus 2 for 0b
2577 $len = int($len/8); # 8-digit parts, w/o '0b'
2578 my $val; my $i = -8;
2581 $val = substr($$bs,$i,8);
2582 $val =~ s/^[+-]?0b// if $len == 0; # for last part only
2583 #$val = oct('0b'.$val); # does not work on Perl prior to 5.6.0
2585 # $val = ('0' x (8-CORE::length($val))).$val if CORE::length($val) < 8;
2586 $val = ord(pack('B8',substr('00000000'.$val,-8,8)));
2588 $x += $mul * $val if $val != 0;
2589 $mul *= $x256 if $len >= 0; # skip last mul
2592 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2598 # (ref to num_str) return num_str
2599 # internal, take apart a string and return the pieces
2600 # strip leading/trailing whitespace, leading zeros, underscore and reject
2604 # strip white space at front, also extranous leading zeros
2605 $$x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2606 $$x =~ s/^\s+//; # but this will
2607 $$x =~ s/\s+$//g; # strip white space at end
2609 # shortcut, if nothing to split, return early
2610 if ($$x =~ /^[+-]?\d+\z/)
2612 $$x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2613 return (\$sign, $x, \'', \'', \0);
2616 # invalid starting char?
2617 return if $$x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2619 return __from_hex($x) if $$x =~ /^[\-\+]?0x/; # hex string
2620 return __from_bin($x) if $$x =~ /^[\-\+]?0b/; # binary string
2622 # strip underscores between digits
2623 $$x =~ s/(\d)_(\d)/$1$2/g;
2624 $$x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3
2626 # some possible inputs:
2627 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2628 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 # 0e999
2630 #return if $$x =~ /[Ee].*[Ee]/; # more than one E => error
2632 my ($m,$e,$last) = split /[Ee]/,$$x;
2633 return if defined $last; # last defined => 1e2E3 or others
2634 $e = '0' if !defined $e || $e eq "";
2636 # sign,value for exponent,mantint,mantfrac
2637 my ($es,$ev,$mis,$miv,$mfv);
2639 if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2643 return if $m eq '.' || $m eq '';
2644 my ($mi,$mf,$lastf) = split /\./,$m;
2645 return if defined $lastf; # lastf defined => 1.2.3 or others
2646 $mi = '0' if !defined $mi;
2647 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2648 $mf = '0' if !defined $mf || $mf eq '';
2649 if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2651 $mis = $1||'+'; $miv = $2;
2652 return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
2654 # handle the 0e999 case here
2655 $ev = 0 if $miv eq '0' && $mfv eq '';
2656 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2659 return; # NaN, not a number
2662 ##############################################################################
2663 # internal calculation routines (others are in Math::BigInt::Calc etc)
2667 # (BINT or num_str, BINT or num_str) return BINT
2668 # does modify first argument
2671 my $x = shift; my $ty = shift;
2672 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2673 return $x * $ty / bgcd($x,$ty);
2678 # (BINT or num_str, BINT or num_str) return BINT
2679 # does modify both arguments
2680 # GCD -- Euclids algorithm E, Knuth Vol 2 pg 296
2683 return $x->bnan() if $x->{sign} !~ /^[+-]$/ || $ty->{sign} !~ /^[+-]$/;
2685 while (!$ty->is_zero())
2687 ($x, $ty) = ($ty,bmod($x,$ty));
2692 ###############################################################################
2693 # this method return 0 if the object can be modified, or 1 for not
2694 # We use a fast constant sub() here, to avoid costly calls. Subclasses
2695 # may override it with special code (f.i. Math::BigInt::Constant does so)
2697 sub modify () { 0; }
2704 Math::BigInt - Arbitrary size integer math package
2710 # or make it faster: install (optional) Math::BigInt::GMP
2711 # and always use (it will fall back to pure Perl if the
2712 # GMP library is not installed):
2714 use Math::BigInt lib => 'GMP';
2717 $x = Math::BigInt->new($str); # defaults to 0
2718 $nan = Math::BigInt->bnan(); # create a NotANumber
2719 $zero = Math::BigInt->bzero(); # create a +0
2720 $inf = Math::BigInt->binf(); # create a +inf
2721 $inf = Math::BigInt->binf('-'); # create a -inf
2722 $one = Math::BigInt->bone(); # create a +1
2723 $one = Math::BigInt->bone('-'); # create a -1
2725 # Testing (don't modify their arguments)
2726 # (return true if the condition is met, otherwise false)
2728 $x->is_zero(); # if $x is +0
2729 $x->is_nan(); # if $x is NaN
2730 $x->is_one(); # if $x is +1
2731 $x->is_one('-'); # if $x is -1
2732 $x->is_odd(); # if $x is odd
2733 $x->is_even(); # if $x is even
2734 $x->is_pos(); # if $x >= 0
2735 $x->is_neg(); # if $x < 0
2736 $x->is_inf(sign); # if $x is +inf, or -inf (sign is default '+')
2737 $x->is_int(); # if $x is an integer (not a float)
2739 # comparing and digit/sign extration
2740 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2741 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2742 $x->sign(); # return the sign, either +,- or NaN
2743 $x->digit($n); # return the nth digit, counting from right
2744 $x->digit(-$n); # return the nth digit, counting from left
2746 # The following all modify their first argument. If you want to preserve
2747 # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
2748 # neccessary when mixing $a = $b assigments with non-overloaded math.
2750 $x->bzero(); # set $x to 0
2751 $x->bnan(); # set $x to NaN
2752 $x->bone(); # set $x to +1
2753 $x->bone('-'); # set $x to -1
2754 $x->binf(); # set $x to inf
2755 $x->binf('-'); # set $x to -inf
2757 $x->bneg(); # negation
2758 $x->babs(); # absolute value
2759 $x->bnorm(); # normalize (no-op in BigInt)
2760 $x->bnot(); # two's complement (bit wise not)
2761 $x->binc(); # increment $x by 1
2762 $x->bdec(); # decrement $x by 1
2764 $x->badd($y); # addition (add $y to $x)
2765 $x->bsub($y); # subtraction (subtract $y from $x)
2766 $x->bmul($y); # multiplication (multiply $x by $y)
2767 $x->bdiv($y); # divide, set $x to quotient
2768 # return (quo,rem) or quo if scalar
2770 $x->bmod($y); # modulus (x % y)
2771 $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
2772 $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
2774 $x->bpow($y); # power of arguments (x ** y)
2775 $x->blsft($y); # left shift
2776 $x->brsft($y); # right shift
2777 $x->blsft($y,$n); # left shift, by base $n (like 10)
2778 $x->brsft($y,$n); # right shift, by base $n (like 10)
2780 $x->band($y); # bitwise and
2781 $x->bior($y); # bitwise inclusive or
2782 $x->bxor($y); # bitwise exclusive or
2783 $x->bnot(); # bitwise not (two's complement)
2785 $x->bsqrt(); # calculate square-root
2786 $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
2787 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2789 $x->round($A,$P,$mode); # round to accuracy or precision using mode $mode
2790 $x->bround($N); # accuracy: preserve $N digits
2791 $x->bfround($N); # round to $Nth digit, no-op for BigInts
2793 # The following do not modify their arguments in BigInt (are no-ops),
2794 # but do so in BigFloat:
2796 $x->bfloor(); # return integer less or equal than $x
2797 $x->bceil(); # return integer greater or equal than $x
2799 # The following do not modify their arguments:
2801 bgcd(@values); # greatest common divisor (no OO style)
2802 blcm(@values); # lowest common multiplicator (no OO style)
2804 $x->length(); # return number of digits in number
2805 ($x,$f) = $x->length(); # length of number and length of fraction part,
2806 # latter is always 0 digits long for BigInt's
2808 $x->exponent(); # return exponent as BigInt
2809 $x->mantissa(); # return (signed) mantissa as BigInt
2810 $x->parts(); # return (mantissa,exponent) as BigInt
2811 $x->copy(); # make a true copy of $x (unlike $y = $x;)
2812 $x->as_int(); # return as BigInt (in BigInt: same as copy())
2813 $x->numify(); # return as scalar (might overflow!)
2815 # conversation to string (do not modify their argument)
2816 $x->bstr(); # normalized string
2817 $x->bsstr(); # normalized string in scientific notation
2818 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
2819 $x->as_bin(); # as signed binary string with prefixed 0b
2822 # precision and accuracy (see section about rounding for more)
2823 $x->precision(); # return P of $x (or global, if P of $x undef)
2824 $x->precision($n); # set P of $x to $n
2825 $x->accuracy(); # return A of $x (or global, if A of $x undef)
2826 $x->accuracy($n); # set A $x to $n
2829 Math::BigInt->precision(); # get/set global P for all BigInt objects
2830 Math::BigInt->accuracy(); # get/set global A for all BigInt objects
2831 Math::BigInt->config(); # return hash containing configuration
2835 All operators (inlcuding basic math operations) are overloaded if you
2836 declare your big integers as
2838 $i = new Math::BigInt '123_456_789_123_456_789';
2840 Operations with overloaded operators preserve the arguments which is
2841 exactly what you expect.
2847 Input values to these routines may be any string, that looks like a number
2848 and results in an integer, including hexadecimal and binary numbers.
2850 Scalars holding numbers may also be passed, but note that non-integer numbers
2851 may already have lost precision due to the conversation to float. Quote
2852 your input if you want BigInt to see all the digits:
2854 $x = Math::BigInt->new(12345678890123456789); # bad
2855 $x = Math::BigInt->new('12345678901234567890'); # good
2857 You can include one underscore between any two digits.
2859 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
2860 Non-integer values result in NaN.
2862 Currently, Math::BigInt::new() defaults to 0, while Math::BigInt::new('')
2863 results in 'NaN'. This might change in the future, so use always the following
2864 explicit forms to get a zero or NaN:
2866 $zero = Math::BigInt->bzero();
2867 $nan = Math::BigInt->bnan();
2869 C<bnorm()> on a BigInt object is now effectively a no-op, since the numbers
2870 are always stored in normalized form. If passed a string, creates a BigInt
2871 object from the input.
2875 Output values are BigInt objects (normalized), except for bstr(), which
2876 returns a string in normalized form.
2877 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
2878 C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
2879 return either undef, <0, 0 or >0 and are suited for sort.
2885 Each of the methods below (except config(), accuracy() and precision())
2886 accepts three additional parameters. These arguments $A, $P and $R are
2887 accuracy, precision and round_mode. Please see the section about
2888 L<ACCURACY and PRECISION> for more information.
2894 print Dumper ( Math::BigInt->config() );
2895 print Math::BigInt->config()->{lib},"\n";
2897 Returns a hash containing the configuration, e.g. the version number, lib
2898 loaded etc. The following hash keys are currently filled in with the
2899 appropriate information.
2903 ============================================================
2904 lib Name of the low-level math library
2906 lib_version Version of low-level math library (see 'lib')
2908 class The class name of config() you just called
2910 upgrade To which class math operations might be upgraded
2912 downgrade To which class math operations might be downgraded
2914 precision Global precision
2916 accuracy Global accuracy
2918 round_mode Global round mode
2920 version version number of the class you used
2922 div_scale Fallback acccuracy for div
2924 trap_nan If true, traps creation of NaN via croak()
2926 trap_inf If true, traps creation of +inf/-inf via croak()
2929 The following values can be set by passing C<config()> a reference to a hash:
2932 upgrade downgrade precision accuracy round_mode div_scale
2936 $new_cfg = Math::BigInt->config( { trap_inf => 1, precision => 5 } );
2940 $x->accuracy(5); # local for $x
2941 CLASS->accuracy(5); # global for all members of CLASS
2942 $A = $x->accuracy(); # read out
2943 $A = CLASS->accuracy(); # read out
2945 Set or get the global or local accuracy, aka how many significant digits the
2948 Please see the section about L<ACCURACY AND PRECISION> for further details.
2950 Value must be greater than zero. Pass an undef value to disable it:
2952 $x->accuracy(undef);
2953 Math::BigInt->accuracy(undef);
2955 Returns the current accuracy. For C<$x->accuracy()> it will return either the
2956 local accuracy, or if not defined, the global. This means the return value
2957 represents the accuracy that will be in effect for $x:
2959 $y = Math::BigInt->new(1234567); # unrounded
2960 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
2961 $x = Math::BigInt->new(123456); # will be automatically rounded
2962 print "$x $y\n"; # '123500 1234567'
2963 print $x->accuracy(),"\n"; # will be 4
2964 print $y->accuracy(),"\n"; # also 4, since global is 4
2965 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
2966 print $x->accuracy(),"\n"; # still 4
2967 print $y->accuracy(),"\n"; # 5, since global is 5
2969 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
2970 globals separated from Math::BigInt, but it is possible to subclass
2971 Math::BigInt and make the globals of the subclass aliases to the ones from
2976 $x->precision(-2); # local for $x, round right of the dot
2977 $x->precision(2); # ditto, but round left of the dot
2978 CLASS->accuracy(5); # global for all members of CLASS
2979 CLASS->precision(-5); # ditto
2980 $P = CLASS->precision(); # read out
2981 $P = $x->precision(); # read out
2983 Set or get the global or local precision, aka how many digits the result has
2984 after the dot (or where to round it when passing a positive number). In
2985 Math::BigInt, passing a negative number precision has no effect since no
2986 numbers have digits after the dot.
2988 Please see the section about L<ACCURACY AND PRECISION> for further details.
2990 Value must be greater than zero. Pass an undef value to disable it:
2992 $x->precision(undef);
2993 Math::BigInt->precision(undef);
2995 Returns the current precision. For C<$x->precision()> it will return either the
2996 local precision of $x, or if not defined, the global. This means the return
2997 value represents the accuracy that will be in effect for $x:
2999 $y = Math::BigInt->new(1234567); # unrounded
3000 print Math::BigInt->precision(4),"\n"; # set 4, print 4
3001 $x = Math::BigInt->new(123456); # will be automatically rounded
3003 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
3004 globals separated from Math::BigInt, but it is possible to subclass
3005 Math::BigInt and make the globals of the subclass aliases to the ones from
3012 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
3013 2, but others work, too.
3015 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
3019 $x = Math::BigInt->new(10);
3020 $x->brsft(1); # same as $x >> 1: 5
3021 $x = Math::BigInt->new(1234);
3022 $x->brsft(2,10); # result 12
3024 There is one exception, and that is base 2 with negative $x:
3027 $x = Math::BigInt->new(-5);
3030 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
3035 $x = Math::BigInt->new($str,$A,$P,$R);
3037 Creates a new BigInt object from a scalar or another BigInt object. The
3038 input is accepted as decimal, hex (with leading '0x') or binary (with leading
3041 See L<Input> for more info on accepted input formats.
3045 $x = Math::BigInt->bnan();
3047 Creates a new BigInt object representing NaN (Not A Number).
3048 If used on an object, it will set it to NaN:
3054 $x = Math::BigInt->bzero();
3056 Creates a new BigInt object representing zero.
3057 If used on an object, it will set it to zero:
3063 $x = Math::BigInt->binf($sign);
3065 Creates a new BigInt object representing infinity. The optional argument is
3066 either '-' or '+', indicating whether you want infinity or minus infinity.
3067 If used on an object, it will set it to infinity:
3074 $x = Math::BigInt->binf($sign);
3076 Creates a new BigInt object representing one. The optional argument is
3077 either '-' or '+', indicating whether you want one or minus one.
3078 If used on an object, it will set it to one:
3083 =head2 is_one()/is_zero()/is_nan()/is_inf()
3086 $x->is_zero(); # true if arg is +0
3087 $x->is_nan(); # true if arg is NaN
3088 $x->is_one(); # true if arg is +1
3089 $x->is_one('-'); # true if arg is -1
3090 $x->is_inf(); # true if +inf
3091 $x->is_inf('-'); # true if -inf (sign is default '+')
3093 These methods all test the BigInt for beeing one specific value and return
3094 true or false depending on the input. These are faster than doing something
3099 =head2 is_pos()/is_neg()
3101 $x->is_pos(); # true if >= 0
3102 $x->is_neg(); # true if < 0
3104 The methods return true if the argument is positive or negative, respectively.
3105 C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
3106 C<-inf> is negative. A C<zero> is positive.
3108 These methods are only testing the sign, and not the value.
3110 C<is_positive()> and C<is_negative()> are aliase to C<is_pos()> and
3111 C<is_neg()>, respectively. C<is_positive()> and C<is_negative()> were
3112 introduced in v1.36, while C<is_pos()> and C<is_neg()> were only introduced
3115 =head2 is_odd()/is_even()/is_int()
3117 $x->is_odd(); # true if odd, false for even
3118 $x->is_even(); # true if even, false for odd
3119 $x->is_int(); # true if $x is an integer
3121 The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
3122 C<-inf> are not integers and are neither odd nor even.
3124 In BigInt, all numbers except C<NaN>, C<+inf> and C<-inf> are integers.
3130 Compares $x with $y and takes the sign into account.
3131 Returns -1, 0, 1 or undef.
3137 Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
3143 Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
3147 $x->digit($n); # return the nth digit, counting from right
3149 If C<$n> is negative, returns the digit counting from left.
3155 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
3156 and '-inf', respectively. Does nothing for NaN or zero.
3162 Set the number to it's absolute value, e.g. change the sign from '-' to '+'
3163 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
3168 $x->bnorm(); # normalize (no-op)
3174 Two's complement (bit wise not). This is equivalent to
3182 $x->binc(); # increment x by 1
3186 $x->bdec(); # decrement x by 1
3190 $x->badd($y); # addition (add $y to $x)
3194 $x->bsub($y); # subtraction (subtract $y from $x)
3198 $x->bmul($y); # multiplication (multiply $x by $y)
3202 $x->bdiv($y); # divide, set $x to quotient
3203 # return (quo,rem) or quo if scalar
3207 $x->bmod($y); # modulus (x % y)
3211 num->bmodinv($mod); # modular inverse
3213 Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
3214 returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
3215 C<bgcd($num, $mod)==1>.
3219 $num->bmodpow($exp,$mod); # modular exponentation
3220 # ($num**$exp % $mod)
3222 Returns the value of C<$num> taken to the power C<$exp> in the modulus
3223 C<$mod> using binary exponentation. C<bmodpow> is far superior to
3228 because it is much faster - it reduces internal variables into
3229 the modulus whenever possible, so it operates on smaller numbers.
3231 C<bmodpow> also supports negative exponents.
3233 bmodpow($num, -1, $mod)
3235 is exactly equivalent to
3241 $x->bpow($y); # power of arguments (x ** y)
3245 $x->blsft($y); # left shift
3246 $x->blsft($y,$n); # left shift, in base $n (like 10)
3250 $x->brsft($y); # right shift
3251 $x->brsft($y,$n); # right shift, in base $n (like 10)
3255 $x->band($y); # bitwise and
3259 $x->bior($y); # bitwise inclusive or
3263 $x->bxor($y); # bitwise exclusive or
3267 $x->bnot(); # bitwise not (two's complement)
3271 $x->bsqrt(); # calculate square-root
3275 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3279 $x->round($A,$P,$round_mode);
3281 Round $x to accuracy C<$A> or precision C<$P> using the round mode
3286 $x->bround($N); # accuracy: preserve $N digits
3290 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3296 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3297 does change $x in BigFloat.
3303 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3304 does change $x in BigFloat.
3308 bgcd(@values); # greatest common divisor (no OO style)
3312 blcm(@values); # lowest common multiplicator (no OO style)
3317 ($xl,$fl) = $x->length();
3319 Returns the number of digits in the decimal representation of the number.
3320 In list context, returns the length of the integer and fraction part. For
3321 BigInt's, the length of the fraction part will always be 0.
3327 Return the exponent of $x as BigInt.
3333 Return the signed mantissa of $x as BigInt.
3337 $x->parts(); # return (mantissa,exponent) as BigInt
3341 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3347 Returns $x as a BigInt (truncated towards zero). In BigInt this is the same as
3350 C<as_number()> is an alias to this method. C<as_number> was introduced in
3351 v1.22, while C<as_int()> was only introduced in v1.68.
3357 Returns a normalized string represantation of C<$x>.
3361 $x->bsstr(); # normalized string in scientific notation
3365 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3369 $x->as_bin(); # as signed binary string with prefixed 0b
3371 =head1 ACCURACY and PRECISION
3373 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3374 accuracy and precision based rounding, both automatically after every
3375 operation, as well as manually.
3377 This section describes the accuracy/precision handling in Math::Big* as it
3378 used to be and as it is now, complete with an explanation of all terms and
3381 Not yet implemented things (but with correct description) are marked with '!',
3382 things that need to be answered are marked with '?'.
3384 In the next paragraph follows a short description of terms used here (because
3385 these may differ from terms used by others people or documentation).
3387 During the rest of this document, the shortcuts A (for accuracy), P (for
3388 precision), F (fallback) and R (rounding mode) will be used.
3392 A fixed number of digits before (positive) or after (negative)
3393 the decimal point. For example, 123.45 has a precision of -2. 0 means an
3394 integer like 123 (or 120). A precision of 2 means two digits to the left
3395 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3396 numbers with zeros before the decimal point may have different precisions,
3397 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3398 was). It could also have p < 0, when the digits after the decimal point
3401 The string output (of floating point numbers) will be padded with zeros:
3403 Initial value P A Result String
3404 ------------------------------------------------------------
3405 1234.01 -3 1000 1000
3408 1234.001 1 1234 1234.0
3410 1234.01 2 1234.01 1234.01
3411 1234.01 5 1234.01 1234.01000
3413 For BigInts, no padding occurs.
3417 Number of significant digits. Leading zeros are not counted. A
3418 number may have an accuracy greater than the non-zero digits
3419 when there are zeros in it or trailing zeros. For example, 123.456 has
3420 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3422 The string output (of floating point numbers) will be padded with zeros:
3424 Initial value P A Result String
3425 ------------------------------------------------------------
3427 1234.01 6 1234.01 1234.01
3428 1234.1 8 1234.1 1234.1000
3430 For BigInts, no padding occurs.
3434 When both A and P are undefined, this is used as a fallback accuracy when
3437 =head2 Rounding mode R
3439 When rounding a number, different 'styles' or 'kinds'
3440 of rounding are possible. (Note that random rounding, as in
3441 Math::Round, is not implemented.)
3447 truncation invariably removes all digits following the
3448 rounding place, replacing them with zeros. Thus, 987.65 rounded
3449 to tens (P=1) becomes 980, and rounded to the fourth sigdig
3450 becomes 987.6 (A=4). 123.456 rounded to the second place after the
3451 decimal point (P=-2) becomes 123.46.
3453 All other implemented styles of rounding attempt to round to the
3454 "nearest digit." If the digit D immediately to the right of the
3455 rounding place (skipping the decimal point) is greater than 5, the
3456 number is incremented at the rounding place (possibly causing a
3457 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3458 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3459 truncated at the rounding place: e.g. when rounding to units, 0.4
3460 rounds to 0, and -19.4 rounds to -19.
3462 However the results of other styles of rounding differ if the
3463 digit immediately to the right of the rounding place (skipping the
3464 decimal point) is 5 and if there are no digits, or no digits other
3465 than 0, after that 5. In such cases:
3469 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3470 if it is not already. E.g., when rounding to the first sigdig, 0.45
3471 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3475 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3476 it is not already. E.g., when rounding to the first sigdig, 0.45
3477 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3481 round to plus infinity, i.e. always round up. E.g., when
3482 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3483 and 0.4501 also becomes 0.5.
3487 round to minus infinity, i.e. always round down. E.g., when
3488 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3489 but 0.4501 becomes 0.5.
3493 round to zero, i.e. positive numbers down, negative ones up.
3494 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3495 becomes -0.5, but 0.4501 becomes 0.5.
3499 The handling of A & P in MBI/MBF (the old core code shipped with Perl
3500 versions <= 5.7.2) is like this:
3506 * ffround($p) is able to round to $p number of digits after the decimal
3508 * otherwise P is unused
3510 =item Accuracy (significant digits)
3512 * fround($a) rounds to $a significant digits
3513 * only fdiv() and fsqrt() take A as (optional) paramater
3514 + other operations simply create the same number (fneg etc), or more (fmul)
3516 + rounding/truncating is only done when explicitly calling one of fround
3517 or ffround, and never for BigInt (not implemented)
3518 * fsqrt() simply hands its accuracy argument over to fdiv.
3519 * the documentation and the comment in the code indicate two different ways
3520 on how fdiv() determines the maximum number of digits it should calculate,
3521 and the actual code does yet another thing
3523 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3525 result has at most max(scale, length(dividend), length(divisor)) digits
3527 scale = max(scale, length(dividend)-1,length(divisor)-1);
3528 scale += length(divisior) - length(dividend);
3529 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3530 Actually, the 'difference' added to the scale is calculated from the
3531 number of "significant digits" in dividend and divisor, which is derived
3532 by looking at the length of the mantissa. Which is wrong, since it includes
3533 the + sign (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops
3534 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3535 assumption that 124 has 3 significant digits, while 120/7 will get you
3536 '17', not '17.1' since 120 is thought to have 2 significant digits.
3537 The rounding after the division then uses the remainder and $y to determine
3538 wether it must round up or down.
3539 ? I have no idea which is the right way. That's why I used a slightly more
3540 ? simple scheme and tweaked the few failing testcases to match it.
3544 This is how it works now:
3548 =item Setting/Accessing
3550 * You can set the A global via C<< Math::BigInt->accuracy() >> or
3551 C<< Math::BigFloat->accuracy() >> or whatever class you are using.
3552 * You can also set P globally by using C<< Math::SomeClass->precision() >>
3554 * Globals are classwide, and not inherited by subclasses.
3555 * to undefine A, use C<< Math::SomeCLass->accuracy(undef); >>
3556 * to undefine P, use C<< Math::SomeClass->precision(undef); >>
3557 * Setting C<< Math::SomeClass->accuracy() >> clears automatically
3558 C<< Math::SomeClass->precision() >>, and vice versa.
3559 * To be valid, A must be > 0, P can have any value.
3560 * If P is negative, this means round to the P'th place to the right of the
3561 decimal point; positive values mean to the left of the decimal point.
3562 P of 0 means round to integer.
3563 * to find out the current global A, use C<< Math::SomeClass->accuracy() >>
3564 * to find out the current global P, use C<< Math::SomeClass->precision() >>
3565 * use C<< $x->accuracy() >> respective C<< $x->precision() >> for the local
3566 setting of C<< $x >>.
3567 * Please note that C<< $x->accuracy() >> respecive C<< $x->precision() >>
3568 return eventually defined global A or P, when C<< $x >>'s A or P is not
3571 =item Creating numbers
3573 * When you create a number, you can give it's desired A or P via:
3574 $x = Math::BigInt->new($number,$A,$P);
3575 * Only one of A or P can be defined, otherwise the result is NaN
3576 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
3577 globals (if set) will be used. Thus changing the global defaults later on
3578 will not change the A or P of previously created numbers (i.e., A and P of
3579 $x will be what was in effect when $x was created)
3580 * If given undef for A and P, B<no> rounding will occur, and the globals will
3581 B<not> be used. This is used by subclasses to create numbers without
3582 suffering rounding in the parent. Thus a subclass is able to have it's own
3583 globals enforced upon creation of a number by using
3584 C<< $x = Math::BigInt->new($number,undef,undef) >>:
3586 use Math::BigInt::SomeSubclass;
3589 Math::BigInt->accuracy(2);
3590 Math::BigInt::SomeSubClass->accuracy(3);
3591 $x = Math::BigInt::SomeSubClass->new(1234);
3593 $x is now 1230, and not 1200. A subclass might choose to implement
3594 this otherwise, e.g. falling back to the parent's A and P.
3598 * If A or P are enabled/defined, they are used to round the result of each
3599 operation according to the rules below
3600 * Negative P is ignored in Math::BigInt, since BigInts never have digits
3601 after the decimal point
3602 * Math::BigFloat uses Math::BigInt internally, but setting A or P inside
3603 Math::BigInt as globals does not tamper with the parts of a BigFloat.
3604 A flag is used to mark all Math::BigFloat numbers as 'never round'.
3608 * It only makes sense that a number has only one of A or P at a time.
3609 If you set either A or P on one object, or globally, the other one will
3610 be automatically cleared.
3611 * If two objects are involved in an operation, and one of them has A in
3612 effect, and the other P, this results in an error (NaN).
3613 * A takes precendence over P (Hint: A comes before P).
3614 If neither of them is defined, nothing is used, i.e. the result will have
3615 as many digits as it can (with an exception for fdiv/fsqrt) and will not
3617 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
3618 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
3619 If either the dividend's or the divisor's mantissa has more digits than
3620 the value of F, the higher value will be used instead of F.
3621 This is to limit the digits (A) of the result (just consider what would
3622 happen with unlimited A and P in the case of 1/3 :-)
3623 * fdiv will calculate (at least) 4 more digits than required (determined by
3624 A, P or F), and, if F is not used, round the result
3625 (this will still fail in the case of a result like 0.12345000000001 with A
3626 or P of 5, but this can not be helped - or can it?)
3627 * Thus you can have the math done by on Math::Big* class in two modi:
3628 + never round (this is the default):
3629 This is done by setting A and P to undef. No math operation
3630 will round the result, with fdiv() and fsqrt() as exceptions to guard
3631 against overflows. You must explicitely call bround(), bfround() or
3632 round() (the latter with parameters).
3633 Note: Once you have rounded a number, the settings will 'stick' on it
3634 and 'infect' all other numbers engaged in math operations with it, since
3635 local settings have the highest precedence. So, to get SaferRound[tm],
3636 use a copy() before rounding like this:
3638 $x = Math::BigFloat->new(12.34);
3639 $y = Math::BigFloat->new(98.76);
3640 $z = $x * $y; # 1218.6984
3641 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
3642 $z = $x * $y; # still 1218.6984, without
3643 # copy would have been 1210!
3645 + round after each op:
3646 After each single operation (except for testing like is_zero()), the
3647 method round() is called and the result is rounded appropriately. By
3648 setting proper values for A and P, you can have all-the-same-A or
3649 all-the-same-P modes. For example, Math::Currency might set A to undef,
3650 and P to -2, globally.
3652 ?Maybe an extra option that forbids local A & P settings would be in order,
3653 ?so that intermediate rounding does not 'poison' further math?
3655 =item Overriding globals
3657 * you will be able to give A, P and R as an argument to all the calculation
3658 routines; the second parameter is A, the third one is P, and the fourth is
3659 R (shift right by one for binary operations like badd). P is used only if
3660 the first parameter (A) is undefined. These three parameters override the
3661 globals in the order detailed as follows, i.e. the first defined value
3663 (local: per object, global: global default, parameter: argument to sub)
3666 + local A (if defined on both of the operands: smaller one is taken)
3667 + local P (if defined on both of the operands: bigger one is taken)
3671 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
3672 arguments (A and P) instead of one
3674 =item Local settings
3676 * You can set A or P locally by using C<< $x->accuracy() >> or
3677 C<< $x->precision() >>
3678 and thus force different A and P for different objects/numbers.
3679 * Setting A or P this way immediately rounds $x to the new value.
3680 * C<< $x->accuracy() >> clears C<< $x->precision() >>, and vice versa.
3684 * the rounding routines will use the respective global or local settings.
3685 fround()/bround() is for accuracy rounding, while ffround()/bfround()
3687 * the two rounding functions take as the second parameter one of the
3688 following rounding modes (R):
3689 'even', 'odd', '+inf', '-inf', 'zero', 'trunc'
3690 * you can set/get the global R by using C<< Math::SomeClass->round_mode() >>
3691 or by setting C<< $Math::SomeClass::round_mode >>
3692 * after each operation, C<< $result->round() >> is called, and the result may
3693 eventually be rounded (that is, if A or P were set either locally,
3694 globally or as parameter to the operation)
3695 * to manually round a number, call C<< $x->round($A,$P,$round_mode); >>
3696 this will round the number by using the appropriate rounding function
3697 and then normalize it.
3698 * rounding modifies the local settings of the number:
3700 $x = Math::BigFloat->new(123.456);
3704 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
3705 will be 4 from now on.
3707 =item Default values
3716 * The defaults are set up so that the new code gives the same results as
3717 the old code (except in a few cases on fdiv):
3718 + Both A and P are undefined and thus will not be used for rounding
3719 after each operation.
3720 + round() is thus a no-op, unless given extra parameters A and P
3726 The actual numbers are stored as unsigned big integers (with seperate sign).
3727 You should neither care about nor depend on the internal representation; it
3728 might change without notice. Use only method calls like C<< $x->sign(); >>
3729 instead relying on the internal hash keys like in C<< $x->{sign}; >>.
3733 Math with the numbers is done (by default) by a module called
3734 C<Math::BigInt::Calc>. This is equivalent to saying:
3736 use Math::BigInt lib => 'Calc';
3738 You can change this by using:
3740 use Math::BigInt lib => 'BitVect';
3742 The following would first try to find Math::BigInt::Foo, then
3743 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
3745 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
3747 Since Math::BigInt::GMP is in almost all cases faster than Calc (especially in
3748 cases involving really big numbers, where it is B<much> faster), and there is
3749 no penalty if Math::BigInt::GMP is not installed, it is a good idea to always
3752 use Math::BigInt lib => 'GMP';
3754 Different low-level libraries use different formats to store the
3755 numbers. You should not depend on the number having a specific format.
3757 See the respective math library module documentation for further details.
3761 The sign is either '+', '-', 'NaN', '+inf' or '-inf' and stored seperately.
3763 A sign of 'NaN' is used to represent the result when input arguments are not
3764 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
3765 minus infinity. You will get '+inf' when dividing a positive number by 0, and
3766 '-inf' when dividing any negative number by 0.
3768 =head2 mantissa(), exponent() and parts()
3770 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
3773 $m = $x->mantissa();
3774 $e = $x->exponent();
3775 $y = $m * ( 10 ** $e );
3776 print "ok\n" if $x == $y;
3778 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
3779 in one go. Both the returned mantissa and exponent have a sign.
3781 Currently, for BigInts C<$e> is always 0, except for NaN, +inf and -inf,
3782 where it is C<NaN>; and for C<$x == 0>, where it is C<1> (to be compatible
3783 with Math::BigFloat's internal representation of a zero as C<0E1>).
3785 C<$m> is currently just a copy of the original number. The relation between
3786 C<$e> and C<$m> will stay always the same, though their real values might
3793 sub bint { Math::BigInt->new(shift); }
3795 $x = Math::BigInt->bstr("1234") # string "1234"
3796 $x = "$x"; # same as bstr()
3797 $x = Math::BigInt->bneg("1234"); # BigInt "-1234"
3798 $x = Math::BigInt->babs("-12345"); # BigInt "12345"
3799 $x = Math::BigInt->bnorm("-0 00"); # BigInt "0"
3800 $x = bint(1) + bint(2); # BigInt "3"
3801 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
3802 $x = bint(1); # BigInt "1"
3803 $x = $x + 5 / 2; # BigInt "3"
3804 $x = $x ** 3; # BigInt "27"
3805 $x *= 2; # BigInt "54"
3806 $x = Math::BigInt->new(0); # BigInt "0"
3808 $x = Math::BigInt->badd(4,5) # BigInt "9"
3809 print $x->bsstr(); # 9e+0
3811 Examples for rounding:
3816 $x = Math::BigFloat->new(123.4567);
3817 $y = Math::BigFloat->new(123.456789);
3818 Math::BigFloat->accuracy(4); # no more A than 4
3820 ok ($x->copy()->fround(),123.4); # even rounding
3821 print $x->copy()->fround(),"\n"; # 123.4
3822 Math::BigFloat->round_mode('odd'); # round to odd
3823 print $x->copy()->fround(),"\n"; # 123.5
3824 Math::BigFloat->accuracy(5); # no more A than 5
3825 Math::BigFloat->round_mode('odd'); # round to odd
3826 print $x->copy()->fround(),"\n"; # 123.46
3827 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
3828 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
3830 Math::BigFloat->accuracy(undef); # A not important now
3831 Math::BigFloat->precision(2); # P important
3832 print $x->copy()->bnorm(),"\n"; # 123.46
3833 print $x->copy()->fround(),"\n"; # 123.46
3835 Examples for converting:
3837 my $x = Math::BigInt->new('0b1'.'01' x 123);
3838 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
3840 =head1 Autocreating constants
3842 After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
3843 and binary constants in the given scope are converted to C<Math::BigInt>.
3844 This conversion happens at compile time.
3848 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
3850 prints the integer value of C<2**100>. Note that without conversion of
3851 constants the expression 2**100 will be calculated as perl scalar.
3853 Please note that strings and floating point constants are not affected,
3856 use Math::BigInt qw/:constant/;
3858 $x = 1234567890123456789012345678901234567890
3859 + 123456789123456789;
3860 $y = '1234567890123456789012345678901234567890'
3861 + '123456789123456789';
3863 do not work. You need an explicit Math::BigInt->new() around one of the
3864 operands. You should also quote large constants to protect loss of precision:
3868 $x = Math::BigInt->new('1234567889123456789123456789123456789');
3870 Without the quotes Perl would convert the large number to a floating point
3871 constant at compile time and then hand the result to BigInt, which results in
3872 an truncated result or a NaN.
3874 This also applies to integers that look like floating point constants:
3876 use Math::BigInt ':constant';
3878 print ref(123e2),"\n";
3879 print ref(123.2e2),"\n";
3881 will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
3882 to get this to work.
3886 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
3887 must be made in the second case. For long numbers, the copy can eat up to 20%
3888 of the work (in the case of addition/subtraction, less for
3889 multiplication/division). If $y is very small compared to $x, the form
3890 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
3891 more time then the actual addition.
3893 With a technique called copy-on-write, the cost of copying with overload could
3894 be minimized or even completely avoided. A test implementation of COW did show
3895 performance gains for overloaded math, but introduced a performance loss due
3896 to a constant overhead for all other operatons. So Math::BigInt does currently
3899 The rewritten version of this module (vs. v0.01) is slower on certain
3900 operations, like C<new()>, C<bstr()> and C<numify()>. The reason are that it
3901 does now more work and handles much more cases. The time spent in these
3902 operations is usually gained in the other math operations so that code on
3903 the average should get (much) faster. If they don't, please contact the author.
3905 Some operations may be slower for small numbers, but are significantly faster
3906 for big numbers. Other operations are now constant (O(1), like C<bneg()>,
3907 C<babs()> etc), instead of O(N) and thus nearly always take much less time.
3908 These optimizations were done on purpose.
3910 If you find the Calc module to slow, try to install any of the replacement
3911 modules and see if they help you.
3913 =head2 Alternative math libraries
3915 You can use an alternative library to drive Math::BigInt via:
3917 use Math::BigInt lib => 'Module';
3919 See L<MATH LIBRARY> for more information.
3921 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
3925 =head1 Subclassing Math::BigInt
3927 The basic design of Math::BigInt allows simple subclasses with very little
3928 work, as long as a few simple rules are followed:
3934 The public API must remain consistent, i.e. if a sub-class is overloading
3935 addition, the sub-class must use the same name, in this case badd(). The
3936 reason for this is that Math::BigInt is optimized to call the object methods
3941 The private object hash keys like C<$x->{sign}> may not be changed, but
3942 additional keys can be added, like C<$x->{_custom}>.
3946 Accessor functions are available for all existing object hash keys and should
3947 be used instead of directly accessing the internal hash keys. The reason for
3948 this is that Math::BigInt itself has a pluggable interface which permits it
3949 to support different storage methods.
3953 More complex sub-classes may have to replicate more of the logic internal of
3954 Math::BigInt if they need to change more basic behaviors. A subclass that
3955 needs to merely change the output only needs to overload C<bstr()>.
3957 All other object methods and overloaded functions can be directly inherited
3958 from the parent class.
3960 At the very minimum, any subclass will need to provide it's own C<new()> and can
3961 store additional hash keys in the object. There are also some package globals
3962 that must be defined, e.g.:
3966 $precision = -2; # round to 2 decimal places
3967 $round_mode = 'even';
3970 Additionally, you might want to provide the following two globals to allow
3971 auto-upgrading and auto-downgrading to work correctly:
3976 This allows Math::BigInt to correctly retrieve package globals from the
3977 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
3978 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
3984 in your subclass to automatically inherit the overloading from the parent. If
3985 you like, you can change part of the overloading, look at Math::String for an
3990 When used like this:
3992 use Math::BigInt upgrade => 'Foo::Bar';
3994 certain operations will 'upgrade' their calculation and thus the result to
3995 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
3997 use Math::BigInt upgrade => 'Math::BigFloat';
3999 As a shortcut, you can use the module C<bignum>:
4003 Also good for oneliners:
4005 perl -Mbignum -le 'print 2 ** 255'
4007 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
4008 as well es preserve accuracy (as in sqrt(3)).
4010 Beware: This feature is not fully implemented yet.
4014 The following methods upgrade themselves unconditionally; that is if upgrade
4015 is in effect, they will always hand up their work:
4027 Beware: This list is not complete.
4029 All other methods upgrade themselves only when one (or all) of their
4030 arguments are of the class mentioned in $upgrade (This might change in later
4031 versions to a more sophisticated scheme):
4037 =item broot() does not work
4039 The broot() function in BigInt may only work for small values. This will be
4040 fixed in a later version.
4042 =item Out of Memory!
4044 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
4045 C<eval()> in your code will crash with "Out of memory". This is probably an
4046 overload/exporter bug. You can workaround by not having C<eval()>
4047 and ':constant' at the same time or upgrade your Perl to a newer version.
4049 =item Fails to load Calc on Perl prior 5.6.0
4051 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
4052 will fall back to eval { require ... } when loading the math lib on Perls
4053 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
4054 filesystems using a different seperator.
4060 Some things might not work as you expect them. Below is documented what is
4061 known to be troublesome:
4065 =item bstr(), bsstr() and 'cmp'
4067 Both C<bstr()> and C<bsstr()> as well as automated stringify via overload now
4068 drop the leading '+'. The old code would return '+3', the new returns '3'.
4069 This is to be consistent with Perl and to make C<cmp> (especially with
4070 overloading) to work as you expect. It also solves problems with C<Test.pm>,
4071 because it's C<ok()> uses 'eq' internally.
4073 Mark Biggar said, when asked about to drop the '+' altogether, or make only
4076 I agree (with the first alternative), don't add the '+' on positive
4077 numbers. It's not as important anymore with the new internal
4078 form for numbers. It made doing things like abs and neg easier,
4079 but those have to be done differently now anyway.
4081 So, the following examples will now work all as expected:
4084 BEGIN { plan tests => 1 }
4087 my $x = new Math::BigInt 3*3;
4088 my $y = new Math::BigInt 3*3;
4091 print "$x eq 9" if $x eq $y;
4092 print "$x eq 9" if $x eq '9';
4093 print "$x eq 9" if $x eq 3*3;
4095 Additionally, the following still works:
4097 print "$x == 9" if $x == $y;
4098 print "$x == 9" if $x == 9;
4099 print "$x == 9" if $x == 3*3;
4101 There is now a C<bsstr()> method to get the string in scientific notation aka
4102 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
4103 for comparisation, but Perl will represent some numbers as 100 and others
4104 as 1e+308. If in doubt, convert both arguments to Math::BigInt before
4105 comparing them as strings:
4108 BEGIN { plan tests => 3 }
4111 $x = Math::BigInt->new('1e56'); $y = 1e56;
4112 ok ($x,$y); # will fail
4113 ok ($x->bsstr(),$y); # okay
4114 $y = Math::BigInt->new($y);
4117 Alternatively, simple use C<< <=> >> for comparisations, this will get it
4118 always right. There is not yet a way to get a number automatically represented
4119 as a string that matches exactly the way Perl represents it.
4123 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
4126 $x = Math::BigInt->new(123);
4127 $y = int($x); # BigInt 123
4128 $x = Math::BigFloat->new(123.45);
4129 $y = int($x); # BigInt 123
4131 In all Perl versions you can use C<as_number()> for the same effect:
4133 $x = Math::BigFloat->new(123.45);
4134 $y = $x->as_number(); # BigInt 123
4136 This also works for other subclasses, like Math::String.
4138 It is yet unlcear whether overloaded int() should return a scalar or a BigInt.
4142 The following will probably not do what you expect:
4144 $c = Math::BigInt->new(123);
4145 print $c->length(),"\n"; # prints 30
4147 It prints both the number of digits in the number and in the fraction part
4148 since print calls C<length()> in list context. Use something like:
4150 print scalar $c->length(),"\n"; # prints 3
4154 The following will probably not do what you expect:
4156 print $c->bdiv(10000),"\n";
4158 It prints both quotient and remainder since print calls C<bdiv()> in list
4159 context. Also, C<bdiv()> will modify $c, so be carefull. You probably want
4162 print $c / 10000,"\n";
4163 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
4167 The quotient is always the greatest integer less than or equal to the
4168 real-valued quotient of the two operands, and the remainder (when it is
4169 nonzero) always has the same sign as the second operand; so, for
4179 As a consequence, the behavior of the operator % agrees with the
4180 behavior of Perl's built-in % operator (as documented in the perlop
4181 manpage), and the equation
4183 $x == ($x / $y) * $y + ($x % $y)
4185 holds true for any $x and $y, which justifies calling the two return
4186 values of bdiv() the quotient and remainder. The only exception to this rule
4187 are when $y == 0 and $x is negative, then the remainder will also be
4188 negative. See below under "infinity handling" for the reasoning behing this.
4190 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
4191 not change BigInt's way to do things. This is because under 'use integer' Perl
4192 will do what the underlying C thinks is right and this is different for each
4193 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
4194 the author to implement it ;)
4196 =item infinity handling
4198 Here are some examples that explain the reasons why certain results occur while
4201 The following table shows the result of the division and the remainder, so that
4202 the equation above holds true. Some "ordinary" cases are strewn in to show more
4203 clearly the reasoning:
4205 A / B = C, R so that C * B + R = A
4206 =========================================================
4207 5 / 8 = 0, 5 0 * 8 + 5 = 5
4208 0 / 8 = 0, 0 0 * 8 + 0 = 0
4209 0 / inf = 0, 0 0 * inf + 0 = 0
4210 0 /-inf = 0, 0 0 * -inf + 0 = 0
4211 5 / inf = 0, 5 0 * inf + 5 = 5
4212 5 /-inf = 0, 5 0 * -inf + 5 = 5
4213 -5/ inf = 0, -5 0 * inf + -5 = -5
4214 -5/-inf = 0, -5 0 * -inf + -5 = -5
4215 inf/ 5 = inf, 0 inf * 5 + 0 = inf
4216 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
4217 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
4218 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
4219 5/ 5 = 1, 0 1 * 5 + 0 = 5
4220 -5/ -5 = 1, 0 1 * -5 + 0 = -5
4221 inf/ inf = 1, 0 1 * inf + 0 = inf
4222 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
4223 inf/-inf = -1, 0 -1 * -inf + 0 = inf
4224 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
4225 8/ 0 = inf, 8 inf * 0 + 8 = 8
4226 inf/ 0 = inf, inf inf * 0 + inf = inf
4229 These cases below violate the "remainder has the sign of the second of the two
4230 arguments", since they wouldn't match up otherwise.
4232 A / B = C, R so that C * B + R = A
4233 ========================================================
4234 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
4235 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
4237 =item Modifying and =
4241 $x = Math::BigFloat->new(5);
4244 It will not do what you think, e.g. making a copy of $x. Instead it just makes
4245 a second reference to the B<same> object and stores it in $y. Thus anything
4246 that modifies $x (except overloaded operators) will modify $y, and vice versa.
4247 Or in other words, C<=> is only safe if you modify your BigInts only via
4248 overloaded math. As soon as you use a method call it breaks:
4251 print "$x, $y\n"; # prints '10, 10'
4253 If you want a true copy of $x, use:
4257 You can also chain the calls like this, this will make first a copy and then
4260 $y = $x->copy()->bmul(2);
4262 See also the documentation for overload.pm regarding C<=>.
4266 C<bpow()> (and the rounding functions) now modifies the first argument and
4267 returns it, unlike the old code which left it alone and only returned the
4268 result. This is to be consistent with C<badd()> etc. The first three will
4269 modify $x, the last one won't:
4271 print bpow($x,$i),"\n"; # modify $x
4272 print $x->bpow($i),"\n"; # ditto
4273 print $x **= $i,"\n"; # the same
4274 print $x ** $i,"\n"; # leave $x alone
4276 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4278 =item Overloading -$x
4288 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4289 needs to preserve $x since it does not know that it later will get overwritten.
4290 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4292 With Copy-On-Write, this issue would be gone, but C-o-W is not implemented
4293 since it is slower for all other things.
4295 =item Mixing different object types
4297 In Perl you will get a floating point value if you do one of the following:
4303 With overloaded math, only the first two variants will result in a BigFloat:
4308 $mbf = Math::BigFloat->new(5);
4309 $mbi2 = Math::BigInteger->new(5);
4310 $mbi = Math::BigInteger->new(2);
4312 # what actually gets called:
4313 $float = $mbf + $mbi; # $mbf->badd()
4314 $float = $mbf / $mbi; # $mbf->bdiv()
4315 $integer = $mbi + $mbf; # $mbi->badd()
4316 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4317 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4319 This is because math with overloaded operators follows the first (dominating)
4320 operand, and the operation of that is called and returns thus the result. So,
4321 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4322 the result should be a Math::BigFloat or the second operant is one.
4324 To get a Math::BigFloat you either need to call the operation manually,
4325 make sure the operands are already of the proper type or casted to that type
4326 via Math::BigFloat->new():
4328 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4330 Beware of simple "casting" the entire expression, this would only convert
4331 the already computed result:
4333 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4335 Beware also of the order of more complicated expressions like:
4337 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4338 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4340 If in doubt, break the expression into simpler terms, or cast all operands
4341 to the desired resulting type.
4343 Scalar values are a bit different, since:
4348 will both result in the proper type due to the way the overloaded math works.
4350 This section also applies to other overloaded math packages, like Math::String.
4352 One solution to you problem might be autoupgrading|upgrading. See the
4353 pragmas L<bignum>, L<bigint> and L<bigrat> for an easy way to do this.
4357 C<bsqrt()> works only good if the result is a big integer, e.g. the square
4358 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4359 mode. The reason is that the result is always truncated to an integer.
4361 If you want a better approximation of the square root, then use:
4363 $x = Math::BigFloat->new(12);
4364 Math::BigFloat->precision(0);
4365 Math::BigFloat->round_mode('even');
4366 print $x->copy->bsqrt(),"\n"; # 4
4368 Math::BigFloat->precision(2);
4369 print $x->bsqrt(),"\n"; # 3.46
4370 print $x->bsqrt(3),"\n"; # 3.464
4374 For negative numbers in base see also L<brsft|brsft>.
4380 This program is free software; you may redistribute it and/or modify it under
4381 the same terms as Perl itself.
4385 L<Math::BigFloat>, L<Math::BigRat> and L<Math::Big> as well as
4386 L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4388 The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest
4389 because they solve the autoupgrading/downgrading issue, at least partly.
4392 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4393 more documentation including a full version history, testcases, empty
4394 subclass files and benchmarks.
4398 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4399 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001, 2002
4400 and still at it in 2003.
4402 Many people contributed in one or more ways to the final beast, see the file
4403 CREDITS for an (uncomplete) list. If you miss your name, please drop me a