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 $class = "Math::BigInt";
23 @ISA = qw( Exporter );
24 @EXPORT_OK = qw( objectify _swap bgcd blcm);
25 use vars qw/$round_mode $accuracy $precision $div_scale $rnd_mode/;
26 use vars qw/$upgrade $downgrade/;
27 # the following are internal and should never be accessed from the outside
28 use vars qw/$_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 indicates this
33 # swapping. To make it work, we use a helper routine which not only reswaps the
34 # params, but also makes a new object in this case. See _swap() for details,
35 # especially the cases of operators with different classes.
37 # For overloaded ops with only one argument we simple use $_[0]->copy() to
38 # preserve the argument.
40 # Thus inheritance of overload operators becomes possible and transparent for
41 # our subclasses without the need to repeat the entire overload section there.
44 '=' => sub { $_[0]->copy(); },
46 # '+' and '-' do not use _swap, since it is a triffle slower. If you want to
47 # override _swap (if ever), then override overload of '+' and '-', too!
48 # for sub it is a bit tricky to keep b: b-a => -a+b
49 '-' => sub { my $c = $_[0]->copy; $_[2] ?
50 $c->bneg()->badd($_[1]) :
52 '+' => sub { $_[0]->copy()->badd($_[1]); },
54 # some shortcuts for speed (assumes that reversed order of arguments is routed
55 # to normal '+' and we thus can always modify first arg. If this is changed,
56 # this breaks and must be adjusted.)
57 '+=' => sub { $_[0]->badd($_[1]); },
58 '-=' => sub { $_[0]->bsub($_[1]); },
59 '*=' => sub { $_[0]->bmul($_[1]); },
60 '/=' => sub { scalar $_[0]->bdiv($_[1]); },
61 '%=' => sub { $_[0]->bmod($_[1]); },
62 '^=' => sub { $_[0]->bxor($_[1]); },
63 '&=' => sub { $_[0]->band($_[1]); },
64 '|=' => sub { $_[0]->bior($_[1]); },
65 '**=' => sub { $_[0]->bpow($_[1]); },
67 # not supported by Perl yet
68 '..' => \&_pointpoint,
70 '<=>' => sub { $_[2] ?
71 ref($_[0])->bcmp($_[1],$_[0]) :
75 "$_[1]" cmp $_[0]->bstr() :
76 $_[0]->bstr() cmp "$_[1]" },
78 'log' => sub { $_[0]->copy()->blog(); },
79 'int' => sub { $_[0]->copy(); },
80 'neg' => sub { $_[0]->copy()->bneg(); },
81 'abs' => sub { $_[0]->copy()->babs(); },
82 'sqrt' => sub { $_[0]->copy()->bsqrt(); },
83 '~' => sub { $_[0]->copy()->bnot(); },
85 '*' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bmul($a[1]); },
86 '/' => sub { my @a = ref($_[0])->_swap(@_);scalar $a[0]->bdiv($a[1]);},
87 '%' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bmod($a[1]); },
88 '**' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bpow($a[1]); },
89 '<<' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->blsft($a[1]); },
90 '>>' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->brsft($a[1]); },
92 '&' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->band($a[1]); },
93 '|' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bior($a[1]); },
94 '^' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bxor($a[1]); },
96 # can modify arg of ++ and --, so avoid a new-copy for speed, but don't
97 # use $_[0]->__one(), it modifies $_[0] to be 1!
98 '++' => sub { $_[0]->binc() },
99 '--' => sub { $_[0]->bdec() },
101 # if overloaded, O(1) instead of O(N) and twice as fast for small numbers
103 # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/
104 # v5.6.1 dumps on that: return !$_[0]->is_zero() || undef; :-(
105 my $t = !$_[0]->is_zero();
110 # the original qw() does not work with the TIESCALAR below, why?
111 # Order of arguments unsignificant
112 '""' => sub { $_[0]->bstr(); },
113 '0+' => sub { $_[0]->numify(); }
116 ##############################################################################
117 # global constants, flags and accessory
119 # these are public, but their usage is not recommended, use the accessor
122 $round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
127 $upgrade = undef; # default is no upgrade
128 $downgrade = undef; # default is no downgrade
130 # these are internally, and not to be used from the outside
132 use constant MB_NEVER_ROUND => 0x0001;
134 $_trap_nan = 0; # are NaNs ok? set w/ config()
135 $_trap_inf = 0; # are infs ok? set w/ config()
136 my $nan = 'NaN'; # constants for easier life
138 my $CALC = 'Math::BigInt::Calc'; # module to do the low level math
139 my $IMPORT = 0; # was import() called yet?
140 # used to make require work
142 ##############################################################################
143 # the old code had $rnd_mode, so we need to support it, too
146 sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; }
147 sub FETCH { return $round_mode; }
148 sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
150 BEGIN { tie $rnd_mode, 'Math::BigInt'; }
152 ##############################################################################
157 # make Class->round_mode() work
159 my $class = ref($self) || $self || __PACKAGE__;
163 if ($m !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
165 require Carp; Carp::croak ("Unknown round mode '$m'");
167 return ${"${class}::round_mode"} = $m;
169 ${"${class}::round_mode"};
175 # make Class->upgrade() work
177 my $class = ref($self) || $self || __PACKAGE__;
178 # need to set new value?
182 return ${"${class}::upgrade"} = $u;
184 ${"${class}::upgrade"};
190 # make Class->downgrade() work
192 my $class = ref($self) || $self || __PACKAGE__;
193 # need to set new value?
197 return ${"${class}::downgrade"} = $u;
199 ${"${class}::downgrade"};
205 # make Class->div_scale() work
207 my $class = ref($self) || $self || __PACKAGE__;
212 require Carp; Carp::croak ('div_scale must be greater than zero');
214 ${"${class}::div_scale"} = shift;
216 ${"${class}::div_scale"};
221 # $x->accuracy($a); ref($x) $a
222 # $x->accuracy(); ref($x)
223 # Class->accuracy(); class
224 # Class->accuracy($a); class $a
227 my $class = ref($x) || $x || __PACKAGE__;
230 # need to set new value?
234 # convert objects to scalars to avoid deep recursion. If object doesn't
235 # have numify(), then hopefully it will have overloading for int() and
236 # boolean test without wandering into a deep recursion path...
237 $a = $a->numify() if ref($a) && $a->can('numify');
241 # also croak on non-numerical
245 Carp::croak ('Argument to accuracy must be greater than zero');
249 require Carp; Carp::croak ('Argument to accuracy must be an integer');
254 # $object->accuracy() or fallback to global
255 $x->bround($a) if $a; # not for undef, 0
256 $x->{_a} = $a; # set/overwrite, even if not rounded
257 $x->{_p} = undef; # clear P
258 $a = ${"${class}::accuracy"} unless defined $a; # proper return value
263 ${"${class}::accuracy"} = $a;
264 ${"${class}::precision"} = undef; # clear P
266 return $a; # shortcut
270 # $object->accuracy() or fallback to global
271 $r = $x->{_a} if ref($x);
272 # but don't return global undef, when $x's accuracy is 0!
273 $r = ${"${class}::accuracy"} if !defined $r;
279 # $x->precision($p); ref($x) $p
280 # $x->precision(); ref($x)
281 # Class->precision(); class
282 # Class->precision($p); class $p
285 my $class = ref($x) || $x || __PACKAGE__;
291 # convert objects to scalars to avoid deep recursion. If object doesn't
292 # have numify(), then hopefully it will have overloading for int() and
293 # boolean test without wandering into a deep recursion path...
294 $p = $p->numify() if ref($p) && $p->can('numify');
295 if ((defined $p) && (int($p) != $p))
297 require Carp; Carp::croak ('Argument to precision must be an integer');
301 # $object->precision() or fallback to global
302 $x->bfround($p) if $p; # not for undef, 0
303 $x->{_p} = $p; # set/overwrite, even if not rounded
304 $x->{_a} = undef; # clear A
305 $p = ${"${class}::precision"} unless defined $p; # proper return value
310 ${"${class}::precision"} = $p;
311 ${"${class}::accuracy"} = undef; # clear A
313 return $p; # shortcut
317 # $object->precision() or fallback to global
318 $r = $x->{_p} if ref($x);
319 # but don't return global undef, when $x's precision is 0!
320 $r = ${"${class}::precision"} if !defined $r;
326 # return (or set) configuration data as hash ref
327 my $class = shift || 'Math::BigInt';
332 # try to set given options as arguments from hash
335 if (ref($args) ne 'HASH')
339 # these values can be "set"
343 upgrade downgrade precision accuracy round_mode div_scale/
346 $set_args->{$key} = $args->{$key} if exists $args->{$key};
347 delete $args->{$key};
352 Carp::croak ("Illegal key(s) '",
353 join("','",keys %$args),"' passed to $class\->config()");
355 foreach my $key (keys %$set_args)
357 if ($key =~ /^trap_(inf|nan)\z/)
359 ${"${class}::_trap_$1"} = ($set_args->{"trap_$1"} ? 1 : 0);
362 # use a call instead of just setting the $variable to check argument
363 $class->$key($set_args->{$key});
367 # now return actual configuration
371 lib_version => ${"${CALC}::VERSION"},
373 trap_nan => ${"${class}::_trap_nan"},
374 trap_inf => ${"${class}::_trap_inf"},
375 version => ${"${class}::VERSION"},
378 upgrade downgrade precision accuracy round_mode div_scale
381 $cfg->{$key} = ${"${class}::$key"};
388 # select accuracy parameter based on precedence,
389 # used by bround() and bfround(), may return undef for scale (means no op)
390 my ($x,$s,$m,$scale,$mode) = @_;
391 $scale = $x->{_a} if !defined $scale;
392 $scale = $s if (!defined $scale);
393 $mode = $m if !defined $mode;
394 return ($scale,$mode);
399 # select precision parameter based on precedence,
400 # used by bround() and bfround(), may return undef for scale (means no op)
401 my ($x,$s,$m,$scale,$mode) = @_;
402 $scale = $x->{_p} if !defined $scale;
403 $scale = $s if (!defined $scale);
404 $mode = $m if !defined $mode;
405 return ($scale,$mode);
408 ##############################################################################
416 # if two arguments, the first one is the class to "swallow" subclasses
424 return unless ref($x); # only for objects
426 my $self = {}; bless $self,$c;
428 foreach my $k (keys %$x)
432 $self->{value} = $CALC->_copy($x->{value}); next;
434 if (!($r = ref($x->{$k})))
436 $self->{$k} = $x->{$k}; next;
440 $self->{$k} = \${$x->{$k}};
442 elsif ($r eq 'ARRAY')
444 $self->{$k} = [ @{$x->{$k}} ];
448 # only one level deep!
449 foreach my $h (keys %{$x->{$k}})
451 $self->{$k}->{$h} = $x->{$k}->{$h};
457 if ($xk->can('copy'))
459 $self->{$k} = $xk->copy();
463 $self->{$k} = $xk->new($xk);
472 # create a new BigInt object from a string or another BigInt object.
473 # see hash keys documented at top
475 # the argument could be an object, so avoid ||, && etc on it, this would
476 # cause costly overloaded code to be called. The only allowed ops are
479 my ($class,$wanted,$a,$p,$r) = @_;
481 # avoid numify-calls by not using || on $wanted!
482 return $class->bzero($a,$p) if !defined $wanted; # default to 0
483 return $class->copy($wanted,$a,$p,$r)
484 if ref($wanted) && $wanted->isa($class); # MBI or subclass
486 $class->import() if $IMPORT == 0; # make require work
488 my $self = bless {}, $class;
490 # shortcut for "normal" numbers
491 if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*\z/))
493 $self->{sign} = $1 || '+';
495 if ($wanted =~ /^[+-]/)
497 # remove sign without touching wanted to make it work with constants
498 my $t = $wanted; $t =~ s/^[+-]//; $ref = \$t;
500 # force to string version (otherwise Pari is unhappy about overflowed
501 # constants, for instance)
502 # not good, BigInt shouldn't need to know about alternative libs:
503 # $ref = \"$$ref" if $CALC eq 'Math::BigInt::Pari';
504 $self->{value} = $CALC->_new($ref);
506 if ( (defined $a) || (defined $p)
507 || (defined ${"${class}::precision"})
508 || (defined ${"${class}::accuracy"})
511 $self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p);
516 # handle '+inf', '-inf' first
517 if ($wanted =~ /^[+-]?inf$/)
519 $self->{value} = $CALC->_zero();
520 $self->{sign} = $wanted; $self->{sign} = '+inf' if $self->{sign} eq 'inf';
523 # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign
524 my ($mis,$miv,$mfv,$es,$ev) = _split(\$wanted);
529 require Carp; Carp::croak("$wanted is not a number in $class");
531 $self->{value} = $CALC->_zero();
532 $self->{sign} = $nan;
537 # _from_hex or _from_bin
538 $self->{value} = $mis->{value};
539 $self->{sign} = $mis->{sign};
540 return $self; # throw away $mis
542 # make integer from mantissa by adjusting exp, then convert to bigint
543 $self->{sign} = $$mis; # store sign
544 $self->{value} = $CALC->_zero(); # for all the NaN cases
545 my $e = int("$$es$$ev"); # exponent (avoid recursion)
548 my $diff = $e - CORE::length($$mfv);
549 if ($diff < 0) # Not integer
553 require Carp; Carp::croak("$wanted not an integer in $class");
556 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
557 $self->{sign} = $nan;
561 # adjust fraction and add it to value
562 #print "diff > 0 $$miv\n";
563 $$miv = $$miv . ($$mfv . '0' x $diff);
568 if ($$mfv ne '') # e <= 0
570 # fraction and negative/zero E => NOI
573 require Carp; Carp::croak("$wanted not an integer in $class");
575 #print "NOI 2 \$\$mfv '$$mfv'\n";
576 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
577 $self->{sign} = $nan;
581 # xE-y, and empty mfv
584 if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
588 require Carp; Carp::croak("$wanted not an integer in $class");
591 return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
592 $self->{sign} = $nan;
596 $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
597 $self->{value} = $CALC->_new($miv) if $self->{sign} =~ /^[+-]$/;
598 # if any of the globals is set, use them to round and store them inside $self
599 # do not round for new($x,undef,undef) since that is used by MBF to signal
601 $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p;
607 # create a bigint 'NaN', if given a BigInt, set it to 'NaN'
609 $self = $class if !defined $self;
612 my $c = $self; $self = {}; bless $self, $c;
615 if (${"${class}::_trap_nan"})
618 Carp::croak ("Tried to set $self to NaN in $class\::bnan()");
620 $self->import() if $IMPORT == 0; # make require work
621 return if $self->modify('bnan');
622 if ($self->can('_bnan'))
624 # use subclass to initialize
629 # otherwise do our own thing
630 $self->{value} = $CALC->_zero();
632 $self->{sign} = $nan;
633 delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly
639 # create a bigint '+-inf', if given a BigInt, set it to '+-inf'
640 # the sign is either '+', or if given, used from there
642 my $sign = shift; $sign = '+' if !defined $sign || $sign !~ /^-(inf)?$/;
643 $self = $class if !defined $self;
646 my $c = $self; $self = {}; bless $self, $c;
649 if (${"${class}::_trap_inf"})
652 Carp::croak ("Tried to set $self to +-inf in $class\::binfn()");
654 $self->import() if $IMPORT == 0; # make require work
655 return if $self->modify('binf');
656 if ($self->can('_binf'))
658 # use subclass to initialize
663 # otherwise do our own thing
664 $self->{value} = $CALC->_zero();
666 $sign = $sign . 'inf' if $sign !~ /inf$/; # - => -inf
667 $self->{sign} = $sign;
668 ($self->{_a},$self->{_p}) = @_; # take over requested rounding
674 # create a bigint '+0', if given a BigInt, set it to 0
676 $self = $class if !defined $self;
680 my $c = $self; $self = {}; bless $self, $c;
682 $self->import() if $IMPORT == 0; # make require work
683 return if $self->modify('bzero');
685 if ($self->can('_bzero'))
687 # use subclass to initialize
692 # otherwise do our own thing
693 $self->{value} = $CALC->_zero();
700 # call like: $x->bzero($a,$p,$r,$y);
701 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
706 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
708 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
716 # create a bigint '+1' (or -1 if given sign '-'),
717 # if given a BigInt, set it to +1 or -1, respecively
719 my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
720 $self = $class if !defined $self;
724 my $c = $self; $self = {}; bless $self, $c;
726 $self->import() if $IMPORT == 0; # make require work
727 return if $self->modify('bone');
729 if ($self->can('_bone'))
731 # use subclass to initialize
736 # otherwise do our own thing
737 $self->{value} = $CALC->_one();
739 $self->{sign} = $sign;
744 # call like: $x->bone($sign,$a,$p,$r,$y);
745 ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
750 if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
752 if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
758 ##############################################################################
759 # string conversation
763 # (ref to BFLOAT or num_str ) return num_str
764 # Convert number from internal format to scientific string format.
765 # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
766 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
767 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
769 if ($x->{sign} !~ /^[+-]$/)
771 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
774 my ($m,$e) = $x->parts();
775 my $sign = 'e+'; # e can only be positive
776 return $m->bstr().$sign.$e->bstr();
781 # make a string from bigint object
782 my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
783 # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
785 if ($x->{sign} !~ /^[+-]$/)
787 return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
790 my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
791 return $es.${$CALC->_str($x->{value})};
796 # Make a "normal" scalar from a BigInt object
797 my $x = shift; $x = $class->new($x) unless ref $x;
799 return $x->bstr() if $x->{sign} !~ /^[+-]$/;
800 my $num = $CALC->_num($x->{value});
801 return -$num if $x->{sign} eq '-';
805 ##############################################################################
806 # public stuff (usually prefixed with "b")
810 # return the sign of the number: +/-/-inf/+inf/NaN
811 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
816 sub _find_round_parameters
818 # After any operation or when calling round(), the result is rounded by
819 # regarding the A & P from arguments, local parameters, or globals.
821 # !!!!!!! If you change this, remember to change round(), too! !!!!!!!!!!
823 # This procedure finds the round parameters, but it is for speed reasons
824 # duplicated in round. Otherwise, it is tested by the testsuite and used
827 # returns ($self) or ($self,$a,$p,$r) - sets $self to NaN of both A and P
828 # were requested/defined (locally or globally or both)
830 my ($self,$a,$p,$r,@args) = @_;
831 # $a accuracy, if given by caller
832 # $p precision, if given by caller
833 # $r round_mode, if given by caller
834 # @args all 'other' arguments (0 for unary, 1 for binary ops)
836 # leave bigfloat parts alone
837 return ($self) if exists $self->{_f} && ($self->{_f} & MB_NEVER_ROUND) != 0;
839 my $c = ref($self); # find out class of argument(s)
842 # now pick $a or $p, but only if we have got "arguments"
845 foreach ($self,@args)
847 # take the defined one, or if both defined, the one that is smaller
848 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
853 # even if $a is defined, take $p, to signal error for both defined
854 foreach ($self,@args)
856 # take the defined one, or if both defined, the one that is bigger
858 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
861 # if still none defined, use globals (#2)
862 $a = ${"$c\::accuracy"} unless defined $a;
863 $p = ${"$c\::precision"} unless defined $p;
865 # A == 0 is useless, so undef it to signal no rounding
866 $a = undef if defined $a && $a == 0;
869 return ($self) unless defined $a || defined $p; # early out
871 # set A and set P is an fatal error
872 return ($self->bnan()) if defined $a && defined $p; # error
874 $r = ${"$c\::round_mode"} unless defined $r;
875 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
877 require Carp; Carp::croak ("Unknown round mode '$r'");
885 # Round $self according to given parameters, or given second argument's
886 # parameters or global defaults
888 # for speed reasons, _find_round_parameters is embeded here:
890 my ($self,$a,$p,$r,@args) = @_;
891 # $a accuracy, if given by caller
892 # $p precision, if given by caller
893 # $r round_mode, if given by caller
894 # @args all 'other' arguments (0 for unary, 1 for binary ops)
896 # leave bigfloat parts alone
897 return ($self) if exists $self->{_f} && ($self->{_f} & MB_NEVER_ROUND) != 0;
899 my $c = ref($self); # find out class of argument(s)
902 # now pick $a or $p, but only if we have got "arguments"
905 foreach ($self,@args)
907 # take the defined one, or if both defined, the one that is smaller
908 $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
913 # even if $a is defined, take $p, to signal error for both defined
914 foreach ($self,@args)
916 # take the defined one, or if both defined, the one that is bigger
918 $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
921 # if still none defined, use globals (#2)
922 $a = ${"$c\::accuracy"} unless defined $a;
923 $p = ${"$c\::precision"} unless defined $p;
925 # A == 0 is useless, so undef it to signal no rounding
926 $a = undef if defined $a && $a == 0;
929 return $self unless defined $a || defined $p; # early out
931 # set A and set P is an fatal error
932 return $self->bnan() if defined $a && defined $p;
934 $r = ${"$c\::round_mode"} unless defined $r;
935 if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
940 # now round, by calling either fround or ffround:
943 $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a;
945 else # both can't be undefined due to early out
947 $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p;
949 $self->bnorm(); # after round, normalize
954 # (numstr or BINT) return BINT
955 # Normalize number -- no-op here
956 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
962 # (BINT or num_str) return BINT
963 # make number absolute, or return absolute BINT from string
964 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
966 return $x if $x->modify('babs');
967 # post-normalized abs for internal use (does nothing for NaN)
968 $x->{sign} =~ s/^-/+/;
974 # (BINT or num_str) return BINT
975 # negate number or make a negated number from string
976 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
978 return $x if $x->modify('bneg');
980 # for +0 dont negate (to have always normalized)
981 $x->{sign} =~ tr/+-/-+/ if !$x->is_zero(); # does nothing for NaN
987 # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
988 # (BINT or num_str, BINT or num_str) return cond_code
991 my ($self,$x,$y) = (ref($_[0]),@_);
993 # objectify is costly, so avoid it
994 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
996 ($self,$x,$y) = objectify(2,@_);
999 return $upgrade->bcmp($x,$y) if defined $upgrade &&
1000 ((!$x->isa($self)) || (!$y->isa($self)));
1002 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1004 # handle +-inf and NaN
1005 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1006 return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
1007 return +1 if $x->{sign} eq '+inf';
1008 return -1 if $x->{sign} eq '-inf';
1009 return -1 if $y->{sign} eq '+inf';
1012 # check sign for speed first
1013 return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
1014 return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
1016 # have same sign, so compare absolute values. Don't make tests for zero here
1017 # because it's actually slower than testin in Calc (especially w/ Pari et al)
1019 # post-normalized compare for internal use (honors signs)
1020 if ($x->{sign} eq '+')
1022 # $x and $y both > 0
1023 return $CALC->_acmp($x->{value},$y->{value});
1027 $CALC->_acmp($y->{value},$x->{value}); # swaped (lib returns 0,1,-1)
1032 # Compares 2 values, ignoring their signs.
1033 # Returns one of undef, <0, =0, >0. (suitable for sort)
1034 # (BINT, BINT) return cond_code
1037 my ($self,$x,$y) = (ref($_[0]),@_);
1038 # objectify is costly, so avoid it
1039 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1041 ($self,$x,$y) = objectify(2,@_);
1044 return $upgrade->bacmp($x,$y) if defined $upgrade &&
1045 ((!$x->isa($self)) || (!$y->isa($self)));
1047 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1049 # handle +-inf and NaN
1050 return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1051 return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
1052 return +1; # inf is always bigger
1054 $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
1059 # add second arg (BINT or string) to first (BINT) (modifies first)
1060 # return result as BINT
1063 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1064 # objectify is costly, so avoid it
1065 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1067 ($self,$x,$y,@r) = objectify(2,@_);
1070 return $x if $x->modify('badd');
1071 return $upgrade->badd($x,$y,@r) if defined $upgrade &&
1072 ((!$x->isa($self)) || (!$y->isa($self)));
1074 $r[3] = $y; # no push!
1075 # inf and NaN handling
1076 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
1079 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1081 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1083 # +inf++inf or -inf+-inf => same, rest is NaN
1084 return $x if $x->{sign} eq $y->{sign};
1087 # +-inf + something => +inf
1088 # something +-inf => +-inf
1089 $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
1093 my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
1097 $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
1102 my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
1105 #print "swapped sub (a=$a)\n";
1106 $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
1111 # speedup, if equal, set result to 0
1112 #print "equal sub, result = 0\n";
1113 $x->{value} = $CALC->_zero();
1118 #print "unswapped sub (a=$a)\n";
1119 $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
1123 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1129 # (BINT or num_str, BINT or num_str) return num_str
1130 # subtract second arg from first, modify first
1133 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1134 # objectify is costly, so avoid it
1135 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1137 ($self,$x,$y,@r) = objectify(2,@_);
1140 return $x if $x->modify('bsub');
1142 # upgrade done by badd():
1143 # return $upgrade->badd($x,$y,@r) if defined $upgrade &&
1144 # ((!$x->isa($self)) || (!$y->isa($self)));
1148 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1152 $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
1153 $x->badd($y,@r); # badd does not leave internal zeros
1154 $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
1155 $x; # already rounded by badd() or no round necc.
1160 # increment arg by one
1161 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1162 return $x if $x->modify('binc');
1164 if ($x->{sign} eq '+')
1166 $x->{value} = $CALC->_inc($x->{value});
1167 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1170 elsif ($x->{sign} eq '-')
1172 $x->{value} = $CALC->_dec($x->{value});
1173 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1174 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1177 # inf, nan handling etc
1178 $x->badd($self->__one(),$a,$p,$r); # badd does round
1183 # decrement arg by one
1184 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1185 return $x if $x->modify('bdec');
1187 my $zero = $CALC->_is_zero($x->{value}) && $x->{sign} eq '+';
1189 if (($x->{sign} eq '-') || $zero)
1191 $x->{value} = $CALC->_inc($x->{value});
1192 $x->{sign} = '-' if $zero; # 0 => 1 => -1
1193 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
1194 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1198 elsif ($x->{sign} eq '+')
1200 $x->{value} = $CALC->_dec($x->{value});
1201 $x->round($a,$p,$r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1204 # inf, nan handling etc
1205 $x->badd($self->__one('-'),$a,$p,$r); # badd does round
1210 # not implemented yet
1211 my ($self,$x,$base,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1213 return $upgrade->blog($upgrade->new($x),$base,$a,$p,$r) if defined $upgrade;
1220 # (BINT or num_str, BINT or num_str) return BINT
1221 # does not modify arguments, but returns new object
1222 # Lowest Common Multiplicator
1224 my $y = shift; my ($x);
1231 $x = $class->new($y);
1233 while (@_) { $x = __lcm($x,shift); }
1239 # (BINT or num_str, BINT or num_str) return BINT
1240 # does not modify arguments, but returns new object
1241 # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
1244 $y = __PACKAGE__->new($y) if !ref($y);
1246 my $x = $y->copy(); # keep arguments
1247 if ($CALC->can('_gcd'))
1251 $y = shift; $y = $self->new($y) if !ref($y);
1252 next if $y->is_zero();
1253 return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
1254 $x->{value} = $CALC->_gcd($x->{value},$y->{value}); last if $x->is_one();
1261 $y = shift; $y = $self->new($y) if !ref($y);
1262 $x = __gcd($x,$y->copy()); last if $x->is_one(); # _gcd handles NaN
1270 # (num_str or BINT) return BINT
1271 # represent ~x as twos-complement number
1272 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1273 my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1275 return $x if $x->modify('bnot');
1276 $x->bneg()->bdec(); # bdec already does round
1279 # is_foo test routines
1283 # return true if arg (BINT or num_str) is zero (array '+', '0')
1284 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1285 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1287 return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
1288 $CALC->_is_zero($x->{value});
1293 # return true if arg (BINT or num_str) is NaN
1294 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
1296 return 1 if $x->{sign} eq $nan;
1302 # return true if arg (BINT or num_str) is +-inf
1303 my ($self,$x,$sign) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1305 $sign = '' if !defined $sign;
1306 return 1 if $sign eq $x->{sign}; # match ("+inf" eq "+inf")
1307 return 0 if $sign !~ /^([+-]|)$/;
1311 return 1 if ($x->{sign} =~ /^[+-]inf$/);
1314 $sign = quotemeta($sign.'inf');
1315 return 1 if ($x->{sign} =~ /^$sign$/);
1321 # return true if arg (BINT or num_str) is +1
1322 # or -1 if sign is given
1323 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1324 my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
1326 $sign = '+' if !defined $sign || $sign ne '-';
1328 return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
1329 $CALC->_is_one($x->{value});
1334 # return true when arg (BINT or num_str) is odd, false for even
1335 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1336 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1338 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1339 $CALC->_is_odd($x->{value});
1344 # return true when arg (BINT or num_str) is even, false for odd
1345 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1346 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1348 return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
1349 $CALC->_is_even($x->{value});
1354 # return true when arg (BINT or num_str) is positive (>= 0)
1355 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1356 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1358 return 1 if $x->{sign} =~ /^\+/;
1364 # return true when arg (BINT or num_str) is negative (< 0)
1365 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1366 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1368 return 1 if ($x->{sign} =~ /^-/);
1374 # return true when arg (BINT or num_str) is an integer
1375 # always true for BigInt, but different for Floats
1376 # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
1377 my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
1379 $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
1382 ###############################################################################
1386 # multiply two numbers -- stolen from Knuth Vol 2 pg 233
1387 # (BINT or num_str, BINT or num_str) return BINT
1390 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1391 # objectify is costly, so avoid it
1392 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1394 ($self,$x,$y,@r) = objectify(2,@_);
1397 return $x if $x->modify('bmul');
1399 return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
1402 if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
1404 return $x->bnan() if $x->is_zero() || $y->is_zero();
1405 # result will always be +-inf:
1406 # +inf * +/+inf => +inf, -inf * -/-inf => +inf
1407 # +inf * -/-inf => -inf, -inf * +/+inf => -inf
1408 return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
1409 return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
1410 return $x->binf('-');
1413 return $upgrade->bmul($x,$y,@r)
1414 if defined $upgrade && $y->isa($upgrade);
1416 $r[3] = $y; # no push here
1418 $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
1420 $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
1421 $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
1423 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1429 # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
1430 my ($self,$x,$y) = @_;
1432 # NaN if x == NaN or y == NaN or x==y==0
1433 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
1434 if (($x->is_nan() || $y->is_nan()) ||
1435 ($x->is_zero() && $y->is_zero()));
1437 # +-inf / +-inf == NaN, reminder also NaN
1438 if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
1440 return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
1442 # x / +-inf => 0, remainder x (works even if x == 0)
1443 if ($y->{sign} =~ /^[+-]inf$/)
1445 my $t = $x->copy(); # bzero clobbers up $x
1446 return wantarray ? ($x->bzero(),$t) : $x->bzero()
1449 # 5 / 0 => +inf, -6 / 0 => -inf
1450 # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
1451 # exception: -8 / 0 has remainder -8, not 8
1452 # exception: -inf / 0 has remainder -inf, not inf
1455 # +-inf / 0 => special case for -inf
1456 return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
1457 if (!$x->is_zero() && !$x->is_inf())
1459 my $t = $x->copy(); # binf clobbers up $x
1461 ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
1465 # last case: +-inf / ordinary number
1467 $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
1469 return wantarray ? ($x,$self->bzero()) : $x;
1474 # (dividend: BINT or num_str, divisor: BINT or num_str) return
1475 # (BINT,BINT) (quo,rem) or BINT (only rem)
1478 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1479 # objectify is costly, so avoid it
1480 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1482 ($self,$x,$y,@r) = objectify(2,@_);
1485 return $x if $x->modify('bdiv');
1487 return $self->_div_inf($x,$y)
1488 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
1490 return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
1491 if defined $upgrade;
1493 $r[3] = $y; # no push!
1495 # calc new sign and in case $y == +/- 1, return $x
1496 my $xsign = $x->{sign}; # keep
1497 $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
1501 my $rem = $self->bzero();
1502 ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
1503 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1504 $rem->{_a} = $x->{_a};
1505 $rem->{_p} = $x->{_p};
1506 $x->round(@r) if !exists $x->{_f} || ($x->{_f} & MB_NEVER_ROUND) == 0;
1507 if (! $CALC->_is_zero($rem->{value}))
1509 $rem->{sign} = $y->{sign};
1510 $rem = $y->copy()->bsub($rem) if $xsign ne $y->{sign}; # one of them '-'
1514 $rem->{sign} = '+'; # dont leave -0
1516 $rem->round(@r) if !exists $rem->{_f} || ($rem->{_f} & MB_NEVER_ROUND) == 0;
1520 $x->{value} = $CALC->_div($x->{value},$y->{value});
1521 $x->{sign} = '+' if $CALC->_is_zero($x->{value});
1523 $x->round(@r) if !exists $x->{_f} || ($x->{_f} & MB_NEVER_ROUND) == 0;
1527 ###############################################################################
1532 # modulus (or remainder)
1533 # (BINT or num_str, BINT or num_str) return BINT
1536 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1537 # objectify is costly, so avoid it
1538 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1540 ($self,$x,$y,@r) = objectify(2,@_);
1543 return $x if $x->modify('bmod');
1544 $r[3] = $y; # no push!
1545 if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
1547 my ($d,$r) = $self->_div_inf($x,$y);
1548 $x->{sign} = $r->{sign};
1549 $x->{value} = $r->{value};
1550 return $x->round(@r);
1553 if ($CALC->can('_mod'))
1555 # calc new sign and in case $y == +/- 1, return $x
1556 $x->{value} = $CALC->_mod($x->{value},$y->{value});
1557 if (!$CALC->_is_zero($x->{value}))
1559 my $xsign = $x->{sign};
1560 $x->{sign} = $y->{sign};
1561 if ($xsign ne $y->{sign})
1563 my $t = $CALC->_copy($x->{value}); # copy $x
1564 $x->{value} = $CALC->_copy($y->{value}); # copy $y to $x
1565 $x->{value} = $CALC->_sub($y->{value},$t,1); # $y-$x
1570 $x->{sign} = '+'; # dont leave -0
1572 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1575 my ($t,$rem) = $self->bdiv($x->copy(),$y,@r); # slow way (also rounds)
1577 foreach (qw/value sign _a _p/)
1579 $x->{$_} = $rem->{$_};
1586 # Modular inverse. given a number which is (hopefully) relatively
1587 # prime to the modulus, calculate its inverse using Euclid's
1588 # alogrithm. If the number is not relatively prime to the modulus
1589 # (i.e. their gcd is not one) then NaN is returned.
1592 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1593 # objectify is costly, so avoid it
1594 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1596 ($self,$x,$y,@r) = objectify(2,@_);
1599 return $x if $x->modify('bmodinv');
1602 if ($y->{sign} ne '+' # -, NaN, +inf, -inf
1603 || $x->is_zero() # or num == 0
1604 || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
1607 # put least residue into $x if $x was negative, and thus make it positive
1608 $x->bmod($y) if $x->{sign} eq '-';
1610 if ($CALC->can('_modinv'))
1613 ($x->{value},$sign) = $CALC->_modinv($x->{value},$y->{value});
1614 $x->bnan() if !defined $x->{value}; # in case no GCD found
1615 return $x if !defined $sign; # already real result
1616 $x->{sign} = $sign; # flip/flop see below
1617 $x->bmod($y); # calc real result
1620 my ($u, $u1) = ($self->bzero(), $self->bone());
1621 my ($a, $b) = ($y->copy(), $x->copy());
1623 # first step need always be done since $num (and thus $b) is never 0
1624 # Note that the loop is aligned so that the check occurs between #2 and #1
1625 # thus saving us one step #2 at the loop end. Typical loop count is 1. Even
1626 # a case with 28 loops still gains about 3% with this layout.
1628 ($a, $q, $b) = ($b, $a->bdiv($b)); # step #1
1629 # Euclid's Algorithm (calculate GCD of ($a,$b) in $a and also calculate
1630 # two values in $u and $u1, we use only $u1 afterwards)
1631 my $sign = 1; # flip-flop
1632 while (!$b->is_zero()) # found GCD if $b == 0
1634 # the original algorithm had:
1635 # ($u, $u1) = ($u1, $u->bsub($u1->copy()->bmul($q))); # step #2
1636 # The following creates exact the same sequence of numbers in $u1,
1637 # except for the sign ($u1 is now always positive). Since formerly
1638 # the sign of $u1 was alternating between '-' and '+', the $sign
1639 # flip-flop will take care of that, so that at the end of the loop
1640 # we have the real sign of $u1. Keeping numbers positive gains us
1641 # speed since badd() is faster than bsub() and makes it possible
1642 # to have the algorithmn in Calc for even more speed.
1644 ($u, $u1) = ($u1, $u->badd($u1->copy()->bmul($q))); # step #2
1645 $sign = - $sign; # flip sign
1647 ($a, $q, $b) = ($b, $a->bdiv($b)); # step #1 again
1650 # If the gcd is not 1, then return NaN! It would be pointless to
1651 # have called bgcd to check this first, because we would then be
1652 # performing the same Euclidean Algorithm *twice*.
1653 return $x->bnan() unless $a->is_one();
1655 $u1->bneg() if $sign != 1; # need to flip?
1657 $u1->bmod($y); # calc result
1658 $x->{value} = $u1->{value}; # and copy over to $x
1659 $x->{sign} = $u1->{sign}; # to modify in place
1665 # takes a very large number to a very large exponent in a given very
1666 # large modulus, quickly, thanks to binary exponentation. supports
1667 # negative exponents.
1668 my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
1670 return $num if $num->modify('bmodpow');
1672 # check modulus for valid values
1673 return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
1674 || $mod->is_zero());
1676 # check exponent for valid values
1677 if ($exp->{sign} =~ /\w/)
1679 # i.e., if it's NaN, +inf, or -inf...
1680 return $num->bnan();
1683 $num->bmodinv ($mod) if ($exp->{sign} eq '-');
1685 # check num for valid values (also NaN if there was no inverse but $exp < 0)
1686 return $num->bnan() if $num->{sign} !~ /^[+-]$/;
1688 if ($CALC->can('_modpow'))
1690 # $mod is positive, sign on $exp is ignored, result also positive
1691 $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
1695 # in the trivial case,
1696 return $num->bzero(@r) if $mod->is_one();
1697 return $num->bone('+',@r) if $num->is_zero() or $num->is_one();
1699 # $num->bmod($mod); # if $x is large, make it smaller first
1700 my $acc = $num->copy(); # but this is not really faster...
1702 $num->bone(); # keep ref to $num
1704 my $expbin = $exp->as_bin(); $expbin =~ s/^[-]?0b//; # ignore sign and prefix
1705 my $len = CORE::length($expbin);
1708 if( substr($expbin,$len,1) eq '1')
1710 $num->bmul($acc)->bmod($mod);
1712 $acc->bmul($acc)->bmod($mod);
1718 ###############################################################################
1722 # (BINT or num_str, BINT or num_str) return BINT
1723 # compute factorial numbers
1724 # modifies first argument
1725 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
1727 return $x if $x->modify('bfac');
1729 return $x->bnan() if $x->{sign} ne '+'; # inf, NnN, <0 etc => NaN
1730 return $x->bone('+',@r) if $x->is_zero() || $x->is_one(); # 0 or 1 => 1
1732 if ($CALC->can('_fac'))
1734 $x->{value} = $CALC->_fac($x->{value});
1735 return $x->round(@r);
1740 # seems we need not to temp. clear A/P of $x since the result is the same
1741 my $f = $self->new(2);
1742 while ($f->bacmp($n) < 0)
1744 $x->bmul($f); $f->binc();
1746 $x->bmul($f,@r); # last step and also round
1751 # (BINT or num_str, BINT or num_str) return BINT
1752 # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
1753 # modifies first argument
1756 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1757 # objectify is costly, so avoid it
1758 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1760 ($self,$x,$y,@r) = objectify(2,@_);
1763 return $x if $x->modify('bpow');
1765 return $upgrade->bpow($upgrade->new($x),$y,@r)
1766 if defined $upgrade && !$y->isa($self);
1768 $r[3] = $y; # no push!
1769 return $x if $x->{sign} =~ /^[+-]inf$/; # -inf/+inf ** x
1770 return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
1771 return $x->bone('+',@r) if $y->is_zero();
1772 return $x->round(@r) if $x->is_one() || $y->is_one();
1773 if ($x->{sign} eq '-' && $CALC->_is_one($x->{value}))
1775 # if $x == -1 and odd/even y => +1/-1
1776 return $y->is_odd() ? $x->round(@r) : $x->babs()->round(@r);
1777 # my Casio FX-5500L has a bug here: -1 ** 2 is -1, but -1 * -1 is 1;
1779 # 1 ** -y => 1 / (1 ** |y|)
1780 # so do test for negative $y after above's clause
1781 return $x->bnan() if $y->{sign} eq '-';
1782 return $x->round(@r) if $x->is_zero(); # 0**y => 0 (if not y <= 0)
1784 if ($CALC->can('_pow'))
1786 $x->{value} = $CALC->_pow($x->{value},$y->{value});
1787 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1791 # based on the assumption that shifting in base 10 is fast, and that mul
1792 # works faster if numbers are small: we count trailing zeros (this step is
1793 # O(1)..O(N), but in case of O(N) we save much more time due to this),
1794 # stripping them out of the multiplication, and add $count * $y zeros
1795 # afterwards like this:
1796 # 300 ** 3 == 300*300*300 == 3*3*3 . '0' x 2 * 3 == 27 . '0' x 6
1797 # creates deep recursion since brsft/blsft use bpow sometimes.
1798 # my $zeros = $x->_trailing_zeros();
1801 # $x->brsft($zeros,10); # remove zeros
1802 # $x->bpow($y); # recursion (will not branch into here again)
1803 # $zeros = $y * $zeros; # real number of zeros to add
1804 # $x->blsft($zeros,10);
1805 # return $x->round(@r);
1808 my $pow2 = $self->__one();
1809 my $y_bin = $y->as_bin(); $y_bin =~ s/^0b//;
1810 my $len = CORE::length($y_bin);
1813 $pow2->bmul($x) if substr($y_bin,$len,1) eq '1'; # is odd?
1817 $x->round(@r) if !exists $x->{_f} || $x->{_f} & MB_NEVER_ROUND == 0;
1823 # (BINT or num_str, BINT or num_str) return BINT
1824 # compute x << y, base n, y >= 0
1827 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1828 # objectify is costly, so avoid it
1829 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1831 ($self,$x,$y,$n,@r) = objectify(2,@_);
1834 return $x if $x->modify('blsft');
1835 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1836 return $x->round(@r) if $y->is_zero();
1838 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1840 my $t; $t = $CALC->_lsft($x->{value},$y->{value},$n) if $CALC->can('_lsft');
1843 $x->{value} = $t; return $x->round(@r);
1846 return $x->bmul( $self->bpow($n, $y, @r), @r );
1851 # (BINT or num_str, BINT or num_str) return BINT
1852 # compute x >> y, base n, y >= 0
1855 my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
1856 # objectify is costly, so avoid it
1857 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1859 ($self,$x,$y,$n,@r) = objectify(2,@_);
1862 return $x if $x->modify('brsft');
1863 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1864 return $x->round(@r) if $y->is_zero();
1865 return $x->bzero(@r) if $x->is_zero(); # 0 => 0
1867 $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
1869 # this only works for negative numbers when shifting in base 2
1870 if (($x->{sign} eq '-') && ($n == 2))
1872 return $x->round(@r) if $x->is_one('-'); # -1 => -1
1875 # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
1876 # but perhaps there is a better emulation for two's complement shift...
1877 # if $y != 1, we must simulate it by doing:
1878 # convert to bin, flip all bits, shift, and be done
1879 $x->binc(); # -3 => -2
1880 my $bin = $x->as_bin();
1881 $bin =~ s/^-0b//; # strip '-0b' prefix
1882 $bin =~ tr/10/01/; # flip bits
1884 if (CORE::length($bin) <= $y)
1886 $bin = '0'; # shifting to far right creates -1
1887 # 0, because later increment makes
1888 # that 1, attached '-' makes it '-1'
1889 # because -1 >> x == -1 !
1893 $bin =~ s/.{$y}$//; # cut off at the right side
1894 $bin = '1' . $bin; # extend left side by one dummy '1'
1895 $bin =~ tr/10/01/; # flip bits back
1897 my $res = $self->new('0b'.$bin); # add prefix and convert back
1898 $res->binc(); # remember to increment
1899 $x->{value} = $res->{value}; # take over value
1900 return $x->round(@r); # we are done now, magic, isn't?
1902 $x->bdec(); # n == 2, but $y == 1: this fixes it
1905 my $t; $t = $CALC->_rsft($x->{value},$y->{value},$n) if $CALC->can('_rsft');
1909 return $x->round(@r);
1912 $x->bdiv($self->bpow($n,$y, @r), @r);
1918 #(BINT or num_str, BINT or num_str) return BINT
1922 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1923 # objectify is costly, so avoid it
1924 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1926 ($self,$x,$y,@r) = objectify(2,@_);
1929 return $x if $x->modify('band');
1931 $r[3] = $y; # no push!
1932 local $Math::BigInt::upgrade = undef;
1934 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1935 return $x->bzero(@r) if $y->is_zero() || $x->is_zero();
1937 my $sign = 0; # sign of result
1938 $sign = 1 if ($x->{sign} eq '-') && ($y->{sign} eq '-');
1939 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1940 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1942 if ($CALC->can('_and') && $sx == 1 && $sy == 1)
1944 $x->{value} = $CALC->_and($x->{value},$y->{value});
1945 return $x->round(@r);
1948 my $m = $self->bone(); my ($xr,$yr);
1949 my $x10000 = $self->new (0x1000);
1950 my $y1 = copy(ref($x),$y); # make copy
1951 $y1->babs(); # and positive
1952 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
1953 use integer; # need this for negative bools
1954 while (!$x1->is_zero() && !$y1->is_zero())
1956 ($x1, $xr) = bdiv($x1, $x10000);
1957 ($y1, $yr) = bdiv($y1, $x10000);
1958 # make both op's numbers!
1959 $x->badd( bmul( $class->new(
1960 abs($sx*int($xr->numify()) & $sy*int($yr->numify()))),
1964 $x->bneg() if $sign;
1970 #(BINT or num_str, BINT or num_str) return BINT
1974 my ($self,$x,$y,@r) = (ref($_[0]),@_);
1975 # objectify is costly, so avoid it
1976 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
1978 ($self,$x,$y,@r) = objectify(2,@_);
1981 return $x if $x->modify('bior');
1982 $r[3] = $y; # no push!
1984 local $Math::BigInt::upgrade = undef;
1986 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
1987 return $x->round(@r) if $y->is_zero();
1989 my $sign = 0; # sign of result
1990 $sign = 1 if ($x->{sign} eq '-') || ($y->{sign} eq '-');
1991 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
1992 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
1994 # don't use lib for negative values
1995 if ($CALC->can('_or') && $sx == 1 && $sy == 1)
1997 $x->{value} = $CALC->_or($x->{value},$y->{value});
1998 return $x->round(@r);
2001 my $m = $self->bone(); my ($xr,$yr);
2002 my $x10000 = $self->new(0x10000);
2003 my $y1 = copy(ref($x),$y); # make copy
2004 $y1->babs(); # and positive
2005 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
2006 use integer; # need this for negative bools
2007 while (!$x1->is_zero() || !$y1->is_zero())
2009 ($x1, $xr) = bdiv($x1,$x10000);
2010 ($y1, $yr) = bdiv($y1,$x10000);
2011 # make both op's numbers!
2012 $x->badd( bmul( $class->new(
2013 abs($sx*int($xr->numify()) | $sy*int($yr->numify()))),
2017 $x->bneg() if $sign;
2023 #(BINT or num_str, BINT or num_str) return BINT
2027 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2028 # objectify is costly, so avoid it
2029 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2031 ($self,$x,$y,@r) = objectify(2,@_);
2034 return $x if $x->modify('bxor');
2035 $r[3] = $y; # no push!
2037 local $Math::BigInt::upgrade = undef;
2039 return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
2040 return $x->round(@r) if $y->is_zero();
2042 my $sign = 0; # sign of result
2043 $sign = 1 if $x->{sign} ne $y->{sign};
2044 my $sx = 1; $sx = -1 if $x->{sign} eq '-';
2045 my $sy = 1; $sy = -1 if $y->{sign} eq '-';
2047 # don't use lib for negative values
2048 if ($CALC->can('_xor') && $sx == 1 && $sy == 1)
2050 $x->{value} = $CALC->_xor($x->{value},$y->{value});
2051 return $x->round(@r);
2054 my $m = $self->bone(); my ($xr,$yr);
2055 my $x10000 = $self->new(0x10000);
2056 my $y1 = copy(ref($x),$y); # make copy
2057 $y1->babs(); # and positive
2058 my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
2059 use integer; # need this for negative bools
2060 while (!$x1->is_zero() || !$y1->is_zero())
2062 ($x1, $xr) = bdiv($x1, $x10000);
2063 ($y1, $yr) = bdiv($y1, $x10000);
2064 # make both op's numbers!
2065 $x->badd( bmul( $class->new(
2066 abs($sx*int($xr->numify()) ^ $sy*int($yr->numify()))),
2070 $x->bneg() if $sign;
2076 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2078 my $e = $CALC->_len($x->{value});
2079 return wantarray ? ($e,0) : $e;
2084 # return the nth decimal digit, negative values count backward, 0 is right
2085 my ($self,$x,$n) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2087 $CALC->_digit($x->{value},$n||0);
2092 # return the amount of trailing zeros in $x
2094 $x = $class->new($x) unless ref $x;
2096 return 0 if $x->is_zero() || $x->is_odd() || $x->{sign} !~ /^[+-]$/;
2098 return $CALC->_zeros($x->{value}) if $CALC->can('_zeros');
2100 # if not: since we do not know underlying internal representation:
2101 my $es = "$x"; $es =~ /([0]*)$/;
2102 return 0 if !defined $1; # no zeros
2103 CORE::length("$1"); # as string, not as +0!
2108 # calculate square root of $x
2109 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2111 return $x if $x->modify('bsqrt');
2113 return $x->bnan() if $x->{sign} !~ /^\+/; # -x or -inf or NaN => NaN
2114 return $x if $x->{sign} eq '+inf'; # sqrt(+inf) == inf
2115 return $x->round(@r) if $x->is_zero() || $x->is_one(); # 0,1 => 0,1
2117 return $upgrade->bsqrt($x,@r) if defined $upgrade;
2119 if ($CALC->can('_sqrt'))
2121 $x->{value} = $CALC->_sqrt($x->{value});
2122 return $x->round(@r);
2125 return $x->bone('+',@r) if $x < 4; # 2,3 => 1
2127 my $l = int($x->length()/2);
2129 $x->bone(); # keep ref($x), but modify it
2130 $x->blsft($l,10) if $l != 0; # first guess: 1.('0' x (l/2))
2132 my $last = $self->bzero();
2133 my $two = $self->new(2);
2134 my $lastlast = $self->bzero();
2135 #my $lastlast = $x+$two;
2136 while ($last != $x && $lastlast != $x)
2138 $lastlast = $last; $last = $x->copy();
2142 $x->bdec() if $x * $x > $y; # overshot?
2148 # calculate $y'th root of $x
2151 my ($self,$x,$y,@r) = (ref($_[0]),@_);
2152 # objectify is costly, so avoid it
2153 if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
2155 ($self,$x,$y,@r) = objectify(2,@_);
2158 return $x if $x->modify('broot');
2160 # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
2161 return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
2162 $y->{sign} !~ /^\+$/;
2164 return $x->round(@r)
2165 if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
2167 return $upgrade->broot($x,@r) if defined $upgrade;
2169 if ($CALC->can('_root'))
2171 $x->{value} = $CALC->_root($x->{value},$y->{value});
2172 return $x->round(@r);
2175 return $x->bsqrt() if $y->bacmp(2) == 0; # 2 => square root
2177 # since we take at least a cubic root, and only 8 ** 1/3 >= 2 (==2):
2178 return $x->bone('+',@r) if $x < 8; # $x=2..7 => 1
2180 my $org = $x->copy();
2181 my $l = int($x->length()/$y->numify());
2183 $x->bone(); # keep ref($x), but modify it
2184 $x->blsft($l,10) if $l != 0; # first guess: 1.('0' x (l/$y))
2186 my $last = $self->bzero();
2187 my $lastlast = $self->bzero();
2188 #my $lastlast = $x+$y;
2189 my $divider = $self->new(2);
2191 #print "start $org divider $divider up $up\n";
2192 while ($last != $x && $lastlast != $x)
2194 #print "at $x ($last $lastlast)\n";
2195 $lastlast = $last; $last = $x->copy();
2196 #print "at $x ($last ",($org / ($x ** $up)),"\n";
2197 $x->badd($org / ($x ** 2));
2200 #print $x ** $y," org ",$org,"\n";
2202 while ($x ** $y < $org)
2204 #print "correcting $x to ";
2206 #print "$x ( $x ** $y == ",$x ** $y,")\n";
2213 # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
2214 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2216 if ($x->{sign} !~ /^[+-]$/)
2218 my $s = $x->{sign}; $s =~ s/^[+-]//;
2219 return $self->new($s); # -inf,+inf => inf
2221 my $e = $class->bzero();
2222 return $e->binc() if $x->is_zero();
2223 $e += $x->_trailing_zeros();
2229 # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
2230 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2232 if ($x->{sign} !~ /^[+-]$/)
2234 return $self->new($x->{sign}); # keep + or - sign
2237 # that's inefficient
2238 my $zeros = $m->_trailing_zeros();
2239 $m->brsft($zeros,10) if $zeros != 0;
2245 # return a copy of both the exponent and the mantissa
2246 my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
2248 return ($x->mantissa(),$x->exponent());
2251 ##############################################################################
2252 # rounding functions
2256 # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
2257 # $n == 0 || $n == 1 => round to integer
2258 my $x = shift; $x = $class->new($x) unless ref $x;
2259 my ($scale,$mode) = $x->_scale_p($x->precision(),$x->round_mode(),@_);
2260 return $x if !defined $scale; # no-op
2261 return $x if $x->modify('bfround');
2263 # no-op for BigInts if $n <= 0
2266 $x->{_a} = undef; # clear an eventual set A
2267 $x->{_p} = $scale; return $x;
2270 $x->bround( $x->length()-$scale, $mode);
2271 $x->{_a} = undef; # bround sets {_a}
2272 $x->{_p} = $scale; # so correct it
2276 sub _scan_for_nonzero
2282 my $len = $x->length();
2283 return 0 if $len == 1; # '5' is trailed by invisible zeros
2284 my $follow = $pad - 1;
2285 return 0 if $follow > $len || $follow < 1;
2287 # since we do not know underlying represention of $x, use decimal string
2288 #my $r = substr ($$xs,-$follow);
2289 my $r = substr ("$x",-$follow);
2290 return 1 if $r =~ /[^0]/;
2296 # to make life easier for switch between MBF and MBI (autoload fxxx()
2297 # like MBF does for bxxx()?)
2299 return $x->bround(@_);
2304 # accuracy: +$n preserve $n digits from left,
2305 # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
2307 # and overwrite the rest with 0's, return normalized number
2308 # do not return $x->bnorm(), but $x
2310 my $x = shift; $x = $class->new($x) unless ref $x;
2311 my ($scale,$mode) = $x->_scale_a($x->accuracy(),$x->round_mode(),@_);
2312 return $x if !defined $scale; # no-op
2313 return $x if $x->modify('bround');
2315 if ($x->is_zero() || $scale == 0)
2317 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2320 return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
2322 # we have fewer digits than we want to scale to
2323 my $len = $x->length();
2324 # convert $scale to a scalar in case it is an object (put's a limit on the
2325 # number length, but this would already limited by memory constraints), makes
2327 $scale = $scale->numify() if ref ($scale);
2329 # scale < 0, but > -len (not >=!)
2330 if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
2332 $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
2336 # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
2337 my ($pad,$digit_round,$digit_after);
2338 $pad = $len - $scale;
2339 $pad = abs($scale-1) if $scale < 0;
2341 # do not use digit(), it is costly for binary => decimal
2343 my $xs = $CALC->_str($x->{value});
2346 # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
2347 # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
2348 $digit_round = '0'; $digit_round = substr($$xs,$pl,1) if $pad <= $len;
2349 $pl++; $pl ++ if $pad >= $len;
2350 $digit_after = '0'; $digit_after = substr($$xs,$pl,1) if $pad > 0;
2352 # in case of 01234 we round down, for 6789 up, and only in case 5 we look
2353 # closer at the remaining digits of the original $x, remember decision
2354 my $round_up = 1; # default round up
2356 ($mode eq 'trunc') || # trunc by round down
2357 ($digit_after =~ /[01234]/) || # round down anyway,
2359 ($digit_after eq '5') && # not 5000...0000
2360 ($x->_scan_for_nonzero($pad,$xs) == 0) &&
2362 ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
2363 ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
2364 ($mode eq '+inf') && ($x->{sign} eq '-') ||
2365 ($mode eq '-inf') && ($x->{sign} eq '+') ||
2366 ($mode eq 'zero') # round down if zero, sign adjusted below
2368 my $put_back = 0; # not yet modified
2370 if (($pad > 0) && ($pad <= $len))
2372 substr($$xs,-$pad,$pad) = '0' x $pad;
2377 $x->bzero(); # round to '0'
2380 if ($round_up) # what gave test above?
2383 $pad = $len, $$xs = '0' x $pad if $scale < 0; # tlr: whack 0.51=>1.0
2385 # we modify directly the string variant instead of creating a number and
2386 # adding it, since that is faster (we already have the string)
2387 my $c = 0; $pad ++; # for $pad == $len case
2388 while ($pad <= $len)
2390 $c = substr($$xs,-$pad,1) + 1; $c = '0' if $c eq '10';
2391 substr($$xs,-$pad,1) = $c; $pad++;
2392 last if $c != 0; # no overflow => early out
2394 $$xs = '1'.$$xs if $c == 0;
2397 $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back in if needed
2399 $x->{_a} = $scale if $scale >= 0;
2402 $x->{_a} = $len+$scale;
2403 $x->{_a} = 0 if $scale < -$len;
2410 # return integer less or equal then number, since it is already integer,
2411 # always returns $self
2412 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2419 # return integer greater or equal then number, since it is already integer,
2420 # always returns $self
2421 my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
2426 ##############################################################################
2427 # private stuff (internal use only)
2431 # internal speedup, set argument to 1, or create a +/- 1
2433 my $x = $self->bone(); # $x->{value} = $CALC->_one();
2434 $x->{sign} = shift || '+';
2440 # Overload will swap params if first one is no object ref so that the first
2441 # one is always an object ref. In this case, third param is true.
2442 # This routine is to overcome the effect of scalar,$object creating an object
2443 # of the class of this package, instead of the second param $object. This
2444 # happens inside overload, when the overload section of this package is
2445 # inherited by sub classes.
2446 # For overload cases (and this is used only there), we need to preserve the
2447 # args, hence the copy().
2448 # You can override this method in a subclass, the overload section will call
2449 # $object->_swap() to make sure it arrives at the proper subclass, with some
2450 # exceptions like '+' and '-'. To make '+' and '-' work, you also need to
2451 # specify your own overload for them.
2453 # object, (object|scalar) => preserve first and make copy
2454 # scalar, object => swapped, re-swap and create new from first
2455 # (using class of second object, not $class!!)
2456 my $self = shift; # for override in subclass
2459 my $c = ref ($_[0]) || $class; # fallback $class should not happen
2460 return ( $c->new($_[1]), $_[0] );
2462 return ( $_[0]->copy(), $_[1] );
2467 # check for strings, if yes, return objects instead
2469 # the first argument is number of args objectify() should look at it will
2470 # return $count+1 elements, the first will be a classname. This is because
2471 # overloaded '""' calls bstr($object,undef,undef) and this would result in
2472 # useless objects beeing created and thrown away. So we cannot simple loop
2473 # over @_. If the given count is 0, all arguments will be used.
2475 # If the second arg is a ref, use it as class.
2476 # If not, try to use it as classname, unless undef, then use $class
2477 # (aka Math::BigInt). The latter shouldn't happen,though.
2480 # $x->badd(1); => ref x, scalar y
2481 # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
2482 # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
2483 # Math::BigInt::badd(1,2); => scalar x, scalar y
2484 # In the last case we check number of arguments to turn it silently into
2485 # $class,1,2. (We can not take '1' as class ;o)
2486 # badd($class,1) is not supported (it should, eventually, try to add undef)
2487 # currently it tries 'Math::BigInt' + 1, which will not work.
2489 # some shortcut for the common cases
2491 return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
2493 my $count = abs(shift || 0);
2495 my (@a,$k,$d); # resulting array, temp, and downgrade
2498 # okay, got object as first
2503 # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
2505 $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
2509 # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
2510 if (defined ${"$a[0]::downgrade"})
2512 $d = ${"$a[0]::downgrade"};
2513 ${"$a[0]::downgrade"} = undef;
2516 my $up = ${"$a[0]::upgrade"};
2517 #print "Now in objectify, my class is today $a[0]\n";
2525 $k = $a[0]->new($k);
2527 elsif (!defined $up && ref($k) ne $a[0])
2529 # foreign object, try to convert to integer
2530 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2543 $k = $a[0]->new($k);
2545 elsif (!defined $up && ref($k) ne $a[0])
2547 # foreign object, try to convert to integer
2548 $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
2552 push @a,@_; # return other params, too
2556 require Carp; Carp::croak ("$class objectify needs list context");
2558 ${"$a[0]::downgrade"} = $d;
2567 my @a; my $l = scalar @_;
2568 for ( my $i = 0; $i < $l ; $i++ )
2570 if ($_[$i] eq ':constant')
2572 # this causes overlord er load to step in
2573 overload::constant integer => sub { $self->new(shift) };
2574 overload::constant binary => sub { $self->new(shift) };
2576 elsif ($_[$i] eq 'upgrade')
2578 # this causes upgrading
2579 $upgrade = $_[$i+1]; # or undef to disable
2582 elsif ($_[$i] =~ /^lib$/i)
2584 # this causes a different low lib to take care...
2585 $CALC = $_[$i+1] || '';
2593 # any non :constant stuff is handled by our parent, Exporter
2594 # even if @_ is empty, to give it a chance
2595 $self->SUPER::import(@a); # need it for subclasses
2596 $self->export_to_level(1,$self,@a); # need it for MBF
2598 # try to load core math lib
2599 my @c = split /\s*,\s*/,$CALC;
2600 push @c,'Calc'; # if all fail, try this
2601 $CALC = ''; # signal error
2602 foreach my $lib (@c)
2604 next if ($lib || '') eq '';
2605 $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
2609 # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
2610 # used in the same script, or eval inside import().
2611 my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
2612 my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
2614 $file = File::Spec->catfile (@parts, $file);
2615 eval { require "$file"; $lib->import( @c ); }
2619 eval "use $lib qw/@c/;";
2621 $CALC = $lib, last if $@ eq ''; # no error in loading lib?
2626 Carp::croak ("Couldn't load any math lib, not even the default");
2632 # convert a (ref to) big hex string to BigInt, return undef for error
2635 my $x = Math::BigInt->bzero();
2638 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2639 $$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
2641 return $x->bnan() if $$hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
2643 my $sign = '+'; $sign = '-' if ($$hs =~ /^-/);
2645 $$hs =~ s/^[+-]//; # strip sign
2646 if ($CALC->can('_from_hex'))
2648 $x->{value} = $CALC->_from_hex($hs);
2652 # fallback to pure perl
2653 my $mul = Math::BigInt->bzero(); $mul++;
2654 my $x65536 = Math::BigInt->new(65536);
2655 my $len = CORE::length($$hs)-2;
2656 $len = int($len/4); # 4-digit parts, w/o '0x'
2657 my $val; my $i = -4;
2660 $val = substr($$hs,$i,4);
2661 $val =~ s/^[+-]?0x// if $len == 0; # for last part only because
2662 $val = hex($val); # hex does not like wrong chars
2664 $x += $mul * $val if $val != 0;
2665 $mul *= $x65536 if $len >= 0; # skip last mul
2668 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2674 # convert a (ref to) big binary string to BigInt, return undef for error
2677 my $x = Math::BigInt->bzero();
2679 $$bs =~ s/([01])_([01])/$1$2/g;
2680 $$bs =~ s/([01])_([01])/$1$2/g;
2681 return $x->bnan() if $$bs !~ /^[+-]?0b[01]+$/;
2683 my $sign = '+'; $sign = '-' if ($$bs =~ /^\-/);
2684 $$bs =~ s/^[+-]//; # strip sign
2685 if ($CALC->can('_from_bin'))
2687 $x->{value} = $CALC->_from_bin($bs);
2691 my $mul = Math::BigInt->bzero(); $mul++;
2692 my $x256 = Math::BigInt->new(256);
2693 my $len = CORE::length($$bs)-2;
2694 $len = int($len/8); # 8-digit parts, w/o '0b'
2695 my $val; my $i = -8;
2698 $val = substr($$bs,$i,8);
2699 $val =~ s/^[+-]?0b// if $len == 0; # for last part only
2700 #$val = oct('0b'.$val); # does not work on Perl prior to 5.6.0
2702 # $val = ('0' x (8-CORE::length($val))).$val if CORE::length($val) < 8;
2703 $val = ord(pack('B8',substr('00000000'.$val,-8,8)));
2705 $x += $mul * $val if $val != 0;
2706 $mul *= $x256 if $len >= 0; # skip last mul
2709 $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
2715 # (ref to num_str) return num_str
2716 # internal, take apart a string and return the pieces
2717 # strip leading/trailing whitespace, leading zeros, underscore and reject
2721 # strip white space at front, also extranous leading zeros
2722 $$x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
2723 $$x =~ s/^\s+//; # but this will
2724 $$x =~ s/\s+$//g; # strip white space at end
2726 # shortcut, if nothing to split, return early
2727 if ($$x =~ /^[+-]?\d+\z/)
2729 $$x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
2730 return (\$sign, $x, \'', \'', \0);
2733 # invalid starting char?
2734 return if $$x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
2736 return __from_hex($x) if $$x =~ /^[\-\+]?0x/; # hex string
2737 return __from_bin($x) if $$x =~ /^[\-\+]?0b/; # binary string
2739 # strip underscores between digits
2740 $$x =~ s/(\d)_(\d)/$1$2/g;
2741 $$x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3
2743 # some possible inputs:
2744 # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
2745 # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2
2747 #return if $$x =~ /[Ee].*[Ee]/; # more than one E => error
2749 my ($m,$e,$last) = split /[Ee]/,$$x;
2750 return if defined $last; # last defined => 1e2E3 or others
2751 $e = '0' if !defined $e || $e eq "";
2753 # sign,value for exponent,mantint,mantfrac
2754 my ($es,$ev,$mis,$miv,$mfv);
2756 if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2760 return if $m eq '.' || $m eq '';
2761 my ($mi,$mf,$lastf) = split /\./,$m;
2762 return if defined $lastf; # last defined => 1.2.3 or others
2763 $mi = '0' if !defined $mi;
2764 $mi .= '0' if $mi =~ /^[\-\+]?$/;
2765 $mf = '0' if !defined $mf || $mf eq '';
2766 if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
2768 $mis = $1||'+'; $miv = $2;
2769 return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
2771 return (\$mis,\$miv,\$mfv,\$es,\$ev);
2774 return; # NaN, not a number
2779 # an object might be asked to return itself as bigint on certain overloaded
2780 # operations, this does exactly this, so that sub classes can simple inherit
2781 # it or override with their own integer conversion routine
2789 # return as hex string, with prefixed 0x
2790 my $x = shift; $x = $class->new($x) if !ref($x);
2792 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2794 my $es = ''; my $s = '';
2795 $s = $x->{sign} if $x->{sign} eq '-';
2796 if ($CALC->can('_as_hex'))
2798 $es = ${$CALC->_as_hex($x->{value})};
2802 return '0x0' if $x->is_zero();
2804 my $x1 = $x->copy()->babs(); my ($xr,$x10000,$h);
2807 $x10000 = Math::BigInt->new (0x10000); $h = 'h4';
2811 $x10000 = Math::BigInt->new (0x1000); $h = 'h3';
2813 while (!$x1->is_zero())
2815 ($x1, $xr) = bdiv($x1,$x10000);
2816 $es .= unpack($h,pack('v',$xr->numify()));
2819 $es =~ s/^[0]+//; # strip leading zeros
2827 # return as binary string, with prefixed 0b
2828 my $x = shift; $x = $class->new($x) if !ref($x);
2830 return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
2832 my $es = ''; my $s = '';
2833 $s = $x->{sign} if $x->{sign} eq '-';
2834 if ($CALC->can('_as_bin'))
2836 $es = ${$CALC->_as_bin($x->{value})};
2840 return '0b0' if $x->is_zero();
2841 my $x1 = $x->copy()->babs(); my ($xr,$x10000,$b);
2844 $x10000 = Math::BigInt->new (0x10000); $b = 'b16';
2848 $x10000 = Math::BigInt->new (0x1000); $b = 'b12';
2850 while (!$x1->is_zero())
2852 ($x1, $xr) = bdiv($x1,$x10000);
2853 $es .= unpack($b,pack('v',$xr->numify()));
2856 $es =~ s/^[0]+//; # strip leading zeros
2862 ##############################################################################
2863 # internal calculation routines (others are in Math::BigInt::Calc etc)
2867 # (BINT or num_str, BINT or num_str) return BINT
2868 # does modify first argument
2871 my $x = shift; my $ty = shift;
2872 return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
2873 return $x * $ty / bgcd($x,$ty);
2878 # (BINT or num_str, BINT or num_str) return BINT
2879 # does modify both arguments
2880 # GCD -- Euclids algorithm E, Knuth Vol 2 pg 296
2883 return $x->bnan() if $x->{sign} !~ /^[+-]$/ || $ty->{sign} !~ /^[+-]$/;
2885 while (!$ty->is_zero())
2887 ($x, $ty) = ($ty,bmod($x,$ty));
2892 ###############################################################################
2893 # this method return 0 if the object can be modified, or 1 for not
2894 # We use a fast use constant statement here, to avoid costly calls. Subclasses
2895 # may override it with special code (f.i. Math::BigInt::Constant does so)
2897 sub modify () { 0; }
2904 Math::BigInt - Arbitrary size integer math package
2910 # or make it faster: install (optional) Math::BigInt::GMP
2911 # and always use (it will fall back to pure Perl if the
2912 # GMP library is not installed):
2914 use Math::BigInt lib => 'GMP';
2917 $x = Math::BigInt->new($str); # defaults to 0
2918 $nan = Math::BigInt->bnan(); # create a NotANumber
2919 $zero = Math::BigInt->bzero(); # create a +0
2920 $inf = Math::BigInt->binf(); # create a +inf
2921 $inf = Math::BigInt->binf('-'); # create a -inf
2922 $one = Math::BigInt->bone(); # create a +1
2923 $one = Math::BigInt->bone('-'); # create a -1
2925 # Testing (don't modify their arguments)
2926 # (return true if the condition is met, otherwise false)
2928 $x->is_zero(); # if $x is +0
2929 $x->is_nan(); # if $x is NaN
2930 $x->is_one(); # if $x is +1
2931 $x->is_one('-'); # if $x is -1
2932 $x->is_odd(); # if $x is odd
2933 $x->is_even(); # if $x is even
2934 $x->is_positive(); # if $x >= 0
2935 $x->is_negative(); # if $x < 0
2936 $x->is_inf(sign); # if $x is +inf, or -inf (sign is default '+')
2937 $x->is_int(); # if $x is an integer (not a float)
2939 # comparing and digit/sign extration
2940 $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
2941 $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
2942 $x->sign(); # return the sign, either +,- or NaN
2943 $x->digit($n); # return the nth digit, counting from right
2944 $x->digit(-$n); # return the nth digit, counting from left
2946 # The following all modify their first argument. If you want to preserve
2947 # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
2948 # neccessary when mixing $a = $b assigments with non-overloaded math.
2950 $x->bzero(); # set $x to 0
2951 $x->bnan(); # set $x to NaN
2952 $x->bone(); # set $x to +1
2953 $x->bone('-'); # set $x to -1
2954 $x->binf(); # set $x to inf
2955 $x->binf('-'); # set $x to -inf
2957 $x->bneg(); # negation
2958 $x->babs(); # absolute value
2959 $x->bnorm(); # normalize (no-op in BigInt)
2960 $x->bnot(); # two's complement (bit wise not)
2961 $x->binc(); # increment $x by 1
2962 $x->bdec(); # decrement $x by 1
2964 $x->badd($y); # addition (add $y to $x)
2965 $x->bsub($y); # subtraction (subtract $y from $x)
2966 $x->bmul($y); # multiplication (multiply $x by $y)
2967 $x->bdiv($y); # divide, set $x to quotient
2968 # return (quo,rem) or quo if scalar
2970 $x->bmod($y); # modulus (x % y)
2971 $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
2972 $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
2974 $x->bpow($y); # power of arguments (x ** y)
2975 $x->blsft($y); # left shift
2976 $x->brsft($y); # right shift
2977 $x->blsft($y,$n); # left shift, by base $n (like 10)
2978 $x->brsft($y,$n); # right shift, by base $n (like 10)
2980 $x->band($y); # bitwise and
2981 $x->bior($y); # bitwise inclusive or
2982 $x->bxor($y); # bitwise exclusive or
2983 $x->bnot(); # bitwise not (two's complement)
2985 $x->bsqrt(); # calculate square-root
2986 $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
2987 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
2989 $x->round($A,$P,$mode); # round to accuracy or precision using mode $mode
2990 $x->bround($N); # accuracy: preserve $N digits
2991 $x->bfround($N); # round to $Nth digit, no-op for BigInts
2993 # The following do not modify their arguments in BigInt (are no-ops),
2994 # but do so in BigFloat:
2996 $x->bfloor(); # return integer less or equal than $x
2997 $x->bceil(); # return integer greater or equal than $x
2999 # The following do not modify their arguments:
3001 bgcd(@values); # greatest common divisor (no OO style)
3002 blcm(@values); # lowest common multiplicator (no OO style)
3004 $x->length(); # return number of digits in number
3005 ($x,$f) = $x->length(); # length of number and length of fraction part,
3006 # latter is always 0 digits long for BigInt's
3008 $x->exponent(); # return exponent as BigInt
3009 $x->mantissa(); # return (signed) mantissa as BigInt
3010 $x->parts(); # return (mantissa,exponent) as BigInt
3011 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3012 $x->as_number(); # return as BigInt (in BigInt: same as copy())
3014 # conversation to string (do not modify their argument)
3015 $x->bstr(); # normalized string
3016 $x->bsstr(); # normalized string in scientific notation
3017 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3018 $x->as_bin(); # as signed binary string with prefixed 0b
3021 # precision and accuracy (see section about rounding for more)
3022 $x->precision(); # return P of $x (or global, if P of $x undef)
3023 $x->precision($n); # set P of $x to $n
3024 $x->accuracy(); # return A of $x (or global, if A of $x undef)
3025 $x->accuracy($n); # set A $x to $n
3028 Math::BigInt->precision(); # get/set global P for all BigInt objects
3029 Math::BigInt->accuracy(); # get/set global A for all BigInt objects
3030 Math::BigInt->config(); # return hash containing configuration
3034 All operators (inlcuding basic math operations) are overloaded if you
3035 declare your big integers as
3037 $i = new Math::BigInt '123_456_789_123_456_789';
3039 Operations with overloaded operators preserve the arguments which is
3040 exactly what you expect.
3044 =item Canonical notation
3046 Big integer values are strings of the form C</^[+-]\d+$/> with leading
3049 '-0' canonical value '-0', normalized '0'
3050 ' -123_123_123' canonical value '-123123123'
3051 '1_23_456_7890' canonical value '1234567890'
3055 Input values to these routines may be either Math::BigInt objects or
3056 strings of the form C</^\s*[+-]?[\d]+\.?[\d]*E?[+-]?[\d]*$/>.
3058 You can include one underscore between any two digits.
3060 This means integer values like 1.01E2 or even 1000E-2 are also accepted.
3061 Non integer values result in NaN.
3063 Math::BigInt::new() defaults to 0, while Math::BigInt::new('') results
3066 bnorm() on a BigInt object is now effectively a no-op, since the numbers
3067 are always stored in normalized form. On a string, it creates a BigInt
3072 Output values are BigInt objects (normalized), except for bstr(), which
3073 returns a string in normalized form.
3074 Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
3075 C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
3076 return either undef, <0, 0 or >0 and are suited for sort.
3082 Each of the methods below (except config(), accuracy() and precision())
3083 accepts three additional parameters. These arguments $A, $P and $R are
3084 accuracy, precision and round_mode. Please see the section about
3085 L<ACCURACY and PRECISION> for more information.
3091 print Dumper ( Math::BigInt->config() );
3092 print Math::BigInt->config()->{lib},"\n";
3094 Returns a hash containing the configuration, e.g. the version number, lib
3095 loaded etc. The following hash keys are currently filled in with the
3096 appropriate information.
3100 ============================================================
3101 lib Name of the Math library
3103 lib_version Version of 'lib'
3105 class The class of config you just called
3107 upgrade To which class numbers are upgraded
3109 downgrade To which class numbers are downgraded
3111 precision Global precision
3113 accuracy Global accuracy
3115 round_mode Global round mode
3117 version version number of the class you used
3119 div_scale Fallback acccuracy for div
3122 It is currently not supported to set the configuration parameters by passing
3123 a hash ref to C<config()>.
3127 $x->accuracy(5); # local for $x
3128 CLASS->accuracy(5); # global for all members of CLASS
3129 $A = $x->accuracy(); # read out
3130 $A = CLASS->accuracy(); # read out
3132 Set or get the global or local accuracy, aka how many significant digits the
3135 Please see the section about L<ACCURACY AND PRECISION> for further details.
3137 Value must be greater than zero. Pass an undef value to disable it:
3139 $x->accuracy(undef);
3140 Math::BigInt->accuracy(undef);
3142 Returns the current accuracy. For C<$x->accuracy()> it will return either the
3143 local accuracy, or if not defined, the global. This means the return value
3144 represents the accuracy that will be in effect for $x:
3146 $y = Math::BigInt->new(1234567); # unrounded
3147 print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
3148 $x = Math::BigInt->new(123456); # will be automatically rounded
3149 print "$x $y\n"; # '123500 1234567'
3150 print $x->accuracy(),"\n"; # will be 4
3151 print $y->accuracy(),"\n"; # also 4, since global is 4
3152 print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
3153 print $x->accuracy(),"\n"; # still 4
3154 print $y->accuracy(),"\n"; # 5, since global is 5
3156 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
3157 globals separated from Math::BigInt, but it is possible to subclass
3158 Math::BigInt and make the globals of the subclass aliases to the ones from
3163 $x->precision(-2); # local for $x, round right of the dot
3164 $x->precision(2); # ditto, but round left of the dot
3165 CLASS->accuracy(5); # global for all members of CLASS
3166 CLASS->precision(-5); # ditto
3167 $P = CLASS->precision(); # read out
3168 $P = $x->precision(); # read out
3170 Set or get the global or local precision, aka how many digits the result has
3171 after the dot (or where to round it when passing a positive number). In
3172 Math::BigInt, passing a negative number precision has no effect since no
3173 numbers have digits after the dot.
3175 Please see the section about L<ACCURACY AND PRECISION> for further details.
3177 Value must be greater than zero. Pass an undef value to disable it:
3179 $x->precision(undef);
3180 Math::BigInt->precision(undef);
3182 Returns the current precision. For C<$x->precision()> it will return either the
3183 local precision of $x, or if not defined, the global. This means the return
3184 value represents the accuracy that will be in effect for $x:
3186 $y = Math::BigInt->new(1234567); # unrounded
3187 print Math::BigInt->precision(4),"\n"; # set 4, print 4
3188 $x = Math::BigInt->new(123456); # will be automatically rounded
3190 Note: Works also for subclasses like Math::BigFloat. Each class has it's own
3191 globals separated from Math::BigInt, but it is possible to subclass
3192 Math::BigInt and make the globals of the subclass aliases to the ones from
3199 Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
3200 2, but others work, too.
3202 Right shifting usually amounts to dividing $x by $n ** $y and truncating the
3206 $x = Math::BigInt->new(10);
3207 $x->brsft(1); # same as $x >> 1: 5
3208 $x = Math::BigInt->new(1234);
3209 $x->brsft(2,10); # result 12
3211 There is one exception, and that is base 2 with negative $x:
3214 $x = Math::BigInt->new(-5);
3217 This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
3222 $x = Math::BigInt->new($str,$A,$P,$R);
3224 Creates a new BigInt object from a string or another BigInt object. The
3225 input is accepted as decimal, hex (with leading '0x') or binary (with leading
3230 $x = Math::BigInt->bnan();
3232 Creates a new BigInt object representing NaN (Not A Number).
3233 If used on an object, it will set it to NaN:
3239 $x = Math::BigInt->bzero();
3241 Creates a new BigInt object representing zero.
3242 If used on an object, it will set it to zero:
3248 $x = Math::BigInt->binf($sign);
3250 Creates a new BigInt object representing infinity. The optional argument is
3251 either '-' or '+', indicating whether you want infinity or minus infinity.
3252 If used on an object, it will set it to infinity:
3259 $x = Math::BigInt->binf($sign);
3261 Creates a new BigInt object representing one. The optional argument is
3262 either '-' or '+', indicating whether you want one or minus one.
3263 If used on an object, it will set it to one:
3268 =head2 is_one()/is_zero()/is_nan()/is_inf()
3271 $x->is_zero(); # true if arg is +0
3272 $x->is_nan(); # true if arg is NaN
3273 $x->is_one(); # true if arg is +1
3274 $x->is_one('-'); # true if arg is -1
3275 $x->is_inf(); # true if +inf
3276 $x->is_inf('-'); # true if -inf (sign is default '+')
3278 These methods all test the BigInt for beeing one specific value and return
3279 true or false depending on the input. These are faster than doing something
3284 =head2 is_positive()/is_negative()
3286 $x->is_positive(); # true if >= 0
3287 $x->is_negative(); # true if < 0
3289 The methods return true if the argument is positive or negative, respectively.
3290 C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
3291 C<-inf> is negative. A C<zero> is positive.
3293 These methods are only testing the sign, and not the value.
3295 =head2 is_odd()/is_even()/is_int()
3297 $x->is_odd(); # true if odd, false for even
3298 $x->is_even(); # true if even, false for odd
3299 $x->is_int(); # true if $x is an integer
3301 The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
3302 C<-inf> are not integers and are neither odd nor even.
3308 Compares $x with $y and takes the sign into account.
3309 Returns -1, 0, 1 or undef.
3315 Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
3321 Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
3325 $x->digit($n); # return the nth digit, counting from right
3331 Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
3332 and '-inf', respectively. Does nothing for NaN or zero.
3338 Set the number to it's absolute value, e.g. change the sign from '-' to '+'
3339 and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
3344 $x->bnorm(); # normalize (no-op)
3348 $x->bnot(); # two's complement (bit wise not)
3352 $x->binc(); # increment x by 1
3356 $x->bdec(); # decrement x by 1
3360 $x->badd($y); # addition (add $y to $x)
3364 $x->bsub($y); # subtraction (subtract $y from $x)
3368 $x->bmul($y); # multiplication (multiply $x by $y)
3372 $x->bdiv($y); # divide, set $x to quotient
3373 # return (quo,rem) or quo if scalar
3377 $x->bmod($y); # modulus (x % y)
3381 num->bmodinv($mod); # modular inverse
3383 Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
3384 returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
3385 C<bgcd($num, $mod)==1>.
3389 $num->bmodpow($exp,$mod); # modular exponentation
3390 # ($num**$exp % $mod)
3392 Returns the value of C<$num> taken to the power C<$exp> in the modulus
3393 C<$mod> using binary exponentation. C<bmodpow> is far superior to
3398 because C<bmodpow> is much faster--it reduces internal variables into
3399 the modulus whenever possible, so it operates on smaller numbers.
3401 C<bmodpow> also supports negative exponents.
3403 bmodpow($num, -1, $mod)
3405 is exactly equivalent to
3411 $x->bpow($y); # power of arguments (x ** y)
3415 $x->blsft($y); # left shift
3416 $x->blsft($y,$n); # left shift, in base $n (like 10)
3420 $x->brsft($y); # right shift
3421 $x->brsft($y,$n); # right shift, in base $n (like 10)
3425 $x->band($y); # bitwise and
3429 $x->bior($y); # bitwise inclusive or
3433 $x->bxor($y); # bitwise exclusive or
3437 $x->bnot(); # bitwise not (two's complement)
3441 $x->bsqrt(); # calculate square-root
3445 $x->bfac(); # factorial of $x (1*2*3*4*..$x)
3449 $x->round($A,$P,$round_mode);
3451 Round $x to accuracy C<$A> or precision C<$P> using the round mode
3456 $x->bround($N); # accuracy: preserve $N digits
3460 $x->bfround($N); # round to $Nth digit, no-op for BigInts
3466 Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
3467 does change $x in BigFloat.
3473 Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
3474 does change $x in BigFloat.
3478 bgcd(@values); # greatest common divisor (no OO style)
3482 blcm(@values); # lowest common multiplicator (no OO style)
3487 ($xl,$fl) = $x->length();
3489 Returns the number of digits in the decimal representation of the number.
3490 In list context, returns the length of the integer and fraction part. For
3491 BigInt's, the length of the fraction part will always be 0.
3497 Return the exponent of $x as BigInt.
3503 Return the signed mantissa of $x as BigInt.
3507 $x->parts(); # return (mantissa,exponent) as BigInt
3511 $x->copy(); # make a true copy of $x (unlike $y = $x;)
3515 $x->as_number(); # return as BigInt (in BigInt: same as copy())
3519 $x->bstr(); # return normalized string
3523 $x->bsstr(); # normalized string in scientific notation
3527 $x->as_hex(); # as signed hexadecimal string with prefixed 0x
3531 $x->as_bin(); # as signed binary string with prefixed 0b
3533 =head1 ACCURACY and PRECISION
3535 Since version v1.33, Math::BigInt and Math::BigFloat have full support for
3536 accuracy and precision based rounding, both automatically after every
3537 operation as well as manually.
3539 This section describes the accuracy/precision handling in Math::Big* as it
3540 used to be and as it is now, complete with an explanation of all terms and
3543 Not yet implemented things (but with correct description) are marked with '!',
3544 things that need to be answered are marked with '?'.
3546 In the next paragraph follows a short description of terms used here (because
3547 these may differ from terms used by others people or documentation).
3549 During the rest of this document, the shortcuts A (for accuracy), P (for
3550 precision), F (fallback) and R (rounding mode) will be used.
3554 A fixed number of digits before (positive) or after (negative)
3555 the decimal point. For example, 123.45 has a precision of -2. 0 means an
3556 integer like 123 (or 120). A precision of 2 means two digits to the left
3557 of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
3558 numbers with zeros before the decimal point may have different precisions,
3559 because 1200 can have p = 0, 1 or 2 (depending on what the inital value
3560 was). It could also have p < 0, when the digits after the decimal point
3563 The string output (of floating point numbers) will be padded with zeros:
3565 Initial value P A Result String
3566 ------------------------------------------------------------
3567 1234.01 -3 1000 1000
3570 1234.001 1 1234 1234.0
3572 1234.01 2 1234.01 1234.01
3573 1234.01 5 1234.01 1234.01000
3575 For BigInts, no padding occurs.
3579 Number of significant digits. Leading zeros are not counted. A
3580 number may have an accuracy greater than the non-zero digits
3581 when there are zeros in it or trailing zeros. For example, 123.456 has
3582 A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
3584 The string output (of floating point numbers) will be padded with zeros:
3586 Initial value P A Result String
3587 ------------------------------------------------------------
3589 1234.01 6 1234.01 1234.01
3590 1234.1 8 1234.1 1234.1000
3592 For BigInts, no padding occurs.
3596 When both A and P are undefined, this is used as a fallback accuracy when
3599 =head2 Rounding mode R
3601 When rounding a number, different 'styles' or 'kinds'
3602 of rounding are possible. (Note that random rounding, as in
3603 Math::Round, is not implemented.)
3609 truncation invariably removes all digits following the
3610 rounding place, replacing them with zeros. Thus, 987.65 rounded
3611 to tens (P=1) becomes 980, and rounded to the fourth sigdig
3612 becomes 987.6 (A=4). 123.456 rounded to the second place after the
3613 decimal point (P=-2) becomes 123.46.
3615 All other implemented styles of rounding attempt to round to the
3616 "nearest digit." If the digit D immediately to the right of the
3617 rounding place (skipping the decimal point) is greater than 5, the
3618 number is incremented at the rounding place (possibly causing a
3619 cascade of incrementation): e.g. when rounding to units, 0.9 rounds
3620 to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
3621 truncated at the rounding place: e.g. when rounding to units, 0.4
3622 rounds to 0, and -19.4 rounds to -19.
3624 However the results of other styles of rounding differ if the
3625 digit immediately to the right of the rounding place (skipping the
3626 decimal point) is 5 and if there are no digits, or no digits other
3627 than 0, after that 5. In such cases:
3631 rounds the digit at the rounding place to 0, 2, 4, 6, or 8
3632 if it is not already. E.g., when rounding to the first sigdig, 0.45
3633 becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
3637 rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
3638 it is not already. E.g., when rounding to the first sigdig, 0.45
3639 becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
3643 round to plus infinity, i.e. always round up. E.g., when
3644 rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
3645 and 0.4501 also becomes 0.5.
3649 round to minus infinity, i.e. always round down. E.g., when
3650 rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
3651 but 0.4501 becomes 0.5.
3655 round to zero, i.e. positive numbers down, negative ones up.
3656 E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
3657 becomes -0.5, but 0.4501 becomes 0.5.
3661 The handling of A & P in MBI/MBF (the old core code shipped with Perl
3662 versions <= 5.7.2) is like this:
3668 * ffround($p) is able to round to $p number of digits after the decimal
3670 * otherwise P is unused
3672 =item Accuracy (significant digits)
3674 * fround($a) rounds to $a significant digits
3675 * only fdiv() and fsqrt() take A as (optional) paramater
3676 + other operations simply create the same number (fneg etc), or more (fmul)
3678 + rounding/truncating is only done when explicitly calling one of fround
3679 or ffround, and never for BigInt (not implemented)
3680 * fsqrt() simply hands its accuracy argument over to fdiv.
3681 * the documentation and the comment in the code indicate two different ways
3682 on how fdiv() determines the maximum number of digits it should calculate,
3683 and the actual code does yet another thing
3685 max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
3687 result has at most max(scale, length(dividend), length(divisor)) digits
3689 scale = max(scale, length(dividend)-1,length(divisor)-1);
3690 scale += length(divisior) - length(dividend);
3691 So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
3692 Actually, the 'difference' added to the scale is calculated from the
3693 number of "significant digits" in dividend and divisor, which is derived
3694 by looking at the length of the mantissa. Which is wrong, since it includes
3695 the + sign (oups) and actually gets 2 for '+100' and 4 for '+101'. Oups
3696 again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
3697 assumption that 124 has 3 significant digits, while 120/7 will get you
3698 '17', not '17.1' since 120 is thought to have 2 significant digits.
3699 The rounding after the division then uses the remainder and $y to determine
3700 wether it must round up or down.
3701 ? I have no idea which is the right way. That's why I used a slightly more
3702 ? simple scheme and tweaked the few failing testcases to match it.
3706 This is how it works now:
3710 =item Setting/Accessing
3712 * You can set the A global via Math::BigInt->accuracy() or
3713 Math::BigFloat->accuracy() or whatever class you are using.
3714 * You can also set P globally by using Math::SomeClass->precision() likewise.
3715 * Globals are classwide, and not inherited by subclasses.
3716 * to undefine A, use Math::SomeCLass->accuracy(undef);
3717 * to undefine P, use Math::SomeClass->precision(undef);
3718 * Setting Math::SomeClass->accuracy() clears automatically
3719 Math::SomeClass->precision(), and vice versa.
3720 * To be valid, A must be > 0, P can have any value.
3721 * If P is negative, this means round to the P'th place to the right of the
3722 decimal point; positive values mean to the left of the decimal point.
3723 P of 0 means round to integer.
3724 * to find out the current global A, take Math::SomeClass->accuracy()
3725 * to find out the current global P, take Math::SomeClass->precision()
3726 * use $x->accuracy() respective $x->precision() for the local setting of $x.
3727 * Please note that $x->accuracy() respecive $x->precision() fall back to the
3728 defined globals, when $x's A or P is not set.
3730 =item Creating numbers
3732 * When you create a number, you can give it's desired A or P via:
3733 $x = Math::BigInt->new($number,$A,$P);
3734 * Only one of A or P can be defined, otherwise the result is NaN
3735 * If no A or P is give ($x = Math::BigInt->new($number) form), then the
3736 globals (if set) will be used. Thus changing the global defaults later on
3737 will not change the A or P of previously created numbers (i.e., A and P of
3738 $x will be what was in effect when $x was created)
3739 * If given undef for A and P, B<no> rounding will occur, and the globals will
3740 B<not> be used. This is used by subclasses to create numbers without
3741 suffering rounding in the parent. Thus a subclass is able to have it's own
3742 globals enforced upon creation of a number by using
3743 $x = Math::BigInt->new($number,undef,undef):
3745 use Math::BigInt::SomeSubclass;
3748 Math::BigInt->accuracy(2);
3749 Math::BigInt::SomeSubClass->accuracy(3);
3750 $x = Math::BigInt::SomeSubClass->new(1234);
3752 $x is now 1230, and not 1200. A subclass might choose to implement
3753 this otherwise, e.g. falling back to the parent's A and P.
3757 * If A or P are enabled/defined, they are used to round the result of each
3758 operation according to the rules below
3759 * Negative P is ignored in Math::BigInt, since BigInts never have digits
3760 after the decimal point
3761 * Math::BigFloat uses Math::BigInts internally, but setting A or P inside
3762 Math::BigInt as globals should not tamper with the parts of a BigFloat.
3763 Thus a flag is used to mark all Math::BigFloat numbers as 'never round'
3767 * It only makes sense that a number has only one of A or P at a time.
3768 Since you can set/get both A and P, there is a rule that will practically
3769 enforce only A or P to be in effect at a time, even if both are set.
3770 This is called precedence.
3771 * If two objects are involved in an operation, and one of them has A in
3772 effect, and the other P, this results in an error (NaN).
3773 * A takes precendence over P (Hint: A comes before P). If A is defined, it
3774 is used, otherwise P is used. If neither of them is defined, nothing is
3775 used, i.e. the result will have as many digits as it can (with an
3776 exception for fdiv/fsqrt) and will not be rounded.
3777 * There is another setting for fdiv() (and thus for fsqrt()). If neither of
3778 A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
3779 If either the dividend's or the divisor's mantissa has more digits than
3780 the value of F, the higher value will be used instead of F.
3781 This is to limit the digits (A) of the result (just consider what would
3782 happen with unlimited A and P in the case of 1/3 :-)
3783 * fdiv will calculate (at least) 4 more digits than required (determined by
3784 A, P or F), and, if F is not used, round the result
3785 (this will still fail in the case of a result like 0.12345000000001 with A
3786 or P of 5, but this can not be helped - or can it?)
3787 * Thus you can have the math done by on Math::Big* class in three modes:
3788 + never round (this is the default):
3789 This is done by setting A and P to undef. No math operation
3790 will round the result, with fdiv() and fsqrt() as exceptions to guard
3791 against overflows. You must explicitely call bround(), bfround() or
3792 round() (the latter with parameters).
3793 Note: Once you have rounded a number, the settings will 'stick' on it
3794 and 'infect' all other numbers engaged in math operations with it, since
3795 local settings have the highest precedence. So, to get SaferRound[tm],
3796 use a copy() before rounding like this:
3798 $x = Math::BigFloat->new(12.34);
3799 $y = Math::BigFloat->new(98.76);
3800 $z = $x * $y; # 1218.6984
3801 print $x->copy()->fround(3); # 12.3 (but A is now 3!)
3802 $z = $x * $y; # still 1218.6984, without
3803 # copy would have been 1210!
3805 + round after each op:
3806 After each single operation (except for testing like is_zero()), the
3807 method round() is called and the result is rounded appropriately. By
3808 setting proper values for A and P, you can have all-the-same-A or
3809 all-the-same-P modes. For example, Math::Currency might set A to undef,
3810 and P to -2, globally.
3812 ?Maybe an extra option that forbids local A & P settings would be in order,
3813 ?so that intermediate rounding does not 'poison' further math?
3815 =item Overriding globals
3817 * you will be able to give A, P and R as an argument to all the calculation
3818 routines; the second parameter is A, the third one is P, and the fourth is
3819 R (shift right by one for binary operations like badd). P is used only if
3820 the first parameter (A) is undefined. These three parameters override the
3821 globals in the order detailed as follows, i.e. the first defined value
3823 (local: per object, global: global default, parameter: argument to sub)
3826 + local A (if defined on both of the operands: smaller one is taken)
3827 + local P (if defined on both of the operands: bigger one is taken)
3831 * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
3832 arguments (A and P) instead of one
3834 =item Local settings
3836 * You can set A and P locally by using $x->accuracy() and $x->precision()
3837 and thus force different A and P for different objects/numbers.
3838 * Setting A or P this way immediately rounds $x to the new value.
3839 * $x->accuracy() clears $x->precision(), and vice versa.
3843 * the rounding routines will use the respective global or local settings.
3844 fround()/bround() is for accuracy rounding, while ffround()/bfround()
3846 * the two rounding functions take as the second parameter one of the
3847 following rounding modes (R):
3848 'even', 'odd', '+inf', '-inf', 'zero', 'trunc'
3849 * you can set and get the global R by using Math::SomeClass->round_mode()
3850 or by setting $Math::SomeClass::round_mode
3851 * after each operation, $result->round() is called, and the result may
3852 eventually be rounded (that is, if A or P were set either locally,
3853 globally or as parameter to the operation)
3854 * to manually round a number, call $x->round($A,$P,$round_mode);
3855 this will round the number by using the appropriate rounding function
3856 and then normalize it.
3857 * rounding modifies the local settings of the number:
3859 $x = Math::BigFloat->new(123.456);
3863 Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
3864 will be 4 from now on.
3866 =item Default values
3875 * The defaults are set up so that the new code gives the same results as
3876 the old code (except in a few cases on fdiv):
3877 + Both A and P are undefined and thus will not be used for rounding
3878 after each operation.
3879 + round() is thus a no-op, unless given extra parameters A and P
3885 The actual numbers are stored as unsigned big integers (with seperate sign).
3886 You should neither care about nor depend on the internal representation; it
3887 might change without notice. Use only method calls like C<< $x->sign(); >>
3888 instead relying on the internal hash keys like in C<< $x->{sign}; >>.
3892 Math with the numbers is done (by default) by a module called
3893 Math::BigInt::Calc. This is equivalent to saying:
3895 use Math::BigInt lib => 'Calc';
3897 You can change this by using:
3899 use Math::BigInt lib => 'BitVect';
3901 The following would first try to find Math::BigInt::Foo, then
3902 Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
3904 use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
3906 Calc.pm uses as internal format an array of elements of some decimal base
3907 (usually 1e5 or 1e7) with the least significant digit first, while BitVect.pm
3908 uses a bit vector of base 2, most significant bit first. Other modules might
3909 use even different means of representing the numbers. See the respective
3910 module documentation for further details.
3914 The sign is either '+', '-', 'NaN', '+inf' or '-inf' and stored seperately.
3916 A sign of 'NaN' is used to represent the result when input arguments are not
3917 numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
3918 minus infinity. You will get '+inf' when dividing a positive number by 0, and
3919 '-inf' when dividing any negative number by 0.
3921 =head2 mantissa(), exponent() and parts()
3923 C<mantissa()> and C<exponent()> return the said parts of the BigInt such
3926 $m = $x->mantissa();
3927 $e = $x->exponent();
3928 $y = $m * ( 10 ** $e );
3929 print "ok\n" if $x == $y;
3931 C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
3932 in one go. Both the returned mantissa and exponent have a sign.
3934 Currently, for BigInts C<$e> will be always 0, except for NaN, +inf and -inf,
3935 where it will be NaN; and for $x == 0, where it will be 1
3936 (to be compatible with Math::BigFloat's internal representation of a zero as
3939 C<$m> will always be a copy of the original number. The relation between $e
3940 and $m might change in the future, but will always be equivalent in a
3941 numerical sense, e.g. $m might get minimized.
3947 sub bint { Math::BigInt->new(shift); }
3949 $x = Math::BigInt->bstr("1234") # string "1234"
3950 $x = "$x"; # same as bstr()
3951 $x = Math::BigInt->bneg("1234"); # BigInt "-1234"
3952 $x = Math::BigInt->babs("-12345"); # BigInt "12345"
3953 $x = Math::BigInt->bnorm("-0 00"); # BigInt "0"
3954 $x = bint(1) + bint(2); # BigInt "3"
3955 $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
3956 $x = bint(1); # BigInt "1"
3957 $x = $x + 5 / 2; # BigInt "3"
3958 $x = $x ** 3; # BigInt "27"
3959 $x *= 2; # BigInt "54"
3960 $x = Math::BigInt->new(0); # BigInt "0"
3962 $x = Math::BigInt->badd(4,5) # BigInt "9"
3963 print $x->bsstr(); # 9e+0
3965 Examples for rounding:
3970 $x = Math::BigFloat->new(123.4567);
3971 $y = Math::BigFloat->new(123.456789);
3972 Math::BigFloat->accuracy(4); # no more A than 4
3974 ok ($x->copy()->fround(),123.4); # even rounding
3975 print $x->copy()->fround(),"\n"; # 123.4
3976 Math::BigFloat->round_mode('odd'); # round to odd
3977 print $x->copy()->fround(),"\n"; # 123.5
3978 Math::BigFloat->accuracy(5); # no more A than 5
3979 Math::BigFloat->round_mode('odd'); # round to odd
3980 print $x->copy()->fround(),"\n"; # 123.46
3981 $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
3982 print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
3984 Math::BigFloat->accuracy(undef); # A not important now
3985 Math::BigFloat->precision(2); # P important
3986 print $x->copy()->bnorm(),"\n"; # 123.46
3987 print $x->copy()->fround(),"\n"; # 123.46
3989 Examples for converting:
3991 my $x = Math::BigInt->new('0b1'.'01' x 123);
3992 print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
3994 =head1 Autocreating constants
3996 After C<use Math::BigInt ':constant'> all the B<integer> decimal, hexadecimal
3997 and binary constants in the given scope are converted to C<Math::BigInt>.
3998 This conversion happens at compile time.
4002 perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
4004 prints the integer value of C<2**100>. Note that without conversion of
4005 constants the expression 2**100 will be calculated as perl scalar.
4007 Please note that strings and floating point constants are not affected,
4010 use Math::BigInt qw/:constant/;
4012 $x = 1234567890123456789012345678901234567890
4013 + 123456789123456789;
4014 $y = '1234567890123456789012345678901234567890'
4015 + '123456789123456789';
4017 do not work. You need an explicit Math::BigInt->new() around one of the
4018 operands. You should also quote large constants to protect loss of precision:
4022 $x = Math::BigInt->new('1234567889123456789123456789123456789');
4024 Without the quotes Perl would convert the large number to a floating point
4025 constant at compile time and then hand the result to BigInt, which results in
4026 an truncated result or a NaN.
4028 This also applies to integers that look like floating point constants:
4030 use Math::BigInt ':constant';
4032 print ref(123e2),"\n";
4033 print ref(123.2e2),"\n";
4035 will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
4036 to get this to work.
4040 Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
4041 must be made in the second case. For long numbers, the copy can eat up to 20%
4042 of the work (in the case of addition/subtraction, less for
4043 multiplication/division). If $y is very small compared to $x, the form
4044 $x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
4045 more time then the actual addition.
4047 With a technique called copy-on-write, the cost of copying with overload could
4048 be minimized or even completely avoided. A test implementation of COW did show
4049 performance gains for overloaded math, but introduced a performance loss due
4050 to a constant overhead for all other operatons.
4052 The rewritten version of this module is slower on certain operations, like
4053 new(), bstr() and numify(). The reason are that it does now more work and
4054 handles more cases. The time spent in these operations is usually gained in
4055 the other operations so that programs on the average should get faster. If
4056 they don't, please contect the author.
4058 Some operations may be slower for small numbers, but are significantly faster
4059 for big numbers. Other operations are now constant (O(1), like bneg(), babs()
4060 etc), instead of O(N) and thus nearly always take much less time. These
4061 optimizations were done on purpose.
4063 If you find the Calc module to slow, try to install any of the replacement
4064 modules and see if they help you.
4066 =head2 Alternative math libraries
4068 You can use an alternative library to drive Math::BigInt via:
4070 use Math::BigInt lib => 'Module';
4072 See L<MATH LIBRARY> for more information.
4074 For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
4078 =head1 Subclassing Math::BigInt
4080 The basic design of Math::BigInt allows simple subclasses with very little
4081 work, as long as a few simple rules are followed:
4087 The public API must remain consistent, i.e. if a sub-class is overloading
4088 addition, the sub-class must use the same name, in this case badd(). The
4089 reason for this is that Math::BigInt is optimized to call the object methods
4094 The private object hash keys like C<$x->{sign}> may not be changed, but
4095 additional keys can be added, like C<$x->{_custom}>.
4099 Accessor functions are available for all existing object hash keys and should
4100 be used instead of directly accessing the internal hash keys. The reason for
4101 this is that Math::BigInt itself has a pluggable interface which permits it
4102 to support different storage methods.
4106 More complex sub-classes may have to replicate more of the logic internal of
4107 Math::BigInt if they need to change more basic behaviors. A subclass that
4108 needs to merely change the output only needs to overload C<bstr()>.
4110 All other object methods and overloaded functions can be directly inherited
4111 from the parent class.
4113 At the very minimum, any subclass will need to provide it's own C<new()> and can
4114 store additional hash keys in the object. There are also some package globals
4115 that must be defined, e.g.:
4119 $precision = -2; # round to 2 decimal places
4120 $round_mode = 'even';
4123 Additionally, you might want to provide the following two globals to allow
4124 auto-upgrading and auto-downgrading to work correctly:
4129 This allows Math::BigInt to correctly retrieve package globals from the
4130 subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
4131 t/Math/BigFloat/SubClass.pm completely functional subclass examples.
4137 in your subclass to automatically inherit the overloading from the parent. If
4138 you like, you can change part of the overloading, look at Math::String for an
4143 When used like this:
4145 use Math::BigInt upgrade => 'Foo::Bar';
4147 certain operations will 'upgrade' their calculation and thus the result to
4148 the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
4150 use Math::BigInt upgrade => 'Math::BigFloat';
4152 As a shortcut, you can use the module C<bignum>:
4156 Also good for oneliners:
4158 perl -Mbignum -le 'print 2 ** 255'
4160 This makes it possible to mix arguments of different classes (as in 2.5 + 2)
4161 as well es preserve accuracy (as in sqrt(3)).
4163 Beware: This feature is not fully implemented yet.
4167 The following methods upgrade themselves unconditionally; that is if upgrade
4168 is in effect, they will always hand up their work:
4180 Beware: This list is not complete.
4182 All other methods upgrade themselves only when one (or all) of their
4183 arguments are of the class mentioned in $upgrade (This might change in later
4184 versions to a more sophisticated scheme):
4190 =item broot() does not work
4192 The broot() function in BigInt may only work for small values. This will be
4193 fixed in a later version.
4195 =item Out of Memory!
4197 Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
4198 C<eval()> in your code will crash with "Out of memory". This is probably an
4199 overload/exporter bug. You can workaround by not having C<eval()>
4200 and ':constant' at the same time or upgrade your Perl to a newer version.
4202 =item Fails to load Calc on Perl prior 5.6.0
4204 Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
4205 will fall back to eval { require ... } when loading the math lib on Perls
4206 prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
4207 filesystems using a different seperator.
4213 Some things might not work as you expect them. Below is documented what is
4214 known to be troublesome:
4218 =item stringify, bstr(), bsstr() and 'cmp'
4220 Both stringify and bstr() now drop the leading '+'. The old code would return
4221 '+3', the new returns '3'. This is to be consistent with Perl and to make
4222 cmp (especially with overloading) to work as you expect. It also solves
4223 problems with Test.pm, it's ok() uses 'eq' internally.
4225 Mark said, when asked about to drop the '+' altogether, or make only cmp work:
4227 I agree (with the first alternative), don't add the '+' on positive
4228 numbers. It's not as important anymore with the new internal
4229 form for numbers. It made doing things like abs and neg easier,
4230 but those have to be done differently now anyway.
4232 So, the following examples will now work all as expected:
4235 BEGIN { plan tests => 1 }
4238 my $x = new Math::BigInt 3*3;
4239 my $y = new Math::BigInt 3*3;
4242 print "$x eq 9" if $x eq $y;
4243 print "$x eq 9" if $x eq '9';
4244 print "$x eq 9" if $x eq 3*3;
4246 Additionally, the following still works:
4248 print "$x == 9" if $x == $y;
4249 print "$x == 9" if $x == 9;
4250 print "$x == 9" if $x == 3*3;
4252 There is now a C<bsstr()> method to get the string in scientific notation aka
4253 C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
4254 for comparisation, but Perl will represent some numbers as 100 and others
4255 as 1e+308. If in doubt, convert both arguments to Math::BigInt before doing eq:
4258 BEGIN { plan tests => 3 }
4261 $x = Math::BigInt->new('1e56'); $y = 1e56;
4262 ok ($x,$y); # will fail
4263 ok ($x->bsstr(),$y); # okay
4264 $y = Math::BigInt->new($y);
4267 Alternatively, simple use <=> for comparisations, that will get it always
4268 right. There is not yet a way to get a number automatically represented as
4269 a string that matches exactly the way Perl represents it.
4273 C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
4276 $x = Math::BigInt->new(123);
4277 $y = int($x); # BigInt 123
4278 $x = Math::BigFloat->new(123.45);
4279 $y = int($x); # BigInt 123
4281 In all Perl versions you can use C<as_number()> for the same effect:
4283 $x = Math::BigFloat->new(123.45);
4284 $y = $x->as_number(); # BigInt 123
4286 This also works for other subclasses, like Math::String.
4288 It is yet unlcear whether overloaded int() should return a scalar or a BigInt.
4292 The following will probably not do what you expect:
4294 $c = Math::BigInt->new(123);
4295 print $c->length(),"\n"; # prints 30
4297 It prints both the number of digits in the number and in the fraction part
4298 since print calls C<length()> in list context. Use something like:
4300 print scalar $c->length(),"\n"; # prints 3
4304 The following will probably not do what you expect:
4306 print $c->bdiv(10000),"\n";
4308 It prints both quotient and remainder since print calls C<bdiv()> in list
4309 context. Also, C<bdiv()> will modify $c, so be carefull. You probably want
4312 print $c / 10000,"\n";
4313 print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
4317 The quotient is always the greatest integer less than or equal to the
4318 real-valued quotient of the two operands, and the remainder (when it is
4319 nonzero) always has the same sign as the second operand; so, for
4329 As a consequence, the behavior of the operator % agrees with the
4330 behavior of Perl's built-in % operator (as documented in the perlop
4331 manpage), and the equation
4333 $x == ($x / $y) * $y + ($x % $y)
4335 holds true for any $x and $y, which justifies calling the two return
4336 values of bdiv() the quotient and remainder. The only exception to this rule
4337 are when $y == 0 and $x is negative, then the remainder will also be
4338 negative. See below under "infinity handling" for the reasoning behing this.
4340 Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
4341 not change BigInt's way to do things. This is because under 'use integer' Perl
4342 will do what the underlying C thinks is right and this is different for each
4343 system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
4344 the author to implement it ;)
4346 =item infinity handling
4348 Here are some examples that explain the reasons why certain results occur while
4351 The following table shows the result of the division and the remainder, so that
4352 the equation above holds true. Some "ordinary" cases are strewn in to show more
4353 clearly the reasoning:
4355 A / B = C, R so that C * B + R = A
4356 =========================================================
4357 5 / 8 = 0, 5 0 * 8 + 5 = 5
4358 0 / 8 = 0, 0 0 * 8 + 0 = 0
4359 0 / inf = 0, 0 0 * inf + 0 = 0
4360 0 /-inf = 0, 0 0 * -inf + 0 = 0
4361 5 / inf = 0, 5 0 * inf + 5 = 5
4362 5 /-inf = 0, 5 0 * -inf + 5 = 5
4363 -5/ inf = 0, -5 0 * inf + -5 = -5
4364 -5/-inf = 0, -5 0 * -inf + -5 = -5
4365 inf/ 5 = inf, 0 inf * 5 + 0 = inf
4366 -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
4367 inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
4368 -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
4369 5/ 5 = 1, 0 1 * 5 + 0 = 5
4370 -5/ -5 = 1, 0 1 * -5 + 0 = -5
4371 inf/ inf = 1, 0 1 * inf + 0 = inf
4372 -inf/-inf = 1, 0 1 * -inf + 0 = -inf
4373 inf/-inf = -1, 0 -1 * -inf + 0 = inf
4374 -inf/ inf = -1, 0 1 * -inf + 0 = -inf
4375 8/ 0 = inf, 8 inf * 0 + 8 = 8
4376 inf/ 0 = inf, inf inf * 0 + inf = inf
4379 These cases below violate the "remainder has the sign of the second of the two
4380 arguments", since they wouldn't match up otherwise.
4382 A / B = C, R so that C * B + R = A
4383 ========================================================
4384 -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
4385 -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
4387 =item Modifying and =
4391 $x = Math::BigFloat->new(5);
4394 It will not do what you think, e.g. making a copy of $x. Instead it just makes
4395 a second reference to the B<same> object and stores it in $y. Thus anything
4396 that modifies $x (except overloaded operators) will modify $y, and vice versa.
4397 Or in other words, C<=> is only safe if you modify your BigInts only via
4398 overloaded math. As soon as you use a method call it breaks:
4401 print "$x, $y\n"; # prints '10, 10'
4403 If you want a true copy of $x, use:
4407 You can also chain the calls like this, this will make first a copy and then
4410 $y = $x->copy()->bmul(2);
4412 See also the documentation for overload.pm regarding C<=>.
4416 C<bpow()> (and the rounding functions) now modifies the first argument and
4417 returns it, unlike the old code which left it alone and only returned the
4418 result. This is to be consistent with C<badd()> etc. The first three will
4419 modify $x, the last one won't:
4421 print bpow($x,$i),"\n"; # modify $x
4422 print $x->bpow($i),"\n"; # ditto
4423 print $x **= $i,"\n"; # the same
4424 print $x ** $i,"\n"; # leave $x alone
4426 The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
4428 =item Overloading -$x
4438 since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
4439 needs to preserve $x since it does not know that it later will get overwritten.
4440 This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
4442 With Copy-On-Write, this issue would be gone, but C-o-W is not implemented
4443 since it is slower for all other things.
4445 =item Mixing different object types
4447 In Perl you will get a floating point value if you do one of the following:
4453 With overloaded math, only the first two variants will result in a BigFloat:
4458 $mbf = Math::BigFloat->new(5);
4459 $mbi2 = Math::BigInteger->new(5);
4460 $mbi = Math::BigInteger->new(2);
4462 # what actually gets called:
4463 $float = $mbf + $mbi; # $mbf->badd()
4464 $float = $mbf / $mbi; # $mbf->bdiv()
4465 $integer = $mbi + $mbf; # $mbi->badd()
4466 $integer = $mbi2 / $mbi; # $mbi2->bdiv()
4467 $integer = $mbi2 / $mbf; # $mbi2->bdiv()
4469 This is because math with overloaded operators follows the first (dominating)
4470 operand, and the operation of that is called and returns thus the result. So,
4471 Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
4472 the result should be a Math::BigFloat or the second operant is one.
4474 To get a Math::BigFloat you either need to call the operation manually,
4475 make sure the operands are already of the proper type or casted to that type
4476 via Math::BigFloat->new():
4478 $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
4480 Beware of simple "casting" the entire expression, this would only convert
4481 the already computed result:
4483 $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
4485 Beware also of the order of more complicated expressions like:
4487 $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
4488 $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
4490 If in doubt, break the expression into simpler terms, or cast all operands
4491 to the desired resulting type.
4493 Scalar values are a bit different, since:
4498 will both result in the proper type due to the way the overloaded math works.
4500 This section also applies to other overloaded math packages, like Math::String.
4502 One solution to you problem might be autoupgrading|upgrading. See the
4503 pragmas L<bignum>, L<bigint> and L<bigrat> for an easy way to do this.
4507 C<bsqrt()> works only good if the result is a big integer, e.g. the square
4508 root of 144 is 12, but from 12 the square root is 3, regardless of rounding
4509 mode. The reason is that the result is always truncated to an integer.
4511 If you want a better approximation of the square root, then use:
4513 $x = Math::BigFloat->new(12);
4514 Math::BigFloat->precision(0);
4515 Math::BigFloat->round_mode('even');
4516 print $x->copy->bsqrt(),"\n"; # 4
4518 Math::BigFloat->precision(2);
4519 print $x->bsqrt(),"\n"; # 3.46
4520 print $x->bsqrt(3),"\n"; # 3.464
4524 For negative numbers in base see also L<brsft|brsft>.
4530 This program is free software; you may redistribute it and/or modify it under
4531 the same terms as Perl itself.
4535 L<Math::BigFloat>, L<Math::BigRat> and L<Math::Big> as well as
4536 L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
4538 The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest
4539 because they solve the autoupgrading/downgrading issue, at least partly.
4542 L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
4543 more documentation including a full version history, testcases, empty
4544 subclass files and benchmarks.
4548 Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
4549 Completely rewritten by Tels http://bloodgate.com in late 2000, 2001, 2002
4550 and still at it in 2003.
4552 Many people contributed in one or more ways to the final beast, see the file
4553 CREDITS for an (uncomplete) list. If you miss your name, please drop me a