+#!/usr/bin/perl -w
+
+# Qs: what exactly happens on numify of HUGE numbers? overflow?
+# $a = -$a is much slower (making copy of $a) than $a->bneg(), hm!?
+# (copy_on_write will help there, but that is not yet implemented)
+
+# The following hash values are used:
+# value: unsigned int with actual value (as a Math::BigInt::Calc or similiar)
+# sign : +,-,NaN,+inf,-inf
+# _a : accuracy
+# _p : precision
+# _f : flags, used by MBF to flag parts of a float as untouchable
+
+# Remember not to take shortcuts ala $xs = $x->{value}; $CALC->foo($xs); since
+# underlying lib might change the reference!
+
package Math::BigInt;
+my $class = "Math::BigInt";
+require 5.005;
+
+$VERSION = '1.47';
+use Exporter;
+@ISA = qw( Exporter );
+@EXPORT_OK = qw( bneg babs bcmp badd bmul bdiv bmod bnorm bsub
+ bgcd blcm bround
+ blsft brsft band bior bxor bnot bpow bnan bzero
+ bacmp bstr bsstr binc bdec binf bfloor bceil
+ is_odd is_even is_zero is_one is_nan is_inf sign
+ is_positive is_negative
+ length as_number objectify _swap
+ );
+#@EXPORT = qw( );
+use vars qw/$round_mode $accuracy $precision $div_scale $rnd_mode/;
+use strict;
+
+# Inside overload, the first arg is always an object. If the original code had
+# it reversed (like $x = 2 * $y), then the third paramater indicates this
+# swapping. To make it work, we use a helper routine which not only reswaps the
+# params, but also makes a new object in this case. See _swap() for details,
+# especially the cases of operators with different classes.
+
+# For overloaded ops with only one argument we simple use $_[0]->copy() to
+# preserve the argument.
+
+# Thus inheritance of overload operators becomes possible and transparent for
+# our subclasses without the need to repeat the entire overload section there.
+
+use overload
+'=' => sub { $_[0]->copy(); },
+
+# '+' and '-' do not use _swap, since it is a triffle slower. If you want to
+# override _swap (if ever), then override overload of '+' and '-', too!
+# for sub it is a bit tricky to keep b: b-a => -a+b
+'-' => sub { my $c = $_[0]->copy; $_[2] ?
+ $c->bneg()->badd($_[1]) :
+ $c->bsub( $_[1]) },
+'+' => sub { $_[0]->copy()->badd($_[1]); },
+
+# some shortcuts for speed (assumes that reversed order of arguments is routed
+# to normal '+' and we thus can always modify first arg. If this is changed,
+# this breaks and must be adjusted.)
+'+=' => sub { $_[0]->badd($_[1]); },
+'-=' => sub { $_[0]->bsub($_[1]); },
+'*=' => sub { $_[0]->bmul($_[1]); },
+'/=' => sub { scalar $_[0]->bdiv($_[1]); },
+'%=' => sub { $_[0]->bmod($_[1]); },
+'^=' => sub { $_[0]->bxor($_[1]); },
+'&=' => sub { $_[0]->band($_[1]); },
+'|=' => sub { $_[0]->bior($_[1]); },
+'**=' => sub { $_[0]->bpow($_[1]); },
+
+'..' => \&_pointpoint,
+
+'<=>' => sub { $_[2] ?
+ ref($_[0])->bcmp($_[1],$_[0]) :
+ ref($_[0])->bcmp($_[0],$_[1])},
+'cmp' => sub {
+ $_[2] ?
+ $_[1] cmp $_[0]->bstr() :
+ $_[0]->bstr() cmp $_[1] },
+
+'int' => sub { $_[0]->copy(); },
+'neg' => sub { $_[0]->copy()->bneg(); },
+'abs' => sub { $_[0]->copy()->babs(); },
+'~' => sub { $_[0]->copy()->bnot(); },
+
+'*' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bmul($a[1]); },
+'/' => sub { my @a = ref($_[0])->_swap(@_);scalar $a[0]->bdiv($a[1]);},
+'%' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bmod($a[1]); },
+'**' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bpow($a[1]); },
+'<<' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->blsft($a[1]); },
+'>>' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->brsft($a[1]); },
+
+'&' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->band($a[1]); },
+'|' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bior($a[1]); },
+'^' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bxor($a[1]); },
+
+# can modify arg of ++ and --, so avoid a new-copy for speed, but don't
+# use $_[0]->__one(), it modifies $_[0] to be 1!
+'++' => sub { $_[0]->binc() },
+'--' => sub { $_[0]->bdec() },
+
+# if overloaded, O(1) instead of O(N) and twice as fast for small numbers
+'bool' => sub {
+ # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/
+ # v5.6.1 dumps on that: return !$_[0]->is_zero() || undef; :-(
+ my $t = !$_[0]->is_zero();
+ undef $t if $t == 0;
+ return $t;
+ },
+
+# the original qw() does not work with the TIESCALAR below, why?
+# Order of arguments unsignificant
+'""' => sub { $_[0]->bstr(); },
+'0+' => sub { $_[0]->numify(); }
+;
+
+##############################################################################
+# global constants, flags and accessory
+
+use constant MB_NEVER_ROUND => 0x0001;
+
+my $NaNOK=1; # are NaNs ok?
+my $nan = 'NaN'; # constants for easier life
+
+my $CALC = 'Math::BigInt::Calc'; # module to do low level math
+sub _core_lib () { return $CALC; } # for test suite
+
+$round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
+$accuracy = undef;
+$precision = undef;
+$div_scale = 40;
+
+##############################################################################
+# the old code had $rnd_mode, so we need to support it, too
+
+$rnd_mode = 'even';
+sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; }
+sub FETCH { return $round_mode; }
+sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
+
+BEGIN { tie $rnd_mode, 'Math::BigInt'; }
+
+##############################################################################
+
+sub round_mode
+ {
+ no strict 'refs';
+ # make Class->round_mode() work
+ my $self = shift;
+ my $class = ref($self) || $self || __PACKAGE__;
+ if (defined $_[0])
+ {
+ my $m = shift;
+ die "Unknown round mode $m"
+ if $m !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/;
+ ${"${class}::round_mode"} = $m; return $m;
+ }
+ return ${"${class}::round_mode"};
+ }
+
+sub div_scale
+ {
+ no strict 'refs';
+ # make Class->round_mode() work
+ my $self = shift;
+ my $class = ref($self) || $self || __PACKAGE__;
+ if (defined $_[0])
+ {
+ die ('div_scale must be greater than zero') if $_[0] < 0;
+ ${"${class}::div_scale"} = shift;
+ }
+ return ${"${class}::div_scale"};
+ }
+
+sub accuracy
+ {
+ # $x->accuracy($a); ref($x) $a
+ # $x->accuracy(); ref($x)
+ # Class->accuracy(); class
+ # Class->accuracy($a); class $a
+
+ my $x = shift;
+ my $class = ref($x) || $x || __PACKAGE__;
+
+ no strict 'refs';
+ # need to set new value?
+ if (@_ > 0)
+ {
+ my $a = shift;
+ die ('accuracy must not be zero') if defined $a && $a == 0;
+ if (ref($x))
+ {
+ # $object->accuracy() or fallback to global
+ $x->bround($a) if defined $a;
+ $x->{_a} = $a; # set/overwrite, even if not rounded
+ $x->{_p} = undef; # clear P
+ }
+ else
+ {
+ # set global
+ ${"${class}::accuracy"} = $a;
+ }
+ return $a; # shortcut
+ }
+
+ if (ref($x))
+ {
+ # $object->accuracy() or fallback to global
+ return $x->{_a} || ${"${class}::accuracy"};
+ }
+ return ${"${class}::accuracy"};
+ }
+
+sub precision
+ {
+ # $x->precision($p); ref($x) $p
+ # $x->precision(); ref($x)
+ # Class->precision(); class
+ # Class->precision($p); class $p
+
+ my $x = shift;
+ my $class = ref($x) || $x || __PACKAGE__;
+
+ no strict 'refs';
+ # need to set new value?
+ if (@_ > 0)
+ {
+ my $p = shift;
+ if (ref($x))
+ {
+ # $object->precision() or fallback to global
+ $x->bfround($p) if defined $p;
+ $x->{_p} = $p; # set/overwrite, even if not rounded
+ $x->{_a} = undef; # clear P
+ }
+ else
+ {
+ # set global
+ ${"${class}::precision"} = $p;
+ }
+ return $p; # shortcut
+ }
+
+ if (ref($x))
+ {
+ # $object->precision() or fallback to global
+ return $x->{_p} || ${"${class}::precision"};
+ }
+ return ${"${class}::precision"};
+ }
+
+sub _scale_a
+ {
+ # select accuracy parameter based on precedence,
+ # used by bround() and bfround(), may return undef for scale (means no op)
+ my ($x,$s,$m,$scale,$mode) = @_;
+ $scale = $x->{_a} if !defined $scale;
+ $scale = $s if (!defined $scale);
+ $mode = $m if !defined $mode;
+ return ($scale,$mode);
+ }
+
+sub _scale_p
+ {
+ # select precision parameter based on precedence,
+ # used by bround() and bfround(), may return undef for scale (means no op)
+ my ($x,$s,$m,$scale,$mode) = @_;
+ $scale = $x->{_p} if !defined $scale;
+ $scale = $s if (!defined $scale);
+ $mode = $m if !defined $mode;
+ return ($scale,$mode);
+ }
+
+##############################################################################
+# constructors
+
+sub copy
+ {
+ my ($c,$x);
+ if (@_ > 1)
+ {
+ # if two arguments, the first one is the class to "swallow" subclasses
+ ($c,$x) = @_;
+ }
+ else
+ {
+ $x = shift;
+ $c = ref($x);
+ }
+ return unless ref($x); # only for objects
+
+ my $self = {}; bless $self,$c;
+ foreach my $k (keys %$x)
+ {
+ if ($k eq 'value')
+ {
+ $self->{value} = $CALC->_copy($x->{value});
+ }
+ elsif (ref($x->{$k}) eq 'SCALAR')
+ {
+ $self->{$k} = \${$x->{$k}};
+ }
+ elsif (ref($x->{$k}) eq 'ARRAY')
+ {
+ $self->{$k} = [ @{$x->{$k}} ];
+ }
+ elsif (ref($x->{$k}) eq 'HASH')
+ {
+ # only one level deep!
+ foreach my $h (keys %{$x->{$k}})
+ {
+ $self->{$k}->{$h} = $x->{$k}->{$h};
+ }
+ }
+ elsif (ref($x->{$k}))
+ {
+ my $c = ref($x->{$k});
+ $self->{$k} = $c->new($x->{$k}); # no copy() due to deep rec
+ }
+ else
+ {
+ $self->{$k} = $x->{$k};
+ }
+ }
+ $self;
+ }
+
+sub new
+ {
+ # create a new BigInt object from a string or another BigInt object.
+ # see hash keys documented at top
+
+ # the argument could be an object, so avoid ||, && etc on it, this would
+ # cause costly overloaded code to be called. The only allowed ops are
+ # ref() and defined.
+
+ my $class = shift;
+
+ my $wanted = shift; # avoid numify call by not using || here
+ return $class->bzero() if !defined $wanted; # default to 0
+ return $class->copy($wanted) if ref($wanted);
+
+ my $self = {}; bless $self, $class;
+ # handle '+inf', '-inf' first
+ if ($wanted =~ /^[+-]?inf$/)
+ {
+ $self->{value} = $CALC->_zero();
+ $self->{sign} = $wanted; $self->{sign} = '+inf' if $self->{sign} eq 'inf';
+ return $self;
+ }
+ # split str in m mantissa, e exponent, i integer, f fraction, v value, s sign
+ my ($mis,$miv,$mfv,$es,$ev) = _split(\$wanted);
+ if (!ref $mis)
+ {
+ die "$wanted is not a number initialized to $class" if !$NaNOK;
+ #print "NaN 1\n";
+ $self->{value} = $CALC->_zero();
+ $self->{sign} = $nan;
+ return $self;
+ }
+ if (!ref $miv)
+ {
+ # _from_hex or _from_bin
+ $self->{value} = $mis->{value};
+ $self->{sign} = $mis->{sign};
+ return $self; # throw away $mis
+ }
+ # make integer from mantissa by adjusting exp, then convert to bigint
+ $self->{sign} = $$mis; # store sign
+ $self->{value} = $CALC->_zero(); # for all the NaN cases
+ my $e = int("$$es$$ev"); # exponent (avoid recursion)
+ if ($e > 0)
+ {
+ my $diff = $e - CORE::length($$mfv);
+ if ($diff < 0) # Not integer
+ {
+ #print "NOI 1\n";
+ $self->{sign} = $nan;
+ }
+ else # diff >= 0
+ {
+ # adjust fraction and add it to value
+ # print "diff > 0 $$miv\n";
+ $$miv = $$miv . ($$mfv . '0' x $diff);
+ }
+ }
+ else
+ {
+ if ($$mfv ne '') # e <= 0
+ {
+ # fraction and negative/zero E => NOI
+ #print "NOI 2 \$\$mfv '$$mfv'\n";
+ $self->{sign} = $nan;
+ }
+ elsif ($e < 0)
+ {
+ # xE-y, and empty mfv
+ #print "xE-y\n";
+ $e = abs($e);
+ if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
+ {
+ #print "NOI 3\n";
+ $self->{sign} = $nan;
+ }
+ }
+ }
+ $self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
+ $self->{value} = $CALC->_new($miv) if $self->{sign} =~ /^[+-]$/;
+ # if any of the globals is set, use them to round and store them inside $self
+ $self->round($accuracy,$precision,$round_mode)
+ if defined $accuracy || defined $precision;
+ return $self;
+ }
+
+sub bnan
+ {
+ # create a bigint 'NaN', if given a BigInt, set it to 'NaN'
+ my $self = shift;
+ $self = $class if !defined $self;
+ if (!ref($self))
+ {
+ my $c = $self; $self = {}; bless $self, $c;
+ }
+ return if $self->modify('bnan');
+ $self->{value} = $CALC->_zero();
+ $self->{sign} = $nan;
+ return $self;
+ }
+
+sub binf
+ {
+ # create a bigint '+-inf', if given a BigInt, set it to '+-inf'
+ # the sign is either '+', or if given, used from there
+ my $self = shift;
+ my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
+ $self = $class if !defined $self;
+ if (!ref($self))
+ {
+ my $c = $self; $self = {}; bless $self, $c;
+ }
+ return if $self->modify('binf');
+ $self->{value} = $CALC->_zero();
+ $self->{sign} = $sign.'inf';
+ return $self;
+ }
+
+sub bzero
+ {
+ # create a bigint '+0', if given a BigInt, set it to 0
+ my $self = shift;
+ $self = $class if !defined $self;
+
+ if (!ref($self))
+ {
+ my $c = $self; $self = {}; bless $self, $c;
+ }
+ return if $self->modify('bzero');
+ $self->{value} = $CALC->_zero();
+ $self->{sign} = '+';
+ return $self;
+ }
+
+sub bone
+ {
+ # create a bigint '+1' (or -1 if given sign '-'),
+ # if given a BigInt, set it to +1 or -1, respecively
+ my $self = shift;
+ my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
+ $self = $class if !defined $self;
+
+ if (!ref($self))
+ {
+ my $c = $self; $self = {}; bless $self, $c;
+ }
+ return if $self->modify('bone');
+ $self->{value} = $CALC->_one();
+ $self->{sign} = $sign;
+ return $self;
+ }
+
+##############################################################################
+# string conversation
+
+sub bsstr
+ {
+ # (ref to BFLOAT or num_str ) return num_str
+ # Convert number from internal format to scientific string format.
+ # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
+ my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
+ # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+
+ if ($x->{sign} !~ /^[+-]$/)
+ {
+ return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
+ return 'inf'; # +inf
+ }
+ my ($m,$e) = $x->parts();
+ # e can only be positive
+ my $sign = 'e+';
+ # MBF: my $s = $e->{sign}; $s = '' if $s eq '-'; my $sep = 'e'.$s;
+ return $m->bstr().$sign.$e->bstr();
+ }
+
+sub bstr
+ {
+ # make a string from bigint object
+ my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
+ # my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+
+ if ($x->{sign} !~ /^[+-]$/)
+ {
+ return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
+ return 'inf'; # +inf
+ }
+ my $es = ''; $es = $x->{sign} if $x->{sign} eq '-';
+ return $es.${$CALC->_str($x->{value})};
+ }
+
+sub numify
+ {
+ # Make a number from a BigInt object
+ my $x = shift; $x = $class->new($x) unless ref $x;
+ return $x->{sign} if $x->{sign} !~ /^[+-]$/;
+ my $num = $CALC->_num($x->{value});
+ return -$num if $x->{sign} eq '-';
+ return $num;
+ }
+
+##############################################################################
+# public stuff (usually prefixed with "b")
+
+sub sign
+ {
+ # return the sign of the number: +/-/NaN
+ my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+
+ return $x->{sign};
+ }
+
+sub _find_round_parameters
+ {
+ # After any operation or when calling round(), the result is rounded by
+ # regarding the A & P from arguments, local parameters, or globals.
+ # The result's A or P are set by the rounding, but not inspected beforehand
+ # (aka only the arguments enter into it). This works because the given
+ # 'first' argument is both the result and true first argument with unchanged
+ # A and P settings.
+ # This does not yet handle $x with A, and $y with P (which should be an
+ # error).
+ my $self = shift;
+ my $a = shift; # accuracy, if given by caller
+ my $p = shift; # precision, if given by caller
+ my $r = shift; # round_mode, if given by caller
+ my @args = @_; # all 'other' arguments (0 for unary, 1 for binary ops)
+
+ $self = new($self) unless ref($self); # if not object, make one
+ my $c = ref($self); # find out class of argument(s)
+ unshift @args,$self; # add 'first' argument
+
+ # leave bigfloat parts alone
+ return ($self) if exists $self->{_f} && $self->{_f} & MB_NEVER_ROUND != 0;
+
+ no strict 'refs';
+
+ # now pick $a or $p, but only if we have got "arguments"
+ if ((!defined $a) && (!defined $p) && (@args > 0))
+ {
+ foreach (@args)
+ {
+ # take the defined one, or if both defined, the one that is smaller
+ $a = $_->{_a} if (defined $_->{_a}) && (!defined $a || $_->{_a} < $a);
+ }
+ if (!defined $a) # if it still is not defined, take p
+ {
+ foreach (@args)
+ {
+ # take the defined one, or if both defined, the one that is bigger
+ # -2 > -3, and 3 > 2
+ $p = $_->{_p} if (defined $_->{_p}) && (!defined $p || $_->{_p} > $p);
+ }
+ # if none defined, use globals (#2)
+ if (!defined $p)
+ {
+ my $z = "$c\::accuracy"; my $a = $$z;
+ if (!defined $a)
+ {
+ $z = "$c\::precision"; $p = $$z;
+ }
+ }
+ } # endif !$a
+ } # endif !$a || !$P && args > 0
+ my @params = ($self);
+ if (defined $a || defined $p)
+ {
+ $r = $r || ${"$c\::round_mode"};
+ die "Unknown round mode '$r'"
+ if $r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/;
+ push @params, ($a,$p,$r);
+ }
+ return @params;
+ }
+
+sub round
+ {
+ # round $self according to given parameters, or given second argument's
+ # parameters or global defaults
+ my $self = shift;
+
+ my @params = $self->_find_round_parameters(@_);
+ return $self->bnorm() if @params == 1; # no-op
+
+ # now round, by calling fround or ffround:
+ if (defined $params[1])
+ {
+ $self->bround($params[1],$params[3]);
+ }
+ else
+ {
+ $self->bfround($params[2],$params[3]);
+ }
+ return $self->bnorm(); # after round, normalize
+ }
+
+sub bnorm
+ {
+ # (numstr or BINT) return BINT
+ # Normalize number -- no-op here
+ my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+ return $x;
+ }
+
+sub babs
+ {
+ # (BINT or num_str) return BINT
+ # make number absolute, or return absolute BINT from string
+ my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+
+ return $x if $x->modify('babs');
+ # post-normalized abs for internal use (does nothing for NaN)
+ $x->{sign} =~ s/^-/+/;
+ $x;
+ }
+
+sub bneg
+ {
+ # (BINT or num_str) return BINT
+ # negate number or make a negated number from string
+ my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+
+ return $x if $x->modify('bneg');
+ # for +0 dont negate (to have always normalized)
+ return $x if $x->is_zero();
+ $x->{sign} =~ tr/+\-/-+/; # does nothing for NaN
+ $x;
+ }
+
+sub bcmp
+ {
+ # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
+ # (BINT or num_str, BINT or num_str) return cond_code
+ my ($self,$x,$y) = objectify(2,@_);
+
+ if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
+ {
+ # handle +-inf and NaN
+ return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
+ return 0 if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
+ return +1 if $x->{sign} eq '+inf';
+ return -1 if $x->{sign} eq '-inf';
+ return -1 if $y->{sign} eq '+inf';
+ return +1 if $y->{sign} eq '-inf';
+ }
+ # check sign for speed first
+ return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
+ return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
+
+ # shortcut
+ my $xz = $x->is_zero();
+ my $yz = $y->is_zero();
+ return 0 if $xz && $yz; # 0 <=> 0
+ return -1 if $xz && $y->{sign} eq '+'; # 0 <=> +y
+ return 1 if $yz && $x->{sign} eq '+'; # +x <=> 0
+
+ # post-normalized compare for internal use (honors signs)
+ if ($x->{sign} eq '+')
+ {
+ return 1 if $y->{sign} eq '-'; # 0 check handled above
+ return $CALC->_acmp($x->{value},$y->{value});
+ }
+
+ # $x->{sign} eq '-'
+ return -1 if $y->{sign} eq '+';
+ return $CALC->_acmp($y->{value},$x->{value}); # swaped
+
+ # &cmp($x->{value},$y->{value},$x->{sign},$y->{sign}) <=> 0;
+ }
+
+sub bacmp
+ {
+ # Compares 2 values, ignoring their signs.
+ # Returns one of undef, <0, =0, >0. (suitable for sort)
+ # (BINT, BINT) return cond_code
+ my ($self,$x,$y) = objectify(2,@_);
+
+ if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
+ {
+ # handle +-inf and NaN
+ return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
+ return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
+ return +1; # inf is always bigger
+ }
+ $CALC->_acmp($x->{value},$y->{value}) <=> 0;
+ }
+
+sub badd
+ {
+ # add second arg (BINT or string) to first (BINT) (modifies first)
+ # return result as BINT
+ my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+
+ return $x if $x->modify('badd');
+
+ # inf and NaN handling
+ if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
+ {
+ # NaN first
+ return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
+ # inf handline
+ if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
+ {
+ # + and + => +, - and - => -, + and - => 0, - and + => 0
+ return $x->bzero() if $x->{sign} ne $y->{sign};
+ return $x;
+ }
+ # +-inf + something => +inf
+ # something +-inf => +-inf
+ $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
+ return $x;
+ }
+
+ my @bn = ($a,$p,$r,$y); # make array for round calls
+ # speed: no add for 0+y or x+0
+ return $x->round(@bn) if $y->is_zero(); # x+0
+ if ($x->is_zero()) # 0+y
+ {
+ # make copy, clobbering up x
+ $x->{value} = $CALC->_copy($y->{value});
+ $x->{sign} = $y->{sign} || $nan;
+ return $x->round(@bn);
+ }
+
+ my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
+
+ if ($sx eq $sy)
+ {
+ $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
+ $x->{sign} = $sx;
+ }
+ else
+ {
+ my $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
+ if ($a > 0)
+ {
+ #print "swapped sub (a=$a)\n";
+ $x->{value} = $CALC->_sub($y->{value},$x->{value},1); # abs sub w/ swap
+ $x->{sign} = $sy;
+ }
+ elsif ($a == 0)
+ {
+ # speedup, if equal, set result to 0
+ #print "equal sub, result = 0\n";
+ $x->{value} = $CALC->_zero();
+ $x->{sign} = '+';
+ }
+ else # a < 0
+ {
+ #print "unswapped sub (a=$a)\n";
+ $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
+ $x->{sign} = $sx;
+ }
+ }
+ return $x->round(@bn);
+ }
+
+sub bsub
+ {
+ # (BINT or num_str, BINT or num_str) return num_str
+ # subtract second arg from first, modify first
+ my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+
+ return $x if $x->modify('bsub');
+
+ if (!$y->is_zero()) # don't need to do anything if $y is 0
+ {
+ $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
+ $x->badd($y,$a,$p,$r); # badd does not leave internal zeros
+ $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
+ }
+ $x; # already rounded by badd()
+ }
+
+sub binc
+ {
+ # increment arg by one
+ my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+ return $x if $x->modify('binc');
+
+ if ($x->{sign} eq '+')
+ {
+ $x->{value} = $CALC->_inc($x->{value});
+ return $x->round($a,$p,$r);
+ }
+ elsif ($x->{sign} eq '-')
+ {
+ $x->{value} = $CALC->_dec($x->{value});
+ $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
+ return $x->round($a,$p,$r);
+ }
+ # inf, nan handling etc
+ $x->badd($self->__one(),$a,$p,$r); # does round
+ }
+
+sub bdec
+ {
+ # decrement arg by one
+ my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+ return $x if $x->modify('bdec');
+
+ my $zero = $CALC->_is_zero($x->{value}) && $x->{sign} eq '+';
+ # <= 0
+ if (($x->{sign} eq '-') || $zero)
+ {
+ $x->{value} = $CALC->_inc($x->{value});
+ $x->{sign} = '-' if $zero; # 0 => 1 => -1
+ $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
+ return $x->round($a,$p,$r);
+ }
+ # > 0
+ elsif ($x->{sign} eq '+')
+ {
+ $x->{value} = $CALC->_dec($x->{value});
+ return $x->round($a,$p,$r);
+ }
+ # inf, nan handling etc
+ $x->badd($self->__one('-'),$a,$p,$r); # does round
+ }
+
+sub blcm
+ {
+ # (BINT or num_str, BINT or num_str) return BINT
+ # does not modify arguments, but returns new object
+ # Lowest Common Multiplicator
+
+ my $y = shift; my ($x);
+ if (ref($y))
+ {
+ $x = $y->copy();
+ }
+ else
+ {
+ $x = $class->new($y);
+ }
+ while (@_) { $x = __lcm($x,shift); }
+ $x;
+ }
+
+sub bgcd
+ {
+ # (BINT or num_str, BINT or num_str) return BINT
+ # does not modify arguments, but returns new object
+ # GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
+
+ my $y = shift;
+ $y = __PACKAGE__->new($y) if !ref($y);
+ my $self = ref($y);
+ my $x = $y->copy(); # keep arguments
+ if ($CALC->can('_gcd'))
+ {
+ while (@_)
+ {
+ $y = shift; $y = $self->new($y) if !ref($y);
+ next if $y->is_zero();
+ return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
+ $x->{value} = $CALC->_gcd($x->{value},$y->{value}); last if $x->is_one();
+ }
+ }
+ else
+ {
+ while (@_)
+ {
+ $y = shift; $y = $self->new($y) if !ref($y);
+ $x = __gcd($x,$y->copy()); last if $x->is_one(); # _gcd handles NaN
+ }
+ }
+ $x->babs();
+ }
+
+sub bnot
+ {
+ # (num_str or BINT) return BINT
+ # represent ~x as twos-complement number
+ # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
+ my ($self,$x,$a,$p,$r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
+
+ return $x if $x->modify('bnot');
+ $x->bneg(); $x->bdec(); # was: bsub(-1,$x);, time it someday
+ return $x->round($a,$p,$r);
+ }
+
+sub is_zero
+ {
+ # return true if arg (BINT or num_str) is zero (array '+', '0')
+ # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
+ $CALC->_is_zero($x->{value});
+ }
+
+sub is_nan
+ {
+ # return true if arg (BINT or num_str) is NaN
+ my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+
+ return 1 if $x->{sign} eq $nan;
+ return 0;
+ }
+
+sub is_inf
+ {
+ # return true if arg (BINT or num_str) is +-inf
+ my ($self,$x,$sign) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+
+ $sign = '' if !defined $sign;
+ return 0 if $sign !~ /^([+-]|)$/;
+
+ if ($sign eq '')
+ {
+ return 1 if ($x->{sign} =~ /^[+-]inf$/);
+ return 0;
+ }
+ $sign = quotemeta($sign.'inf');
+ return 1 if ($x->{sign} =~ /^$sign$/);
+ return 0;
+ }
+
+sub is_one
+ {
+ # return true if arg (BINT or num_str) is +1
+ # or -1 if sign is given
+ # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
+ my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
+
+ $sign = '' if !defined $sign; $sign = '+' if $sign ne '-';
+
+ return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
+ return $CALC->_is_one($x->{value});
+ }
+
+sub is_odd
+ {
+ # return true when arg (BINT or num_str) is odd, false for even
+ # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
+ return $CALC->_is_odd($x->{value});
+ }
+
+sub is_even
+ {
+ # return true when arg (BINT or num_str) is even, false for odd
+ # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
+ return $CALC->_is_even($x->{value});
+ }
+
+sub is_positive
+ {
+ # return true when arg (BINT or num_str) is positive (>= 0)
+ # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ return 1 if $x->{sign} =~ /^\+/;
+ return 0;
+ }
+
+sub is_negative
+ {
+ # return true when arg (BINT or num_str) is negative (< 0)
+ # we don't need $self, so undef instead of ref($_[0]) make it slightly faster
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ return 1 if ($x->{sign} =~ /^-/);
+ return 0;
+ }
+
+###############################################################################
+
+sub bmul
+ {
+ # multiply two numbers -- stolen from Knuth Vol 2 pg 233
+ # (BINT or num_str, BINT or num_str) return BINT
+ my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+
+ return $x if $x->modify('bmul');
+ return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
+ # handle result = 0
+ return $x if $x->is_zero();
+ return $x->bzero() if $y->is_zero();
+ # inf handling
+ if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
+ {
+ # result will always be +-inf:
+ # +inf * +/+inf => +inf, -inf * -/-inf => +inf
+ # +inf * -/-inf => -inf, -inf * +/+inf => -inf
+ return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
+ return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
+ return $x->binf('-');
+ }
+
+ $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
+
+ $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
+ return $x->round($a,$p,$r,$y);
+ }
+
+sub _div_inf
+ {
+ # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
+ my ($self,$x,$y) = @_;
+
+ # NaN if x == NaN or y == NaN or x==y==0
+ return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan()
+ if (($x->is_nan() || $y->is_nan()) ||
+ ($x->is_zero() && $y->is_zero()));
+
+ # +inf / +inf == -inf / -inf == 1, remainder is 0 (A / A = 1, remainder 0)
+ if (($x->{sign} eq $y->{sign}) &&
+ ($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
+ {
+ return wantarray ? ($x->bone(),$self->bzero()) : $x->bone();
+ }
+ # +inf / -inf == -inf / +inf == -1, remainder 0
+ if (($x->{sign} ne $y->{sign}) &&
+ ($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
+ {
+ return wantarray ? ($x->bone('-'),$self->bzero()) : $x->bone('-');
+ }
+ # x / +-inf => 0, remainder x (works even if x == 0)
+ if ($y->{sign} =~ /^[+-]inf$/)
+ {
+ my $t = $x->copy(); # binf clobbers up $x
+ return wantarray ? ($x->bzero(),$t) : $x->bzero()
+ }
+
+ # 5 / 0 => +inf, -6 / 0 => -inf
+ # +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
+ # exception: -8 / 0 has remainder -8, not 8
+ # exception: -inf / 0 has remainder -inf, not inf
+ if ($y->is_zero())
+ {
+ # +-inf / 0 => special case for -inf
+ return wantarray ? ($x,$x->copy()) : $x if $x->is_inf();
+ if (!$x->is_zero() && !$x->is_inf())
+ {
+ my $t = $x->copy(); # binf clobbers up $x
+ return wantarray ?
+ ($x->binf($x->{sign}),$t) : $x->binf($x->{sign})
+ }
+ }
+
+ # last case: +-inf / ordinary number
+ my $sign = '+inf';
+ $sign = '-inf' if substr($x->{sign},0,1) ne $y->{sign};
+ $x->{sign} = $sign;
+ return wantarray ? ($x,$self->bzero()) : $x;
+ }
+
+sub bdiv
+ {
+ # (dividend: BINT or num_str, divisor: BINT or num_str) return
+ # (BINT,BINT) (quo,rem) or BINT (only rem)
+ my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+
+ return $x if $x->modify('bdiv');
+
+ return $self->_div_inf($x,$y)
+ if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
+
+ # 0 / something
+ return wantarray ? ($x,$self->bzero()) : $x if $x->is_zero();
+
+ # Is $x in the interval [0, $y) ?
+ my $cmp = $CALC->_acmp($x->{value},$y->{value});
+ if (($cmp < 0) and ($x->{sign} eq $y->{sign}))
+ {
+ return $x->bzero() unless wantarray;
+ my $t = $x->copy(); # make copy first, because $x->bzero() clobbers $x
+ return ($x->bzero(),$t);
+ }
+ elsif ($cmp == 0)
+ {
+ # shortcut, both are the same, so set to +/- 1
+ $x->__one( ($x->{sign} ne $y->{sign} ? '-' : '+') );
+ return $x unless wantarray;
+ return ($x,$self->bzero());
+ }
+
+ # calc new sign and in case $y == +/- 1, return $x
+ my $xsign = $x->{sign}; # keep
+ $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
+ # check for / +-1 (cant use $y->is_one due to '-'
+ if (($y == 1) || ($y == -1)) # slow!
+ {
+ return wantarray ? ($x,$self->bzero()) : $x;
+ }
+
+ # call div here
+ my $rem = $self->bzero();
+ ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
+ # do not leave result "-0";
+ $x->{sign} = '+' if $CALC->_is_zero($x->{value});
+ $x->round($a,$p,$r,$y);
+
+ if (wantarray)
+ {
+ if (! $CALC->_is_zero($rem->{value}))
+ {
+ $rem->{sign} = $y->{sign};
+ $rem = $y-$rem if $xsign ne $y->{sign}; # one of them '-'
+ }
+ else
+ {
+ $rem->{sign} = '+'; # dont leave -0
+ }
+ $rem->round($a,$p,$r,$x,$y);
+ return ($x,$rem);
+ }
+ return $x;
+ }
+
+sub bmod
+ {
+ # modulus (or remainder)
+ # (BINT or num_str, BINT or num_str) return BINT
+ my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+
+ return $x if $x->modify('bmod');
+ if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
+ {
+ my ($d,$r) = $self->_div_inf($x,$y);
+ return $r;
+ }
+
+ if ($CALC->can('_mod'))
+ {
+ # calc new sign and in case $y == +/- 1, return $x
+ $x->{value} = $CALC->_mod($x->{value},$y->{value});
+ my $xsign = $x->{sign};
+ if (!$CALC->_is_zero($x->{value}))
+ {
+ $x->{sign} = $y->{sign};
+ $x = $y-$x if $xsign ne $y->{sign}; # one of them '-'
+ }
+ else
+ {
+ $x->{sign} = '+'; # dont leave -0
+ }
+ }
+ else
+ {
+ $x = (&bdiv($self,$x,$y))[1]; # slow way
+ }
+ $x->bround($a,$p,$r);
+ }
+
+sub bpow
+ {
+ # (BINT or num_str, BINT or num_str) return BINT
+ # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
+ # modifies first argument
+ my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+
+ return $x if $x->modify('bpow');
+
+ return $x if $x->{sign} =~ /^[+-]inf$/; # -inf/+inf ** x
+ return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
+ return $x->__one() if $y->is_zero();
+ return $x if $x->is_one() || $y->is_one();
+ #if ($x->{sign} eq '-' && @{$x->{value}} == 1 && $x->{value}->[0] == 1)
+ if ($x->{sign} eq '-' && $CALC->_is_one($x->{value}))
+ {
+ # if $x == -1 and odd/even y => +1/-1
+ return $y->is_odd() ? $x : $x->babs();
+ # my Casio FX-5500L has a bug here: -1 ** 2 is -1, but -1 * -1 is 1;
+ }
+ # 1 ** -y => 1 / (1 ** |y|)
+ # so do test for negative $y after above's clause
+ return $x->bnan() if $y->{sign} eq '-';
+ return $x if $x->is_zero(); # 0**y => 0 (if not y <= 0)
+
+ if ($CALC->can('_pow'))
+ {
+ $x->{value} = $CALC->_pow($x->{value},$y->{value});
+ return $x->round($a,$p,$r);
+ }
+
+# based on the assumption that shifting in base 10 is fast, and that mul
+# works faster if numbers are small: we count trailing zeros (this step is
+# O(1)..O(N), but in case of O(N) we save much more time due to this),
+# stripping them out of the multiplication, and add $count * $y zeros
+# afterwards like this:
+# 300 ** 3 == 300*300*300 == 3*3*3 . '0' x 2 * 3 == 27 . '0' x 6
+# creates deep recursion?
+# my $zeros = $x->_trailing_zeros();
+# if ($zeros > 0)
+# {
+# $x->brsft($zeros,10); # remove zeros
+# $x->bpow($y); # recursion (will not branch into here again)
+# $zeros = $y * $zeros; # real number of zeros to add
+# $x->blsft($zeros,10);
+# return $x->round($a,$p,$r);
+# }
+
+ my $pow2 = $self->__one();
+ my $y1 = $class->new($y);
+ my $two = $self->new(2);
+ while (!$y1->is_one())
+ {
+ $pow2->bmul($x) if $y1->is_odd();
+ $y1->bdiv($two);
+ $x->bmul($x);
+ }
+ $x->bmul($pow2) unless $pow2->is_one();
+ return $x->round($a,$p,$r);
+ }
+
+sub blsft
+ {
+ # (BINT or num_str, BINT or num_str) return BINT
+ # compute x << y, base n, y >= 0
+ my ($self,$x,$y,$n) = objectify(2,@_);
+
+ return $x if $x->modify('blsft');
+ return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
+
+ $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
+
+ my $t; $t = $CALC->_lsft($x->{value},$y->{value},$n) if $CALC->can('_lsft');
+ if (defined $t)
+ {
+ $x->{value} = $t; return $x;
+ }
+ # fallback
+ return $x->bmul( $self->bpow($n, $y) );
+ }
+
+sub brsft
+ {
+ # (BINT or num_str, BINT or num_str) return BINT
+ # compute x >> y, base n, y >= 0
+ my ($self,$x,$y,$n) = objectify(2,@_);
+
+ return $x if $x->modify('brsft');
+ return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
+
+ $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
+
+ my $t; $t = $CALC->_rsft($x->{value},$y->{value},$n) if $CALC->can('_rsft');
+ if (defined $t)
+ {
+ $x->{value} = $t; return $x;
+ }
+ # fallback
+ return scalar bdiv($x, $self->bpow($n, $y));
+ }
+
+sub band
+ {
+ #(BINT or num_str, BINT or num_str) return BINT
+ # compute x & y
+ my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+
+ return $x if $x->modify('band');
+
+ return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
+ return $x->bzero() if $y->is_zero();
+
+ my $sign = 0; # sign of result
+ $sign = 1 if ($x->{sign} eq '-') && ($y->{sign} eq '-');
+ my $sx = 1; $sx = -1 if $x->{sign} eq '-';
+ my $sy = 1; $sy = -1 if $y->{sign} eq '-';
+
+ if ($CALC->can('_and') && $sx == 1 && $sy == 1)
+ {
+ $x->{value} = $CALC->_and($x->{value},$y->{value});
+ return $x->round($a,$p,$r);
+ }
+
+ my $m = new Math::BigInt 1; my ($xr,$yr);
+ my $x10000 = new Math::BigInt (0x1000);
+ my $y1 = copy(ref($x),$y); # make copy
+ $y1->babs(); # and positive
+ my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
+ use integer; # need this for negative bools
+ while (!$x1->is_zero() && !$y1->is_zero())
+ {
+ ($x1, $xr) = bdiv($x1, $x10000);
+ ($y1, $yr) = bdiv($y1, $x10000);
+ # make both op's numbers!
+ $x->badd( bmul( $class->new(
+ abs($sx*int($xr->numify()) & $sy*int($yr->numify()))),
+ $m));
+ $m->bmul($x10000);
+ }
+ $x->bneg() if $sign;
+ return $x->round($a,$p,$r);
+ }
+
+sub bior
+ {
+ #(BINT or num_str, BINT or num_str) return BINT
+ # compute x | y
+ my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+
+ return $x if $x->modify('bior');
+
+ return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
+ return $x if $y->is_zero();
+
+ my $sign = 0; # sign of result
+ $sign = 1 if ($x->{sign} eq '-') || ($y->{sign} eq '-');
+ my $sx = 1; $sx = -1 if $x->{sign} eq '-';
+ my $sy = 1; $sy = -1 if $y->{sign} eq '-';
+
+ # don't use lib for negative values
+ if ($CALC->can('_or') && $sx == 1 && $sy == 1)
+ {
+ $x->{value} = $CALC->_or($x->{value},$y->{value});
+ return $x->round($a,$p,$r);
+ }
+
+ my $m = new Math::BigInt 1; my ($xr,$yr);
+ my $x10000 = new Math::BigInt (0x10000);
+ my $y1 = copy(ref($x),$y); # make copy
+ $y1->babs(); # and positive
+ my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
+ use integer; # need this for negative bools
+ while (!$x1->is_zero() || !$y1->is_zero())
+ {
+ ($x1, $xr) = bdiv($x1,$x10000);
+ ($y1, $yr) = bdiv($y1,$x10000);
+ # make both op's numbers!
+ $x->badd( bmul( $class->new(
+ abs($sx*int($xr->numify()) | $sy*int($yr->numify()))),
+ $m));
+ $m->bmul($x10000);
+ }
+ $x->bneg() if $sign;
+ return $x->round($a,$p,$r);
+ }
+
+sub bxor
+ {
+ #(BINT or num_str, BINT or num_str) return BINT
+ # compute x ^ y
+ my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+
+ return $x if $x->modify('bxor');
+
+ return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
+ return $x if $y->is_zero();
+ return $x->bzero() if $x == $y; # shortcut
+
+ my $sign = 0; # sign of result
+ $sign = 1 if $x->{sign} ne $y->{sign};
+ my $sx = 1; $sx = -1 if $x->{sign} eq '-';
+ my $sy = 1; $sy = -1 if $y->{sign} eq '-';
+
+ # don't use lib for negative values
+ if ($CALC->can('_xor') && $sx == 1 && $sy == 1)
+ {
+ $x->{value} = $CALC->_xor($x->{value},$y->{value});
+ return $x->round($a,$p,$r);
+ }
+
+ my $m = new Math::BigInt 1; my ($xr,$yr);
+ my $x10000 = new Math::BigInt (0x10000);
+ my $y1 = copy(ref($x),$y); # make copy
+ $y1->babs(); # and positive
+ my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
+ use integer; # need this for negative bools
+ while (!$x1->is_zero() || !$y1->is_zero())
+ {
+ ($x1, $xr) = bdiv($x1, $x10000);
+ ($y1, $yr) = bdiv($y1, $x10000);
+ # make both op's numbers!
+ $x->badd( bmul( $class->new(
+ abs($sx*int($xr->numify()) ^ $sy*int($yr->numify()))),
+ $m));
+ $m->bmul($x10000);
+ }
+ $x->bneg() if $sign;
+ return $x->round($a,$p,$r);
+ }
+
+sub length
+ {
+ my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+
+ my $e = $CALC->_len($x->{value});
+ return wantarray ? ($e,0) : $e;
+ }
+
+sub digit
+ {
+ # return the nth decimal digit, negative values count backward, 0 is right
+ my $x = shift;
+ my $n = shift || 0;
+
+ return $CALC->_digit($x->{value},$n);
+ }
+
+sub _trailing_zeros
+ {
+ # return the amount of trailing zeros in $x
+ my $x = shift;
+ $x = $class->new($x) unless ref $x;
+
+ return 0 if $x->is_zero() || $x->is_odd() || $x->{sign} !~ /^[+-]$/;
+
+ return $CALC->_zeros($x->{value}) if $CALC->can('_zeros');
+
+ # if not: since we do not know underlying internal representation:
+ my $es = "$x"; $es =~ /([0]*)$/;
+
+ return 0 if !defined $1; # no zeros
+ return CORE::length("$1"); # as string, not as +0!
+ }
+
+sub bsqrt
+ {
+ my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+
+ return $x->bnan() if $x->{sign} =~ /\-|$nan/; # -x or NaN => NaN
+ return $x->bzero() if $x->is_zero(); # 0 => 0
+ return $x if $x == 1; # 1 => 1
+
+ my $y = $x->copy(); # give us one more digit accur.
+ my $l = int($x->length()/2);
+
+ $x->bzero();
+ $x->binc(); # keep ref($x), but modify it
+ $x *= 10 ** $l;
+
+ # print "x: $y guess $x\n";
+
+ my $last = $self->bzero();
+ while ($last != $x)
+ {
+ $last = $x;
+ $x += $y / $x;
+ $x /= 2;
+ }
+ return $x;
+ }
+
+sub exponent
+ {
+ # return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
+ my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+
+ if ($x->{sign} !~ /^[+-]$/)
+ {
+ my $s = $x->{sign}; $s =~ s/^[+-]//;
+ return $self->new($s); # -inf,+inf => inf
+ }
+ my $e = $class->bzero();
+ return $e->binc() if $x->is_zero();
+ $e += $x->_trailing_zeros();
+ return $e;
+ }
+
+sub mantissa
+ {
+ # return the mantissa (compatible to Math::BigFloat, e.g. reduced)
+ my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+
+ if ($x->{sign} !~ /^[+-]$/)
+ {
+ my $s = $x->{sign}; $s =~ s/^[+]//;
+ return $self->new($s); # +inf => inf
+ }
+ my $m = $x->copy();
+ # that's inefficient
+ my $zeros = $m->_trailing_zeros();
+ $m /= 10 ** $zeros if $zeros != 0;
+ return $m;
+ }
+
+sub parts
+ {
+ # return a copy of both the exponent and the mantissa
+ my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+
+ return ($x->mantissa(),$x->exponent());
+ }
+
+##############################################################################
+# rounding functions
+
+sub bfround
+ {
+ # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
+ # $n == 0 || $n == 1 => round to integer
+ my $x = shift; $x = $class->new($x) unless ref $x;
+ my ($scale,$mode) = $x->_scale_p($x->precision(),$x->round_mode(),@_);
+ return $x if !defined $scale; # no-op
+
+ # no-op for BigInts if $n <= 0
+ if ($scale <= 0)
+ {
+ $x->{_p} = $scale; return $x;
+ }
+
+ $x->bround( $x->length()-$scale, $mode);
+ $x->{_a} = undef; # bround sets {_a}
+ $x->{_p} = $scale; # so correct it
+ $x;
+ }
+
+sub _scan_for_nonzero
+ {
+ my $x = shift;
+ my $pad = shift;
+ my $xs = shift;
+
+ my $len = $x->length();
+ return 0 if $len == 1; # '5' is trailed by invisible zeros
+ my $follow = $pad - 1;
+ return 0 if $follow > $len || $follow < 1;
+ #print "checking $x $r\n";
+
+ # since we do not know underlying represention of $x, use decimal string
+ #my $r = substr ($$xs,-$follow);
+ my $r = substr ("$x",-$follow);
+ return 1 if $r =~ /[^0]/; return 0;
+ }
+
+sub fround
+ {
+ # to make life easier for switch between MBF and MBI (autoload fxxx()
+ # like MBF does for bxxx()?)
+ my $x = shift;
+ return $x->bround(@_);
+ }
+
+sub bround
+ {
+ # accuracy: +$n preserve $n digits from left,
+ # -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
+ # no-op for $n == 0
+ # and overwrite the rest with 0's, return normalized number
+ # do not return $x->bnorm(), but $x
+ my $x = shift; $x = $class->new($x) unless ref $x;
+ my ($scale,$mode) = $x->_scale_a($x->accuracy(),$x->round_mode(),@_);
+ return $x if !defined $scale; # no-op
+
+ # print "MBI round: $x to $scale $mode\n";
+ return $x if $x->{sign} !~ /^[+-]$/ || $x->is_zero() || $scale == 0;
+
+ # we have fewer digits than we want to scale to
+ my $len = $x->length();
+ # print "$scale $len\n";
+ # scale < 0, but > -len (not >=!)
+ if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
+ {
+ $x->{_a} = $scale if !defined $x->{_a}; # if not yet defined overwrite
+ return $x;
+ }
+
+ # count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
+ my ($pad,$digit_round,$digit_after);
+ $pad = $len - $scale;
+ $pad = abs($scale-1) if $scale < 0;
+
+ # do not use digit(), it is costly for binary => decimal
+ #$digit_round = '0'; $digit_round = $x->digit($pad) if $pad < $len;
+ #$digit_after = '0'; $digit_after = $x->digit($pad-1) if $pad > 0;
+
+ my $xs = $CALC->_str($x->{value});
+ my $pl = -$pad-1;
+
+ # print "pad $pad pl $pl scale $scale len $len\n";
+ # pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
+ # pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
+ $digit_round = '0'; $digit_round = substr($$xs,$pl,1) if $pad <= $len;
+ $pl++; $pl ++ if $pad >= $len;
+ $digit_after = '0'; $digit_after = substr($$xs,$pl,1)
+ if $pad > 0;
+
+ # print "$pad $pl $$xs dr $digit_round da $digit_after\n";
+
+ # in case of 01234 we round down, for 6789 up, and only in case 5 we look
+ # closer at the remaining digits of the original $x, remember decision
+ my $round_up = 1; # default round up
+ $round_up -- if
+ ($mode eq 'trunc') || # trunc by round down
+ ($digit_after =~ /[01234]/) || # round down anyway,
+ # 6789 => round up
+ ($digit_after eq '5') && # not 5000...0000
+ ($x->_scan_for_nonzero($pad,$xs) == 0) &&
+ (
+ ($mode eq 'even') && ($digit_round =~ /[24680]/) ||
+ ($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
+ ($mode eq '+inf') && ($x->{sign} eq '-') ||
+ ($mode eq '-inf') && ($x->{sign} eq '+') ||
+ ($mode eq 'zero') # round down if zero, sign adjusted below
+ );
+ # allow rounding one place left of mantissa
+ #print "$pad $len $scale\n";
+ # this is triggering warnings, and buggy for $scale < 0
+ #if (-$scale != $len)
+ {
+ # old code, depend on internal representation
+ # split mantissa at $pad and then pad with zeros
+ #my $s5 = int($pad / 5);
+ #my $i = 0;
+ #while ($i < $s5)
+ # {
+ # $x->{value}->[$i++] = 0; # replace with 5 x 0
+ # }
+ #$x->{value}->[$s5] = '00000'.$x->{value}->[$s5]; # pad with 0
+ #my $rem = $pad % 5; # so much left over
+ #if ($rem > 0)
+ # {
+ # #print "remainder $rem\n";
+ ## #print "elem $x->{value}->[$s5]\n";
+ # substr($x->{value}->[$s5],-$rem,$rem) = '0' x $rem; # stamp w/ '0'
+ # }
+ #$x->{value}->[$s5] = int ($x->{value}->[$s5]); # str '05' => int '5'
+ #print ${$CALC->_str($pad->{value})}," $len\n";
+ if (($pad > 0) && ($pad <= $len))
+ {
+ substr($$xs,-$pad,$pad) = '0' x $pad;
+ $x->{value} = $CALC->_new($xs); # put back in
+ }
+ elsif ($pad > $len)
+ {
+ $x->bzero(); # round to '0'
+ }
+ # print "res $pad $len $x $$xs\n";
+ }
+ # move this later on after the inc of the string
+ #$x->{value} = $CALC->_new($xs); # put back in
+ if ($round_up) # what gave test above?
+ {
+ #print " $pad => ";
+ $pad = $len if $scale < 0; # tlr: whack 0.51=>1.0
+ # modify $x in place, undef, undef to avoid rounding
+ # str creation much faster than 10 ** something
+ #print " $pad, $x => ";
+ $x->badd( Math::BigInt->new($x->{sign}.'1'.'0'x$pad) );
+ #print "$x\n";
+ # increment string in place, to avoid dec=>hex for the '1000...000'
+ # $xs ...blah foo
+ }
+ # to here:
+ #$x->{value} = $CALC->_new($xs); # put back in
+
+ $x->{_a} = $scale if $scale >= 0;
+ if ($scale < 0)
+ {
+ $x->{_a} = $len+$scale;
+ $x->{_a} = 0 if $scale < -$len;
+ }
+ $x;
+ }
+
+sub bfloor
+ {
+ # return integer less or equal then number, since it is already integer,
+ # always returns $self
+ my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+
+ # not needed: return $x if $x->modify('bfloor');
+ return $x->round($a,$p,$r);
+ }
+
+sub bceil
+ {
+ # return integer greater or equal then number, since it is already integer,
+ # always returns $self
+ my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+
+ # not needed: return $x if $x->modify('bceil');
+ return $x->round($a,$p,$r);
+ }
+
+##############################################################################
+# private stuff (internal use only)
+
+sub __one
+ {
+ # internal speedup, set argument to 1, or create a +/- 1
+ my $self = shift;
+ my $x = $self->bone(); # $x->{value} = $CALC->_one();
+ $x->{sign} = shift || '+';
+ return $x;
+ }
+
+sub _swap
+ {
+ # Overload will swap params if first one is no object ref so that the first
+ # one is always an object ref. In this case, third param is true.
+ # This routine is to overcome the effect of scalar,$object creating an object
+ # of the class of this package, instead of the second param $object. This
+ # happens inside overload, when the overload section of this package is
+ # inherited by sub classes.
+ # For overload cases (and this is used only there), we need to preserve the
+ # args, hence the copy().
+ # You can override this method in a subclass, the overload section will call
+ # $object->_swap() to make sure it arrives at the proper subclass, with some
+ # exceptions like '+' and '-'.
+
+ # object, (object|scalar) => preserve first and make copy
+ # scalar, object => swapped, re-swap and create new from first
+ # (using class of second object, not $class!!)
+ my $self = shift; # for override in subclass
+ #print "swap $self 0:$_[0] 1:$_[1] 2:$_[2]\n";
+ if ($_[2])
+ {
+ my $c = ref ($_[0]) || $class; # fallback $class should not happen
+ return ( $c->new($_[1]), $_[0] );
+ }
+ return ( $_[0]->copy(), $_[1] );
+ }
+
+sub objectify
+ {
+ # check for strings, if yes, return objects instead
+
+ # the first argument is number of args objectify() should look at it will
+ # return $count+1 elements, the first will be a classname. This is because
+ # overloaded '""' calls bstr($object,undef,undef) and this would result in
+ # useless objects beeing created and thrown away. So we cannot simple loop
+ # over @_. If the given count is 0, all arguments will be used.
+
+ # If the second arg is a ref, use it as class.
+ # If not, try to use it as classname, unless undef, then use $class
+ # (aka Math::BigInt). The latter shouldn't happen,though.
+
+ # caller: gives us:
+ # $x->badd(1); => ref x, scalar y
+ # Class->badd(1,2); => classname x (scalar), scalar x, scalar y
+ # Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
+ # Math::BigInt::badd(1,2); => scalar x, scalar y
+ # In the last case we check number of arguments to turn it silently into
+ # $class,1,2. (We can not take '1' as class ;o)
+ # badd($class,1) is not supported (it should, eventually, try to add undef)
+ # currently it tries 'Math::BigInt' + 1, which will not work.
+
+ # some shortcut for the common cases
+
+ # $x->unary_op();
+ return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
+ # $x->binary_op($y);
+ #return (ref($_[1]),$_[1],$_[2]) if (@_ == 3) && ($_[0]||0 == 2)
+ # && ref($_[1]) && ref($_[2]);
+
+# print "obj '",join ("' '", @_),"'\n";
+
+ my $count = abs(shift || 0);
+
+# print "MBI ",caller(),"\n";
+
+ my @a; # resulting array
+ if (ref $_[0])
+ {
+ # okay, got object as first
+ $a[0] = ref $_[0];
+ }
+ else
+ {
+ # nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
+ $a[0] = $class;
+ #print "@_\n"; sleep(1);
+ $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
+ }
+ #print caller(),"\n";
+ # print "Now in objectify, my class is today $a[0]\n";
+ my $k;
+ if ($count == 0)
+ {
+ while (@_)
+ {
+ $k = shift;
+ if (!ref($k))
+ {
+ $k = $a[0]->new($k);
+ }
+ elsif (ref($k) ne $a[0])
+ {
+ # foreign object, try to convert to integer
+ $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
+ }
+ push @a,$k;
+ }
+ }
+ else
+ {
+ while ($count > 0)
+ {
+ #print "$count\n";
+ $count--;
+ $k = shift;
+# print "$k (",ref($k),") => \n";
+ if (!ref($k))
+ {
+ $k = $a[0]->new($k);
+ }
+ elsif (ref($k) ne $a[0])
+ {
+ # foreign object, try to convert to integer
+ $k->can('as_number') ? $k = $k->as_number() : $k = $a[0]->new($k);
+ }
+ # print "$k (",ref($k),")\n";
+ push @a,$k;
+ }
+ push @a,@_; # return other params, too
+ }
+ #my $i = 0;
+ #foreach (@a)
+ # {
+ # print "o $i $a[0]\n" if $i == 0;
+ # print "o $i ",ref($_),"\n" if $i != 0; $i++;
+ # }
+ #print "objectify done: would return ",scalar @a," values\n";
+ #print caller(1),"\n" unless wantarray;
+ die "$class objectify needs list context" unless wantarray;
+ @a;
+ }
+
+sub import
+ {
+ my $self = shift;
+ #print "import $self @_\n";
+ my @a = @_; my $l = scalar @_; my $j = 0;
+ for ( my $i = 0; $i < $l ; $i++,$j++ )
+ {
+ if ($_[$i] eq ':constant')
+ {
+ # this causes overlord er load to step in
+ overload::constant integer => sub { $self->new(shift) };
+ splice @a, $j, 1; $j --;
+ }
+ elsif ($_[$i] =~ /^lib$/i)
+ {
+ # this causes a different low lib to take care...
+ $CALC = $_[$i+1] || $CALC;
+ my $s = 2; $s = 1 if @a-$j < 2; # avoid "can not modify non-existant..."
+ splice @a, $j, $s; $j -= $s;
+ }
+ }
+ # any non :constant stuff is handled by our parent, Exporter
+ # even if @_ is empty, to give it a chance
+ $self->SUPER::import(@a); # need it for subclasses
+ $self->export_to_level(1,$self,@a); # need it for MBF
+
+ # try to load core math lib
+ my @c = split /\s*,\s*/,$CALC;
+ push @c,'Calc'; # if all fail, try this
+ foreach my $lib (@c)
+ {
+ $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
+ $lib =~ s/\.pm$//;
+ if ($] < 5.6)
+ {
+ # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
+ # used in the same script, or eval inside import().
+ (my $mod = $lib . '.pm') =~ s!::!/!g;
+ # require does not automatically :: => /, so portability problems arise
+ eval { require $mod; $lib->import( @c ); }
+ }
+ else
+ {
+ eval "use $lib @c;";
+ }
+ $CALC = $lib, last if $@ eq ''; # no error in loading lib?
+ }
+ }
+
+sub __from_hex
+ {
+ # convert a (ref to) big hex string to BigInt, return undef for error
+ my $hs = shift;
+
+ my $x = Math::BigInt->bzero();
+ return $x->bnan() if $$hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
+
+ my $sign = '+'; $sign = '-' if ($$hs =~ /^-/);
+
+ $$hs =~ s/^[+-]//; # strip sign
+ if ($CALC->can('_from_hex'))
+ {
+ $x->{value} = $CALC->_from_hex($hs);
+ }
+ else
+ {
+ # fallback to pure perl
+ my $mul = Math::BigInt->bzero(); $mul++;
+ my $x65536 = Math::BigInt->new(65536);
+ my $len = CORE::length($$hs)-2;
+ $len = int($len/4); # 4-digit parts, w/o '0x'
+ my $val; my $i = -4;
+ while ($len >= 0)
+ {
+ $val = substr($$hs,$i,4);
+ $val =~ s/^[+-]?0x// if $len == 0; # for last part only because
+ $val = hex($val); # hex does not like wrong chars
+ # print "$val ",substr($$hs,$i,4),"\n";
+ $i -= 4; $len --;
+ $x += $mul * $val if $val != 0;
+ $mul *= $x65536 if $len >= 0; # skip last mul
+ }
+ }
+ $x->{sign} = $sign if !$x->is_zero(); # no '-0'
+ return $x;
+ }
+
+sub __from_bin
+ {
+ # convert a (ref to) big binary string to BigInt, return undef for error
+ my $bs = shift;
+
+ my $x = Math::BigInt->bzero();
+ return $x->bnan() if $$bs !~ /^[+-]?0b[01]+$/;
+
+ my $mul = Math::BigInt->bzero(); $mul++;
+ my $x256 = Math::BigInt->new(256);
+
+ my $sign = '+'; $sign = '-' if ($$bs =~ /^\-/);
+ $$bs =~ s/^[+-]//; # strip sign
+ if ($CALC->can('_from_bin'))
+ {
+ $x->{value} = $CALC->_from_bin($bs);
+ }
+ else
+ {
+ my $len = CORE::length($$bs)-2;
+ $len = int($len/8); # 8-digit parts, w/o '0b'
+ my $val; my $i = -8;
+ while ($len >= 0)
+ {
+ $val = substr($$bs,$i,8);
+ $val =~ s/^[+-]?0b// if $len == 0; # for last part only
+ #$val = oct('0b'.$val); # does not work on Perl prior to 5.6.0
+ $val = ('0' x (8-CORE::length($val))).$val if CORE::length($val) < 8;
+ $val = ord(pack('B8',$val));
+ # print "$val ",substr($$bs,$i,16),"\n";
+ $i -= 8; $len --;
+ $x += $mul * $val if $val != 0;
+ $mul *= $x256 if $len >= 0; # skip last mul
+ }
+ }
+ $x->{sign} = $sign if !$x->is_zero();
+ return $x;
+ }
+
+sub _split
+ {
+ # (ref to num_str) return num_str
+ # internal, take apart a string and return the pieces
+ # strip leading/trailing whitespace, leading zeros, underscore and reject
+ # invalid input
+ my $x = shift;
+
+ # strip white space at front, also extranous leading zeros
+ $$x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
+ $$x =~ s/^\s+//; # but this will
+ $$x =~ s/\s+$//g; # strip white space at end
+
+ # shortcut, if nothing to split, return early
+ if ($$x =~ /^[+-]?\d+$/)
+ {
+ $$x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
+ return (\$sign, $x, \'', \'', \0);
+ }
+
+ # invalid starting char?
+ return if $$x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
+
+ $$x =~ s/(\d)_(\d)/$1$2/g; # strip underscores between digits
+ $$x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3
+
+ return __from_hex($x) if $$x =~ /^[\-\+]?0x/; # hex string
+ return __from_bin($x) if $$x =~ /^[\-\+]?0b/; # binary string
+
+ # some possible inputs:
+ # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
+ # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2
+
+ return if $$x =~ /[Ee].*[Ee]/; # more than one E => error
+
+ my ($m,$e) = split /[Ee]/,$$x;
+ $e = '0' if !defined $e || $e eq "";
+ # print "m '$m' e '$e'\n";
+ # sign,value for exponent,mantint,mantfrac
+ my ($es,$ev,$mis,$miv,$mfv);
+ # valid exponent?
+ if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
+ {
+ $es = $1; $ev = $2;
+ #print "'$m' '$e' e: $es $ev ";
+ # valid mantissa?
+ return if $m eq '.' || $m eq '';
+ my ($mi,$mf) = split /\./,$m;
+ $mi = '0' if !defined $mi;
+ $mi .= '0' if $mi =~ /^[\-\+]?$/;
+ $mf = '0' if !defined $mf || $mf eq '';
+ if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
+ {
+ $mis = $1||'+'; $miv = $2;
+ # print "$mis $miv";
+ # valid, existing fraction part of mantissa?
+ return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
+ $mfv = $1;
+ #print " split: $mis $miv . $mfv E $es $ev\n";
+ return (\$mis,\$miv,\$mfv,\$es,\$ev);
+ }
+ }
+ return; # NaN, not a number
+ }
+
+sub as_number
+ {
+ # an object might be asked to return itself as bigint on certain overloaded
+ # operations, this does exactly this, so that sub classes can simple inherit
+ # it or override with their own integer conversion routine
+ my $self = shift;
+
+ $self->copy();
+ }
+
+sub as_hex
+ {
+ # return as hex string, with prefixed 0x
+ my $x = shift; $x = $class->new($x) if !ref($x);
+
+ return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
+ return '0x0' if $x->is_zero();
-use overload
-'+' => sub {new Math::BigInt &badd},
-'-' => sub {new Math::BigInt
- $_[2]? bsub($_[1],${$_[0]}) : bsub(${$_[0]},$_[1])},
-'<=>' => sub {new Math::BigInt
- $_[2]? bcmp($_[1],${$_[0]}) : bcmp(${$_[0]},$_[1])},
-'cmp' => sub {new Math::BigInt
- $_[2]? ($_[1] cmp ${$_[0]}) : (${$_[0]} cmp $_[1])},
-'*' => sub {new Math::BigInt &bmul},
-'/' => sub {new Math::BigInt
- $_[2]? scalar bdiv($_[1],${$_[0]}) :
- scalar bdiv(${$_[0]},$_[1])},
-'%' => sub {new Math::BigInt
- $_[2]? bmod($_[1],${$_[0]}) : bmod(${$_[0]},$_[1])},
-'**' => sub {new Math::BigInt
- $_[2]? bpow($_[1],${$_[0]}) : bpow(${$_[0]},$_[1])},
-'neg' => sub {new Math::BigInt &bneg},
-'abs' => sub {new Math::BigInt &babs},
-
-qw(
-"" stringify
-0+ numify) # Order of arguments unsignificant
-;
+ my $es = ''; my $s = '';
+ $s = $x->{sign} if $x->{sign} eq '-';
+ if ($CALC->can('_as_hex'))
+ {
+ $es = ${$CALC->_as_hex($x->{value})};
+ }
+ else
+ {
+ my $x1 = $x->copy()->babs(); my $xr;
+ my $x100 = Math::BigInt->new (0x100);
+ while (!$x1->is_zero())
+ {
+ ($x1, $xr) = bdiv($x1,$x100);
+ $es .= unpack('h2',pack('C',$xr->numify()));
+ }
+ $es = reverse $es;
+ $es =~ s/^[0]+//; # strip leading zeros
+ $s .= '0x';
+ }
+ $s . $es;
+ }
-$NaNOK=1;
-
-sub new {
- my($class) = shift;
- my($foo) = bnorm(shift);
- die "Not a number initialized to Math::BigInt" if !$NaNOK && $foo eq "NaN";
- bless \$foo, $class;
-}
-sub stringify { "${$_[0]}" }
-sub numify { 0 + "${$_[0]}" } # Not needed, additional overhead
- # comparing to direct compilation based on
- # stringify
-sub import {
- shift;
- return unless @_;
- die "unknown import: @_" unless @_ == 1 and $_[0] eq ':constant';
- overload::constant integer => sub {Math::BigInt->new(shift)};
-}
-
-$zero = 0;
-
-
-# normalize string form of number. Strip leading zeros. Strip any
-# white space and add a sign, if missing.
-# Strings that are not numbers result the value 'NaN'.
-
-sub bnorm { #(num_str) return num_str
- local($_) = @_;
- s/\s+//g; # strip white space
- if (s/^([+-]?)0*(\d+)$/$1$2/) { # test if number
- substr($_,$[,0) = '+' unless $1; # Add missing sign
- s/^-0/+0/;
- $_;
- } else {
- 'NaN';
- }
-}
-
-# Convert a number from string format to internal base 100000 format.
-# Assumes normalized value as input.
-sub internal { #(num_str) return int_num_array
- local($d) = @_;
- ($is,$il) = (substr($d,$[,1),length($d)-2);
- substr($d,$[,1) = '';
- ($is, reverse(unpack("a" . ($il%5+1) . ("a5" x ($il/5)), $d)));
-}
-
-# Convert a number from internal base 100000 format to string format.
-# This routine scribbles all over input array.
-sub external { #(int_num_array) return num_str
- $es = shift;
- grep($_ > 9999 || ($_ = substr('0000'.$_,-5)), @_); # zero pad
- &bnorm(join('', $es, reverse(@_))); # reverse concat and normalize
-}
-
-# Negate input value.
-sub bneg { #(num_str) return num_str
- local($_) = &bnorm(@_);
- return $_ if $_ eq '+0' or $_ eq 'NaN';
- vec($_,0,8) ^= ord('+') ^ ord('-');
- $_;
-}
-
-# Returns the absolute value of the input.
-sub babs { #(num_str) return num_str
- &abs(&bnorm(@_));
-}
-
-sub abs { # post-normalized abs for internal use
- local($_) = @_;
- s/^-/+/;
- $_;
-}
-
-# Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
-sub bcmp { #(num_str, num_str) return cond_code
- local($x,$y) = (&bnorm($_[$[]),&bnorm($_[$[+1]));
- if ($x eq 'NaN') {
- undef;
- } elsif ($y eq 'NaN') {
- undef;
- } else {
- &cmp($x,$y) <=> 0;
- }
-}
-
-sub cmp { # post-normalized compare for internal use
- local($cx, $cy) = @_;
-
- return 0 if ($cx eq $cy);
-
- local($sx, $sy) = (substr($cx, 0, 1), substr($cy, 0, 1));
- local($ld);
-
- if ($sx eq '+') {
- return 1 if ($sy eq '-' || $cy eq '+0');
- $ld = length($cx) - length($cy);
- return $ld if ($ld);
- return $cx cmp $cy;
- } else { # $sx eq '-'
- return -1 if ($sy eq '+');
- $ld = length($cy) - length($cx);
- return $ld if ($ld);
- return $cy cmp $cx;
- }
-}
-
-sub badd { #(num_str, num_str) return num_str
- local(*x, *y); ($x, $y) = (&bnorm($_[$[]),&bnorm($_[$[+1]));
- if ($x eq 'NaN') {
- 'NaN';
- } elsif ($y eq 'NaN') {
- 'NaN';
- } else {
- @x = &internal($x); # convert to internal form
- @y = &internal($y);
- local($sx, $sy) = (shift @x, shift @y); # get signs
- if ($sx eq $sy) {
- &external($sx, &add(*x, *y)); # if same sign add
- } else {
- ($x, $y) = (&abs($x),&abs($y)); # make abs
- if (&cmp($y,$x) > 0) {
- &external($sy, &sub(*y, *x));
- } else {
- &external($sx, &sub(*x, *y));
- }
- }
+sub as_bin
+ {
+ # return as binary string, with prefixed 0b
+ my $x = shift; $x = $class->new($x) if !ref($x);
+
+ return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
+ return '0b0' if $x->is_zero();
+
+ my $es = ''; my $s = '';
+ $s = $x->{sign} if $x->{sign} eq '-';
+ if ($CALC->can('_as_bin'))
+ {
+ $es = ${$CALC->_as_bin($x->{value})};
}
-}
-
-sub bsub { #(num_str, num_str) return num_str
- &badd($_[$[],&bneg($_[$[+1]));
-}
-
-# GCD -- Euclids algorithm Knuth Vol 2 pg 296
-sub bgcd { #(num_str, num_str) return num_str
- local($x,$y) = (&bnorm($_[$[]),&bnorm($_[$[+1]));
- if ($x eq 'NaN' || $y eq 'NaN') {
- 'NaN';
- } else {
- ($x, $y) = ($y,&bmod($x,$y)) while $y ne '+0';
- $x;
- }
-}
-
-# routine to add two base 1e5 numbers
-# stolen from Knuth Vol 2 Algorithm A pg 231
-# there are separate routines to add and sub as per Kunth pg 233
-sub add { #(int_num_array, int_num_array) return int_num_array
- local(*x, *y) = @_;
- $car = 0;
- for $x (@x) {
- last unless @y || $car;
- $x -= 1e5 if $car = (($x += (@y ? shift(@y) : 0) + $car) >= 1e5) ? 1 : 0;
- }
- for $y (@y) {
- last unless $car;
- $y -= 1e5 if $car = (($y += $car) >= 1e5) ? 1 : 0;
- }
- (@x, @y, $car);
-}
-
-# subtract base 1e5 numbers -- stolen from Knuth Vol 2 pg 232, $x > $y
-sub sub { #(int_num_array, int_num_array) return int_num_array
- local(*sx, *sy) = @_;
- $bar = 0;
- for $sx (@sx) {
- last unless @sy || $bar;
- $sx += 1e5 if $bar = (($sx -= (@sy ? shift(@sy) : 0) + $bar) < 0);
- }
- @sx;
-}
-
-# multiply two numbers -- stolen from Knuth Vol 2 pg 233
-sub bmul { #(num_str, num_str) return num_str
- local(*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1]));
- if ($x eq 'NaN') {
- 'NaN';
- } elsif ($y eq 'NaN') {
- 'NaN';
- } else {
- @x = &internal($x);
- @y = &internal($y);
- &external(&mul(*x,*y));
- }
-}
-
-# multiply two numbers in internal representation
-# destroys the arguments, supposes that two arguments are different
-sub mul { #(*int_num_array, *int_num_array) return int_num_array
- local(*x, *y) = (shift, shift);
- local($signr) = (shift @x ne shift @y) ? '-' : '+';
- @prod = ();
- for $x (@x) {
- ($car, $cty) = (0, $[);
- for $y (@y) {
- $prod = $x * $y + ($prod[$cty] || 0) + $car;
- $prod[$cty++] =
- $prod - ($car = int($prod * 1e-5)) * 1e5;
- }
- $prod[$cty] += $car if $car;
- $x = shift @prod;
- }
- ($signr, @x, @prod);
-}
-
-# modulus
-sub bmod { #(num_str, num_str) return num_str
- (&bdiv(@_))[$[+1];
-}
-
-sub bdiv { #(dividend: num_str, divisor: num_str) return num_str
- local (*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1]));
- return wantarray ? ('NaN','NaN') : 'NaN'
- if ($x eq 'NaN' || $y eq 'NaN' || $y eq '+0');
- return wantarray ? ('+0',$x) : '+0' if (&cmp(&abs($x),&abs($y)) < 0);
- @x = &internal($x); @y = &internal($y);
- $srem = $y[$[];
- $sr = (shift @x ne shift @y) ? '-' : '+';
- $car = $bar = $prd = 0;
- if (($dd = int(1e5/($y[$#y]+1))) != 1) {
- for $x (@x) {
- $x = $x * $dd + $car;
- $x -= ($car = int($x * 1e-5)) * 1e5;
- }
- push(@x, $car); $car = 0;
- for $y (@y) {
- $y = $y * $dd + $car;
- $y -= ($car = int($y * 1e-5)) * 1e5;
- }
+ else
+ {
+ my $x1 = $x->copy()->babs(); my $xr;
+ my $x100 = Math::BigInt->new (0x100);
+ while (!$x1->is_zero())
+ {
+ ($x1, $xr) = bdiv($x1,$x100);
+ $es .= unpack('b8',pack('C',$xr->numify()));
+ }
+ $es = reverse $es;
+ $es =~ s/^[0]+//; # strip leading zeros
+ $s .= '0b';
}
- else {
- push(@x, 0);
- }
- @q = (); ($v2,$v1) = ($y[-2] || 0, $y[-1]);
- while ($#x > $#y) {
- ($u2,$u1,$u0) = ($x[-3] || 0, $x[-2] || 0, $x[-1]);
- $q = (($u0 == $v1) ? 99999 : int(($u0*1e5+$u1)/$v1));
- --$q while ($v2*$q > ($u0*1e5+$u1-$q*$v1)*1e5+$u2);
- if ($q) {
- ($car, $bar) = (0,0);
- for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) {
- $prd = $q * $y[$y] + $car;
- $prd -= ($car = int($prd * 1e-5)) * 1e5;
- $x[$x] += 1e5 if ($bar = (($x[$x] -= $prd + $bar) < 0));
- }
- if ($x[$#x] < $car + $bar) {
- $car = 0; --$q;
- for ($y = $[, $x = $#x-$#y+$[-1; $y <= $#y; ++$y,++$x) {
- $x[$x] -= 1e5
- if ($car = (($x[$x] += $y[$y] + $car) > 1e5));
- }
- }
- }
- pop(@x); unshift(@q, $q);
- }
- if (wantarray) {
- @d = ();
- if ($dd != 1) {
- $car = 0;
- for $x (reverse @x) {
- $prd = $car * 1e5 + $x;
- $car = $prd - ($tmp = int($prd / $dd)) * $dd;
- unshift(@d, $tmp);
- }
- }
- else {
- @d = @x;
- }
- (&external($sr, @q), &external($srem, @d, $zero));
- } else {
- &external($sr, @q);
- }
-}
-
-# compute power of two numbers -- stolen from Knuth Vol 2 pg 233
-sub bpow { #(num_str, num_str) return num_str
- local(*x, *y); ($x, $y) = (&bnorm($_[$[]), &bnorm($_[$[+1]));
- if ($x eq 'NaN') {
- 'NaN';
- } elsif ($y eq 'NaN') {
- 'NaN';
- } elsif ($x eq '+1') {
- '+1';
- } elsif ($x eq '-1') {
- &bmod($x,2) ? '-1': '+1';
- } elsif ($y =~ /^-/) {
- 'NaN';
- } elsif ($x eq '+0' && $y eq '+0') {
- 'NaN';
- } else {
- @x = &internal($x);
- local(@pow2)=@x;
- local(@pow)=&internal("+1");
- local($y1,$res,@tmp1,@tmp2)=(1); # need tmp to send to mul
- while ($y ne '+0') {
- ($y,$res)=&bdiv($y,2);
- if ($res ne '+0') {@tmp=@pow2; @pow=&mul(*pow,*tmp);}
- if ($y ne '+0') {@tmp=@pow2;@pow2=&mul(*pow2,*tmp);}
- }
- &external(@pow);
+ $s . $es;
+ }
+
+##############################################################################
+# internal calculation routines (others are in Math::BigInt::Calc etc)
+
+sub __lcm
+ {
+ # (BINT or num_str, BINT or num_str) return BINT
+ # does modify first argument
+ # LCM
+
+ my $x = shift; my $ty = shift;
+ return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
+ return $x * $ty / bgcd($x,$ty);
+ }
+
+sub __gcd
+ {
+ # (BINT or num_str, BINT or num_str) return BINT
+ # does modify both arguments
+ # GCD -- Euclids algorithm E, Knuth Vol 2 pg 296
+ my ($x,$ty) = @_;
+
+ return $x->bnan() if $x->{sign} !~ /^[+-]$/ || $ty->{sign} !~ /^[+-]$/;
+
+ while (!$ty->is_zero())
+ {
+ ($x, $ty) = ($ty,bmod($x,$ty));
}
-}
+ $x;
+ }
+
+###############################################################################
+# this method return 0 if the object can be modified, or 1 for not
+# We use a fast use constant statement here, to avoid costly calls. Subclasses
+# may override it with special code (f.i. Math::BigInt::Constant does so)
+
+sub modify () { 0; }
1;
__END__
=head1 SYNOPSIS
use Math::BigInt;
- $i = Math::BigInt->new($string);
-
- $i->bneg return BINT negation
- $i->babs return BINT absolute value
- $i->bcmp(BINT) return CODE compare numbers (undef,<0,=0,>0)
- $i->badd(BINT) return BINT addition
- $i->bsub(BINT) return BINT subtraction
- $i->bmul(BINT) return BINT multiplication
- $i->bdiv(BINT) return (BINT,BINT) division (quo,rem) just quo if scalar
- $i->bmod(BINT) return BINT modulus
- $i->bgcd(BINT) return BINT greatest common divisor
- $i->bnorm return BINT normalization
+
+ # Number creation
+ $x = Math::BigInt->new($str); # defaults to 0
+ $nan = Math::BigInt->bnan(); # create a NotANumber
+ $zero = Math::BigInt->bzero(); # create a +0
+ $inf = Math::BigInt->binf(); # create a +inf
+ $inf = Math::BigInt->binf('-'); # create a -inf
+ $one = Math::BigInt->bone(); # create a +1
+ $one = Math::BigInt->bone('-'); # create a -1
+
+ # Testing
+ $x->is_zero(); # true if arg is +0
+ $x->is_nan(); # true if arg is NaN
+ $x->is_one(); # true if arg is +1
+ $x->is_one('-'); # true if arg is -1
+ $x->is_odd(); # true if odd, false for even
+ $x->is_even(); # true if even, false for odd
+ $x->is_positive(); # true if >= 0
+ $x->is_negative(); # true if < 0
+ $x->is_inf(sign); # true if +inf, or -inf (sign is default '+')
+
+ $x->bcmp($y); # compare numbers (undef,<0,=0,>0)
+ $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
+ $x->sign(); # return the sign, either +,- or NaN
+ $x->digit($n); # return the nth digit, counting from right
+ $x->digit(-$n); # return the nth digit, counting from left
+
+ # The following all modify their first argument:
+
+ # set
+ $x->bzero(); # set $x to 0
+ $x->bnan(); # set $x to NaN
+ $x->bone(); # set $x to +1
+ $x->bone('-'); # set $x to -1
+
+ $x->bneg(); # negation
+ $x->babs(); # absolute value
+ $x->bnorm(); # normalize (no-op)
+ $x->bnot(); # two's complement (bit wise not)
+ $x->binc(); # increment x by 1
+ $x->bdec(); # decrement x by 1
+
+ $x->badd($y); # addition (add $y to $x)
+ $x->bsub($y); # subtraction (subtract $y from $x)
+ $x->bmul($y); # multiplication (multiply $x by $y)
+ $x->bdiv($y); # divide, set $x to quotient
+ # return (quo,rem) or quo if scalar
+
+ $x->bmod($y); # modulus (x % y)
+ $x->bpow($y); # power of arguments (x ** y)
+ $x->blsft($y); # left shift
+ $x->brsft($y); # right shift
+ $x->blsft($y,$n); # left shift, by base $n (like 10)
+ $x->brsft($y,$n); # right shift, by base $n (like 10)
+
+ $x->band($y); # bitwise and
+ $x->bior($y); # bitwise inclusive or
+ $x->bxor($y); # bitwise exclusive or
+ $x->bnot(); # bitwise not (two's complement)
+
+ $x->bsqrt(); # calculate square-root
+
+ $x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r
+ $x->bround($N); # accuracy: preserve $N digits
+ $x->bfround($N); # round to $Nth digit, no-op for BigInts
+
+ # The following do not modify their arguments in BigInt, but do in BigFloat:
+ $x->bfloor(); # return integer less or equal than $x
+ $x->bceil(); # return integer greater or equal than $x
+
+ # The following do not modify their arguments:
+
+ bgcd(@values); # greatest common divisor (no OO style)
+ blcm(@values); # lowest common multiplicator (no OO style)
+
+ $x->length(); # return number of digits in number
+ ($x,$f) = $x->length(); # length of number and length of fraction part,
+ # latter is always 0 digits long for BigInt's
+
+ $x->exponent(); # return exponent as BigInt
+ $x->mantissa(); # return (signed) mantissa as BigInt
+ $x->parts(); # return (mantissa,exponent) as BigInt
+ $x->copy(); # make a true copy of $x (unlike $y = $x;)
+ $x->as_number(); # return as BigInt (in BigInt: same as copy())
+
+ # conversation to string
+ $x->bstr(); # normalized string
+ $x->bsstr(); # normalized string in scientific notation
+ $x->as_hex(); # as signed hexadecimal string with prefixed 0x
+ $x->as_bin(); # as signed binary string with prefixed 0b
=head1 DESCRIPTION
-All basic math operations are overloaded if you declare your big
-integers as
+All operators (inlcuding basic math operations) are overloaded if you
+declare your big integers as
- $i = new Math::BigInt '123 456 789 123 456 789';
+ $i = new Math::BigInt '123_456_789_123_456_789';
+Operations with overloaded operators preserve the arguments which is
+exactly what you expect.
=over 2
=item Canonical notation
-Big integer value are strings of the form C</^[+-]\d+$/> with leading
+Big integer values are strings of the form C</^[+-]\d+$/> with leading
zeros suppressed.
+ '-0' canonical value '-0', normalized '0'
+ ' -123_123_123' canonical value '-123123123'
+ '1_23_456_7890' canonical value '1234567890'
+
=item Input
-Input values to these routines may be strings of the form
-C</^\s*[+-]?[\d\s]+$/>.
+Input values to these routines may be either Math::BigInt objects or
+strings of the form C</^\s*[+-]?[\d]+\.?[\d]*E?[+-]?[\d]*$/>.
+
+You can include one underscore between any two digits.
+
+This means integer values like 1.01E2 or even 1000E-2 are also accepted.
+Non integer values result in NaN.
+
+Math::BigInt::new() defaults to 0, while Math::BigInt::new('') results
+in 'NaN'.
+
+bnorm() on a BigInt object is now effectively a no-op, since the numbers
+are always stored in normalized form. On a string, it creates a BigInt
+object.
=item Output
-Output values always always in canonical form
+Output values are BigInt objects (normalized), except for bstr(), which
+returns a string in normalized form.
+Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
+C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
+return either undef, <0, 0 or >0 and are suited for sort.
+
+=back
+
+=head1 ACCURACY and PRECISION
+
+Since version v1.33, Math::BigInt and Math::BigFloat have full support for
+accuracy and precision based rounding, both automatically after every
+operation as well as manually.
+
+This section describes the accuracy/precision handling in Math::Big* as it
+used to be and as it is now, complete with an explanation of all terms and
+abbreviations.
+
+Not yet implemented things (but with correct description) are marked with '!',
+things that need to be answered are marked with '?'.
+
+In the next paragraph follows a short description of terms used here (because
+these may differ from terms used by others people or documentation).
+
+During the rest of this document, the shortcuts A (for accuracy), P (for
+precision), F (fallback) and R (rounding mode) will be used.
+
+=head2 Precision P
+
+A fixed number of digits before (positive) or after (negative)
+the decimal point. For example, 123.45 has a precision of -2. 0 means an
+integer like 123 (or 120). A precision of 2 means two digits to the left
+of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
+numbers with zeros before the decimal point may have different precisions,
+because 1200 can have p = 0, 1 or 2 (depending on what the inital value
+was). It could also have p < 0, when the digits after the decimal point
+are zero.
+
+The string output (of floating point numbers) will be padded with zeros:
+
+ Initial value P A Result String
+ ------------------------------------------------------------
+ 1234.01 -3 1000 1000
+ 1234 -2 1200 1200
+ 1234.5 -1 1230 1230
+ 1234.001 1 1234 1234.0
+ 1234.01 0 1234 1234
+ 1234.01 2 1234.01 1234.01
+ 1234.01 5 1234.01 1234.01000
+
+For BigInts, no padding occurs.
+
+=head2 Accuracy A
+
+Number of significant digits. Leading zeros are not counted. A
+number may have an accuracy greater than the non-zero digits
+when there are zeros in it or trailing zeros. For example, 123.456 has
+A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
+
+The string output (of floating point numbers) will be padded with zeros:
+
+ Initial value P A Result String
+ ------------------------------------------------------------
+ 1234.01 3 1230 1230
+ 1234.01 6 1234.01 1234.01
+ 1234.1 8 1234.1 1234.1000
+
+For BigInts, no padding occurs.
+
+=head2 Fallback F
+
+When both A and P are undefined, this is used as a fallback accuracy when
+dividing numbers.
+
+=head2 Rounding mode R
+
+When rounding a number, different 'styles' or 'kinds'
+of rounding are possible. (Note that random rounding, as in
+Math::Round, is not implemented.)
+
+=over 2
+
+=item 'trunc'
+
+truncation invariably removes all digits following the
+rounding place, replacing them with zeros. Thus, 987.65 rounded
+to tens (P=1) becomes 980, and rounded to the fourth sigdig
+becomes 987.6 (A=4). 123.456 rounded to the second place after the
+decimal point (P=-2) becomes 123.46.
+
+All other implemented styles of rounding attempt to round to the
+"nearest digit." If the digit D immediately to the right of the
+rounding place (skipping the decimal point) is greater than 5, the
+number is incremented at the rounding place (possibly causing a
+cascade of incrementation): e.g. when rounding to units, 0.9 rounds
+to 1, and -19.9 rounds to -20. If D < 5, the number is similarly
+truncated at the rounding place: e.g. when rounding to units, 0.4
+rounds to 0, and -19.4 rounds to -19.
+
+However the results of other styles of rounding differ if the
+digit immediately to the right of the rounding place (skipping the
+decimal point) is 5 and if there are no digits, or no digits other
+than 0, after that 5. In such cases:
+
+=item 'even'
+
+rounds the digit at the rounding place to 0, 2, 4, 6, or 8
+if it is not already. E.g., when rounding to the first sigdig, 0.45
+becomes 0.4, -0.55 becomes -0.6, but 0.4501 becomes 0.5.
+
+=item 'odd'
+
+rounds the digit at the rounding place to 1, 3, 5, 7, or 9 if
+it is not already. E.g., when rounding to the first sigdig, 0.45
+becomes 0.5, -0.55 becomes -0.5, but 0.5501 becomes 0.6.
+
+=item '+inf'
+
+round to plus infinity, i.e. always round up. E.g., when
+rounding to the first sigdig, 0.45 becomes 0.5, -0.55 becomes -0.5,
+and 0.4501 also becomes 0.5.
+
+=item '-inf'
+
+round to minus infinity, i.e. always round down. E.g., when
+rounding to the first sigdig, 0.45 becomes 0.4, -0.55 becomes -0.6,
+but 0.4501 becomes 0.5.
+
+=item 'zero'
+
+round to zero, i.e. positive numbers down, negative ones up.
+E.g., when rounding to the first sigdig, 0.45 becomes 0.4, -0.55
+becomes -0.5, but 0.4501 becomes 0.5.
+
+=back
+
+The handling of A & P in MBI/MBF (the old core code shipped with Perl
+versions <= 5.7.2) is like this:
+
+=over 2
+
+=item Precision
+
+ * ffround($p) is able to round to $p number of digits after the decimal
+ point
+ * otherwise P is unused
+
+=item Accuracy (significant digits)
+
+ * fround($a) rounds to $a significant digits
+ * only fdiv() and fsqrt() take A as (optional) paramater
+ + other operations simply create the same number (fneg etc), or more (fmul)
+ of digits
+ + rounding/truncating is only done when explicitly calling one of fround
+ or ffround, and never for BigInt (not implemented)
+ * fsqrt() simply hands its accuracy argument over to fdiv.
+ * the documentation and the comment in the code indicate two different ways
+ on how fdiv() determines the maximum number of digits it should calculate,
+ and the actual code does yet another thing
+ POD:
+ max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
+ Comment:
+ result has at most max(scale, length(dividend), length(divisor)) digits
+ Actual code:
+ scale = max(scale, length(dividend)-1,length(divisor)-1);
+ scale += length(divisior) - length(dividend);
+ So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
+ Actually, the 'difference' added to the scale is calculated from the
+ number of "significant digits" in dividend and divisor, which is derived
+ by looking at the length of the mantissa. Which is wrong, since it includes
+ the + sign (oups) and actually gets 2 for '+100' and 4 for '+101'. Oups
+ again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
+ assumption that 124 has 3 significant digits, while 120/7 will get you
+ '17', not '17.1' since 120 is thought to have 2 significant digits.
+ The rounding after the division then uses the remainder and $y to determine
+ wether it must round up or down.
+ ? I have no idea which is the right way. That's why I used a slightly more
+ ? simple scheme and tweaked the few failing testcases to match it.
+
+=back
+
+This is how it works now:
+
+=over 2
+
+=item Setting/Accessing
+
+ * You can set the A global via $Math::BigInt::accuracy or
+ $Math::BigFloat::accuracy or whatever class you are using.
+ * You can also set P globally by using $Math::SomeClass::precision likewise.
+ * Globals are classwide, and not inherited by subclasses.
+ * to undefine A, use $Math::SomeCLass::accuracy = undef
+ * to undefine P, use $Math::SomeClass::precision = undef
+ * To be valid, A must be > 0, P can have any value.
+ * If P is negative, this means round to the P'th place to the right of the
+ decimal point; positive values mean to the left of the decimal point.
+ P of 0 means round to integer.
+ * to find out the current global A, take $Math::SomeClass::accuracy
+ * use $x->accuracy() for the local setting of $x.
+ * to find out the current global P, take $Math::SomeClass::precision
+ * use $x->precision() for the local setting
+
+=item Creating numbers
+
+ !* When you create a number, there should be a way to define its A & P
+ * When a number without specific A or P is created, but the globals are
+ defined, these should be used to round the number immediately and also
+ stored locally with the number. Thus changing the global defaults later on
+ will not change the A or P of previously created numbers (i.e., A and P of
+ $x will be what was in effect when $x was created)
+
+=item Usage
+
+ * If A or P are enabled/defined, they are used to round the result of each
+ operation according to the rules below
+ * Negative P is ignored in Math::BigInt, since BigInts never have digits
+ after the decimal point
+ * Math::BigFloat uses Math::BigInts internally, but setting A or P inside
+ Math::BigInt as globals should not tamper with the parts of a BigFloat.
+ Thus a flag is used to mark all Math::BigFloat numbers as 'never round'
+
+=item Precedence
+
+ * It only makes sense that a number has only one of A or P at a time.
+ Since you can set/get both A and P, there is a rule that will practically
+ enforce only A or P to be in effect at a time, even if both are set.
+ This is called precedence.
+ !* If two objects are involved in an operation, and one of them has A in
+ ! effect, and the other P, this should result in a warning or an error,
+ ! probably in NaN.
+ * A takes precendence over P (Hint: A comes before P). If A is defined, it
+ is used, otherwise P is used. If neither of them is defined, nothing is
+ used, i.e. the result will have as many digits as it can (with an
+ exception for fdiv/fsqrt) and will not be rounded.
+ * There is another setting for fdiv() (and thus for fsqrt()). If neither of
+ A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
+ If either the dividend's or the divisor's mantissa has more digits than
+ the value of F, the higher value will be used instead of F.
+ This is to limit the digits (A) of the result (just consider what would
+ happen with unlimited A and P in the case of 1/3 :-)
+ * fdiv will calculate 1 more digit than required (determined by
+ A, P or F), and, if F is not used, round the result
+ (this will still fail in the case of a result like 0.12345000000001 with A
+ or P of 5, but this can not be helped - or can it?)
+ * Thus you can have the math done by on Math::Big* class in three modes:
+ + never round (this is the default):
+ This is done by setting A and P to undef. No math operation
+ will round the result, with fdiv() and fsqrt() as exceptions to guard
+ against overflows. You must explicitely call bround(), bfround() or
+ round() (the latter with parameters).
+ Note: Once you have rounded a number, the settings will 'stick' on it
+ and 'infect' all other numbers engaged in math operations with it, since
+ local settings have the highest precedence. So, to get SaferRound[tm],
+ use a copy() before rounding like this:
+
+ $x = Math::BigFloat->new(12.34);
+ $y = Math::BigFloat->new(98.76);
+ $z = $x * $y; # 1218.6984
+ print $x->copy()->fround(3); # 12.3 (but A is now 3!)
+ $z = $x * $y; # still 1218.6984, without
+ # copy would have been 1210!
+
+ + round after each op:
+ After each single operation (except for testing like is_zero()), the
+ method round() is called and the result is rounded appropriately. By
+ setting proper values for A and P, you can have all-the-same-A or
+ all-the-same-P modes. For example, Math::Currency might set A to undef,
+ and P to -2, globally.
+
+ ?Maybe an extra option that forbids local A & P settings would be in order,
+ ?so that intermediate rounding does not 'poison' further math?
+
+=item Overriding globals
+
+ * you will be able to give A, P and R as an argument to all the calculation
+ routines; the second parameter is A, the third one is P, and the fourth is
+ R (shift place by one for binary operations like add). P is used only if
+ the first parameter (A) is undefined. These three parameters override the
+ globals in the order detailed as follows, i.e. the first defined value
+ wins:
+ (local: per object, global: global default, parameter: argument to sub)
+ + parameter A
+ + parameter P
+ + local A (if defined on both of the operands: smaller one is taken)
+ + local P (if defined on both of the operands: smaller one is taken)
+ + global A
+ + global P
+ + global F
+ * fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
+ arguments (A and P) instead of one
+
+=item Local settings
+
+ * You can set A and P locally by using $x->accuracy() and $x->precision()
+ and thus force different A and P for different objects/numbers.
+ * Setting A or P this way immediately rounds $x to the new value.
+
+=item Rounding
+
+ * the rounding routines will use the respective global or local settings.
+ fround()/bround() is for accuracy rounding, while ffround()/bfround()
+ is for precision
+ * the two rounding functions take as the second parameter one of the
+ following rounding modes (R):
+ 'even', 'odd', '+inf', '-inf', 'zero', 'trunc'
+ * you can set and get the global R by using Math::SomeClass->round_mode()
+ or by setting $Math::SomeClass::round_mode
+ * after each operation, $result->round() is called, and the result may
+ eventually be rounded (that is, if A or P were set either locally,
+ globally or as parameter to the operation)
+ * to manually round a number, call $x->round($A,$P,$round_mode);
+ this will round the number by using the appropriate rounding function
+ and then normalize it.
+ * rounding modifies the local settings of the number:
+
+ $x = Math::BigFloat->new(123.456);
+ $x->accuracy(5);
+ $x->bround(4);
+
+ Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
+ will be 4 from now on.
+
+=item Default values
+
+ * R: 'even'
+ * F: 40
+ * A: undef
+ * P: undef
+
+=item Remarks
+
+ * The defaults are set up so that the new code gives the same results as
+ the old code (except in a few cases on fdiv):
+ + Both A and P are undefined and thus will not be used for rounding
+ after each operation.
+ + round() is thus a no-op, unless given extra parameters A and P
=back
-Actual math is done in an internal format consisting of an array
-whose first element is the sign (/^[+-]$/) and whose remaining
-elements are base 100000 digits with the least significant digit first.
-The string 'NaN' is used to represent the result when input arguments
-are not numbers, as well as the result of dividing by zero.
+=head1 INTERNALS
+
+The actual numbers are stored as unsigned big integers (with seperate sign).
+You should neither care about nor depend on the internal representation; it
+might change without notice. Use only method calls like C<< $x->sign(); >>
+instead relying on the internal hash keys like in C<< $x->{sign}; >>.
+
+=head2 MATH LIBRARY
+
+Math with the numbers is done (by default) by a module called
+Math::BigInt::Calc. This is equivalent to saying:
+
+ use Math::BigInt lib => 'Calc';
+
+You can change this by using:
+
+ use Math::BigInt lib => 'BitVect';
+
+The following would first try to find Math::BigInt::Foo, then
+Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
+
+ use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
+
+Calc.pm uses as internal format an array of elements of some decimal base
+(usually 1e5, but this might change to 1e7) with the least significant digit
+first, while BitVect.pm uses a bit vector of base 2, most significant bit
+first. Other modules might use even different means of representing the
+numbers. See the respective module documentation for further details.
+
+=head2 SIGN
+
+The sign is either '+', '-', 'NaN', '+inf' or '-inf' and stored seperately.
+
+A sign of 'NaN' is used to represent the result when input arguments are not
+numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
+minus infinity. You will get '+inf' when dividing a positive number by 0, and
+'-inf' when dividing any negative number by 0.
+
+=head2 mantissa(), exponent() and parts()
+
+C<mantissa()> and C<exponent()> return the said parts of the BigInt such
+that:
+
+ $m = $x->mantissa();
+ $e = $x->exponent();
+ $y = $m * ( 10 ** $e );
+ print "ok\n" if $x == $y;
+
+C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
+in one go. Both the returned mantissa and exponent have a sign.
+
+Currently, for BigInts C<$e> will be always 0, except for NaN, +inf and -inf,
+where it will be NaN; and for $x == 0, where it will be 1
+(to be compatible with Math::BigFloat's internal representation of a zero as
+C<0E1>).
+
+C<$m> will always be a copy of the original number. The relation between $e
+and $m might change in the future, but will always be equivalent in a
+numerical sense, e.g. $m might get minimized.
=head1 EXAMPLES
+
+ use Math::BigInt qw(bstr);
+
+ sub bint { Math::BigInt->new(shift); }
+
+ $x = bstr("1234") # string "1234"
+ $x = "$x"; # same as bstr()
+ $x = bneg("1234") # Bigint "-1234"
+ $x = Math::BigInt->bneg("1234"); # Bigint "-1234"
+ $x = Math::BigInt->babs("-12345"); # Bigint "12345"
+ $x = Math::BigInt->bnorm("-0 00"); # BigInt "0"
+ $x = bint(1) + bint(2); # BigInt "3"
+ $x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
+ $x = bint(1); # BigInt "1"
+ $x = $x + 5 / 2; # BigInt "3"
+ $x = $x ** 3; # BigInt "27"
+ $x *= 2; # BigInt "54"
+ $x = new Math::BigInt; # BigInt "0"
+ $x--; # BigInt "-1"
+ $x = Math::BigInt->badd(4,5) # BigInt "9"
+ $x = Math::BigInt::badd(4,5) # BigInt "9"
+ print $x->bsstr(); # 9e+0
+
+Examples for rounding:
+
+ use Math::BigFloat;
+ use Test;
- '+0' canonical zero value
- ' -123 123 123' canonical value '-123123123'
- '1 23 456 7890' canonical value '+1234567890'
+ $x = Math::BigFloat->new(123.4567);
+ $y = Math::BigFloat->new(123.456789);
+ $Math::BigFloat::accuracy = 4; # no more A than 4
+ ok ($x->copy()->fround(),123.4); # even rounding
+ print $x->copy()->fround(),"\n"; # 123.4
+ Math::BigFloat->round_mode('odd'); # round to odd
+ print $x->copy()->fround(),"\n"; # 123.5
+ $Math::BigFloat::accuracy = 5; # no more A than 5
+ Math::BigFloat->round_mode('odd'); # round to odd
+ print $x->copy()->fround(),"\n"; # 123.46
+ $y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
+ print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
+
+ $Math::BigFloat::accuracy = undef; # A not important
+ $Math::BigFloat::precision = 2; # P important
+ print $x->copy()->bnorm(),"\n"; # 123.46
+ print $x->copy()->fround(),"\n"; # 123.46
+
+Examples for converting:
+
+ my $x = Math::BigInt->new('0b1'.'01' x 123);
+ print "bin: ",$x->as_bin()," hex:",$x->as_hex()," dec: ",$x,"\n";
=head1 Autocreating constants
-After C<use Math::BigInt ':constant'> all the integer decimal constants
-in the given scope are converted to C<Math::BigInt>. This conversion
+After C<use Math::BigInt ':constant'> all the B<integer> decimal constants
+in the given scope are converted to C<Math::BigInt>. This conversion
happens at compile time.
-In particular
+In particular,
+
+ perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
+
+prints the integer value of C<2**100>. Note that without conversion of
+constants the expression 2**100 will be calculated as perl scalar.
+
+Please note that strings and floating point constants are not affected,
+so that
+
+ use Math::BigInt qw/:constant/;
+
+ $x = 1234567890123456789012345678901234567890
+ + 123456789123456789;
+ $y = '1234567890123456789012345678901234567890'
+ + '123456789123456789';
- perl -MMath::BigInt=:constant -e 'print 2**100'
+do not work. You need an explicit Math::BigInt->new() around one of the
+operands.
-print the integer value of C<2**100>. Note that without conversion of
-constants the expression 2**100 will be calculated as floating point number.
+=head1 PERFORMANCE
+
+Using the form $x += $y; etc over $x = $x + $y is faster, since a copy of $x
+must be made in the second case. For long numbers, the copy can eat up to 20%
+of the work (in the case of addition/subtraction, less for
+multiplication/division). If $y is very small compared to $x, the form
+$x += $y is MUCH faster than $x = $x + $y since making the copy of $x takes
+more time then the actual addition.
+
+With a technique called copy-on-write, the cost of copying with overload could
+be minimized or even completely avoided. This is currently not implemented.
+
+The new version of this module is slower on new(), bstr() and numify(). Some
+operations may be slower for small numbers, but are significantly faster for
+big numbers. Other operations are now constant (O(1), like bneg(), babs()
+etc), instead of O(N) and thus nearly always take much less time.
+
+If you find the Calc module to slow, try to install any of the replacement
+modules and see if they help you.
+
+=head2 Alternative math libraries
+
+You can use an alternative library to drive Math::BigInt via:
+
+ use Math::BigInt lib => 'Module';
+
+The default is called Math::BigInt::Calc and is a pure-perl implementation
+that consists mainly of the standard routine present in earlier versions of
+Math::BigInt.
+
+There are also Math::BigInt::Scalar (primarily for testing) and
+Math::BigInt::BitVect; as well as Math::BigInt::Pari and likely others.
+All these can be found via L<http://search.cpan.org/>:
+
+ use Math::BigInt lib => 'BitVect';
+
+ my $x = Math::BigInt->new(2);
+ print $x ** (1024*1024);
+
+For more benchmark results see http://bloodgate.com/perl/benchmarks.html
=head1 BUGS
-The current version of this module is a preliminary version of the
-real thing that is currently (as of perl5.002) under development.
+=over 2
+
+=item Out of Memory!
+
+Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
+C<eval()> in your code will crash with "Out of memory". This is probably an
+overload/exporter bug. You can workaround by not having C<eval()>
+and ':constant' at the same time or upgrade your Perl to a newer version.
+
+=item Fails to load Calc on Perl prior 5.6.0
+
+Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
+will fall back to eval { require ... } when loading the math lib on Perls
+prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
+filesystems using a different seperator.
+
+=back
+
+=head1 CAVEATS
+
+Some things might not work as you expect them. Below is documented what is
+known to be troublesome:
+
+=over 1
+
+=item stringify, bstr(), bsstr() and 'cmp'
+
+Both stringify and bstr() now drop the leading '+'. The old code would return
+'+3', the new returns '3'. This is to be consistent with Perl and to make
+cmp (especially with overloading) to work as you expect. It also solves
+problems with Test.pm, it's ok() uses 'eq' internally.
+
+Mark said, when asked about to drop the '+' altogether, or make only cmp work:
+
+ I agree (with the first alternative), don't add the '+' on positive
+ numbers. It's not as important anymore with the new internal
+ form for numbers. It made doing things like abs and neg easier,
+ but those have to be done differently now anyway.
+
+So, the following examples will now work all as expected:
+
+ use Test;
+ BEGIN { plan tests => 1 }
+ use Math::BigInt;
+
+ my $x = new Math::BigInt 3*3;
+ my $y = new Math::BigInt 3*3;
+
+ ok ($x,3*3);
+ print "$x eq 9" if $x eq $y;
+ print "$x eq 9" if $x eq '9';
+ print "$x eq 9" if $x eq 3*3;
+
+Additionally, the following still works:
+
+ print "$x == 9" if $x == $y;
+ print "$x == 9" if $x == 9;
+ print "$x == 9" if $x == 3*3;
+
+There is now a C<bsstr()> method to get the string in scientific notation aka
+C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
+for comparisation, but Perl will represent some numbers as 100 and others
+as 1e+308. If in doubt, convert both arguments to Math::BigInt before doing eq:
+
+ use Test;
+ BEGIN { plan tests => 3 }
+ use Math::BigInt;
+
+ $x = Math::BigInt->new('1e56'); $y = 1e56;
+ ok ($x,$y); # will fail
+ ok ($x->bsstr(),$y); # okay
+ $y = Math::BigInt->new($y);
+ ok ($x,$y); # okay
+
+There is not yet a way to get a number automatically represented in exactly
+the way Perl represents it.
+
+=item int()
+
+C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
+Perl scalar:
+
+ $x = Math::BigInt->new(123);
+ $y = int($x); # BigInt 123
+ $x = Math::BigFloat->new(123.45);
+ $y = int($x); # BigInt 123
+
+In all Perl versions you can use C<as_number()> for the same effect:
+
+ $x = Math::BigFloat->new(123.45);
+ $y = $x->as_number(); # BigInt 123
+
+This also works for other subclasses, like Math::String.
+
+It is yet unlcear whether overloaded int() should return a scalar or a BigInt.
+
+=item length
+
+The following will probably not do what you expect:
+
+ $c = Math::BigInt->new(123);
+ print $c->length(),"\n"; # prints 30
+
+It prints both the number of digits in the number and in the fraction part
+since print calls C<length()> in list context. Use something like:
+
+ print scalar $c->length(),"\n"; # prints 3
+
+=item bdiv
+
+The following will probably not do what you expect:
+
+ print $c->bdiv(10000),"\n";
+
+It prints both quotient and remainder since print calls C<bdiv()> in list
+context. Also, C<bdiv()> will modify $c, so be carefull. You probably want
+to use
+
+ print $c / 10000,"\n";
+ print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
+
+instead.
+
+The quotient is always the greatest integer less than or equal to the
+real-valued quotient of the two operands, and the remainder (when it is
+nonzero) always has the same sign as the second operand; so, for
+example,
+
+ 1 / 4 => ( 0, 1)
+ 1 / -4 => (-1,-3)
+ -3 / 4 => (-1, 1)
+ -3 / -4 => ( 0,-3)
+ -11 / 2 => (-5,1)
+ 11 /-2 => (-5,-1)
+
+As a consequence, the behavior of the operator % agrees with the
+behavior of Perl's built-in % operator (as documented in the perlop
+manpage), and the equation
+
+ $x == ($x / $y) * $y + ($x % $y)
+
+holds true for any $x and $y, which justifies calling the two return
+values of bdiv() the quotient and remainder. The only exception to this rule
+are when $y == 0 and $x is negative, then the remainder will also be
+negative. See below under "infinity handling" for the reasoning behing this.
+
+Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
+not change BigInt's way to do things. This is because under 'use integer' Perl
+will do what the underlying C thinks is right and this is different for each
+system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
+the author to implement it ;)
+
+=item infinity handling
+
+Here are some examples that explain the reasons why certain results occur while
+handling infinity:
+
+The following table shows the result of the division and the remainder, so that
+the equation above holds true. Some "ordinary" cases are strewn in to show more
+clearly the reasoning:
+
+ A / B = C, R so that C * B + R = A
+ =========================================================
+ 5 / 8 = 0, 5 0 * 8 + 5 = 5
+ 0 / 8 = 0, 0 0 * 8 + 0 = 0
+ 0 / inf = 0, 0 0 * inf + 0 = 0
+ 0 /-inf = 0, 0 0 * -inf + 0 = 0
+ 5 / inf = 0, 5 0 * inf + 5 = 5
+ 5 /-inf = 0, 5 0 * -inf + 5 = 5
+ -5/ inf = 0, -5 0 * inf + -5 = -5
+ -5/-inf = 0, -5 0 * -inf + -5 = -5
+ inf/ 5 = inf, 0 inf * 5 + 0 = inf
+ -inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
+ inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
+ -inf/ -5 = inf, 0 inf * -5 + 0 = -inf
+ 5/ 5 = 1, 0 1 * 5 + 0 = 5
+ -5/ -5 = 1, 0 1 * -5 + 0 = -5
+ inf/ inf = 1, 0 1 * inf + 0 = inf
+ -inf/-inf = 1, 0 1 * -inf + 0 = -inf
+ inf/-inf = -1, 0 -1 * -inf + 0 = inf
+ -inf/ inf = -1, 0 1 * -inf + 0 = -inf
+ 8/ 0 = inf, 8 inf * 0 + 8 = 8
+ inf/ 0 = inf, inf inf * 0 + inf = inf
+ 0/ 0 = NaN
+
+These cases below violate the "remainder has the sign of the second of the two
+arguments", since they wouldn't match up otherwise.
+
+ A / B = C, R so that C * B + R = A
+ ========================================================
+ -inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
+ -8/ 0 = -inf, -8 -inf * 0 + 8 = -8
+
+=item Modifying and =
+
+Beware of:
+
+ $x = Math::BigFloat->new(5);
+ $y = $x;
+
+It will not do what you think, e.g. making a copy of $x. Instead it just makes
+a second reference to the B<same> object and stores it in $y. Thus anything
+that modifies $x (except overloaded operators) will modify $y, and vice versa.
+Or in other words, C<=> is only safe if you modify your BigInts only via
+overloaded math. As soon as you use a method call it breaks:
+
+ $x->bmul(2);
+ print "$x, $y\n"; # prints '10, 10'
+
+If you want a true copy of $x, use:
+
+ $y = $x->copy();
+
+You can also chain the calls like this, this will make first a copy and then
+multiply it by 2:
+
+ $y = $x->copy()->bmul(2);
+
+See also the documentation for overload.pm regarding C<=>.
+
+=item bpow
+
+C<bpow()> (and the rounding functions) now modifies the first argument and
+returns it, unlike the old code which left it alone and only returned the
+result. This is to be consistent with C<badd()> etc. The first three will
+modify $x, the last one won't:
+
+ print bpow($x,$i),"\n"; # modify $x
+ print $x->bpow($i),"\n"; # ditto
+ print $x **= $i,"\n"; # the same
+ print $x ** $i,"\n"; # leave $x alone
+
+The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
+
+=item Overloading -$x
+
+The following:
+
+ $x = -$x;
+
+is slower than
+
+ $x->bneg();
+
+since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
+needs to preserve $x since it does not know that it later will get overwritten.
+This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
+
+With Copy-On-Write, this issue will be gone. Stay tuned...
+
+=item Mixing different object types
+
+In Perl you will get a floating point value if you do one of the following:
+
+ $float = 5.0 + 2;
+ $float = 2 + 5.0;
+ $float = 5 / 2;
+
+With overloaded math, only the first two variants will result in a BigFloat:
+
+ use Math::BigInt;
+ use Math::BigFloat;
+
+ $mbf = Math::BigFloat->new(5);
+ $mbi2 = Math::BigInteger->new(5);
+ $mbi = Math::BigInteger->new(2);
+
+ # what actually gets called:
+ $float = $mbf + $mbi; # $mbf->badd()
+ $float = $mbf / $mbi; # $mbf->bdiv()
+ $integer = $mbi + $mbf; # $mbi->badd()
+ $integer = $mbi2 / $mbi; # $mbi2->bdiv()
+ $integer = $mbi2 / $mbf; # $mbi2->bdiv()
+
+This is because math with overloaded operators follows the first (dominating)
+operand, this one's operation is called and returns thus the result. So,
+Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
+the result should be a Math::BigFloat or the second operant is one.
+
+To get a Math::BigFloat you either need to call the operation manually,
+make sure the operands are already of the proper type or casted to that type
+via Math::BigFloat->new():
+
+ $float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
+
+Beware of simple "casting" the entire expression, this would only convert
+the already computed result:
+
+ $float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
+
+Beware also of the order of more complicated expressions like:
+
+ $integer = ($mbi2 + $mbi) / $mbf; # int / float => int
+ $integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
+
+If in doubt, break the expression into simpler terms, or cast all operands
+to the desired resulting type.
+
+Scalar values are a bit different, since:
+
+ $float = 2 + $mbf;
+ $float = $mbf + 2;
+
+will both result in the proper type due to the way the overloaded math works.
+
+This section also applies to other overloaded math packages, like Math::String.
+
+=item bsqrt()
+
+C<bsqrt()> works only good if the result is an big integer, e.g. the square
+root of 144 is 12, but from 12 the square root is 3, regardless of rounding
+mode.
+
+If you want a better approximation of the square root, then use:
+
+ $x = Math::BigFloat->new(12);
+ $Math::BigFloat::precision = 0;
+ Math::BigFloat->round_mode('even');
+ print $x->copy->bsqrt(),"\n"; # 4
+
+ $Math::BigFloat::precision = 2;
+ print $x->bsqrt(),"\n"; # 3.46
+ print $x->bsqrt(3),"\n"; # 3.464
+
+=back
+
+=head1 LICENSE
+
+This program is free software; you may redistribute it and/or modify it under
+the same terms as Perl itself.
+
+=head1 SEE ALSO
+
+L<Math::BigFloat> and L<Math::Big> as well as L<Math::BigInt::BitVect>,
+L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
+
+The package at
+L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
+more documentation including a full version history, testcases, empty
+subclass files and benchmarks.
-=head1 AUTHOR
+=head1 AUTHORS
-Mark Biggar, overloaded interface by Ilya Zakharevich.
+Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
+Completely rewritten by Tels http://bloodgate.com in late 2000, 2001.
=cut
projects / p5sagit/p5-mst-13.2.git / blobdiff