package Math::BigInt;
+#
+# "Mike had an infinite amount to do and a negative amount of time in which
+# to do it." - Before and After
+#
+
+# 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!
+
+my $class = "Math::BigInt";
+require 5.005;
+
+$VERSION = '1.77';
+
+@ISA = qw(Exporter);
+@EXPORT_OK = qw(objectify bgcd blcm);
+
+# _trap_inf and _trap_nan are internal and should never be accessed from the
+# outside
+use vars qw/$round_mode $accuracy $precision $div_scale $rnd_mode
+ $upgrade $downgrade $_trap_nan $_trap_inf/;
+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 is true.
+# In some cases (like add, $x = $x + 2 is the same as $x = 2 + $x) this makes
+# no difference, but in some cases it does.
+
+# 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 {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
+'=' => sub { $_[0]->copy(); },
+
+# 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]); },
+'<<=' => sub { $_[0]->blsft($_[1]); },
+'>>=' => sub { $_[0]->brsft($_[1]); },
+
+# not supported by Perl yet
+'..' => \&_pointpoint,
+
+# we might need '==' and '!=' to get things like "NaN == NaN" right
+'<=>' => sub { $_[2] ?
+ ref($_[0])->bcmp($_[1],$_[0]) :
+ $_[0]->bcmp($_[1]); },
+'cmp' => sub {
+ $_[2] ?
+ "$_[1]" cmp $_[0]->bstr() :
+ $_[0]->bstr() cmp "$_[1]" },
+
+# make cos()/sin()/exp() "work" with BigInt's or subclasses
+'cos' => sub { cos($_[0]->numify()) },
+'sin' => sub { sin($_[0]->numify()) },
+'exp' => sub { exp($_[0]->numify()) },
+'atan2' => sub { $_[2] ?
+ atan2($_[1],$_[0]->numify()) :
+ atan2($_[0]->numify(),$_[1]) },
+
+# are not yet overloadable
+#'hex' => sub { print "hex"; $_[0]; },
+#'oct' => sub { print "oct"; $_[0]; },
+
+'log' => sub { $_[0]->copy()->blog($_[1]); },
+'int' => sub { $_[0]->copy(); },
+'neg' => sub { $_[0]->copy()->bneg(); },
+'abs' => sub { $_[0]->copy()->babs(); },
+'sqrt' => sub { $_[0]->copy()->bsqrt(); },
+'~' => sub { $_[0]->copy()->bnot(); },
+
+# for subtract it's a bit tricky to not modify b: b-a => -a+b
+'-' => sub { my $c = $_[0]->copy; $_[2] ?
+ $c->bneg()->badd( $_[1]) :
+ $c->bsub( $_[1]) },
+'+' => sub { $_[0]->copy()->badd($_[1]); },
+'*' => sub { $_[0]->copy()->bmul($_[1]); },
+
+'/' => sub {
+ $_[2] ? ref($_[0])->new($_[1])->bdiv($_[0]) : $_[0]->copy->bdiv($_[1]);
+ },
+'%' => sub {
+ $_[2] ? ref($_[0])->new($_[1])->bmod($_[0]) : $_[0]->copy->bmod($_[1]);
+ },
+'**' => sub {
+ $_[2] ? ref($_[0])->new($_[1])->bpow($_[0]) : $_[0]->copy->bpow($_[1]);
+ },
+'<<' => sub {
+ $_[2] ? ref($_[0])->new($_[1])->blsft($_[0]) : $_[0]->copy->blsft($_[1]);
+ },
+'>>' => sub {
+ $_[2] ? ref($_[0])->new($_[1])->brsft($_[0]) : $_[0]->copy->brsft($_[1]);
+ },
+'&' => sub {
+ $_[2] ? ref($_[0])->new($_[1])->band($_[0]) : $_[0]->copy->band($_[1]);
+ },
+'|' => sub {
+ $_[2] ? ref($_[0])->new($_[1])->bior($_[0]) : $_[0]->copy->bior($_[1]);
+ },
+'^' => sub {
+ $_[2] ? ref($_[0])->new($_[1])->bxor($_[0]) : $_[0]->copy->bxor($_[1]);
+ },
+
+# can modify arg of ++ and --, so avoid a copy() for speed, but don't
+# use $_[0]->bone(), it would modify $_[0] to be 1!
+'++' => sub { $_[0]->binc() },
+'--' => sub { $_[0]->bdec() },
+
+# if overloaded, O(1) instead of O(N) and twice as fast for small numbers
+'bool' => sub {
+ # this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/
+ # v5.6.1 dumps on this: return !$_[0]->is_zero() || undef; :-(
+ my $t = undef;
+ $t = 1 if !$_[0]->is_zero();
+ $t;
+ },
+
+# the original qw() does not work with the TIESCALAR below, why?
+# Order of arguments unsignificant
+'""' => sub { $_[0]->bstr(); },
+'0+' => sub { $_[0]->numify(); }
;
-$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(@_);
- vec($_,0,8) ^= ord('+') ^ ord('-') unless $_ eq '+0';
- s/^H/N/;
- $_;
-}
-
-# Returns the absolute value of the input.
-sub babs { #(num_str) return num_str
- &abs(&bnorm(@_));
-}
-
-sub abs { # post-normalized abs for internal use
- local($_) = @_;
- 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,$y);
- }
-}
-
-sub cmp { # post-normalized compare for internal use
- local($cx, $cy) = @_;
+##############################################################################
+# global constants, flags and accessory
+
+# These vars are public, but their direct usage is not recommended, use the
+# accessor methods instead
+
+$round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
+$accuracy = undef;
+$precision = undef;
+$div_scale = 40;
+
+$upgrade = undef; # default is no upgrade
+$downgrade = undef; # default is no downgrade
+
+# These are internally, and not to be used from the outside at all
+
+$_trap_nan = 0; # are NaNs ok? set w/ config()
+$_trap_inf = 0; # are infs ok? set w/ config()
+my $nan = 'NaN'; # constants for easier life
+
+my $CALC = 'Math::BigInt::FastCalc'; # module to do the low level math
+ # default is FastCalc.pm
+my $IMPORT = 0; # was import() called yet?
+ # used to make require work
+my %WARN; # warn only once for low-level libs
+my %CAN; # cache for $CALC->can(...)
+my %CALLBACKS; # callbacks to notify on lib loads
+my $EMU_LIB = 'Math/BigInt/CalcEmu.pm'; # emulate low-level math
+
+##############################################################################
+# 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 to enable $rnd_mode to work transparently
+ tie $rnd_mode, 'Math::BigInt';
+
+ # set up some handy alias names
+ *as_int = \&as_number;
+ *is_pos = \&is_positive;
+ *is_neg = \&is_negative;
+ }
+
+##############################################################################
+
+sub round_mode
+ {
+ no strict 'refs';
+ # make Class->round_mode() work
+ my $self = shift;
+ my $class = ref($self) || $self || __PACKAGE__;
+ if (defined $_[0])
+ {
+ my $m = shift;
+ if ($m !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
+ {
+ require Carp; Carp::croak ("Unknown round mode '$m'");
+ }
+ return ${"${class}::round_mode"} = $m;
+ }
+ ${"${class}::round_mode"};
+ }
+
+sub upgrade
+ {
+ no strict 'refs';
+ # make Class->upgrade() work
+ my $self = shift;
+ my $class = ref($self) || $self || __PACKAGE__;
+ # need to set new value?
+ if (@_ > 0)
+ {
+ return ${"${class}::upgrade"} = $_[0];
+ }
+ ${"${class}::upgrade"};
+ }
+
+sub downgrade
+ {
+ no strict 'refs';
+ # make Class->downgrade() work
+ my $self = shift;
+ my $class = ref($self) || $self || __PACKAGE__;
+ # need to set new value?
+ if (@_ > 0)
+ {
+ return ${"${class}::downgrade"} = $_[0];
+ }
+ ${"${class}::downgrade"};
+ }
+
+sub div_scale
+ {
+ no strict 'refs';
+ # make Class->div_scale() work
+ my $self = shift;
+ my $class = ref($self) || $self || __PACKAGE__;
+ if (defined $_[0])
+ {
+ if ($_[0] < 0)
+ {
+ require Carp; Carp::croak ('div_scale must be greater than zero');
+ }
+ ${"${class}::div_scale"} = $_[0];
+ }
+ ${"${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;
+ # convert objects to scalars to avoid deep recursion. If object doesn't
+ # have numify(), then hopefully it will have overloading for int() and
+ # boolean test without wandering into a deep recursion path...
+ $a = $a->numify() if ref($a) && $a->can('numify');
+
+ if (defined $a)
+ {
+ # also croak on non-numerical
+ if (!$a || $a <= 0)
+ {
+ require Carp;
+ Carp::croak ('Argument to accuracy must be greater than zero');
+ }
+ if (int($a) != $a)
+ {
+ require Carp; Carp::croak ('Argument to accuracy must be an integer');
+ }
+ }
+ if (ref($x))
+ {
+ # $object->accuracy() or fallback to global
+ $x->bround($a) if $a; # not for undef, 0
+ $x->{_a} = $a; # set/overwrite, even if not rounded
+ delete $x->{_p}; # clear P
+ $a = ${"${class}::accuracy"} unless defined $a; # proper return value
+ }
+ else
+ {
+ ${"${class}::accuracy"} = $a; # set global A
+ ${"${class}::precision"} = undef; # clear global P
+ }
+ return $a; # shortcut
+ }
+
+ my $a;
+ # $object->accuracy() or fallback to global
+ $a = $x->{_a} if ref($x);
+ # but don't return global undef, when $x's accuracy is 0!
+ $a = ${"${class}::accuracy"} if !defined $a;
+ $a;
+ }
+
+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';
+ if (@_ > 0)
+ {
+ my $p = shift;
+ # convert objects to scalars to avoid deep recursion. If object doesn't
+ # have numify(), then hopefully it will have overloading for int() and
+ # boolean test without wandering into a deep recursion path...
+ $p = $p->numify() if ref($p) && $p->can('numify');
+ if ((defined $p) && (int($p) != $p))
+ {
+ require Carp; Carp::croak ('Argument to precision must be an integer');
+ }
+ if (ref($x))
+ {
+ # $object->precision() or fallback to global
+ $x->bfround($p) if $p; # not for undef, 0
+ $x->{_p} = $p; # set/overwrite, even if not rounded
+ delete $x->{_a}; # clear A
+ $p = ${"${class}::precision"} unless defined $p; # proper return value
+ }
+ else
+ {
+ ${"${class}::precision"} = $p; # set global P
+ ${"${class}::accuracy"} = undef; # clear global A
+ }
+ return $p; # shortcut
+ }
+
+ my $p;
+ # $object->precision() or fallback to global
+ $p = $x->{_p} if ref($x);
+ # but don't return global undef, when $x's precision is 0!
+ $p = ${"${class}::precision"} if !defined $p;
+ $p;
+ }
+
+sub config
+ {
+ # return (or set) configuration data as hash ref
+ my $class = shift || 'Math::BigInt';
+
+ no strict 'refs';
+ if (@_ > 0)
+ {
+ # try to set given options as arguments from hash
+
+ my $args = $_[0];
+ if (ref($args) ne 'HASH')
+ {
+ $args = { @_ };
+ }
+ # these values can be "set"
+ my $set_args = {};
+ foreach my $key (
+ qw/trap_inf trap_nan
+ upgrade downgrade precision accuracy round_mode div_scale/
+ )
+ {
+ $set_args->{$key} = $args->{$key} if exists $args->{$key};
+ delete $args->{$key};
+ }
+ if (keys %$args > 0)
+ {
+ require Carp;
+ Carp::croak ("Illegal key(s) '",
+ join("','",keys %$args),"' passed to $class\->config()");
+ }
+ foreach my $key (keys %$set_args)
+ {
+ if ($key =~ /^trap_(inf|nan)\z/)
+ {
+ ${"${class}::_trap_$1"} = ($set_args->{"trap_$1"} ? 1 : 0);
+ next;
+ }
+ # use a call instead of just setting the $variable to check argument
+ $class->$key($set_args->{$key});
+ }
+ }
+
+ # now return actual configuration
+
+ my $cfg = {
+ lib => $CALC,
+ lib_version => ${"${CALC}::VERSION"},
+ class => $class,
+ trap_nan => ${"${class}::_trap_nan"},
+ trap_inf => ${"${class}::_trap_inf"},
+ version => ${"${class}::VERSION"},
+ };
+ foreach my $key (qw/
+ upgrade downgrade precision accuracy round_mode div_scale
+ /)
+ {
+ $cfg->{$key} = ${"${class}::$key"};
+ };
+ $cfg;
+ }
+
+sub _scale_a
+ {
+ # select accuracy parameter based on precedence,
+ # used by bround() and bfround(), may return undef for scale (means no op)
+ my ($x,$scale,$mode) = @_;
+
+ $scale = $x->{_a} unless defined $scale;
+
+ no strict 'refs';
+ my $class = ref($x);
+
+ $scale = ${ $class . '::accuracy' } unless defined $scale;
+ $mode = ${ $class . '::round_mode' } unless defined $mode;
+
+ ($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,$scale,$mode) = @_;
+
+ $scale = $x->{_p} unless defined $scale;
+
+ no strict 'refs';
+ my $class = ref($x);
+
+ $scale = ${ $class . '::precision' } unless defined $scale;
+ $mode = ${ $class . '::round_mode' } unless defined $mode;
+
+ ($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 {}, $c;
+
+ $self->{sign} = $x->{sign};
+ $self->{value} = $CALC->_copy($x->{value});
+ $self->{_a} = $x->{_a} if defined $x->{_a};
+ $self->{_p} = $x->{_p} if defined $x->{_p};
+ $self;
+ }
+
+sub new
+ {
+ # create a new BigInt object from a string or another BigInt object.
+ # see hash keys documented at top
+
+ # the argument could be an object, so avoid ||, && etc on it, this would
+ # cause costly overloaded code to be called. The only allowed ops are
+ # ref() and defined.
+
+ my ($class,$wanted,$a,$p,$r) = @_;
+
+ # avoid numify-calls by not using || on $wanted!
+ return $class->bzero($a,$p) if !defined $wanted; # default to 0
+ return $class->copy($wanted,$a,$p,$r)
+ if ref($wanted) && $wanted->isa($class); # MBI or subclass
+
+ $class->import() if $IMPORT == 0; # make require work
+
+ my $self = bless {}, $class;
+
+ # shortcut for "normal" numbers
+ if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*\z/))
+ {
+ $self->{sign} = $1 || '+';
+
+ if ($wanted =~ /^[+-]/)
+ {
+ # remove sign without touching wanted to make it work with constants
+ my $t = $wanted; $t =~ s/^[+-]//;
+ $self->{value} = $CALC->_new($t);
+ }
+ else
+ {
+ $self->{value} = $CALC->_new($wanted);
+ }
+ no strict 'refs';
+ if ( (defined $a) || (defined $p)
+ || (defined ${"${class}::precision"})
+ || (defined ${"${class}::accuracy"})
+ )
+ {
+ $self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p);
+ }
+ return $self;
+ }
+
+ # handle '+inf', '-inf' first
+ if ($wanted =~ /^[+-]?inf\z/)
+ {
+ $self->{sign} = $wanted; # set a default sign for bstr()
+ return $self->binf($wanted);
+ }
+ # 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 $mis)
+ {
+ if ($_trap_nan)
+ {
+ require Carp; Carp::croak("$wanted is not a number in $class");
+ }
+ $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 ($diff < 0) # Not integer
+ {
+ if ($_trap_nan)
+ {
+ require Carp; Carp::croak("$wanted not an integer in $class");
+ }
+ #print "NOI 1\n";
+ return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
+ $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 ($$mfv ne '') # e <= 0
+ {
+ # fraction and negative/zero E => NOI
+ if ($_trap_nan)
+ {
+ require Carp; Carp::croak("$wanted not an integer in $class");
+ }
+ #print "NOI 2 \$\$mfv '$$mfv'\n";
+ return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
+ $self->{sign} = $nan;
+ }
+ elsif ($e < 0)
+ {
+ # xE-y, and empty mfv
+ #print "xE-y\n";
+ $e = abs($e);
+ if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
+ {
+ if ($_trap_nan)
+ {
+ require Carp; Carp::croak("$wanted not an integer in $class");
+ }
+ #print "NOI 3\n";
+ return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
+ $self->{sign} = $nan;
+ }
+ }
+ }
+ $self->{sign} = '+' if $$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 $self
+ # do not round for new($x,undef,undef) since that is used by MBF to signal
+ # no rounding
+ $self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p;
+ $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, $c;
+ }
+ no strict 'refs';
+ if (${"${class}::_trap_nan"})
+ {
+ require Carp;
+ Carp::croak ("Tried to set $self to NaN in $class\::bnan()");
+ }
+ $self->import() if $IMPORT == 0; # make require work
+ return if $self->modify('bnan');
+ if ($self->can('_bnan'))
+ {
+ # use subclass to initialize
+ $self->_bnan();
+ }
+ else
+ {
+ # otherwise do our own thing
+ $self->{value} = $CALC->_zero();
+ }
+ $self->{sign} = $nan;
+ delete $self->{_a}; delete $self->{_p}; # rounding NaN is silly
+ $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 || $sign !~ /^-(inf)?$/;
+ $self = $class if !defined $self;
+ if (!ref($self))
+ {
+ my $c = $self; $self = {}; bless $self, $c;
+ }
+ no strict 'refs';
+ if (${"${class}::_trap_inf"})
+ {
+ require Carp;
+ Carp::croak ("Tried to set $self to +-inf in $class\::binf()");
+ }
+ $self->import() if $IMPORT == 0; # make require work
+ return if $self->modify('binf');
+ if ($self->can('_binf'))
+ {
+ # use subclass to initialize
+ $self->_binf();
+ }
+ else
+ {
+ # otherwise do our own thing
+ $self->{value} = $CALC->_zero();
+ }
+ $sign = $sign . 'inf' if $sign !~ /inf$/; # - => -inf
+ $self->{sign} = $sign;
+ ($self->{_a},$self->{_p}) = @_; # take over requested rounding
+ $self;
+ }
+
+sub bzero
+ {
+ # create a bigint '+0', if given a BigInt, set it to 0
+ my $self = shift;
+ $self = __PACKAGE__ if !defined $self;
+
+ if (!ref($self))
+ {
+ my $c = $self; $self = {}; bless $self, $c;
+ }
+ $self->import() if $IMPORT == 0; # make require work
+ return if $self->modify('bzero');
+
+ if ($self->can('_bzero'))
+ {
+ # use subclass to initialize
+ $self->_bzero();
+ }
+ else
+ {
+ # otherwise do our own thing
+ $self->{value} = $CALC->_zero();
+ }
+ $self->{sign} = '+';
+ if (@_ > 0)
+ {
+ if (@_ > 3)
+ {
+ # call like: $x->bzero($a,$p,$r,$y);
+ ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
+ }
+ else
+ {
+ $self->{_a} = $_[0]
+ if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
+ $self->{_p} = $_[1]
+ if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
+ }
+ }
+ $self;
+ }
+
+sub bone
+ {
+ # create a bigint '+1' (or -1 if given sign '-'),
+ # if given a BigInt, set it to +1 or -1, respectively
+ 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;
+ }
+ $self->import() if $IMPORT == 0; # make require work
+ return if $self->modify('bone');
+
+ if ($self->can('_bone'))
+ {
+ # use subclass to initialize
+ $self->_bone();
+ }
+ else
+ {
+ # otherwise do our own thing
+ $self->{value} = $CALC->_one();
+ }
+ $self->{sign} = $sign;
+ if (@_ > 0)
+ {
+ if (@_ > 3)
+ {
+ # call like: $x->bone($sign,$a,$p,$r,$y);
+ ($self,$self->{_a},$self->{_p}) = $self->_find_round_parameters(@_);
+ }
+ else
+ {
+ # call like: $x->bone($sign,$a,$p,$r);
+ $self->{_a} = $_[0]
+ if ( (!defined $self->{_a}) || (defined $_[0] && $_[0] > $self->{_a}));
+ $self->{_p} = $_[1]
+ if ( (!defined $self->{_p}) || (defined $_[1] && $_[1] > $self->{_p}));
+ }
+ }
+ $self;
+ }
+
+##############################################################################
+# string conversation
+
+sub bsstr
+ {
+ # (ref to BFLOAT or num_str ) return num_str
+ # Convert number from internal format to scientific string format.
+ # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ if ($x->{sign} !~ /^[+-]$/)
+ {
+ return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
+ return 'inf'; # +inf
+ }
+ my ($m,$e) = $x->parts();
+ #$m->bstr() . 'e+' . $e->bstr(); # e can only be positive in BigInt
+ # 'e+' because E can only be positive in BigInt
+ $m->bstr() . 'e+' . $CALC->_str($e->{value});
+ }
+
+sub bstr
+ {
+ # make a string from bigint object
+ my ($self,$x) = ref($_[0]) ? (undef,$_[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 '-';
+ $es.$CALC->_str($x->{value});
+ }
+
+sub numify
+ {
+ # Make a "normal" scalar from a BigInt object
+ my $x = shift; $x = $class->new($x) unless ref $x;
+
+ return $x->bstr() if $x->{sign} !~ /^[+-]$/;
+ my $num = $CALC->_num($x->{value});
+ return -$num if $x->{sign} eq '-';
+ $num;
+ }
+
+##############################################################################
+# public stuff (usually prefixed with "b")
+
+sub sign
+ {
+ # return the sign of the number: +/-/-inf/+inf/NaN
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ $x->{sign};
+ }
+
+sub _find_round_parameters
+ {
+ # After any operation or when calling round(), the result is rounded by
+ # regarding the A & P from arguments, local parameters, or globals.
+
+ # !!!!!!! If you change this, remember to change round(), too! !!!!!!!!!!
+
+ # This procedure finds the round parameters, but it is for speed reasons
+ # duplicated in round. Otherwise, it is tested by the testsuite and used
+ # by fdiv().
+
+ # returns ($self) or ($self,$a,$p,$r) - sets $self to NaN of both A and P
+ # were requested/defined (locally or globally or both)
+
+ my ($self,$a,$p,$r,@args) = @_;
+ # $a accuracy, if given by caller
+ # $p precision, if given by caller
+ # $r round_mode, if given by caller
+ # @args all 'other' arguments (0 for unary, 1 for binary ops)
+
+ my $c = ref($self); # find out class of argument(s)
+ no strict 'refs';
+
+ # now pick $a or $p, but only if we have got "arguments"
+ if (!defined $a)
+ {
+ foreach ($self,@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 $p)
+ {
+ # even if $a is defined, take $p, to signal error for both defined
+ foreach ($self,@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 still none defined, use globals (#2)
+ $a = ${"$c\::accuracy"} unless defined $a;
+ $p = ${"$c\::precision"} unless defined $p;
+
+ # A == 0 is useless, so undef it to signal no rounding
+ $a = undef if defined $a && $a == 0;
+
+ # no rounding today?
+ return ($self) unless defined $a || defined $p; # early out
+
+ # set A and set P is an fatal error
+ return ($self->bnan()) if defined $a && defined $p; # error
+
+ $r = ${"$c\::round_mode"} unless defined $r;
+ if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
+ {
+ require Carp; Carp::croak ("Unknown round mode '$r'");
+ }
+
+ ($self,$a,$p,$r);
+ }
+
+sub round
+ {
+ # Round $self according to given parameters, or given second argument's
+ # parameters or global defaults
+
+ # for speed reasons, _find_round_parameters is embeded here:
+
+ my ($self,$a,$p,$r,@args) = @_;
+ # $a accuracy, if given by caller
+ # $p precision, if given by caller
+ # $r round_mode, if given by caller
+ # @args all 'other' arguments (0 for unary, 1 for binary ops)
+
+ my $c = ref($self); # find out class of argument(s)
+ no strict 'refs';
+
+ # now pick $a or $p, but only if we have got "arguments"
+ if (!defined $a)
+ {
+ foreach ($self,@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 $p)
+ {
+ # even if $a is defined, take $p, to signal error for both defined
+ foreach ($self,@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 still none defined, use globals (#2)
+ $a = ${"$c\::accuracy"} unless defined $a;
+ $p = ${"$c\::precision"} unless defined $p;
+
+ # A == 0 is useless, so undef it to signal no rounding
+ $a = undef if defined $a && $a == 0;
+
+ # no rounding today?
+ return $self unless defined $a || defined $p; # early out
+
+ # set A and set P is an fatal error
+ return $self->bnan() if defined $a && defined $p;
+
+ $r = ${"$c\::round_mode"} unless defined $r;
+ if ($r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/)
+ {
+ require Carp; Carp::croak ("Unknown round mode '$r'");
+ }
+
+ # now round, by calling either fround or ffround:
+ if (defined $a)
+ {
+ $self->bround($a,$r) if !defined $self->{_a} || $self->{_a} >= $a;
+ }
+ else # both can't be undefined due to early out
+ {
+ $self->bfround($p,$r) if !defined $self->{_p} || $self->{_p} <= $p;
+ }
+ # bround() or bfround() already callled bnorm() if necc.
+ $self;
+ }
+
+sub bnorm
+ {
+ # (numstr or BINT) return BINT
+ # Normalize number -- no-op here
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+ $x;
+ }
+
+sub babs
+ {
+ # (BINT or num_str) return BINT
+ # make number absolute, or return absolute BINT from string
+ my ($self,$x) = ref($_[0]) ? (undef,$_[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]) ? (undef,$_[0]) : objectify(1,@_);
+
+ return $x if $x->modify('bneg');
+
+ # for +0 dont negate (to have always normalized +0). Does nothing for 'NaN'
+ $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $CALC->_is_zero($x->{value}));
+ $x;
+ }
+
+sub bcmp
+ {
+ # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
+ # (BINT or num_str, BINT or num_str) return cond_code
+
+ # set up parameters
+ my ($self,$x,$y) = (ref($_[0]),@_);
+
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y) = objectify(2,@_);
+ }
+
+ return $upgrade->bcmp($x,$y) if defined $upgrade &&
+ ((!$x->isa($self)) || (!$y->isa($self)));
+
+ if (($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;
+ }
+ # check sign for speed first
+ return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
+ return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
+
+ # have same sign, so compare absolute values. Don't make tests for zero here
+ # because it's actually slower than testin in Calc (especially w/ Pari et al)
+
+ # post-normalized compare for internal use (honors signs)
+ if ($x->{sign} eq '+')
+ {
+ # $x and $y both > 0
+ return $CALC->_acmp($x->{value},$y->{value});
+ }
+
+ # $x && $y both < 0
+ $CALC->_acmp($y->{value},$x->{value}); # swaped acmp (lib returns 0,1,-1)
+ }
+
+sub bacmp
+ {
+ # Compares 2 values, ignoring their signs.
+ # Returns one of undef, <0, =0, >0. (suitable for sort)
+ # (BINT, BINT) return cond_code
+
+ # set up parameters
+ my ($self,$x,$y) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y) = objectify(2,@_);
+ }
+
+ return $upgrade->bacmp($x,$y) if defined $upgrade &&
+ ((!$x->isa($self)) || (!$y->isa($self)));
+
+ if (($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 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} !~ /^[+-]inf$/;
+ return -1;
+ }
+ $CALC->_acmp($x->{value},$y->{value}); # lib does only 0,1,-1
+ }
+
+sub badd
+ {
+ # add second arg (BINT or string) to first (BINT) (modifies first)
+ # return result as BINT
+
+ # set up parameters
+ my ($self,$x,$y,@r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,@r) = objectify(2,@_);
+ }
+
+ return $x if $x->modify('badd');
+ return $upgrade->badd($upgrade->new($x),$upgrade->new($y),@r) if defined $upgrade &&
+ ((!$x->isa($self)) || (!$y->isa($self)));
+
+ $r[3] = $y; # no push!
+ # inf and NaN handling
+ if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
+ {
+ # NaN first
+ return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
+ # inf handling
+ if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
+ {
+ # +inf++inf or -inf+-inf => same, rest is NaN
+ return $x if $x->{sign} eq $y->{sign};
+ return $x->bnan();
+ }
+ # +-inf + something => +inf
+ # something +-inf => +-inf
+ $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
+ return $x;
+ }
- return 0 if ($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($sx, &sub(*x, *y));
- }
- }
+ my ($sx, $sy) = ( $x->{sign}, $y->{sign} ); # get signs
+
+ if ($sx eq $sy)
+ {
+ $x->{value} = $CALC->_add($x->{value},$y->{value}); # same sign, abs add
}
-}
-
-sub bsub { #(num_str, num_str) return num_str
- &badd($_[$[],&bneg($_[$[+1]));
-}
-
-# GCD -- Euclids algorithm Knuth Vol 2 pg 296
-sub bgcd { #(num_str, num_str) return num_str
- local($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 $a = $CALC->_acmp ($y->{value},$x->{value}); # absolute compare
+ if ($a > 0)
+ {
+ $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
+ $x->{value} = $CALC->_zero();
+ $x->{sign} = '+';
+ }
+ else # a < 0
+ {
+ $x->{value} = $CALC->_sub($x->{value}, $y->{value}); # abs sub
+ }
}
- else {
- push(@x, 0);
- }
- @q = (); ($v2,$v1) = @y[-2,-1];
- while ($#x > $#y) {
- ($u2,$u1,$u0) = @x[-3..-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);
+ $x->round(@r);
+ }
+
+sub bsub
+ {
+ # (BINT or num_str, BINT or num_str) return BINT
+ # subtract second arg from first, modify first
+
+ # set up parameters
+ my ($self,$x,$y,@r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,@r) = objectify(2,@_);
}
-}
-1;
-__END__
+ return $x if $x->modify('bsub');
-=head1 NAME
+ return $upgrade->new($x)->bsub($upgrade->new($y),@r) if defined $upgrade &&
+ ((!$x->isa($self)) || (!$y->isa($self)));
-Math::BigInt - Arbitrary size integer math package
+ return $x->round(@r) if $y->is_zero();
-=head1 SYNOPSIS
+ # To correctly handle the lone special case $x->bsub($x), we note the sign
+ # of $x, then flip the sign from $y, and if the sign of $x did change, too,
+ # then we caught the special case:
+ my $xsign = $x->{sign};
+ $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
+ if ($xsign ne $x->{sign})
+ {
+ # special case of $x->bsub($x) results in 0
+ return $x->bzero(@r) if $xsign =~ /^[+-]$/;
+ return $x->bnan(); # NaN, -inf, +inf
+ }
+ $x->badd($y,@r); # badd does not leave internal zeros
+ $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
+ $x; # already rounded by badd() or no round necc.
+ }
- 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
- $i->bnorm return BINT normalization
+sub binc
+ {
+ # increment arg by one
+ my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+ return $x if $x->modify('binc');
-=head1 DESCRIPTION
+ if ($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->bone(),$a,$p,$r); # badd does round
+ }
+
+sub bdec
+ {
+ # decrement arg by one
+ my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+ return $x if $x->modify('bdec');
+
+ if ($x->{sign} eq '-')
+ {
+ # x already < 0
+ $x->{value} = $CALC->_inc($x->{value});
+ }
+ else
+ {
+ return $x->badd($self->bone('-'),@r) unless $x->{sign} eq '+'; # inf or NaN
+ # >= 0
+ if ($CALC->_is_zero($x->{value}))
+ {
+ # == 0
+ $x->{value} = $CALC->_one(); $x->{sign} = '-'; # 0 => -1
+ }
+ else
+ {
+ # > 0
+ $x->{value} = $CALC->_dec($x->{value});
+ }
+ }
+ $x->round(@r);
+ }
-All basic math operations are overloaded if you declare your big
-integers as
+sub blog
+ {
+ # calculate $x = $a ** $base + $b and return $a (e.g. the log() to base
+ # $base of $x)
- $i = new Math::BigInt '123 456 789 123 456 789';
+ # set up parameters
+ my ($self,$x,$base,@r) = (undef,@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$base,@r) = objectify(1,ref($x),@_);
+ }
+
+ return $x if $x->modify('blog');
+ # inf, -inf, NaN, <0 => NaN
+ return $x->bnan()
+ if $x->{sign} ne '+' || (defined $base && $base->{sign} ne '+');
-=over 2
+ return $upgrade->blog($upgrade->new($x),$base,@r) if
+ defined $upgrade;
-=item Canonical notation
+ my ($rc,$exact) = $CALC->_log_int($x->{value},$base->{value});
+ return $x->bnan() unless defined $rc; # not possible to take log?
+ $x->{value} = $rc;
+ $x->round(@r);
+ }
-Big integer value are strings of the form C</^[+-]\d+$/> with leading
-zeros suppressed.
+sub blcm
+ {
+ # (BINT or num_str, BINT or num_str) return BINT
+ # does not modify arguments, but returns new object
+ # Lowest Common Multiplicator
-=item Input
+ my $y = shift; my ($x);
+ if (ref($y))
+ {
+ $x = $y->copy();
+ }
+ else
+ {
+ $x = $class->new($y);
+ }
+ my $self = ref($x);
+ while (@_)
+ {
+ my $y = shift; $y = $self->new($y) if !ref ($y);
+ $x = __lcm($x,$y);
+ }
+ $x;
+ }
-Input values to these routines may be strings of the form
-C</^\s*[+-]?[\d\s]+$/>.
+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)
-=item Output
+ my $y = shift;
+ $y = $class->new($y) if !ref($y);
+ my $self = ref($y);
+ my $x = $y->copy()->babs(); # keep arguments
+ return $x->bnan() if $x->{sign} !~ /^[+-]$/; # x NaN?
-Output values always always in canonical form
+ while (@_)
+ {
+ $y = shift; $y = $self->new($y) if !ref($y);
+ return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
+ $x->{value} = $CALC->_gcd($x->{value},$y->{value});
+ last if $CALC->_is_one($x->{value});
+ }
+ $x;
+ }
-=back
+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');
+ $x->binc()->bneg(); # binc already does round
+ }
-Actual math is done in an internal format consisting of an array
-whose first element is the sign (/^[+-]$/) 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.
+##############################################################################
+# is_foo test routines
+# we don't need $self, so undef instead of ref($_[0]) make it slightly faster
-=head1 EXAMPLES
+sub is_zero
+ {
+ # return true if arg (BINT or num_str) is zero (array '+', '0')
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
+ $CALC->_is_zero($x->{value});
+ }
- '+0' canonical zero value
- ' -123 123 123' canonical value '-123123123'
- '1 23 456 7890' canonical value '+1234567890'
+sub is_nan
+ {
+ # return true if arg (BINT or num_str) is NaN
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+ $x->{sign} eq $nan ? 1 : 0;
+ }
-=head1 Autocreating constants
+sub is_inf
+ {
+ # return true if arg (BINT or num_str) is +-inf
+ my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
-After C<use Math::BigInt ':constant'> all the integer decimal constants
-in the given scope are converted to C<Math::BigInt>. This convertion
-happens at compile time.
+ if (defined $sign)
+ {
+ $sign = '[+-]inf' if $sign eq ''; # +- doesn't matter, only that's inf
+ $sign = "[$1]inf" if $sign =~ /^([+-])(inf)?$/; # extract '+' or '-'
+ return $x->{sign} =~ /^$sign$/ ? 1 : 0;
+ }
+ $x->{sign} =~ /^[+-]inf$/ ? 1 : 0; # only +-inf is infinity
+ }
+
+sub is_one
+ {
+ # return true if arg (BINT or num_str) is +1, or -1 if sign is given
+ my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
+
+ $sign = '+' if !defined $sign || $sign ne '-';
+
+ return 0 if $x->{sign} ne $sign; # -1 != +1, NaN, +-inf aren't either
+ $CALC->_is_one($x->{value});
+ }
-In particular
+sub is_odd
+ {
+ # return true when arg (BINT or num_str) is odd, false for even
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
- perl -MMath::BigInt=:constant -e 'print 2**100'
+ return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
+ $CALC->_is_odd($x->{value});
+ }
-print the integer value of C<2**100>. Note that without convertion of
-constants the expression 2**100 will be calculatted as floating point number.
+sub is_even
+ {
+ # return true when arg (BINT or num_str) is even, false for odd
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
-=head1 BUGS
+ return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
+ $CALC->_is_even($x->{value});
+ }
+
+sub is_positive
+ {
+ # return true when arg (BINT or num_str) is positive (>= 0)
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ return 1 if $x->{sign} eq '+inf'; # +inf is positive
+
+ # 0+ is neither positive nor negative
+ ($x->{sign} eq '+' && !$x->is_zero()) ? 1 : 0;
+ }
+
+sub is_negative
+ {
+ # return true when arg (BINT or num_str) is negative (< 0)
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ $x->{sign} =~ /^-/ ? 1 : 0; # -inf is negative, but NaN is not
+ }
+
+sub is_int
+ {
+ # return true when arg (BINT or num_str) is an integer
+ # always true for BigInt, but different for BigFloats
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ $x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
+ }
+
+###############################################################################
+
+sub bmul
+ {
+ # multiply two numbers -- stolen from Knuth Vol 2 pg 233
+ # (BINT or num_str, BINT or num_str) return BINT
+
+ # set up parameters
+ my ($self,$x,$y,@r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,@r) = objectify(2,@_);
+ }
+
+ return $x if $x->modify('bmul');
+
+ return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
+
+ # inf handling
+ if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
+ {
+ return $x->bnan() if $x->is_zero() || $y->is_zero();
+ # 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('-');
+ }
+
+ return $upgrade->bmul($x,$upgrade->new($y),@r)
+ if defined $upgrade && !$y->isa($self);
+
+ $r[3] = $y; # no push here
+
+ $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-'; # +1 * +1 or -1 * -1 => +
+
+ $x->{value} = $CALC->_mul($x->{value},$y->{value}); # do actual math
+ $x->{sign} = '+' if $CALC->_is_zero($x->{value}); # no -0
+
+ $x->round(@r);
+ }
+
+sub _div_inf
+ {
+ # helper function that handles +-inf cases for bdiv()/bmod() to reuse code
+ my ($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() && $y->is_zero()));
+
+ # +-inf / +-inf == NaN, reminder also NaN
+ if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
+ {
+ return wantarray ? ($x->bnan(),$self->bnan()) : $x->bnan();
+ }
+ # x / +-inf => 0, remainder x (works even if x == 0)
+ if ($y->{sign} =~ /^[+-]inf$/)
+ {
+ my $t = $x->copy(); # bzero 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()) : $x;
+ }
+
+sub bdiv
+ {
+ # (dividend: BINT or num_str, divisor: BINT or num_str) return
+ # (BINT,BINT) (quo,rem) or BINT (only rem)
+
+ # set up parameters
+ my ($self,$x,$y,@r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,@r) = objectify(2,@_);
+ }
+
+ return $x if $x->modify('bdiv');
+
+ return $self->_div_inf($x,$y)
+ if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
+
+ return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
+ if defined $upgrade;
+
+ $r[3] = $y; # no push!
+
+ # calc new sign and in case $y == +/- 1, return $x
+ my $xsign = $x->{sign}; # keep
+ $x->{sign} = ($x->{sign} ne $y->{sign} ? '-' : '+');
+
+ if (wantarray)
+ {
+ my $rem = $self->bzero();
+ ($x->{value},$rem->{value}) = $CALC->_div($x->{value},$y->{value});
+ $x->{sign} = '+' if $CALC->_is_zero($x->{value});
+ $rem->{_a} = $x->{_a};
+ $rem->{_p} = $x->{_p};
+ $x->round(@r);
+ if (! $CALC->_is_zero($rem->{value}))
+ {
+ $rem->{sign} = $y->{sign};
+ $rem = $y->copy()->bsub($rem) if $xsign ne $y->{sign}; # one of them '-'
+ }
+ else
+ {
+ $rem->{sign} = '+'; # dont leave -0
+ }
+ $rem->round(@r);
+ return ($x,$rem);
+ }
+
+ $x->{value} = $CALC->_div($x->{value},$y->{value});
+ $x->{sign} = '+' if $CALC->_is_zero($x->{value});
+
+ $x->round(@r);
+ }
+
+###############################################################################
+# modulus functions
+
+sub bmod
+ {
+ # modulus (or remainder)
+ # (BINT or num_str, BINT or num_str) return BINT
+
+ # set up parameters
+ my ($self,$x,$y,@r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,@r) = objectify(2,@_);
+ }
+
+ return $x if $x->modify('bmod');
+ $r[3] = $y; # no push!
+ if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero())
+ {
+ my ($d,$r) = $self->_div_inf($x,$y);
+ $x->{sign} = $r->{sign};
+ $x->{value} = $r->{value};
+ return $x->round(@r);
+ }
+
+ # calc new sign and in case $y == +/- 1, return $x
+ $x->{value} = $CALC->_mod($x->{value},$y->{value});
+ if (!$CALC->_is_zero($x->{value}))
+ {
+ $x->{value} = $CALC->_sub($y->{value},$x->{value},1) # $y-$x
+ if ($x->{sign} ne $y->{sign});
+ $x->{sign} = $y->{sign};
+ }
+ else
+ {
+ $x->{sign} = '+'; # dont leave -0
+ }
+ $x->round(@r);
+ }
+
+sub bmodinv
+ {
+ # Modular inverse. given a number which is (hopefully) relatively
+ # prime to the modulus, calculate its inverse using Euclid's
+ # alogrithm. If the number is not relatively prime to the modulus
+ # (i.e. their gcd is not one) then NaN is returned.
+
+ # set up parameters
+ my ($self,$x,$y,@r) = (undef,@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,@r) = objectify(2,@_);
+ }
+
+ return $x if $x->modify('bmodinv');
+
+ return $x->bnan()
+ if ($y->{sign} ne '+' # -, NaN, +inf, -inf
+ || $x->is_zero() # or num == 0
+ || $x->{sign} !~ /^[+-]$/ # or num NaN, inf, -inf
+ );
+
+ # put least residue into $x if $x was negative, and thus make it positive
+ $x->bmod($y) if $x->{sign} eq '-';
+
+ my $sign;
+ ($x->{value},$sign) = $CALC->_modinv($x->{value},$y->{value});
+ return $x->bnan() if !defined $x->{value}; # in case no GCD found
+ return $x if !defined $sign; # already real result
+ $x->{sign} = $sign; # flip/flop see below
+ $x->bmod($y); # calc real result
+ $x;
+ }
+
+sub bmodpow
+ {
+ # takes a very large number to a very large exponent in a given very
+ # large modulus, quickly, thanks to binary exponentation. supports
+ # negative exponents.
+ my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
+
+ return $num if $num->modify('bmodpow');
+
+ # check modulus for valid values
+ return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
+ || $mod->is_zero());
+
+ # check exponent for valid values
+ if ($exp->{sign} =~ /\w/)
+ {
+ # i.e., if it's NaN, +inf, or -inf...
+ return $num->bnan();
+ }
+
+ $num->bmodinv ($mod) if ($exp->{sign} eq '-');
+
+ # check num for valid values (also NaN if there was no inverse but $exp < 0)
+ return $num->bnan() if $num->{sign} !~ /^[+-]$/;
+
+ # $mod is positive, sign on $exp is ignored, result also positive
+ $num->{value} = $CALC->_modpow($num->{value},$exp->{value},$mod->{value});
+ $num;
+ }
+
+###############################################################################
+
+sub bfac
+ {
+ # (BINT or num_str, BINT or num_str) return BINT
+ # compute factorial number from $x, modify $x in place
+ my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
+
+ return $x if $x->modify('bfac') || $x->{sign} eq '+inf'; # inf => inf
+ return $x->bnan() if $x->{sign} ne '+'; # NaN, <0 etc => NaN
+
+ $x->{value} = $CALC->_fac($x->{value});
+ $x->round(@r);
+ }
+
+sub bpow
+ {
+ # (BINT or num_str, BINT or num_str) return BINT
+ # compute power of two numbers -- stolen from Knuth Vol 2 pg 233
+ # modifies first argument
+
+ # set up parameters
+ my ($self,$x,$y,@r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,@r) = objectify(2,@_);
+ }
+
+ return $x if $x->modify('bpow');
+
+ return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
+
+ # inf handling
+ if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
+ {
+ if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
+ {
+ # +-inf ** +-inf
+ return $x->bnan();
+ }
+ # +-inf ** Y
+ if ($x->{sign} =~ /^[+-]inf/)
+ {
+ # +inf ** 0 => NaN
+ return $x->bnan() if $y->is_zero();
+ # -inf ** -1 => 1/inf => 0
+ return $x->bzero() if $y->is_one('-') && $x->is_negative();
+
+ # +inf ** Y => inf
+ return $x if $x->{sign} eq '+inf';
+
+ # -inf ** Y => -inf if Y is odd
+ return $x if $y->is_odd();
+ return $x->babs();
+ }
+ # X ** +-inf
+
+ # 1 ** +inf => 1
+ return $x if $x->is_one();
+
+ # 0 ** inf => 0
+ return $x if $x->is_zero() && $y->{sign} =~ /^[+]/;
+
+ # 0 ** -inf => inf
+ return $x->binf() if $x->is_zero();
+
+ # -1 ** -inf => NaN
+ return $x->bnan() if $x->is_one('-') && $y->{sign} =~ /^[-]/;
+
+ # -X ** -inf => 0
+ return $x->bzero() if $x->{sign} eq '-' && $y->{sign} =~ /^[-]/;
+
+ # -1 ** inf => NaN
+ return $x->bnan() if $x->{sign} eq '-';
+
+ # X ** inf => inf
+ return $x->binf() if $y->{sign} =~ /^[+]/;
+ # X ** -inf => 0
+ return $x->bzero();
+ }
+
+ return $upgrade->bpow($upgrade->new($x),$y,@r)
+ if defined $upgrade && !$y->isa($self);
+
+ $r[3] = $y; # no push!
+
+ # cases 0 ** Y, X ** 0, X ** 1, 1 ** Y are handled by Calc or Emu
+
+ my $new_sign = '+';
+ $new_sign = $y->is_odd() ? '-' : '+' if ($x->{sign} ne '+');
+
+ # 0 ** -7 => ( 1 / (0 ** 7)) => 1 / 0 => +inf
+ return $x->binf()
+ if $y->{sign} eq '-' && $x->{sign} eq '+' && $CALC->_is_zero($x->{value});
+ # 1 ** -y => 1 / (1 ** |y|)
+ # so do test for negative $y after above's clause
+ return $x->bnan() if $y->{sign} eq '-' && !$CALC->_is_one($x->{value});
+
+ $x->{value} = $CALC->_pow($x->{value},$y->{value});
+ $x->{sign} = $new_sign;
+ $x->{sign} = '+' if $CALC->_is_zero($y->{value});
+ $x->round(@r);
+ }
+
+sub blsft
+ {
+ # (BINT or num_str, BINT or num_str) return BINT
+ # compute x << y, base n, y >= 0
+
+ # set up parameters
+ my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,$n,@r) = objectify(2,@_);
+ }
+
+ return $x if $x->modify('blsft');
+ return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
+ return $x->round(@r) if $y->is_zero();
+
+ $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
+
+ $x->{value} = $CALC->_lsft($x->{value},$y->{value},$n);
+ $x->round(@r);
+ }
+
+sub brsft
+ {
+ # (BINT or num_str, BINT or num_str) return BINT
+ # compute x >> y, base n, y >= 0
+
+ # set up parameters
+ my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,$n,@r) = objectify(2,@_);
+ }
+
+ return $x if $x->modify('brsft');
+ return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
+ return $x->round(@r) if $y->is_zero();
+ return $x->bzero(@r) if $x->is_zero(); # 0 => 0
+
+ $n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign} eq '-';
+
+ # this only works for negative numbers when shifting in base 2
+ if (($x->{sign} eq '-') && ($n == 2))
+ {
+ return $x->round(@r) if $x->is_one('-'); # -1 => -1
+ if (!$y->is_one())
+ {
+ # although this is O(N*N) in calc (as_bin!) it is O(N) in Pari et al
+ # but perhaps there is a better emulation for two's complement shift...
+ # if $y != 1, we must simulate it by doing:
+ # convert to bin, flip all bits, shift, and be done
+ $x->binc(); # -3 => -2
+ my $bin = $x->as_bin();
+ $bin =~ s/^-0b//; # strip '-0b' prefix
+ $bin =~ tr/10/01/; # flip bits
+ # now shift
+ if (CORE::length($bin) <= $y)
+ {
+ $bin = '0'; # shifting to far right creates -1
+ # 0, because later increment makes
+ # that 1, attached '-' makes it '-1'
+ # because -1 >> x == -1 !
+ }
+ else
+ {
+ $bin =~ s/.{$y}$//; # cut off at the right side
+ $bin = '1' . $bin; # extend left side by one dummy '1'
+ $bin =~ tr/10/01/; # flip bits back
+ }
+ my $res = $self->new('0b'.$bin); # add prefix and convert back
+ $res->binc(); # remember to increment
+ $x->{value} = $res->{value}; # take over value
+ return $x->round(@r); # we are done now, magic, isn't?
+ }
+ # x < 0, n == 2, y == 1
+ $x->bdec(); # n == 2, but $y == 1: this fixes it
+ }
+
+ $x->{value} = $CALC->_rsft($x->{value},$y->{value},$n);
+ $x->round(@r);
+ }
+
+sub band
+ {
+ #(BINT or num_str, BINT or num_str) return BINT
+ # compute x & y
+
+ # set up parameters
+ my ($self,$x,$y,@r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,@r) = objectify(2,@_);
+ }
+
+ return $x if $x->modify('band');
+
+ $r[3] = $y; # no push!
+
+ return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
+
+ my $sx = $x->{sign} eq '+' ? 1 : -1;
+ my $sy = $y->{sign} eq '+' ? 1 : -1;
+
+ if ($sx == 1 && $sy == 1)
+ {
+ $x->{value} = $CALC->_and($x->{value},$y->{value});
+ return $x->round(@r);
+ }
+
+ if ($CAN{signed_and})
+ {
+ $x->{value} = $CALC->_signed_and($x->{value},$y->{value},$sx,$sy);
+ return $x->round(@r);
+ }
+
+ require $EMU_LIB;
+ __emu_band($self,$x,$y,$sx,$sy,@r);
+ }
+
+sub bior
+ {
+ #(BINT or num_str, BINT or num_str) return BINT
+ # compute x | y
+
+ # set up parameters
+ my ($self,$x,$y,@r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,@r) = objectify(2,@_);
+ }
+
+ return $x if $x->modify('bior');
+ $r[3] = $y; # no push!
+
+ return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
+
+ my $sx = $x->{sign} eq '+' ? 1 : -1;
+ my $sy = $y->{sign} eq '+' ? 1 : -1;
+
+ # the sign of X follows the sign of X, e.g. sign of Y irrelevant for bior()
+
+ # don't use lib for negative values
+ if ($sx == 1 && $sy == 1)
+ {
+ $x->{value} = $CALC->_or($x->{value},$y->{value});
+ return $x->round(@r);
+ }
+
+ # if lib can do negative values, let it handle this
+ if ($CAN{signed_or})
+ {
+ $x->{value} = $CALC->_signed_or($x->{value},$y->{value},$sx,$sy);
+ return $x->round(@r);
+ }
+
+ require $EMU_LIB;
+ __emu_bior($self,$x,$y,$sx,$sy,@r);
+ }
+
+sub bxor
+ {
+ #(BINT or num_str, BINT or num_str) return BINT
+ # compute x ^ y
+
+ # set up parameters
+ my ($self,$x,$y,@r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,@r) = objectify(2,@_);
+ }
+
+ return $x if $x->modify('bxor');
+ $r[3] = $y; # no push!
+
+ return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);
+
+ my $sx = $x->{sign} eq '+' ? 1 : -1;
+ my $sy = $y->{sign} eq '+' ? 1 : -1;
+
+ # don't use lib for negative values
+ if ($sx == 1 && $sy == 1)
+ {
+ $x->{value} = $CALC->_xor($x->{value},$y->{value});
+ return $x->round(@r);
+ }
+
+ # if lib can do negative values, let it handle this
+ if ($CAN{signed_xor})
+ {
+ $x->{value} = $CALC->_signed_xor($x->{value},$y->{value},$sx,$sy);
+ return $x->round(@r);
+ }
+
+ require $EMU_LIB;
+ __emu_bxor($self,$x,$y,$sx,$sy,@r);
+ }
+
+sub length
+ {
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ my $e = $CALC->_len($x->{value});
+ wantarray ? ($e,0) : $e;
+ }
+
+sub digit
+ {
+ # return the nth decimal digit, negative values count backward, 0 is right
+ my ($self,$x,$n) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
+
+ $n = $n->numify() if ref($n);
+ $CALC->_digit($x->{value},$n||0);
+ }
+
+sub _trailing_zeros
+ {
+ # return the amount of trailing zeros in $x (as scalar)
+ my $x = shift;
+ $x = $class->new($x) unless ref $x;
+
+ return 0 if $x->{sign} !~ /^[+-]$/; # NaN, inf, -inf etc
+
+ $CALC->_zeros($x->{value}); # must handle odd values, 0 etc
+ }
+
+sub bsqrt
+ {
+ # calculate square root of $x
+ my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
+
+ return $x if $x->modify('bsqrt');
+
+ return $x->bnan() if $x->{sign} !~ /^\+/; # -x or -inf or NaN => NaN
+ return $x if $x->{sign} eq '+inf'; # sqrt(+inf) == inf
+
+ return $upgrade->bsqrt($x,@r) if defined $upgrade;
+
+ $x->{value} = $CALC->_sqrt($x->{value});
+ $x->round(@r);
+ }
+
+sub broot
+ {
+ # calculate $y'th root of $x
+
+ # set up parameters
+ my ($self,$x,$y,@r) = (ref($_[0]),@_);
+
+ $y = $self->new(2) unless defined $y;
+
+ # objectify is costly, so avoid it
+ if ((!ref($x)) || (ref($x) ne ref($y)))
+ {
+ ($self,$x,$y,@r) = objectify(2,$self || $class,@_);
+ }
+
+ return $x if $x->modify('broot');
+
+ # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
+ return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
+ $y->{sign} !~ /^\+$/;
+
+ return $x->round(@r)
+ if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
+
+ return $upgrade->new($x)->broot($upgrade->new($y),@r) if defined $upgrade;
+
+ $x->{value} = $CALC->_root($x->{value},$y->{value});
+ $x->round(@r);
+ }
+
+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/^[+-]//; # NaN, -inf,+inf => NaN or inf
+ return $self->new($s);
+ }
+ return $self->bone() if $x->is_zero();
+
+ $self->new($x->_trailing_zeros());
+ }
+
+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} !~ /^[+-]$/)
+ {
+ # for NaN, +inf, -inf: keep the sign
+ return $self->new($x->{sign});
+ }
+ my $m = $x->copy(); delete $m->{_p}; delete $m->{_a};
+ # that's a bit inefficient:
+ my $zeros = $m->_trailing_zeros();
+ $m->brsft($zeros,10) if $zeros != 0;
+ $m;
+ }
+
+sub parts
+ {
+ # return a copy of both the exponent and the mantissa
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ ($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; my $self = ref($x) || $x; $x = $self->new($x) unless ref $x;
+
+ my ($scale,$mode) = $x->_scale_p(@_);
+
+ return $x if !defined $scale || $x->modify('bfround'); # no-op
+
+ # no-op for BigInts if $n <= 0
+ $x->bround( $x->length()-$scale, $mode) if $scale > 0;
+
+ delete $x->{_a}; # delete to save memory
+ $x->{_p} = $scale; # store new _p
+ $x;
+ }
+
+sub _scan_for_nonzero
+ {
+ # internal, used by bround() to scan for non-zeros after a '5'
+ my ($x,$pad,$xs,$len) = @_;
+
+ return 0 if $len == 1; # "5" is trailed by invisible zeros
+ my $follow = $pad - 1;
+ return 0 if $follow > $len || $follow < 1;
+
+ # use the string form to check whether only '0's follow or not
+ substr ($xs,-$follow) =~ /[^0]/ ? 1 : 0;
+ }
+
+sub fround
+ {
+ # Exists to make life easier for switch between MBF and MBI (should we
+ # autoload fxxx() like MBF does for bxxx()?)
+ my $x = shift; $x = $class->new($x) unless ref $x;
+ $x->bround(@_);
+ }
+
+sub bround
+ {
+ # accuracy: +$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(@_);
+ return $x if !defined $scale || $x->modify('bround'); # no-op
+
+ if ($x->is_zero() || $scale == 0)
+ {
+ $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
+ return $x;
+ }
+ return $x if $x->{sign} !~ /^[+-]$/; # inf, NaN
+
+ # we have fewer digits than we want to scale to
+ my $len = $x->length();
+ # convert $scale to a scalar in case it is an object (put's a limit on the
+ # number length, but this would already limited by memory constraints), makes
+ # it faster
+ $scale = $scale->numify() if ref ($scale);
+
+ # scale < 0, but > -len (not >=!)
+ if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
+ {
+ $x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale; # 3 > 2
+ 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 very costly for binary => decimal
+ # getting the entire string is also costly, but we need to do it only once
+ my $xs = $CALC->_str($x->{value});
+ my $pl = -$pad-1;
+
+ # 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;
+
+ # 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,$xs,$len) == 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
+ );
+ my $put_back = 0; # not yet modified
+
+ if (($pad > 0) && ($pad <= $len))
+ {
+ substr($xs,-$pad,$pad) = '0' x $pad; # replace with '00...'
+ $put_back = 1; # need to put back
+ }
+ elsif ($pad > $len)
+ {
+ $x->bzero(); # round to '0'
+ }
+
+ if ($round_up) # what gave test above?
+ {
+ $put_back = 1; # need to put back
+ $pad = $len, $xs = '0' x $pad if $scale < 0; # tlr: whack 0.51=>1.0
+
+ # we modify directly the string variant instead of creating a number and
+ # adding it, since that is faster (we already have the string)
+ my $c = 0; $pad ++; # for $pad == $len case
+ while ($pad <= $len)
+ {
+ $c = substr($xs,-$pad,1) + 1; $c = '0' if $c eq '10';
+ substr($xs,-$pad,1) = $c; $pad++;
+ last if $c != 0; # no overflow => early out
+ }
+ $xs = '1'.$xs if $c == 0;
+
+ }
+ $x->{value} = $CALC->_new($xs) if $put_back == 1; # put back, if needed
+
+ $x->{_a} = $scale if $scale >= 0;
+ if ($scale < 0)
+ {
+ $x->{_a} = $len+$scale;
+ $x->{_a} = 0 if $scale < -$len;
+ }
+ $x;
+ }
+
+sub bfloor
+ {
+ # return integer less or equal then number; no-op since it's already integer
+ my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
+
+ $x->round(@r);
+ }
+
+sub bceil
+ {
+ # return integer greater or equal then number; no-op since it's already int
+ my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
+
+ $x->round(@r);
+ }
+
+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.
+ $_[0]->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
+
+ my $s = '';
+ $s = $x->{sign} if $x->{sign} eq '-';
+ $s . $CALC->_as_hex($x->{value});
+ }
+
+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
+
+ my $s = ''; $s = $x->{sign} if $x->{sign} eq '-';
+ return $s . $CALC->_as_bin($x->{value});
+ }
+
+##############################################################################
+# private stuff (internal use only)
+
+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 being 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]);
+
+ my $count = abs(shift || 0);
+
+ my (@a,$k,$d); # resulting array, temp, and downgrade
+ 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;
+ $a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
+ }
+
+ no strict 'refs';
+ # disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
+ if (defined ${"$a[0]::downgrade"})
+ {
+ $d = ${"$a[0]::downgrade"};
+ ${"$a[0]::downgrade"} = undef;
+ }
+
+ my $up = ${"$a[0]::upgrade"};
+ #print "Now in objectify, my class is today $a[0], count = $count\n";
+ if ($count == 0)
+ {
+ while (@_)
+ {
+ $k = shift;
+ if (!ref($k))
+ {
+ $k = $a[0]->new($k);
+ }
+ elsif (!defined $up && ref($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)
+ {
+ $count--;
+ $k = shift;
+ if (!ref($k))
+ {
+ $k = $a[0]->new($k);
+ }
+ elsif (!defined $up && ref($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,$k;
+ }
+ push @a,@_; # return other params, too
+ }
+ if (! wantarray)
+ {
+ require Carp; Carp::croak ("$class objectify needs list context");
+ }
+ ${"$a[0]::downgrade"} = $d;
+ @a;
+ }
+
+sub _register_callback
+ {
+ my ($class,$callback) = @_;
+
+ if (ref($callback) ne 'CODE')
+ {
+ require Carp;
+ Carp::croak ("$callback is not a coderef");
+ }
+ $CALLBACKS{$class} = $callback;
+ }
+
+sub import
+ {
+ my $self = shift;
+
+ $IMPORT++; # remember we did import()
+ my @a; my $l = scalar @_;
+ for ( my $i = 0; $i < $l ; $i++ )
+ {
+ if ($_[$i] eq ':constant')
+ {
+ # this causes overlord er load to step in
+ overload::constant
+ integer => sub { $self->new(shift) },
+ binary => sub { $self->new(shift) };
+ }
+ elsif ($_[$i] eq 'upgrade')
+ {
+ # this causes upgrading
+ $upgrade = $_[$i+1]; # or undef to disable
+ $i++;
+ }
+ elsif ($_[$i] =~ /^lib$/i)
+ {
+ # this causes a different low lib to take care...
+ $CALC = $_[$i+1] || '';
+ $i++;
+ }
+ else
+ {
+ push @a, $_[$i];
+ }
+ }
+ # any non :constant stuff is handled by our parent, Exporter
+ if (@a > 0)
+ {
+ require Exporter;
+
+ $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;
+ foreach (@c)
+ {
+ $_ =~ tr/a-zA-Z0-9://cd; # limit to sane characters
+ }
+ push @c, 'FastCalc', 'Calc'; # if all fail, try these
+ $CALC = ''; # signal error
+ foreach my $lib (@c)
+ {
+ next if ($lib || '') eq '';
+ $lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
+ $lib =~ s/\.pm$//;
+ if ($] < 5.006)
+ {
+ # Perl < 5.6.0 dies with "out of memory!" when eval("") and ':constant' is
+ # used in the same script, or eval("") inside import().
+ my @parts = split /::/, $lib; # Math::BigInt => Math BigInt
+ my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
+ require File::Spec;
+ $file = File::Spec->catfile (@parts, $file);
+ eval { require "$file"; $lib->import( @c ); }
+ }
+ else
+ {
+ eval "use $lib qw/@c/;";
+ }
+ if ($@ eq '')
+ {
+ my $ok = 1;
+ # loaded it ok, see if the api_version() is high enough
+ if ($lib->can('api_version') && $lib->api_version() >= 1.0)
+ {
+ $ok = 0;
+ # api_version matches, check if it really provides anything we need
+ for my $method (qw/
+ one two ten
+ str num
+ add mul div sub dec inc
+ acmp len digit is_one is_zero is_even is_odd
+ is_two is_ten
+ new copy check from_hex from_bin as_hex as_bin zeros
+ rsft lsft xor and or
+ mod sqrt root fac pow modinv modpow log_int gcd
+ /)
+ {
+ if (!$lib->can("_$method"))
+ {
+ if (($WARN{$lib}||0) < 2)
+ {
+ require Carp;
+ Carp::carp ("$lib is missing method '_$method'");
+ $WARN{$lib} = 1; # still warn about the lib
+ }
+ $ok++; last;
+ }
+ }
+ }
+ if ($ok == 0)
+ {
+ $CALC = $lib;
+ last; # found a usable one, break
+ }
+ else
+ {
+ if (($WARN{$lib}||0) < 2)
+ {
+ my $ver = eval "\$$lib\::VERSION" || 'unknown';
+ require Carp;
+ Carp::carp ("Cannot load outdated $lib v$ver, please upgrade");
+ $WARN{$lib} = 2; # never warn again
+ }
+ }
+ }
+ }
+ if ($CALC eq '')
+ {
+ require Carp;
+ Carp::croak ("Couldn't load any math lib, not even 'Calc.pm'");
+ }
+
+ # notify callbacks
+ foreach my $class (keys %CALLBACKS)
+ {
+ &{$CALLBACKS{$class}}($CALC);
+ }
+
+ # Fill $CAN with the results of $CALC->can(...) for emulating lower math lib
+ # functions
+
+ %CAN = ();
+ for my $method (qw/ signed_and signed_or signed_xor /)
+ {
+ $CAN{$method} = $CALC->can("_$method") ? 1 : 0;
+ }
+
+ # import done
+ }
+
+sub __from_hex
+ {
+ # internal
+ # convert a (ref to) big hex string to BigInt, return undef for error
+ my $hs = shift;
+
+ my $x = Math::BigInt->bzero();
+
+ # strip underscores
+ $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
+ $hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
+
+ return $x->bnan() if $hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
+
+ my $sign = '+'; $sign = '-' if $hs =~ /^-/;
+
+ $hs =~ s/^[+-]//; # strip sign
+ $x->{value} = $CALC->_from_hex($hs);
+ $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
+ $x;
+ }
+
+sub __from_bin
+ {
+ # internal
+ # convert a (ref to) big binary string to BigInt, return undef for error
+ my $bs = shift;
+
+ my $x = Math::BigInt->bzero();
+ # strip underscores
+ $bs =~ s/([01])_([01])/$1$2/g;
+ $bs =~ s/([01])_([01])/$1$2/g;
+ return $x->bnan() if $bs !~ /^[+-]?0b[01]+$/;
+
+ my $sign = '+'; $sign = '-' if $bs =~ /^\-/;
+ $bs =~ s/^[+-]//; # strip sign
+
+ $x->{value} = $CALC->_from_bin($bs);
+ $x->{sign} = $sign unless $CALC->_is_zero($x->{value}); # no '-0'
+ $x;
+ }
+
+sub _split
+ {
+ # input: num_str; output: undef for invalid or
+ # (\$mantissa_sign,\$mantissa_value,\$mantissa_fraction,\$exp_sign,\$exp_value)
+ # 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+\z/)
+ {
+ $x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
+ return (\$sign, \$x, \'', \'', \0);
+ }
+
+ # invalid starting char?
+ return if $x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
+
+ return __from_hex($x) if $x =~ /^[\-\+]?0x/; # hex string
+ return __from_bin($x) if $x =~ /^[\-\+]?0b/; # binary string
+
+ # strip underscores between digits
+ $x =~ s/(\d)_(\d)/$1$2/g;
+ $x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3
+
+ # some possible inputs:
+ # 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
+ # .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2 # 0e999
+
+ my ($m,$e,$last) = split /[Ee]/,$x;
+ return if defined $last; # last defined => 1e2E3 or others
+ $e = '0' if !defined $e || $e eq "";
+
+ # sign,value for exponent,mantint,mantfrac
+ my ($es,$ev,$mis,$miv,$mfv);
+ # valid exponent?
+ if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
+ {
+ $es = $1; $ev = $2;
+ # valid mantissa?
+ return if $m eq '.' || $m eq '';
+ my ($mi,$mf,$lastf) = split /\./,$m;
+ return if defined $lastf; # lastf defined => 1.2.3 or others
+ $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 = $2;
+ return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
+ $mfv = $1;
+ # handle the 0e999 case here
+ $ev = 0 if $miv eq '0' && $mfv eq '';
+ return (\$mis,\$miv,\$mfv,\$es,\$ev);
+ }
+ }
+ return; # NaN, not a number
+ }
+
+##############################################################################
+# internal calculation routines (others are in Math::BigInt::Calc etc)
+
+sub __lcm
+ {
+ # (BINT or num_str, BINT or num_str) return BINT
+ # does modify first argument
+ # LCM
+
+ my ($x,$ty) = @_;
+ return $x->bnan() if ($x->{sign} eq $nan) || ($ty->{sign} eq $nan);
+ my $method = ref($x) . '::bgcd';
+ no strict 'refs';
+ $x * $ty / &$method($x,$ty);
+ }
+
+###############################################################################
+# this method returns 0 if the object can be modified, or 1 if not.
+# We use a fast constant sub() here, to avoid costly calls. Subclasses
+# may override it with special code (f.i. Math::BigInt::Constant does so)
+
+sub modify () { 0; }
+
+1;
+__END__
+
+=pod
+
+=head1 NAME
+
+Math::BigInt - Arbitrary size integer/float math package
+
+=head1 SYNOPSIS
+
+ use Math::BigInt;
+
+ # or make it faster: install (optional) Math::BigInt::GMP
+ # and always use (it will fall back to pure Perl if the
+ # GMP library is not installed):
+
+ use Math::BigInt lib => 'GMP';
+
+ my $str = '1234567890';
+ my @values = (64,74,18);
+ my $n = 1; my $sign = '-';
+
+ # Number creation
+ $x = Math::BigInt->new($str); # defaults to 0
+ $y = $x->copy(); # make a true copy
+ $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 (don't modify their arguments)
+ # (return true if the condition is met, otherwise false)
+
+ $x->is_zero(); # if $x is +0
+ $x->is_nan(); # if $x is NaN
+ $x->is_one(); # if $x is +1
+ $x->is_one('-'); # if $x is -1
+ $x->is_odd(); # if $x is odd
+ $x->is_even(); # if $x is even
+ $x->is_pos(); # if $x >= 0
+ $x->is_neg(); # if $x < 0
+ $x->is_inf($sign); # if $x is +inf, or -inf (sign is default '+')
+ $x->is_int(); # if $x is an integer (not a float)
+
+ # comparing and digit/sign extraction
+ $x->bcmp($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. If you want to preserve
+ # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
+ # necessary when mixing $a = $b assignments with non-overloaded math.
+
+ $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->binf(); # set $x to inf
+ $x->binf('-'); # set $x to -inf
+
+ $x->bneg(); # negation
+ $x->babs(); # absolute value
+ $x->bnorm(); # normalize (no-op in BigInt)
+ $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)
+ $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
+ $x->bmodinv($mod); # the inverse of $x in the given modulus $mod
+
+ $x->bpow($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
+ $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
+ $x->bfac(); # factorial of $x (1*2*3*4*..$x)
+
+ $x->round($A,$P,$mode); # round to accuracy or precision using mode $mode
+ $x->bround($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 (are no-ops),
+ # but do so 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:
+
+ # greatest common divisor (no OO style)
+ my $gcd = Math::BigInt::bgcd(@values);
+ # lowest common multiplicator (no OO style)
+ my $lcm = Math::BigInt::blcm(@values);
+
+ $x->length(); # return number of digits in number
+ ($xl,$f) = $x->length(); # length of number and length of fraction part,
+ # latter is always 0 digits long for BigInts
+
+ $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_int(); # return as BigInt (in BigInt: same as copy())
+ $x->numify(); # return as scalar (might overflow!)
+
+ # conversation to string (do not modify their argument)
+ $x->bstr(); # normalized string (e.g. '3')
+ $x->bsstr(); # norm. string in scientific notation (e.g. '3E0')
+ $x->as_hex(); # as signed hexadecimal string with prefixed 0x
+ $x->as_bin(); # as signed binary string with prefixed 0b
+
+
+ # precision and accuracy (see section about rounding for more)
+ $x->precision(); # return P of $x (or global, if P of $x undef)
+ $x->precision($n); # set P of $x to $n
+ $x->accuracy(); # return A of $x (or global, if A of $x undef)
+ $x->accuracy($n); # set A $x to $n
+
+ # Global methods
+ Math::BigInt->precision(); # get/set global P for all BigInt objects
+ Math::BigInt->accuracy(); # get/set global A for all BigInt objects
+ Math::BigInt->round_mode(); # get/set global round mode, one of
+ # 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
+ Math::BigInt->config(); # return hash containing configuration
+
+=head1 DESCRIPTION
+
+All operators (including basic math operations) are overloaded if you
+declare your big integers as
+
+ $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 Input
+
+Input values to these routines may be any string, that looks like a number
+and results in an integer, including hexadecimal and binary numbers.
+
+Scalars holding numbers may also be passed, but note that non-integer numbers
+may already have lost precision due to the conversation to float. Quote
+your input if you want BigInt to see all the digits:
+
+ $x = Math::BigInt->new(12345678890123456789); # bad
+ $x = Math::BigInt->new('12345678901234567890'); # good
+
+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.
+
+Currently, Math::BigInt::new() defaults to 0, while Math::BigInt::new('')
+results in 'NaN'. This might change in the future, so use always the following
+explicit forms to get a zero or NaN:
+
+ $zero = Math::BigInt->bzero();
+ $nan = Math::BigInt->bnan();
+
+C<bnorm()> on a BigInt object is now effectively a no-op, since the numbers
+are always stored in normalized form. If passed a string, creates a BigInt
+object from the input.
+
+=item Output
+
+Output values are BigInt objects (normalized), except for the methods which
+return a string (see L<SYNOPSIS>).
+
+Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
+C<is_nan()>, etc.) return true or false, while others (C<bcmp()>, C<bacmp()>)
+return either undef (if NaN is involved), <0, 0 or >0 and are suited for sort.
+
+=back
+
+=head1 METHODS
+
+Each of the methods below (except config(), accuracy() and precision())
+accepts three additional parameters. These arguments C<$A>, C<$P> and C<$R>
+are C<accuracy>, C<precision> and C<round_mode>. Please see the section about
+L<ACCURACY and PRECISION> for more information.
+
+=head2 config
+
+ use Data::Dumper;
+
+ print Dumper ( Math::BigInt->config() );
+ print Math::BigInt->config()->{lib},"\n";
+
+Returns a hash containing the configuration, e.g. the version number, lib
+loaded etc. The following hash keys are currently filled in with the
+appropriate information.
+
+ key Description
+ Example
+ ============================================================
+ lib Name of the low-level math library
+ Math::BigInt::Calc
+ lib_version Version of low-level math library (see 'lib')
+ 0.30
+ class The class name of config() you just called
+ Math::BigInt
+ upgrade To which class math operations might be upgraded
+ Math::BigFloat
+ downgrade To which class math operations might be downgraded
+ undef
+ precision Global precision
+ undef
+ accuracy Global accuracy
+ undef
+ round_mode Global round mode
+ even
+ version version number of the class you used
+ 1.61
+ div_scale Fallback accuracy for div
+ 40
+ trap_nan If true, traps creation of NaN via croak()
+ 1
+ trap_inf If true, traps creation of +inf/-inf via croak()
+ 1
+
+The following values can be set by passing C<config()> a reference to a hash:
+
+ trap_inf trap_nan
+ upgrade downgrade precision accuracy round_mode div_scale
+
+Example:
+
+ $new_cfg = Math::BigInt->config( { trap_inf => 1, precision => 5 } );
+
+=head2 accuracy
+
+ $x->accuracy(5); # local for $x
+ CLASS->accuracy(5); # global for all members of CLASS
+ # Note: This also applies to new()!
+
+ $A = $x->accuracy(); # read out accuracy that affects $x
+ $A = CLASS->accuracy(); # read out global accuracy
+
+Set or get the global or local accuracy, aka how many significant digits the
+results have. If you set a global accuracy, then this also applies to new()!
+
+Warning! The accuracy I<sticks>, e.g. once you created a number under the
+influence of C<< CLASS->accuracy($A) >>, all results from math operations with
+that number will also be rounded.
+
+In most cases, you should probably round the results explicitly using one of
+L<round()>, L<bround()> or L<bfround()> or by passing the desired accuracy
+to the math operation as additional parameter:
+
+ my $x = Math::BigInt->new(30000);
+ my $y = Math::BigInt->new(7);
+ print scalar $x->copy()->bdiv($y, 2); # print 4300
+ print scalar $x->copy()->bdiv($y)->bround(2); # print 4300
+
+Please see the section about L<ACCURACY AND PRECISION> for further details.
+
+Value must be greater than zero. Pass an undef value to disable it:
+
+ $x->accuracy(undef);
+ Math::BigInt->accuracy(undef);
+
+Returns the current accuracy. For C<$x->accuracy()> it will return either the
+local accuracy, or if not defined, the global. This means the return value
+represents the accuracy that will be in effect for $x:
+
+ $y = Math::BigInt->new(1234567); # unrounded
+ print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
+ $x = Math::BigInt->new(123456); # $x will be automatically rounded!
+ print "$x $y\n"; # '123500 1234567'
+ print $x->accuracy(),"\n"; # will be 4
+ print $y->accuracy(),"\n"; # also 4, since global is 4
+ print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
+ print $x->accuracy(),"\n"; # still 4
+ print $y->accuracy(),"\n"; # 5, since global is 5
+
+Note: Works also for subclasses like Math::BigFloat. Each class has it's own
+globals separated from Math::BigInt, but it is possible to subclass
+Math::BigInt and make the globals of the subclass aliases to the ones from
+Math::BigInt.
+
+=head2 precision
+
+ $x->precision(-2); # local for $x, round at the second digit right of the dot
+ $x->precision(2); # ditto, round at the second digit left of the dot
+
+ CLASS->precision(5); # Global for all members of CLASS
+ # This also applies to new()!
+ CLASS->precision(-5); # ditto
+
+ $P = CLASS->precision(); # read out global precision
+ $P = $x->precision(); # read out precision that affects $x
+
+Note: You probably want to use L<accuracy()> instead. With L<accuracy> you
+set the number of digits each result should have, with L<precision> you
+set the place where to round!
+
+C<precision()> sets or gets the global or local precision, aka at which digit
+before or after the dot to round all results. A set global precision also
+applies to all newly created numbers!
+
+In Math::BigInt, passing a negative number precision has no effect since no
+numbers have digits after the dot. In L<Math::BigFloat>, it will round all
+results to P digits after the dot.
+
+Please see the section about L<ACCURACY AND PRECISION> for further details.
+
+Pass an undef value to disable it:
+
+ $x->precision(undef);
+ Math::BigInt->precision(undef);
+
+Returns the current precision. For C<$x->precision()> it will return either the
+local precision of $x, or if not defined, the global. This means the return
+value represents the prevision that will be in effect for $x:
+
+ $y = Math::BigInt->new(1234567); # unrounded
+ print Math::BigInt->precision(4),"\n"; # set 4, print 4
+ $x = Math::BigInt->new(123456); # will be automatically rounded
+ print $x; # print "120000"!
+
+Note: Works also for subclasses like L<Math::BigFloat>. Each class has its
+own globals separated from Math::BigInt, but it is possible to subclass
+Math::BigInt and make the globals of the subclass aliases to the ones from
+Math::BigInt.
+
+=head2 brsft
+
+ $x->brsft($y,$n);
+
+Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
+2, but others work, too.
+
+Right shifting usually amounts to dividing $x by $n ** $y and truncating the
+result:
+
+
+ $x = Math::BigInt->new(10);
+ $x->brsft(1); # same as $x >> 1: 5
+ $x = Math::BigInt->new(1234);
+ $x->brsft(2,10); # result 12
+
+There is one exception, and that is base 2 with negative $x:
+
+
+ $x = Math::BigInt->new(-5);
+ print $x->brsft(1);
+
+This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
+result).
+
+=head2 new
+
+ $x = Math::BigInt->new($str,$A,$P,$R);
+
+Creates a new BigInt object from a scalar or another BigInt object. The
+input is accepted as decimal, hex (with leading '0x') or binary (with leading
+'0b').
+
+See L<Input> for more info on accepted input formats.
+
+=head2 bnan
+
+ $x = Math::BigInt->bnan();
+
+Creates a new BigInt object representing NaN (Not A Number).
+If used on an object, it will set it to NaN:
+
+ $x->bnan();
+
+=head2 bzero
+
+ $x = Math::BigInt->bzero();
+
+Creates a new BigInt object representing zero.
+If used on an object, it will set it to zero:
+
+ $x->bzero();
+
+=head2 binf
+
+ $x = Math::BigInt->binf($sign);
+
+Creates a new BigInt object representing infinity. The optional argument is
+either '-' or '+', indicating whether you want infinity or minus infinity.
+If used on an object, it will set it to infinity:
+
+ $x->binf();
+ $x->binf('-');
+
+=head2 bone
+
+ $x = Math::BigInt->binf($sign);
+
+Creates a new BigInt object representing one. The optional argument is
+either '-' or '+', indicating whether you want one or minus one.
+If used on an object, it will set it to one:
+
+ $x->bone(); # +1
+ $x->bone('-'); # -1
+
+=head2 is_one()/is_zero()/is_nan()/is_inf()
+
+
+ $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_inf(); # true if +inf
+ $x->is_inf('-'); # true if -inf (sign is default '+')
+
+These methods all test the BigInt for being one specific value and return
+true or false depending on the input. These are faster than doing something
+like:
+
+ if ($x == 0)
+
+=head2 is_pos()/is_neg()
+
+ $x->is_pos(); # true if > 0
+ $x->is_neg(); # true if < 0
+
+The methods return true if the argument is positive or negative, respectively.
+C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
+C<-inf> is negative. A C<zero> is neither positive nor negative.
+
+These methods are only testing the sign, and not the value.
+
+C<is_positive()> and C<is_negative()> are aliases to C<is_pos()> and
+C<is_neg()>, respectively. C<is_positive()> and C<is_negative()> were
+introduced in v1.36, while C<is_pos()> and C<is_neg()> were only introduced
+in v1.68.
+
+=head2 is_odd()/is_even()/is_int()
+
+ $x->is_odd(); # true if odd, false for even
+ $x->is_even(); # true if even, false for odd
+ $x->is_int(); # true if $x is an integer
+
+The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
+C<-inf> are not integers and are neither odd nor even.
+
+In BigInt, all numbers except C<NaN>, C<+inf> and C<-inf> are integers.
+
+=head2 bcmp
+
+ $x->bcmp($y);
+
+Compares $x with $y and takes the sign into account.
+Returns -1, 0, 1 or undef.
+
+=head2 bacmp
+
+ $x->bacmp($y);
+
+Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
+
+=head2 sign
+
+ $x->sign();
+
+Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
+
+If you want $x to have a certain sign, use one of the following methods:
+
+ $x->babs(); # '+'
+ $x->babs()->bneg(); # '-'
+ $x->bnan(); # 'NaN'
+ $x->binf(); # '+inf'
+ $x->binf('-'); # '-inf'
+
+=head2 digit
+
+ $x->digit($n); # return the nth digit, counting from right
+
+If C<$n> is negative, returns the digit counting from left.
+
+=head2 bneg
+
+ $x->bneg();
+
+Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
+and '-inf', respectively. Does nothing for NaN or zero.
+
+=head2 babs
+
+ $x->babs();
+
+Set the number to it's absolute value, e.g. change the sign from '-' to '+'
+and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
+numbers.
+
+=head2 bnorm
+
+ $x->bnorm(); # normalize (no-op)
+
+=head2 bnot
+
+ $x->bnot();
+
+Two's complement (bit wise not). This is equivalent to
+
+ $x->binc()->bneg();
+
+but faster.
+
+=head2 binc
+
+ $x->binc(); # increment x by 1
+
+=head2 bdec
+
+ $x->bdec(); # decrement x by 1
+
+=head2 badd
+
+ $x->badd($y); # addition (add $y to $x)
+
+=head2 bsub
+
+ $x->bsub($y); # subtraction (subtract $y from $x)
+
+=head2 bmul
+
+ $x->bmul($y); # multiplication (multiply $x by $y)
+
+=head2 bdiv
+
+ $x->bdiv($y); # divide, set $x to quotient
+ # return (quo,rem) or quo if scalar
+
+=head2 bmod
+
+ $x->bmod($y); # modulus (x % y)
+
+=head2 bmodinv
+
+ num->bmodinv($mod); # modular inverse
+
+Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
+returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
+C<bgcd($num, $mod)==1>.
+
+=head2 bmodpow
+
+ $num->bmodpow($exp,$mod); # modular exponentation
+ # ($num**$exp % $mod)
+
+Returns the value of C<$num> taken to the power C<$exp> in the modulus
+C<$mod> using binary exponentation. C<bmodpow> is far superior to
+writing
+
+ $num ** $exp % $mod
+
+because it is much faster - it reduces internal variables into
+the modulus whenever possible, so it operates on smaller numbers.
+
+C<bmodpow> also supports negative exponents.
+
+ bmodpow($num, -1, $mod)
+
+is exactly equivalent to
+
+ bmodinv($num, $mod)
+
+=head2 bpow
+
+ $x->bpow($y); # power of arguments (x ** y)
+
+=head2 blsft
+
+ $x->blsft($y); # left shift
+ $x->blsft($y,$n); # left shift, in base $n (like 10)
+
+=head2 brsft
+
+ $x->brsft($y); # right shift
+ $x->brsft($y,$n); # right shift, in base $n (like 10)
+
+=head2 band
+
+ $x->band($y); # bitwise and
+
+=head2 bior
+
+ $x->bior($y); # bitwise inclusive or
+
+=head2 bxor
+
+ $x->bxor($y); # bitwise exclusive or
+
+=head2 bnot
+
+ $x->bnot(); # bitwise not (two's complement)
+
+=head2 bsqrt
+
+ $x->bsqrt(); # calculate square-root
+
+=head2 bfac
+
+ $x->bfac(); # factorial of $x (1*2*3*4*..$x)
+
+=head2 round
+
+ $x->round($A,$P,$round_mode);
+
+Round $x to accuracy C<$A> or precision C<$P> using the round mode
+C<$round_mode>.
+
+=head2 bround
+
+ $x->bround($N); # accuracy: preserve $N digits
+
+=head2 bfround
+
+ $x->bfround($N); # round to $Nth digit, no-op for BigInts
+
+=head2 bfloor
+
+ $x->bfloor();
+
+Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
+does change $x in BigFloat.
+
+=head2 bceil
+
+ $x->bceil();
+
+Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
+does change $x in BigFloat.
+
+=head2 bgcd
+
+ bgcd(@values); # greatest common divisor (no OO style)
+
+=head2 blcm
+
+ blcm(@values); # lowest common multiplicator (no OO style)
+
+head2 length
+
+ $x->length();
+ ($xl,$fl) = $x->length();
+
+Returns the number of digits in the decimal representation of the number.
+In list context, returns the length of the integer and fraction part. For
+BigInt's, the length of the fraction part will always be 0.
+
+=head2 exponent
+
+ $x->exponent();
+
+Return the exponent of $x as BigInt.
+
+=head2 mantissa
+
+ $x->mantissa();
+
+Return the signed mantissa of $x as BigInt.
+
+=head2 parts
+
+ $x->parts(); # return (mantissa,exponent) as BigInt
+
+=head2 copy
+
+ $x->copy(); # make a true copy of $x (unlike $y = $x;)
+
+=head2 as_int
+
+ $x->as_int();
+
+Returns $x as a BigInt (truncated towards zero). In BigInt this is the same as
+C<copy()>.
+
+C<as_number()> is an alias to this method. C<as_number> was introduced in
+v1.22, while C<as_int()> was only introduced in v1.68.
+
+=head2 bstr
+
+ $x->bstr();
+
+Returns a normalized string representation of C<$x>.
+
+=head2 bsstr
+
+ $x->bsstr(); # normalized string in scientific notation
+
+=head2 as_hex
+
+ $x->as_hex(); # as signed hexadecimal string with prefixed 0x
+
+=head2 as_bin
+
+ $x->as_bin(); # as signed binary string with prefixed 0b
+
+=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(divisor) - 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 (oops) and actually gets 2 for '+100' and 4 for '+101'. Oops
+ 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 C<< Math::BigInt->accuracy() >> or
+ C<< Math::BigFloat->accuracy() >> or whatever class you are using.
+ * You can also set P globally by using C<< Math::SomeClass->precision() >>
+ likewise.
+ * Globals are classwide, and not inherited by subclasses.
+ * to undefine A, use C<< Math::SomeCLass->accuracy(undef); >>
+ * to undefine P, use C<< Math::SomeClass->precision(undef); >>
+ * Setting C<< Math::SomeClass->accuracy() >> clears automatically
+ C<< Math::SomeClass->precision() >>, and vice versa.
+ * 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, use C<< Math::SomeClass->accuracy() >>
+ * to find out the current global P, use C<< Math::SomeClass->precision() >>
+ * use C<< $x->accuracy() >> respective C<< $x->precision() >> for the local
+ setting of C<< $x >>.
+ * Please note that C<< $x->accuracy() >> respective C<< $x->precision() >>
+ return eventually defined global A or P, when C<< $x >>'s A or P is not
+ set.
+
+=item Creating numbers
+
+ * When you create a number, you can give it's desired A or P via:
+ $x = Math::BigInt->new($number,$A,$P);
+ * Only one of A or P can be defined, otherwise the result is NaN
+ * If no A or P is give ($x = Math::BigInt->new($number) form), then the
+ globals (if set) will be used. 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)
+ * If given undef for A and P, B<no> rounding will occur, and the globals will
+ B<not> be used. This is used by subclasses to create numbers without
+ suffering rounding in the parent. Thus a subclass is able to have it's own
+ globals enforced upon creation of a number by using
+ C<< $x = Math::BigInt->new($number,undef,undef) >>:
+
+ use Math::BigInt::SomeSubclass;
+ use Math::BigInt;
+
+ Math::BigInt->accuracy(2);
+ Math::BigInt::SomeSubClass->accuracy(3);
+ $x = Math::BigInt::SomeSubClass->new(1234);
+
+ $x is now 1230, and not 1200. A subclass might choose to implement
+ this otherwise, e.g. falling back to the parent's A and P.
+
+=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::BigInt internally, but setting A or P inside
+ Math::BigInt as globals does not tamper with the parts of a BigFloat.
+ 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.
+ If you set either A or P on one object, or globally, the other one will
+ be automatically cleared.
+ * If two objects are involved in an operation, and one of them has A in
+ effect, and the other P, this results in an error (NaN).
+ * A takes precedence over P (Hint: A comes before P).
+ 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 (at least) 4 more digits 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 two modi:
+ + 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 explicitly 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 right by one for binary operations like badd). 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: bigger 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 or P locally by using C<< $x->accuracy() >> or
+ C<< $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.
+ * C<< $x->accuracy() >> clears C<< $x->precision() >>, and vice versa.
+
+=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/get the global R by using C<< Math::SomeClass->round_mode() >>
+ or by setting C<< $Math::SomeClass::round_mode >>
+ * after each operation, C<< $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 C<< $x->round($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
+
+=head1 Infinity and Not a Number
+
+While BigInt has extensive handling of inf and NaN, certain quirks remain.
+
+=over 2
+
+=item oct()/hex()
+
+These perl routines currently (as of Perl v.5.8.6) cannot handle passed
+inf.
+
+ te@linux:~> perl -wle 'print 2 ** 3333'
+ inf
+ te@linux:~> perl -wle 'print 2 ** 3333 == 2 ** 3333'
+ 1
+ te@linux:~> perl -wle 'print oct(2 ** 3333)'
+ 0
+ te@linux:~> perl -wle 'print hex(2 ** 3333)'
+ Illegal hexadecimal digit 'i' ignored at -e line 1.
+ 0
+
+The same problems occur if you pass them Math::BigInt->binf() objects. Since
+overloading these routines is not possible, this cannot be fixed from BigInt.
+
+=item ==, !=, <, >, <=, >= with NaNs
+
+BigInt's bcmp() routine currently returns undef to signal that a NaN was
+involved in a comparison. However, the overload code turns that into
+either 1 or '' and thus operations like C<< NaN != NaN >> might return
+wrong values.
+
+=item log(-inf)
+
+C<< log(-inf) >> is highly weird. Since log(-x)=pi*i+log(x), then
+log(-inf)=pi*i+inf. However, since the imaginary part is finite, the real
+infinity "overshadows" it, so the number might as well just be infinity.
+However, the result is a complex number, and since BigInt/BigFloat can only
+have real numbers as results, the result is NaN.
+
+=item exp(), cos(), sin(), atan2()
+
+These all might have problems handling infinity right.
+
+=back
+
+=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 B<ONLY> method calls like C<< $x->sign(); >>
+instead relying on the internal representation.
+
+=head2 MATH LIBRARY
+
+Math with the numbers is done (by default) by a module called
+C<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';
+
+Since Math::BigInt::GMP is in almost all cases faster than Calc (especially in
+math involving really big numbers, where it is B<much> faster), and there is
+no penalty if Math::BigInt::GMP is not installed, it is a good idea to always
+use the following:
+
+ use Math::BigInt lib => 'GMP';
+
+Different low-level libraries use different formats to store the
+numbers. You should B<NOT> depend on the number having a specific format
+internally.
+
+See the respective math library module documentation for further details.
+
+=head2 SIGN
+
+The sign is either '+', '-', 'NaN', '+inf' or '-inf'.
+
+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> is always 0, except for NaN, +inf and -inf,
+where it is C<NaN>; and for C<$x == 0>, where it is C<1> (to be compatible
+with Math::BigFloat's internal representation of a zero as C<0E1>).
+
+C<$m> is currently just a copy of the original number. The relation between
+C<$e> and C<$m> will stay always the same, though their real values might
+change.
+
+=head1 EXAMPLES
+
+ use Math::BigInt;
+
+ sub bint { Math::BigInt->new(shift); }
+
+ $x = Math::BigInt->bstr("1234") # string "1234"
+ $x = "$x"; # same as bstr()
+ $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 = Math::BigInt->new(0); # BigInt "0"
+ $x--; # BigInt "-1"
+ $x = Math::BigInt->badd(4,5) # BigInt "9"
+ print $x->bsstr(); # 9e+0
+
+Examples for rounding:
+
+ use Math::BigFloat;
+ use Test;
+
+ $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 now
+ 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 B<integer> decimal, hexadecimal
+and binary constants in the given scope are converted to C<Math::BigInt>.
+This conversion happens at compile time.
+
+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';
+
+do not work. You need an explicit Math::BigInt->new() around one of the
+operands. You should also quote large constants to protect loss of precision:
+
+ use Math::BigInt;
+
+ $x = Math::BigInt->new('1234567889123456789123456789123456789');
+
+Without the quotes Perl would convert the large number to a floating point
+constant at compile time and then hand the result to BigInt, which results in
+an truncated result or a NaN.
+
+This also applies to integers that look like floating point constants:
+
+ use Math::BigInt ':constant';
+
+ print ref(123e2),"\n";
+ print ref(123.2e2),"\n";
+
+will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
+to get this to work.
+
+=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 $x takes
+more time then the actual addition.
+
+With a technique called copy-on-write, the cost of copying with overload could
+be minimized or even completely avoided. A test implementation of COW did show
+performance gains for overloaded math, but introduced a performance loss due
+to a constant overhead for all other operations. So Math::BigInt does currently
+not COW.
+
+The rewritten version of this module (vs. v0.01) is slower on certain
+operations, like C<new()>, C<bstr()> and C<numify()>. The reason are that it
+does now more work and handles much more cases. The time spent in these
+operations is usually gained in the other math operations so that code on
+the average should get (much) faster. If they don't, please contact the author.
+
+Some operations may be slower for small numbers, but are significantly faster
+for big numbers. Other operations are now constant (O(1), like C<bneg()>,
+C<babs()> etc), instead of O(N) and thus nearly always take much less time.
+These optimizations were done on purpose.
+
+If you find the Calc module to slow, try to install any of the replacement
+modules and see if they help you.
+
+=head2 Alternative math libraries
+
+You can use an alternative library to drive Math::BigInt via:
+
+ use Math::BigInt lib => 'Module';
+
+See L<MATH LIBRARY> for more information.
+
+For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
+
+=head2 SUBCLASSING
+
+=head1 Subclassing Math::BigInt
+
+The basic design of Math::BigInt allows simple subclasses with very little
+work, as long as a few simple rules are followed:
+
+=over 2
+
+=item *
+
+The public API must remain consistent, i.e. if a sub-class is overloading
+addition, the sub-class must use the same name, in this case badd(). The
+reason for this is that Math::BigInt is optimized to call the object methods
+directly.
+
+=item *
+
+The private object hash keys like C<$x->{sign}> may not be changed, but
+additional keys can be added, like C<$x->{_custom}>.
+
+=item *
+
+Accessor functions are available for all existing object hash keys and should
+be used instead of directly accessing the internal hash keys. The reason for
+this is that Math::BigInt itself has a pluggable interface which permits it
+to support different storage methods.
+
+=back
+
+More complex sub-classes may have to replicate more of the logic internal of
+Math::BigInt if they need to change more basic behaviors. A subclass that
+needs to merely change the output only needs to overload C<bstr()>.
+
+All other object methods and overloaded functions can be directly inherited
+from the parent class.
+
+At the very minimum, any subclass will need to provide it's own C<new()> and can
+store additional hash keys in the object. There are also some package globals
+that must be defined, e.g.:
+
+ # Globals
+ $accuracy = undef;
+ $precision = -2; # round to 2 decimal places
+ $round_mode = 'even';
+ $div_scale = 40;
+
+Additionally, you might want to provide the following two globals to allow
+auto-upgrading and auto-downgrading to work correctly:
+
+ $upgrade = undef;
+ $downgrade = undef;
+
+This allows Math::BigInt to correctly retrieve package globals from the
+subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
+t/Math/BigFloat/SubClass.pm completely functional subclass examples.
+
+Don't forget to
+
+ use overload;
+
+in your subclass to automatically inherit the overloading from the parent. If
+you like, you can change part of the overloading, look at Math::String for an
+example.
+
+=head1 UPGRADING
+
+When used like this:
+
+ use Math::BigInt upgrade => 'Foo::Bar';
+
+certain operations will 'upgrade' their calculation and thus the result to
+the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
+
+ use Math::BigInt upgrade => 'Math::BigFloat';
+
+As a shortcut, you can use the module C<bignum>:
+
+ use bignum;
+
+Also good for oneliners:
+
+ perl -Mbignum -le 'print 2 ** 255'
+
+This makes it possible to mix arguments of different classes (as in 2.5 + 2)
+as well es preserve accuracy (as in sqrt(3)).
+
+Beware: This feature is not fully implemented yet.
+
+=head2 Auto-upgrade
+
+The following methods upgrade themselves unconditionally; that is if upgrade
+is in effect, they will always hand up their work:
+
+=over 2
+
+=item bsqrt()
+
+=item div()
+
+=item blog()
+
+=back
+
+Beware: This list is not complete.
+
+All other methods upgrade themselves only when one (or all) of their
+arguments are of the class mentioned in $upgrade (This might change in later
+versions to a more sophisticated scheme):
+
+=head1 BUGS
+
+=over 2
+
+=item broot() does not work
+
+The broot() function in BigInt may only work for small values. This will be
+fixed in a later version.
+
+=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 bstr(), bsstr() and 'cmp'
+
+Both C<bstr()> and C<bsstr()> as well as automated stringify via overload now
+drop the leading '+'. The old code would return '+3', the new returns '3'.
+This is to be consistent with Perl and to make C<cmp> (especially with
+overloading) to work as you expect. It also solves problems with C<Test.pm>,
+because it's C<ok()> uses 'eq' internally.
+
+Mark Biggar said, when asked about to drop the '+' altogether, or make only
+C<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 comparison, but Perl will represent some numbers as 100 and others
+as 1e+308. If in doubt, convert both arguments to Math::BigInt before
+comparing them as strings:
+
+ 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
+
+Alternatively, simple use C<< <=> >> for comparisons, this will get it
+always right. There is not yet a way to get a number automatically represented
+as a string that matches exactly the way Perl represents it.
+
+See also the section about L<Infinity and Not a Number> for problems in
+comparing NaNs.
+
+=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()> or C<as_int> for the same
+effect:
+
+ $x = Math::BigFloat->new(123.45);
+ $y = $x->as_number(); # BigInt 123
+ $y = $x->as_int(); # ditto
+
+This also works for other subclasses, like Math::String.
+
+It is yet unclear whether overloaded int() should return a scalar or a BigInt.
+
+If you want a real Perl scalar, use C<numify()>:
+
+ $y = $x->numify(); # 123 as scalar
+
+This is seldom necessary, though, because this is done automatically, like
+when you access an array:
+
+ $z = $array[$x]; # does work automatically
+
+=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 careful. 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 behind 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).
+
+=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, and the operation of that 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.
+
+One solution to you problem might be autoupgrading|upgrading. See the
+pragmas L<bignum>, L<bigint> and L<bigrat> for an easy way to do this.
+
+=item bsqrt()
+
+C<bsqrt()> works only good if the result is a big integer, e.g. the square
+root of 144 is 12, but from 12 the square root is 3, regardless of rounding
+mode. The reason is that the result is always truncated to an integer.
+
+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
+
+=item brsft()
+
+For negative numbers in base see also L<brsft|brsft>.
+
+=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>, L<Math::BigRat> and L<Math::Big> as well as
+L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
+
+The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest
+because they solve the autoupgrading/downgrading issue, at least partly.
+
+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.
-The current version of this module is a preliminary version of the
-real thing that is currently (as of perl5.002) under development.
+=head1 AUTHORS
-=head1 AUTHOR
+Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
+Completely rewritten by Tels http://bloodgate.com in late 2000, 2001 - 2004
+and still at it in 2005.
-Mark Biggar, overloaded interface by Ilya Zakharevich.
+Many people contributed in one or more ways to the final beast, see the file
+CREDITS for an (incomplete) list. If you miss your name, please drop me a
+mail. Thank you!
=cut
https://perl5.git.perl.org / perl5.git / blobdiff