Mass Deface
{value}; $CALC->foo($xs); since
# underlying lib might change the reference!
my $class = "Math::BigInt";
use 5.006002;
$VERSION = '1.998';
@ISA = qw(Exporter);
@EXPORT_OK = qw(objectify bgcd blcm);
# _trap_inf and _trap_nan are internal and should never be accessed from the
# outside
use vars qw/$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 parameter 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.
# We register ops that are not registerable yet, so suppress warnings
{ no warnings;
use overload
'=' => 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
'<=>' => sub { my $rc = $_[2] ?
ref($_[0])->bcmp($_[1],$_[0]) :
$_[0]->bcmp($_[1]);
$rc = 1 unless defined $rc;
$rc <=> 0;
},
# we need '>=' to get things like "1 >= NaN" right:
'>=' => sub { my $rc = $_[2] ?
ref($_[0])->bcmp($_[1],$_[0]) :
$_[0]->bcmp($_[1]);
# if there was a NaN involved, return false
return '' unless defined $rc;
$rc >= 0;
},
'cmp' => sub {
$_[2] ?
"$_[1]" cmp $_[0]->bstr() :
$_[0]->bstr() cmp "$_[1]" },
'cos' => sub { $_[0]->copy->bcos(); },
'sin' => sub { $_[0]->copy->bsin(); },
'atan2' => sub { $_[2] ?
ref($_[0])->new($_[1])->batan2($_[0]) :
$_[0]->copy()->batan2($_[1]) },
# are not yet overloadable
#'hex' => sub { print "hex"; $_[0]; },
#'oct' => sub { print "oct"; $_[0]; },
# log(N) is log(N, e), where e is Euler's number
'log' => sub { $_[0]->copy()->blog($_[1], undef); },
'exp' => sub { $_[0]->copy()->bexp($_[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(); }
;
} # no warnings scope
##############################################################################
# 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', 'trunc' or 'common'
$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::Calc'; # module to do the low level math
# default is Calc.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|common)$/)
{
require Carp; Carp::croak ("Unknown round mode '$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 $class = shift || 'Math::BigInt';
no strict 'refs';
if (@_ > 1 || (@_ == 1 && (ref($_[0]) eq 'HASH')))
{
# 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}::$key"};
};
if (@_ == 1 && (ref($_[0]) ne 'HASH'))
{
# calls of the style config('lib') return just this value
return $cfg->{$_[0]};
}
$cfg;
}
sub _scale_a
{
# select accuracy parameter based on precedence,
# used by bround() and bfround(), may return undef for scale (means no op)
my ($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;
if (defined $scale)
{
$scale = $scale->can('numify') ? $scale->numify() : "$scale" if ref($scale);
$scale = int($scale);
}
($scale,$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;
if (defined $scale)
{
$scale = $scale->can('numify') ? $scale->numify() : "$scale" if ref($scale);
$scale = int($scale);
}
($scale,$mode);
}
##############################################################################
# constructors
sub copy
{
# if two arguments, the first one is the class to "swallow" subclasses
if (@_ > 1)
{
my $self = bless {
sign => $_[1]->{sign},
value => $CALC->_copy($_[1]->{value}),
}, $_[0] if @_ > 1;
$self->{_a} = $_[1]->{_a} if defined $_[1]->{_a};
$self->{_p} = $_[1]->{_p} if defined $_[1]->{_p};
return $self;
}
my $self = bless {
sign => $_[0]->{sign},
value => $CALC->_copy($_[0]->{value}),
}, ref($_[0]);
$self->{_a} = $_[0]->{_a} if defined $_[0]->{_a};
$self->{_p} = $_[0]->{_p} if defined $_[0]->{_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 conversion
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';
# convert to normal scalar for speed and correctness in inner parts
$a = $a->can('numify') ? $a->numify() : "$a" if defined $a && ref($a);
$p = $p->can('numify') ? $p->numify() : "$p" if defined $p && ref($p);
# 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|common)$/)
{
require Carp; Carp::croak ("Unknown round mode '$r'");
}
$a = int($a) if defined $a;
$p = int($p) if defined $p;
($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 embedded 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|common)$/)
{
require Carp; Carp::croak ("Unknown round mode '$r'");
}
# now round, by calling either fround or ffround:
if (defined $a)
{
$self->bround(int($a),$r) if !defined $self->{_a} || $self->{_a} >= $a;
}
else # both can't be undefined due to early out
{
$self->bfround(int($p),$r) if !defined $self->{_p} || $self->{_p} <= $p;
}
# bround() or bfround() already called bnorm() if nec.
$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 bsgn {
# Signum function.
my $self = shift;
return $self if $self->modify('bsgn');
return $self -> bone("+") if $self -> is_pos();
return $self -> bone("-") if $self -> is_neg();
return $self; # zero or NaN
}
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;
}
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
}
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
}
}
$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,@_);
}
return $x if $x->modify('bsub');
return $upgrade->new($x)->bsub($upgrade->new($y),@r) if defined $upgrade &&
((!$x->isa($self)) || (!$y->isa($self)));
return $x->round(@r) if $y->is_zero();
# 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 nec.
}
sub binc
{
# increment arg by one
my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
return $x if $x->modify('binc');
if ($x->{sign} eq '+')
{
$x->{value} = $CALC->_inc($x->{value});
return $x->round($a,$p,$r);
}
elsif ($x->{sign} eq '-')
{
$x->{value} = $CALC->_dec($x->{value});
$x->{sign} = '+' if $CALC->_is_zero($x->{value}); # -1 +1 => -0 => +0
return $x->round($a,$p,$r);
}
# inf, nan handling etc
$x->badd($self->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);
}
sub blog
{
# calculate $x = $a ** $base + $b and return $a (e.g. the log() to base
# $base of $x)
# 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(2,@_);
}
return $x if $x->modify('blog');
$base = $self->new($base) if defined $base && !ref $base;
# inf, -inf, NaN, <0 => NaN
return $x->bnan()
if $x->{sign} ne '+' || (defined $base && $base->{sign} ne '+');
return $upgrade->blog($upgrade->new($x),$base,@r) if
defined $upgrade;
# fix for bug #24969:
# the default base is e (Euler's number) which is not an integer
if (!defined $base)
{
require Math::BigFloat;
my $u = Math::BigFloat->blog(Math::BigFloat->new($x))->as_int();
# modify $x in place
$x->{value} = $u->{value};
$x->{sign} = $u->{sign};
return $x;
}
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);
}
sub bnok
{
# Calculate n over k (binomial coefficient or "choose" function) as integer.
# 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('bnok');
return $x->bnan() if $x->{sign} eq 'NaN' || $y->{sign} eq 'NaN';
return $x->binf() if $x->{sign} eq '+inf';
# k > n or k < 0 => 0
my $cmp = $x->bacmp($y);
return $x->bzero() if $cmp < 0 || $y->{sign} =~ /^-/;
# k == n => 1
return $x->bone(@r) if $cmp == 0;
if ($CALC->can('_nok'))
{
$x->{value} = $CALC->_nok($x->{value},$y->{value});
}
else
{
# ( 7 ) 7! 1*2*3*4 * 5*6*7 5 * 6 * 7 6 7
# ( - ) = --------- = --------------- = --------- = 5 * - * -
# ( 3 ) (7-3)! 3! 1*2*3*4 * 1*2*3 1 * 2 * 3 2 3
if (!$y->is_zero())
{
my $z = $x - $y;
$z->binc();
my $r = $z->copy(); $z->binc();
my $d = $self->new(2);
while ($z->bacmp($x) <= 0) # f <= x ?
{
$r->bmul($z); $r->bdiv($d);
$z->binc(); $d->binc();
}
$x->{value} = $r->{value}; $x->{sign} = '+';
}
else { $x->bone(); }
}
$x->round(@r);
}
sub bexp
{
# Calculate e ** $x (Euler's number to the power of X), truncated to
# an integer value.
my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
return $x if $x->modify('bexp');
# inf, -inf, NaN, <0 => NaN
return $x->bnan() if $x->{sign} eq 'NaN';
return $x->bone() if $x->is_zero();
return $x if $x->{sign} eq '+inf';
return $x->bzero() if $x->{sign} eq '-inf';
my $u;
{
# run through Math::BigFloat unless told otherwise
require Math::BigFloat unless defined $upgrade;
local $upgrade = 'Math::BigFloat' unless defined $upgrade;
# calculate result, truncate it to integer
$u = $upgrade->bexp($upgrade->new($x),@r);
}
if (!defined $upgrade)
{
$u = $u->as_int();
# modify $x in place
$x->{value} = $u->{value};
$x->round(@r);
}
else { $x = $u; }
}
sub blcm
{
# (BINT or num_str, BINT or num_str) return BINT
# does not modify arguments, but returns new object
# Lowest Common Multiple
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;
}
sub bgcd
{
# (BINT or num_str, BINT or num_str) return BINT
# does not modify arguments, but returns new object
# GCD -- Euclid's algorithm, variant C (Knuth Vol 3, pg 341 ff)
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?
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;
}
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
}
##############################################################################
# is_foo test routines
# we don't need $self, so undef instead of ref($_[0]) make it slightly faster
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});
}
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;
}
sub is_inf
{
# return true if arg (BINT or num_str) is +-inf
my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
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});
}
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,@_);
return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
$CALC->_is_odd($x->{value});
}
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,@_);
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 the first number by the second number
# (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 bmuladd
{
# multiply two numbers and then add the third to the result
# (BINT or num_str, BINT or num_str, BINT or num_str) return BINT
# set up parameters
my ($self,$x,$y,$z,@r) = objectify(3,@_);
return $x if $x->modify('bmuladd');
return $x->bnan() if ($x->{sign} eq $nan) ||
($y->{sign} eq $nan) ||
($z->{sign} eq $nan);
# inf handling of x and y
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('-');
}
# inf handling x*y and z
if (($z->{sign} =~ /^[+-]inf$/))
{
# something +-inf => +-inf
$x->{sign} = $z->{sign}, return $x if $z->{sign} =~ /^[+-]inf$/;
}
return $upgrade->bmuladd($x,$upgrade->new($y),$upgrade->new($z),@r)
if defined $upgrade && (!$y->isa($self) || !$z->isa($self) || !$x->isa($self));
# TODO: what if $y and $z have A or P set?
$r[3] = $z; # 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
my ($sx, $sz) = ( $x->{sign}, $z->{sign} ); # get signs
if ($sx eq $sz)
{
$x->{value} = $CALC->_add($x->{value},$z->{value}); # same sign, abs add
}
else
{
my $a = $CALC->_acmp ($z->{value},$x->{value}); # absolute compare
if ($a > 0)
{
$x->{value} = $CALC->_sub($z->{value},$x->{value},1); # abs sub w/ swap
$x->{sign} = $sz;
}
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}, $z->{value}); # abs sub
}
}
$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, remainder 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
{
# Return modular multiplicative inverse: z is the modular inverse of x (mod
# y) if and only if x*z (mod y) = 1 (mod y). If the modulus y is larger than
# one, x and z are relative primes (i.e., their greatest common divisor is
# one).
#
# If no modular multiplicative inverse exists, 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 NaN if one or both arguments is +inf, -inf, or nan.
return $x->bnan() if ($y->{sign} !~ /^[+-]$/ ||
$x->{sign} !~ /^[+-]$/);
# Return NaN if $y is zero; 1 % 0 makes no sense.
return $x->bnan() if $y->is_zero();
# Return 0 in the trivial case. $x % 1 or $x % -1 is zero for all finite
# integers $x.
return $x->bzero() if ($y->is_one() ||
$y->is_one('-'));
# Return NaN if $x = 0, or $x modulo $y is zero. The only valid case when
# $x = 0 is when $y = 1 or $y = -1, but that was covered above.
#
# Note that computing $x modulo $y here affects the value we'll feed to
# $CALC->_modinv() below when $x and $y have opposite signs. E.g., if $x =
# 5 and $y = 7, those two values are fed to _modinv(), but if $x = -5 and
# $y = 7, the values fed to _modinv() are $x = 2 (= -5 % 7) and $y = 7.
# The value if $x is affected only when $x and $y have opposite signs.
$x->bmod($y);
return $x->bnan() if $x->is_zero();
# Compute the modular multiplicative inverse of the absolute values. We'll
# correct for the signs of $x and $y later. Return NaN if no GCD is found.
($x->{value}, $x->{sign}) = $CALC->_modinv($x->{value}, $y->{value});
return $x->bnan() if !defined $x->{value};
# Library inconsistency workaround: _modinv() in Math::BigInt::GMP versions
# <= 1.32 return undef rather than a "+" for the sign.
$x->{sign} = '+' unless defined $x->{sign};
# When one or both arguments are negative, we have the following
# relations. If x and y are positive:
#
# modinv(-x, -y) = -modinv(x, y)
# modinv(-x, y) = y - modinv(x, y) = -modinv(x, y) (mod y)
# modinv( x, -y) = modinv(x, y) - y = modinv(x, y) (mod -y)
# We must swap the sign of the result if the original $x is negative.
# However, we must compensate for ignoring the signs when computing the
# inverse modulo. The net effect is that we must swap the sign of the
# result if $y is negative.
$x -> bneg() if $y->{sign} eq '-';
# Compute $x modulo $y again after correcting the sign.
$x -> bmod($y) if $x->{sign} ne $y->{sign};
return $x;
}
sub bmodpow
{
# Modular exponentiation. Raises a very large number to a very large exponent
# in a given very large modulus quickly, thanks to binary exponentiation.
# Supports negative exponents.
my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
return $num if $num->modify('bmodpow');
# When the exponent 'e' is negative, use the following relation, which is
# based on finding the multiplicative inverse 'd' of 'b' modulo 'm':
#
# b^(-e) (mod m) = d^e (mod m) where b*d = 1 (mod m)
$num->bmodinv($mod) if ($exp->{sign} eq '-');
# Check for valid input. All operands must be finite, and the modulus must be
# non-zero.
return $num->bnan() if ($num->{sign} =~ /NaN|inf/ || # NaN, -inf, +inf
$exp->{sign} =~ /NaN|inf/ || # NaN, -inf, +inf
$mod->{sign} =~ /NaN|inf/ || # NaN, -inf, +inf
$mod->is_zero());
# Compute 'a (mod m)', ignoring the signs on 'a' and 'm'. If the resulting
# value is zero, the output is also zero, regardless of the signs on 'a' and
# 'm'.
my $value = $CALC->_modpow($num->{value}, $exp->{value}, $mod->{value});
my $sign = '+';
# If the resulting value is non-zero, we have four special cases, depending
# on the signs on 'a' and 'm'.
unless ($CALC->_is_zero($value)) {
# There is a negative sign on 'a' (= $num**$exp) only if the number we
# are exponentiating ($num) is negative and the exponent ($exp) is odd.
if ($num->{sign} eq '-' && $exp->is_odd()) {
# When both the number 'a' and the modulus 'm' have a negative sign,
# use this relation:
#
# -a (mod -m) = -(a (mod m))
if ($mod->{sign} eq '-') {
$sign = '-';
}
# When only the number 'a' has a negative sign, use this relation:
#
# -a (mod m) = m - (a (mod m))
else {
# Use copy of $mod since _sub() modifies the first argument.
my $mod = $CALC->_copy($mod->{value});
$value = $CALC->_sub($mod, $value);
$sign = '+';
}
} else {
# When only the modulus 'm' has a negative sign, use this relation:
#
# a (mod -m) = (a (mod m)) - m
# = -(m - (a (mod m)))
if ($mod->{sign} eq '-') {
# Use copy of $mod since _sub() modifies the first argument.
my $mod = $CALC->_copy($mod->{value});
$value = $CALC->_sub($mod, $value);
$sign = '-';
}
# When neither the number 'a' nor the modulus 'm' have a negative
# sign, directly return the already computed value.
#
# (a (mod m))
}
}
$num->{value} = $value;
$num->{sign} = $sign;
return $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) || $y->{sign} eq '-');
$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 ($y >= CORE::length($bin))
{
$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();
# 12300 => 2 trailing zeros => exponent is 2
$self->new( $CALC->_zeros($x->{value}) );
}
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 = $CALC->_zeros($m->{value});
$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});
}
sub as_oct
{
# return as octal string, with prefixed 0
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_oct($x->{value});
}
##############################################################################
# private stuff (internal use only)
sub objectify {
# Convert strings and "foreign objects" to the objects we want.
# The first argument, $count, is the number of following arguments that
# objectify() looks at and converts to objects. The first is a classname.
# If the given count is 0, all arguments will be used.
# After the count is read, objectify obtains the name of the class to which
# the following arguments are converted. If the second argument is a
# reference, use the reference type as the class name. Otherwise, if it is
# a string that looks like a class name, use that. Otherwise, use $class.
# 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
# A shortcut for the common case $x->unary_op():
return (ref($_[1]), $_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
# Check the context.
unless (wantarray) {
require Carp;
Carp::croak ("${class}::objectify() needs list context");
}
# Get the number of arguments to objectify.
my $count = shift;
$count ||= @_;
# Initialize the output array.
my @a = @_;
# If the first argument is a reference, use that reference type as our
# class name. Otherwise, if the first argument looks like a class name,
# then use that as our class name. Otherwise, use the default class name.
{
if (ref($a[0])) { # reference?
unshift @a, ref($a[0]);
last;
}
if ($a[0] =~ /^[A-Z].*::/) { # string with class name?
last;
}
unshift @a, $class; # default class name
}
no strict 'refs';
# What we upgrade to, if anything.
my $up = ${"$a[0]::upgrade"};
# Disable downgrading, because Math::BigFloat -> foo('1.0','2.0') needs
# floats.
my $down;
if (defined ${"$a[0]::downgrade"}) {
$down = ${"$a[0]::downgrade"};
${"$a[0]::downgrade"} = undef;
}
for my $i (1 .. $count) {
my $ref = ref $a[$i];
# If it is an object of the right class, all is fine.
if ($ref eq $a[0]) {
next;
}
# Don't do anything with undefs.
unless (defined($a[$i])) {
next;
}
# Perl scalars are fed to the appropriate constructor.
unless ($ref) {
$a[$i] = $a[0] -> new($a[$i]);
next;
}
# Upgrading is OK, so skip further tests if the argument is upgraded.
if (defined $up && $ref eq $up) {
next;
}
# If we want a Math::BigInt, see if the object can become one.
# Support the old misnomer as_number().
if ($a[0] eq 'Math::BigInt') {
if ($a[$i] -> can('as_int')) {
$a[$i] = $a[$i] -> as_int();
next;
}
if ($a[$i] -> can('as_number')) {
$a[$i] = $a[$i] -> as_number();
next;
}
}
# If we want a Math::BigFloat, see if the object can become one.
if ($a[0] eq 'Math::BigFloat') {
if ($a[$i] -> can('as_float')) {
$a[$i] = $a[$i] -> as_float();
next;
}
}
# Last resort.
$a[$i] = $a[0] -> new($a[$i]);
}
# Reset the downgrading.
${"$a[0]::downgrade"} = $down;
return @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 @_;
my $warn_or_die = 0; # 0 - no warn, 1 - warn, 2 - die
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|try|only)\z/)
{
# this causes a different low lib to take care...
$CALC = $_[$i+1] || '';
# lib => 1 (warn on fallback), try => 0 (no warn), only => 2 (die on fallback)
$warn_or_die = 1 if $_[$i] eq 'lib';
$warn_or_die = 2 if $_[$i] eq 'only';
$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, \'Calc' # if all fail, try these
if $warn_or_die < 2; # but not for "only"
$CALC = ''; # signal error
foreach my $l (@c)
{
# fallback libraries are "marked" as \'string', extract string if nec.
my $lib = $l; $lib = $$l if ref($l);
next if ($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 $method (qw/
one two ten
str num
add mul div sub dec inc
acmp len digit is_one is_zero is_even is_odd
is_two is_ten
zeros new copy check
from_hex from_oct from_bin as_hex as_bin as_oct
rsft lsft xor and or
mod sqrt root fac pow modinv modpow log_int gcd
/)
{
if (!$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 = $lib;
if ($warn_or_die > 0 && ref($l))
{
require Carp;
my $msg = "Math::BigInt: couldn't load specified math lib(s), fallback to $lib";
Carp::carp ($msg) if $warn_or_die == 1;
Carp::croak ($msg) if $warn_or_die == 2;
}
last; # found a usable one, break
}
else
{
if (($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 ($CALC eq '')
{
require Carp;
if ($warn_or_die == 2)
{
Carp::croak ("Couldn't load specified math lib(s) and fallback disallowed");
}
else
{
Carp::croak ("Couldn't load any math lib(s), not even fallback to 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 {
# Create a bigint from a hexadecimal string.
my ($self, $str) = @_;
if ($str =~ s/
^
( [+-]? )
(0?x)?
(
[0-9a-fA-F]*
( _ [0-9a-fA-F]+ )*
)
$
//x)
{
# Get a "clean" version of the string, i.e., non-emtpy and with no
# underscores or invalid characters.
my $sign = $1;
my $chrs = $3;
$chrs =~ tr/_//d;
$chrs = '0' unless CORE::length $chrs;
# Initialize output.
my $x = Math::BigInt->bzero();
# The library method requires a prefix.
$x->{value} = $CALC->_from_hex('0x' . $chrs);
# Place the sign.
if ($sign eq '-' && ! $CALC->_is_zero($x->{value})) {
$x->{sign} = '-';
}
return $x;
}
# CORE::hex() parses as much as it can, and ignores any trailing garbage.
# For backwards compatibility, we return NaN.
return $self->bnan();
}
sub from_oct {
# Create a bigint from an octal string.
my ($self, $str) = @_;
if ($str =~ s/
^
( [+-]? )
(
[0-7]*
( _ [0-7]+ )*
)
$
//x)
{
# Get a "clean" version of the string, i.e., non-emtpy and with no
# underscores or invalid characters.
my $sign = $1;
my $chrs = $2;
$chrs =~ tr/_//d;
$chrs = '0' unless CORE::length $chrs;
# Initialize output.
my $x = Math::BigInt->bzero();
# The library method requires a prefix.
$x->{value} = $CALC->_from_oct('0' . $chrs);
# Place the sign.
if ($sign eq '-' && ! $CALC->_is_zero($x->{value})) {
$x->{sign} = '-';
}
return $x;
}
# CORE::oct() parses as much as it can, and ignores any trailing garbage.
# For backwards compatibility, we return NaN.
return $self->bnan();
}
sub from_bin {
# Create a bigint from a binary string.
my ($self, $str) = @_;
if ($str =~ s/
^
( [+-]? )
(0?b)?
(
[01]*
( _ [01]+ )*
)
$
//x)
{
# Get a "clean" version of the string, i.e., non-emtpy and with no
# underscores or invalid characters.
my $sign = $1;
my $chrs = $3;
$chrs =~ tr/_//d;
$chrs = '0' unless CORE::length $chrs;
# Initialize output.
my $x = Math::BigInt->bzero();
# The library method requires a prefix.
$x->{value} = $CALC->_from_bin('0b' . $chrs);
# Place the sign.
if ($sign eq '-' && ! $CALC->_is_zero($x->{value})) {
$x->{sign} = '-';
}
return $x;
}
# For consistency with from_hex() and from_oct(), we return NaN when the
# input is invalid.
return $self->bnan();
}
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 extraneous 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 =~ /^[+-]?[0-9]+\z/)
{
$x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
return (\$sign, \$x, \'', \'', \0);
}
# invalid starting char?
return if $x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
return Math::BigInt->from_hex($x) if $x =~ /^[+-]?0x/; # hex string
return Math::BigInt->from_bin($x) if $x =~ /^[+-]?0b/; # binary string
# strip underscores between digits
$x =~ s/([0-9])_([0-9])/$1$2/g;
$x =~ s/([0-9])_([0-9])/$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 ($e =~ /^([+-]?)0*([0-9]+)$/) # 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*([0-9]+)$/) # strip leading zeros
{
$mis = $1||'+'; $miv = $2;
return unless ($mf =~ /^([0-9]*?)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);
}
###############################################################################
# trigonometric functions
sub bpi
{
# Calculate PI to N digits. Unless upgrading is in effect, returns the
# result truncated to an integer, that is, always returns '3'.
my ($self,$n) = @_;
if (@_ == 1)
{
# called like Math::BigInt::bpi(10);
$n = $self; $self = $class;
}
$self = ref($self) if ref($self);
return $upgrade->new($n) if defined $upgrade;
# hard-wired to "3"
$self->new(3);
}
sub bcos
{
# Calculate cosinus(x) to N digits. Unless upgrading is in effect, returns the
# result truncated to an integer.
my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
return $x if $x->modify('bcos');
return $x->bnan() if $x->{sign} !~ /^[+-]\z/; # -inf +inf or NaN => NaN
return $upgrade->new($x)->bcos(@r) if defined $upgrade;
require Math::BigFloat;
# calculate the result and truncate it to integer
my $t = Math::BigFloat->new($x)->bcos(@r)->as_int();
$x->bone() if $t->is_one();
$x->bzero() if $t->is_zero();
$x->round(@r);
}
sub bsin
{
# Calculate sinus(x) to N digits. Unless upgrading is in effect, returns the
# result truncated to an integer.
my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
return $x if $x->modify('bsin');
return $x->bnan() if $x->{sign} !~ /^[+-]\z/; # -inf +inf or NaN => NaN
return $upgrade->new($x)->bsin(@r) if defined $upgrade;
require Math::BigFloat;
# calculate the result and truncate it to integer
my $t = Math::BigFloat->new($x)->bsin(@r)->as_int();
$x->bone() if $t->is_one();
$x->bzero() if $t->is_zero();
$x->round(@r);
}
sub batan2
{
# calculate arcus tangens of ($y/$x)
# set up parameters
my ($self,$y,$x,@r) = (ref($_[0]),@_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
{
($self,$y,$x,@r) = objectify(2,@_);
}
return $y if $y->modify('batan2');
return $y->bnan() if ($y->{sign} eq $nan) || ($x->{sign} eq $nan);
# Y X
# != 0 -inf result is +- pi
if ($x->is_inf() || $y->is_inf())
{
# upgrade to BigFloat etc.
return $upgrade->new($y)->batan2($upgrade->new($x),@r) if defined $upgrade;
if ($y->is_inf())
{
if ($x->{sign} eq '-inf')
{
# calculate 3 pi/4 => 2.3.. => 2
$y->bone( substr($y->{sign},0,1) );
$y->bmul($self->new(2));
}
elsif ($x->{sign} eq '+inf')
{
# calculate pi/4 => 0.7 => 0
$y->bzero();
}
else
{
# calculate pi/2 => 1.5 => 1
$y->bone( substr($y->{sign},0,1) );
}
}
else
{
if ($x->{sign} eq '+inf')
{
# calculate pi/4 => 0.7 => 0
$y->bzero();
}
else
{
# PI => 3.1415.. => 3
$y->bone( substr($y->{sign},0,1) );
$y->bmul($self->new(3));
}
}
return $y;
}
return $upgrade->new($y)->batan2($upgrade->new($x),@r) if defined $upgrade;
require Math::BigFloat;
my $r = Math::BigFloat->new($y)->batan2(Math::BigFloat->new($x),@r)->as_int();
$x->{value} = $r->{value};
$x->{sign} = $r->{sign};
$x;
}
sub batan
{
# Calculate arcus tangens of x to N digits. Unless upgrading is in effect, returns the
# result truncated to an integer.
my ($self,$x,@r) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
return $x if $x->modify('batan');
return $x->bnan() if $x->{sign} !~ /^[+-]\z/; # -inf +inf or NaN => NaN
return $upgrade->new($x)->batan(@r) if defined $upgrade;
# calculate the result and truncate it to integer
my $t = Math::BigFloat->new($x)->batan(@r);
$x->{value} = $CALC->_new( $x->as_int()->bstr() );
$x->round(@r);
}
###############################################################################
# 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 with huge numbers: install (optional)
# Math::BigInt::GMP and always use (it will fall back to
# pure Perl if the GMP library is not installed):
# (See also the L for more information.
For more benchmark results see L