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 similar)
# sign : +, -, NaN, +inf, -inf
# _a : accuracy
# _p : precision
# Remember not to take shortcuts ala $xs = $x->{value}; $CALC->foo($xs); since
# underlying lib might change the reference!
use 5.006001;
use strict;
use warnings;
use Carp ();
our $VERSION = '1.999806';
our @ISA = qw(Exporter);
our @EXPORT_OK = qw(objectify bgcd blcm);
my $class = "Math::BigInt";
# 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.
use overload
# overload key: with_assign
'+' => sub { $_[0] -> copy() -> badd($_[1]); },
'-' => sub { my $c = $_[0] -> copy;
$_[2] ? $c -> bneg() -> badd($_[1])
: $c -> bsub($_[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]); },
# overload key: assign
'+=' => sub { $_[0]->badd($_[1]); },
'-=' => sub { $_[0]->bsub($_[1]); },
'*=' => sub { $_[0]->bmul($_[1]); },
'/=' => sub { scalar $_[0]->bdiv($_[1]); },
'%=' => sub { $_[0]->bmod($_[1]); },
'**=' => sub { $_[0]->bpow($_[1]); },
'<<=' => sub { $_[0]->blsft($_[1]); },
'>>=' => sub { $_[0]->brsft($_[1]); },
# 'x=' => sub { },
# '.=' => sub { },
# overload key: num_comparison
'<' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> blt($_[0])
: $_[0] -> blt($_[1]); },
'<=' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> ble($_[0])
: $_[0] -> ble($_[1]); },
'>' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bgt($_[0])
: $_[0] -> bgt($_[1]); },
'>=' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bge($_[0])
: $_[0] -> bge($_[1]); },
'==' => sub { $_[0] -> beq($_[1]); },
'!=' => sub { $_[0] -> bne($_[1]); },
# overload key: 3way_comparison
'<=>' => sub { my $cmp = $_[0] -> bcmp($_[1]);
defined($cmp) && $_[2] ? -$cmp : $cmp; },
'cmp' => sub { $_[2] ? "$_[1]" cmp $_[0] -> bstr()
: $_[0] -> bstr() cmp "$_[1]"; },
# overload key: str_comparison
# 'lt' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bstrlt($_[0])
# : $_[0] -> bstrlt($_[1]); },
#
# 'le' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bstrle($_[0])
# : $_[0] -> bstrle($_[1]); },
#
# 'gt' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bstrgt($_[0])
# : $_[0] -> bstrgt($_[1]); },
#
# 'ge' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bstrge($_[0])
# : $_[0] -> bstrge($_[1]); },
#
# 'eq' => sub { $_[0] -> bstreq($_[1]); },
#
# 'ne' => sub { $_[0] -> bstrne($_[1]); },
# overload key: binary
'&' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> band($_[0])
: $_[0] -> copy -> band($_[1]); },
'&=' => sub { $_[0] -> band($_[1]); },
'|' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bior($_[0])
: $_[0] -> copy -> bior($_[1]); },
'|=' => sub { $_[0] -> bior($_[1]); },
'^' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bxor($_[0])
: $_[0] -> copy -> bxor($_[1]); },
'^=' => sub { $_[0] -> bxor($_[1]); },
# '&.' => sub { },
# '&.=' => sub { },
# '|.' => sub { },
# '|.=' => sub { },
# '^.' => sub { },
# '^.=' => sub { },
# overload key: unary
'neg' => sub { $_[0] -> copy() -> bneg(); },
# '!' => sub { },
'~' => sub { $_[0] -> copy() -> bnot(); },
# '~.' => sub { },
# overload key: mutators
'++' => sub { $_[0] -> binc() },
'--' => sub { $_[0] -> bdec() },
# overload key: func
'atan2' => sub { $_[2] ? ref($_[0]) -> new($_[1]) -> batan2($_[0])
: $_[0] -> copy() -> batan2($_[1]); },
'cos' => sub { $_[0] -> copy -> bcos(); },
'sin' => sub { $_[0] -> copy -> bsin(); },
'exp' => sub { $_[0] -> copy() -> bexp($_[1]); },
'abs' => sub { $_[0] -> copy() -> babs(); },
'log' => sub { $_[0] -> copy() -> blog(); },
'sqrt' => sub { $_[0] -> copy() -> bsqrt(); },
'int' => sub { $_[0] -> copy() -> bint(); },
# overload key: conversion
'bool' => sub { $_[0] -> is_zero() ? '' : 1; },
'""' => sub { $_[0] -> bstr(); },
'0+' => sub { $_[0] -> numify(); },
'=' => sub { $_[0]->copy(); },
;
##############################################################################
# global constants, flags and accessory
# These vars are public, but their direct usage is not recommended, use the
# accessor methods instead
our $round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'
our $accuracy = undef;
our $precision = undef;
our $div_scale = 40;
our $upgrade = undef; # default is no upgrade
our $downgrade = undef; # default is no downgrade
# These are internally, and not to be used from the outside at all
our $_trap_nan = 0; # are NaNs ok? set w/ config()
our $_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
our $rnd_mode = 'even';
sub TIESCALAR {
my ($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;
}
###############################################################################
# Configuration methods
###############################################################################
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)$/) {
Carp::croak("Unknown round mode '$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) {
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 || $a <= 0) {
Carp::croak('Argument to accuracy must be greater than zero');
}
if (int($a) != $a) {
Carp::croak('Argument to accuracy must be an integer');
}
}
if (ref($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) != $p)) {
Carp::croak('Argument to precision must be an integer');
}
if (ref($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 $class = shift || __PACKAGE__;
no strict 'refs';
if (@_ > 1 || (@_ == 1 && (ref($_[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/
accuracy precision
round_mode div_scale
upgrade downgrade
trap_inf trap_nan
/)
{
$set_args->{$key} = $args->{$key} if exists $args->{$key};
delete $args->{$key};
}
if (keys %$args > 0) {
Carp::croak("Illegal key(s) '", join("', '", keys %$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/
accuracy precision
round_mode div_scale
upgrade downgrade
/)
{
$cfg->{$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);
}
###############################################################################
# Constructor methods
###############################################################################
sub new {
# Create a new Math::BigInt object from a string or another Math::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 $self = shift;
my $selfref = ref $self;
my $class = $selfref || $self;
# The POD says:
#
# "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();
#
# But although this use has been discouraged for more than 10 years, people
# apparently still use it, so we still support it.
return $self->bzero() unless @_;
my ($wanted, $a, $p, $r) = @_;
# Always return a new object, so it called as an instance method, copy the
# invocand, and if called as a class method, initialize a new object.
$self = $selfref ? $self -> copy()
: bless {}, $class;
unless (defined $wanted) {
#Carp::carp("Use of uninitialized value in new()");
return $self->bzero($a, $p, $r);
}
if (ref($wanted) && $wanted->isa($class)) { # MBI or subclass
# Using "$copy = $wanted -> copy()" here fails some tests. Fixme!
my $copy = $class -> copy($wanted);
if ($selfref) {
%$self = %$copy;
} else {
$self = $copy;
}
return $self;
}
$class->import() if $IMPORT == 0; # make require work
# Shortcut for non-zero scalar integers with no non-zero exponent.
if (!ref($wanted) &&
$wanted =~ / ^
([+-]?) # optional sign
([1-9][0-9]*) # non-zero significand
(\.0*)? # ... with optional zero fraction
([Ee][+-]?0+)? # optional zero exponent
\z
/x)
{
my $sgn = $1;
my $abs = $2;
$self->{sign} = $sgn || '+';
$self->{value} = $CALC->_new($abs);
no strict 'refs';
if (defined($a) || defined($p)
|| defined(${"${class}::precision"})
|| defined(${"${class}::accuracy"}))
{
$self->round($a, $p, $r)
unless @_ >= 3 && !defined $a && !defined $p;
}
return $self;
}
# Handle Infs.
if ($wanted =~ /^\s*([+-]?)inf(inity)?\s*\z/i) {
my $sgn = $1 || '+';
$self->{sign} = $sgn . 'inf'; # set a default sign for bstr()
return $class->binf($sgn);
}
# Handle explicit NaNs (not the ones returned due to invalid input).
if ($wanted =~ /^\s*([+-]?)nan\s*\z/i) {
$self = $class -> bnan();
$self->round($a, $p, $r) unless @_ >= 3 && !defined $a && !defined $p;
return $self;
}
# Handle hexadecimal numbers.
if ($wanted =~ /^\s*[+-]?0[Xx]/) {
$self = $class -> from_hex($wanted);
$self->round($a, $p, $r) unless @_ >= 3 && !defined $a && !defined $p;
return $self;
}
# Handle binary numbers.
if ($wanted =~ /^\s*[+-]?0[Bb]/) {
$self = $class -> from_bin($wanted);
$self->round($a, $p, $r) unless @_ >= 3 && !defined $a && !defined $p;
return $self;
}
# Split string into mantissa, exponent, integer, fraction, value, and sign.
my ($mis, $miv, $mfv, $es, $ev) = _split($wanted);
if (!ref $mis) {
if ($_trap_nan) {
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 exponent, then convert to a
# Math::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) {
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 ($$mfv ne '') { # e <= 0
# fraction and negative/zero E => NOI
if ($_trap_nan) {
Carp::croak("$wanted not an integer in $class");
}
#print "NOI 2 \$\$mfv '$$mfv'\n";
return $upgrade->new($wanted, $a, $p, $r) if defined $upgrade;
$self->{sign} = $nan;
} elsif ($e < 0) {
# xE-y, and empty mfv
# Split the mantissa at the decimal point. E.g., if
# $$miv = 12345 and $e = -2, then $frac = 45 and $$miv = 123.
my $frac = substr($$miv, $e); # $frac is fraction part
substr($$miv, $e) = ""; # $$miv is now integer part
if ($frac =~ /[^0]/) {
if ($_trap_nan) {
Carp::croak("$wanted not an integer in $class");
}
#print "NOI 3\n";
return $upgrade->new($wanted, $a, $p, $r) if defined $upgrade;
$self->{sign} = $nan;
}
}
}
unless ($self->{sign} eq $nan) {
$self->{sign} = '+' if $$miv eq '0'; # normalize -0 => +0
$self->{value} = $CALC->_new($$miv) if $self->{sign} =~ /^[+-]$/;
}
# If any of the globals are 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 @_ >= 3 && !defined $a && !defined $p;
$self;
}
# Create a Math::BigInt from a hexadecimal string.
sub from_hex {
my $self = shift;
my $selfref = ref $self;
my $class = $selfref || $self;
# Don't modify constant (read-only) objects.
return if $selfref && $self->modify('from_hex');
my $str = shift;
# If called as a class method, initialize a new object.
$self = $class -> bzero() unless $selfref;
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;
# The library method requires a prefix.
$self->{value} = $CALC->_from_hex('0x' . $chrs);
# Place the sign.
$self->{sign} = $sign eq '-' && ! $CALC->_is_zero($self->{value})
? '-' : '+';
return $self;
}
# CORE::hex() parses as much as it can, and ignores any trailing garbage.
# For backwards compatibility, we return NaN.
return $self->bnan();
}
# Create a Math::BigInt from an octal string.
sub from_oct {
my $self = shift;
my $selfref = ref $self;
my $class = $selfref || $self;
# Don't modify constant (read-only) objects.
return if $selfref && $self->modify('from_oct');
my $str = shift;
# If called as a class method, initialize a new object.
$self = $class -> bzero() unless $selfref;
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;
# The library method requires a prefix.
$self->{value} = $CALC->_from_oct('0' . $chrs);
# Place the sign.
$self->{sign} = $sign eq '-' && ! $CALC->_is_zero($self->{value})
? '-' : '+';
return $self;
}
# CORE::oct() parses as much as it can, and ignores any trailing garbage.
# For backwards compatibility, we return NaN.
return $self->bnan();
}
# Create a Math::BigInt from a binary string.
sub from_bin {
my $self = shift;
my $selfref = ref $self;
my $class = $selfref || $self;
# Don't modify constant (read-only) objects.
return if $selfref && $self->modify('from_bin');
my $str = shift;
# If called as a class method, initialize a new object.
$self = $class -> bzero() unless $selfref;
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;
# The library method requires a prefix.
$self->{value} = $CALC->_from_bin('0b' . $chrs);
# Place the sign.
$self->{sign} = $sign eq '-' && ! $CALC->_is_zero($self->{value})
? '-' : '+';
return $self;
}
# For consistency with from_hex() and from_oct(), we return NaN when the
# input is invalid.
return $self->bnan();
}
# Create a Math::BigInt from a byte string.
sub from_bytes {
my $self = shift;
my $selfref = ref $self;
my $class = $selfref || $self;
# Don't modify constant (read-only) objects.
return if $selfref && $self->modify('from_bytes');
my $str = shift;
# If called as a class method, initialize a new object.
$self = $class -> bzero() unless $selfref;
$self -> {sign} = '+';
$self -> {value} = $CALC -> _from_bytes($str);
return $self;
}
sub bzero {
# create/assign '+0'
if (@_ == 0) {
#Carp::carp("Using bzero() as a function is deprecated;",
# " use bzero() as a method instead");
unshift @_, __PACKAGE__;
}
my $self = shift;
my $selfref = ref $self;
my $class = $selfref || $self;
$self->import() if $IMPORT == 0; # make require work
# Don't modify constant (read-only) objects.
return if $selfref && $self->modify('bzero');
$self = bless {}, $class unless $selfref;
$self->{sign} = '+';
$self->{value} = $CALC->_zero();
if (@_ > 0) {
if (@_ > 3) {
# call like: $x->bzero($a, $p, $r, $y, ...);
($self, $self->{_a}, $self->{_p}) = $self->_find_round_parameters(@_);
} else {
# call like: $x->bzero($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});
}
}
return $self;
}
sub bone {
# Create or assign '+1' (or -1 if given sign '-').
if (@_ == 0 || (defined($_[0]) && ($_[0] eq '+' || $_[0] eq '-'))) {
#Carp::carp("Using bone() as a function is deprecated;",
# " use bone() as a method instead");
unshift @_, __PACKAGE__;
}
my $self = shift;
my $selfref = ref $self;
my $class = $selfref || $self;
$self->import() if $IMPORT == 0; # make require work
# Don't modify constant (read-only) objects.
return if $selfref && $self->modify('bone');
my $sign = shift;
$sign = defined $sign && $sign =~ /^\s*-/ ? "-" : "+";
$self = bless {}, $class unless $selfref;
$self->{sign} = $sign;
$self->{value} = $CALC->_one();
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});
}
}
return $self;
}
sub binf {
# create/assign a '+inf' or '-inf'
if (@_ == 0 || (defined($_[0]) && !ref($_[0]) &&
$_[0] =~ /^\s*[+-](inf(inity)?)?\s*$/))
{
#Carp::carp("Using binf() as a function is deprecated;",
# " use binf() as a method instead");
unshift @_, __PACKAGE__;
}
my $self = shift;
my $selfref = ref $self;
my $class = $selfref || $self;
{
no strict 'refs';
if (${"${class}::_trap_inf"}) {
Carp::croak("Tried to create +-inf in $class->binf()");
}
}
$self->import() if $IMPORT == 0; # make require work
# Don't modify constant (read-only) objects.
return if $selfref && $self->modify('binf');
my $sign = shift;
$sign = defined $sign && $sign =~ /^\s*-/ ? "-" : "+";
$self = bless {}, $class unless $selfref;
$self -> {sign} = $sign . 'inf';
$self -> {value} = $CALC -> _zero();
return $self;
}
sub bnan {
# create/assign a 'NaN'
if (@_ == 0) {
#Carp::carp("Using bnan() as a function is deprecated;",
# " use bnan() as a method instead");
unshift @_, __PACKAGE__;
}
my $self = shift;
my $selfref = ref($self);
my $class = $selfref || $self;
{
no strict 'refs';
if (${"${class}::_trap_nan"}) {
Carp::croak("Tried to create NaN in $class->bnan()");
}
}
$self->import() if $IMPORT == 0; # make require work
# Don't modify constant (read-only) objects.
return if $selfref && $self->modify('bnan');
$self = bless {}, $class unless $selfref;
$self -> {sign} = $nan;
$self -> {value} = $CALC -> _zero();
return $self;
}
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 copy {
my $self = shift;
my $selfref = ref $self;
my $class = $selfref || $self;
# If called as a class method, the object to copy is the next argument.
$self = shift() unless $selfref;
my $copy = bless {}, $class;
$copy->{sign} = $self->{sign};
$copy->{value} = $CALC->_copy($self->{value});
$copy->{_a} = $self->{_a} if exists $self->{_a};
$copy->{_p} = $self->{_p} if exists $self->{_p};
return $copy;
}
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();
}
###############################################################################
# Boolean methods
###############################################################################
sub is_zero {
# return true if arg (BINT or num_str) is zero (array '+', '0')
my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);
return 0 if $x->{sign} !~ /^\+$/; # -, NaN & +-inf aren't
$CALC->_is_zero($x->{value});
}
sub is_one {
# return true if arg (BINT or num_str) is +1, or -1 if sign is given
my ($class, $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_finite {
my $x = shift;
return $x->{sign} eq '+' || $x->{sign} eq '-';
}
sub is_inf {
# return true if arg (BINT or num_str) is +-inf
my ($class, $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_nan {
# return true if arg (BINT or num_str) is NaN
my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);
$x->{sign} eq $nan ? 1 : 0;
}
sub is_positive {
# return true when arg (BINT or num_str) is positive (> 0)
my ($class, $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 ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);
$x->{sign} =~ /^-/ ? 1 : 0; # -inf is negative, but NaN is not
}
sub is_odd {
# return true when arg (BINT or num_str) is odd, false for even
my ($class, $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 ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);
return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
$CALC->_is_even($x->{value});
}
sub is_int {
# return true when arg (BINT or num_str) is an integer
# always true for Math::BigInt, but different for Math::BigFloat objects
my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);
$x->{sign} =~ /^[+-]$/ ? 1 : 0; # inf/-inf/NaN aren't
}
###############################################################################
# Comparison methods
###############################################################################
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 ($class, $x, $y) = ref($_[0]) && ref($_[0]) eq ref($_[1])
? (ref($_[0]), @_)
: objectify(2, @_);
return $upgrade->bcmp($x, $y) if defined $upgrade &&
((!$x->isa($class)) || (!$y->isa($class)));
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 testing 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}); # swapped 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 ($class, $x, $y) = ref($_[0]) && ref($_[0]) eq ref($_[1])
? (ref($_[0]), @_)
: objectify(2, @_);
return $upgrade->bacmp($x, $y) if defined $upgrade &&
((!$x->isa($class)) || (!$y->isa($class)));
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 beq {
my $self = shift;
my $selfref = ref $self;
my $class = $selfref || $self;
Carp::croak 'beq() is an instance method, not a class method' unless $selfref;
Carp::croak 'Wrong number of arguments for beq()' unless @_ == 1;
my $cmp = $self -> bcmp(shift);
return defined($cmp) && ! $cmp;
}
sub bne {
my $self = shift;
my $selfref = ref $self;
my $class = $selfref || $self;
Carp::croak 'bne() is an instance method, not a class method' unless $selfref;
Carp::croak 'Wrong number of arguments for bne()' unless @_ == 1;
my $cmp = $self -> bcmp(shift);
return defined($cmp) && ! $cmp ? '' : 1;
}
sub blt {
my $self = shift;
my $selfref = ref $self;
my $class = $selfref || $self;
Carp::croak 'blt() is an instance method, not a class method' unless $selfref;
Carp::croak 'Wrong number of arguments for blt()' unless @_ == 1;
my $cmp = $self -> bcmp(shift);
return defined($cmp) && $cmp < 0;
}
sub ble {
my $self = shift;
my $selfref = ref $self;
my $class = $selfref || $self;
Carp::croak 'ble() is an instance method, not a class method' unless $selfref;
Carp::croak 'Wrong number of arguments for ble()' unless @_ == 1;
my $cmp = $self -> bcmp(shift);
return defined($cmp) && $cmp <= 0;
}
sub bgt {
my $self = shift;
my $selfref = ref $self;
my $class = $selfref || $self;
Carp::croak 'bgt() is an instance method, not a class method' unless $selfref;
Carp::croak 'Wrong number of arguments for bgt()' unless @_ == 1;
my $cmp = $self -> bcmp(shift);
return defined($cmp) && $cmp > 0;
}
sub bge {
my $self = shift;
my $selfref = ref $self;
my $class = $selfref || $self;
Carp::croak 'bge() is an instance method, not a class method'
unless $selfref;
Carp::croak 'Wrong number of arguments for bge()' unless @_ == 1;
my $cmp = $self -> bcmp(shift);
return defined($cmp) && $cmp >= 0;
}
###############################################################################
# Arithmetic methods
###############################################################################
sub bneg {
# (BINT or num_str) return BINT
# negate number or make a negated number from string
my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);
return $x if $x->modify('bneg');
# for +0 do not negate (to have always normalized +0). Does nothing for 'NaN'
$x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $CALC->_is_zero($x->{value}));
$x;
}
sub babs {
# (BINT or num_str) return BINT
# make number absolute, or return absolute BINT from string
my ($class, $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 bnorm {
# (numstr or BINT) return BINT
# Normalize number -- no-op here
my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);
$x;
}
sub binc {
# increment arg by one
my ($class, $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($class->bone(), $a, $p, $r); # badd does round
}
sub bdec {
# decrement arg by one
my ($class, $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($class->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 bstrcmp {
# my $self = shift;
# my $selfref = ref $self;
# my $class = $selfref || $self;
#
# Carp::croak 'bstrcmp() is an instance method, not a class method'
# unless $selfref;
# Carp::croak 'Wrong number of arguments for bstrcmp()' unless @_ == 1;
#
# return $self -> bstr() CORE::cmp shift;
#}
#
#sub bstreq {
# my $self = shift;
# my $selfref = ref $self;
# my $class = $selfref || $self;
#
# Carp::croak 'bstreq() is an instance method, not a class method'
# unless $selfref;
# Carp::croak 'Wrong number of arguments for bstreq()' unless @_ == 1;
#
# my $cmp = $self -> bstrcmp(shift);
# return defined($cmp) && ! $cmp;
#}
#
#sub bstrne {
# my $self = shift;
# my $selfref = ref $self;
# my $class = $selfref || $self;
#
# Carp::croak 'bstrne() is an instance method, not a class method'
# unless $selfref;
# Carp::croak 'Wrong number of arguments for bstrne()' unless @_ == 1;
#
# my $cmp = $self -> bstrcmp(shift);
# return defined($cmp) && ! $cmp ? '' : 1;
#}
#
#sub bstrlt {
# my $self = shift;
# my $selfref = ref $self;
# my $class = $selfref || $self;
#
# Carp::croak 'bstrlt() is an instance method, not a class method'
# unless $selfref;
# Carp::croak 'Wrong number of arguments for bstrlt()' unless @_ == 1;
#
# my $cmp = $self -> bstrcmp(shift);
# return defined($cmp) && $cmp < 0;
#}
#
#sub bstrle {
# my $self = shift;
# my $selfref = ref $self;
# my $class = $selfref || $self;
#
# Carp::croak 'bstrle() is an instance method, not a class method'
# unless $selfref;
# Carp::croak 'Wrong number of arguments for bstrle()' unless @_ == 1;
#
# my $cmp = $self -> bstrcmp(shift);
# return defined($cmp) && $cmp <= 0;
#}
#
#sub bstrgt {
# my $self = shift;
# my $selfref = ref $self;
# my $class = $selfref || $self;
#
# Carp::croak 'bstrgt() is an instance method, not a class method'
# unless $selfref;
# Carp::croak 'Wrong number of arguments for bstrgt()' unless @_ == 1;
#
# my $cmp = $self -> bstrcmp(shift);
# return defined($cmp) && $cmp > 0;
#}
#
#sub bstrge {
# my $self = shift;
# my $selfref = ref $self;
# my $class = $selfref || $self;
#
# Carp::croak 'bstrge() is an instance method, not a class method'
# unless $selfref;
# Carp::croak 'Wrong number of arguments for bstrge()' unless @_ == 1;
#
# my $cmp = $self -> bstrcmp(shift);
# return defined($cmp) && $cmp >= 0;
#}
sub badd {
# add second arg (BINT or string) to first (BINT) (modifies first)
# return result as BINT
# set up parameters
my ($class, $x, $y, @r) = (ref($_[0]), @_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
($class, $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($class)) || (!$y->isa($class)));
$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 ($class, $x, $y, @r) = (ref($_[0]), @_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
($class, $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($class) || !$y -> isa($class));
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 rounding
}
sub bmul {
# multiply the first number by the second number
# (BINT or num_str, BINT or num_str) return BINT
# set up parameters
my ($class, $x, $y, @r) = (ref($_[0]), @_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
($class, $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($class);
$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 ($class, $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($class) || !$z->isa($class) || !$x->isa($class));
# 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 bdiv {
# This does floored division, where the quotient is floored, i.e., rounded
# towards negative infinity. As a consequence, the remainder has the same
# sign as the divisor.
# Set up parameters.
my ($class, $x, $y, @r) = (ref($_[0]), @_);
# objectify() is costly, so avoid it if we can.
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
($class, $x, $y, @r) = objectify(2, @_);
}
return $x if $x -> modify('bdiv');
my $wantarray = wantarray; # call only once
# At least one argument is NaN. Return NaN for both quotient and the
# modulo/remainder.
if ($x -> is_nan() || $y -> is_nan()) {
return $wantarray ? ($x -> bnan(), $class -> bnan()) : $x -> bnan();
}
# Divide by zero and modulo zero.
#
# Division: Use the common convention that x / 0 is inf with the same sign
# as x, except when x = 0, where we return NaN. This is also what earlier
# versions did.
#
# Modulo: In modular arithmetic, the congruence relation z = x (mod y)
# means that there is some integer k such that z - x = k y. If y = 0, we
# get z - x = 0 or z = x. This is also what earlier versions did, except
# that 0 % 0 returned NaN.
#
# inf / 0 = inf inf % 0 = inf
# 5 / 0 = inf 5 % 0 = 5
# 0 / 0 = NaN 0 % 0 = 0
# -5 / 0 = -inf -5 % 0 = -5
# -inf / 0 = -inf -inf % 0 = -inf
if ($y -> is_zero()) {
my $rem;
if ($wantarray) {
$rem = $x -> copy();
}
if ($x -> is_zero()) {
$x -> bnan();
} else {
$x -> binf($x -> {sign});
}
return $wantarray ? ($x, $rem) : $x;
}
# Numerator (dividend) is +/-inf, and denominator is finite and non-zero.
# The divide by zero cases are covered above. In all of the cases listed
# below we return the same as core Perl.
#
# inf / -inf = NaN inf % -inf = NaN
# inf / -5 = -inf inf % -5 = NaN
# inf / 5 = inf inf % 5 = NaN
# inf / inf = NaN inf % inf = NaN
#
# -inf / -inf = NaN -inf % -inf = NaN
# -inf / -5 = inf -inf % -5 = NaN
# -inf / 5 = -inf -inf % 5 = NaN
# -inf / inf = NaN -inf % inf = NaN
if ($x -> is_inf()) {
my $rem;
$rem = $class -> bnan() if $wantarray;
if ($y -> is_inf()) {
$x -> bnan();
} else {
my $sign = $x -> bcmp(0) == $y -> bcmp(0) ? '+' : '-';
$x -> binf($sign);
}
return $wantarray ? ($x, $rem) : $x;
}
# Denominator (divisor) is +/-inf. The cases when the numerator is +/-inf
# are covered above. In the modulo cases (in the right column) we return
# the same as core Perl, which does floored division, so for consistency we
# also do floored division in the division cases (in the left column).
#
# -5 / inf = -1 -5 % inf = inf
# 0 / inf = 0 0 % inf = 0
# 5 / inf = 0 5 % inf = 5
#
# -5 / -inf = 0 -5 % -inf = -5
# 0 / -inf = 0 0 % -inf = 0
# 5 / -inf = -1 5 % -inf = -inf
if ($y -> is_inf()) {
my $rem;
if ($x -> is_zero() || $x -> bcmp(0) == $y -> bcmp(0)) {
$rem = $x -> copy() if $wantarray;
$x -> bzero();
} else {
$rem = $class -> binf($y -> {sign}) if $wantarray;
$x -> bone('-');
}
return $wantarray ? ($x, $rem) : $x;
}
# At this point, both the numerator and denominator are finite numbers, and
# the denominator (divisor) is non-zero.
return $upgrade -> bdiv($upgrade -> new($x), $upgrade -> new($y), @r)
if defined $upgrade;
$r[3] = $y; # no push!
# Inialize remainder.
my $rem = $class -> bzero();
# Are both operands the same object, i.e., like $x -> bdiv($x)? If so,
# flipping the sign of $y also flips the sign of $x.
my $xsign = $x -> {sign};
my $ysign = $y -> {sign};
$y -> {sign} =~ tr/+-/-+/; # Flip the sign of $y, and see ...
my $same = $xsign ne $x -> {sign}; # ... if that changed the sign of $x.
$y -> {sign} = $ysign; # Re-insert the original sign.
if ($same) {
$x -> bone();
} else {
($x -> {value}, $rem -> {value}) =
$CALC -> _div($x -> {value}, $y -> {value});
if ($CALC -> _is_zero($rem -> {value})) {
if ($xsign eq $ysign || $CALC -> _is_zero($x -> {value})) {
$x -> {sign} = '+';
} else {
$x -> {sign} = '-';
}
} else {
if ($xsign eq $ysign) {
$x -> {sign} = '+';
} else {
if ($xsign eq '+') {
$x -> badd(1);
} else {
$x -> bsub(1);
}
$x -> {sign} = '-';
}
}
}
$x -> round(@r);
if ($wantarray) {
unless ($CALC -> _is_zero($rem -> {value})) {
if ($xsign ne $ysign) {
$rem = $y -> copy() -> babs() -> bsub($rem);
}
$rem -> {sign} = $ysign;
}
$rem -> {_a} = $x -> {_a};
$rem -> {_p} = $x -> {_p};
$rem -> round(@r);
return ($x, $rem);
}
return $x;
}
sub btdiv {
# This does truncated division, where the quotient is truncted, i.e.,
# rounded towards zero.
#
# ($q, $r) = $x -> btdiv($y) returns $q and $r so that $q is int($x / $y)
# and $q * $y + $r = $x.
# Set up parameters
my ($class, $x, $y, @r) = (ref($_[0]), @_);
# objectify is costly, so avoid it if we can.
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
($class, $x, $y, @r) = objectify(2, @_);
}
return $x if $x -> modify('btdiv');
my $wantarray = wantarray; # call only once
# At least one argument is NaN. Return NaN for both quotient and the
# modulo/remainder.
if ($x -> is_nan() || $y -> is_nan()) {
return $wantarray ? ($x -> bnan(), $class -> bnan()) : $x -> bnan();
}
# Divide by zero and modulo zero.
#
# Division: Use the common convention that x / 0 is inf with the same sign
# as x, except when x = 0, where we return NaN. This is also what earlier
# versions did.
#
# Modulo: In modular arithmetic, the congruence relation z = x (mod y)
# means that there is some integer k such that z - x = k y. If y = 0, we
# get z - x = 0 or z = x. This is also what earlier versions did, except
# that 0 % 0 returned NaN.
#
# inf / 0 = inf inf % 0 = inf
# 5 / 0 = inf 5 % 0 = 5
# 0 / 0 = NaN 0 % 0 = 0
# -5 / 0 = -inf -5 % 0 = -5
# -inf / 0 = -inf -inf % 0 = -inf
if ($y -> is_zero()) {
my $rem;
if ($wantarray) {
$rem = $x -> copy();
}
if ($x -> is_zero()) {
$x -> bnan();
} else {
$x -> binf($x -> {sign});
}
return $wantarray ? ($x, $rem) : $x;
}
# Numerator (dividend) is +/-inf, and denominator is finite and non-zero.
# The divide by zero cases are covered above. In all of the cases listed
# below we return the same as core Perl.
#
# inf / -inf = NaN inf % -inf = NaN
# inf / -5 = -inf inf % -5 = NaN
# inf / 5 = inf inf % 5 = NaN
# inf / inf = NaN inf % inf = NaN
#
# -inf / -inf = NaN -inf % -inf = NaN
# -inf / -5 = inf -inf % -5 = NaN
# -inf / 5 = -inf -inf % 5 = NaN
# -inf / inf = NaN -inf % inf = NaN
if ($x -> is_inf()) {
my $rem;
$rem = $class -> bnan() if $wantarray;
if ($y -> is_inf()) {
$x -> bnan();
} else {
my $sign = $x -> bcmp(0) == $y -> bcmp(0) ? '+' : '-';
$x -> binf($sign);
}
return $wantarray ? ($x, $rem) : $x;
}
# Denominator (divisor) is +/-inf. The cases when the numerator is +/-inf
# are covered above. In the modulo cases (in the right column) we return
# the same as core Perl, which does floored division, so for consistency we
# also do floored division in the division cases (in the left column).
#
# -5 / inf = 0 -5 % inf = -5
# 0 / inf = 0 0 % inf = 0
# 5 / inf = 0 5 % inf = 5
#
# -5 / -inf = 0 -5 % -inf = -5
# 0 / -inf = 0 0 % -inf = 0
# 5 / -inf = 0 5 % -inf = 5
if ($y -> is_inf()) {
my $rem;
$rem = $x -> copy() if $wantarray;
$x -> bzero();
return $wantarray ? ($x, $rem) : $x;
}
return $upgrade -> btdiv($upgrade -> new($x), $upgrade -> new($y), @r)
if defined $upgrade;
$r[3] = $y; # no push!
# Inialize remainder.
my $rem = $class -> bzero();
# Are both operands the same object, i.e., like $x -> bdiv($x)? If so,
# flipping the sign of $y also flips the sign of $x.
my $xsign = $x -> {sign};
my $ysign = $y -> {sign};
$y -> {sign} =~ tr/+-/-+/; # Flip the sign of $y, and see ...
my $same = $xsign ne $x -> {sign}; # ... if that changed the sign of $x.
$y -> {sign} = $ysign; # Re-insert the original sign.
if ($same) {
$x -> bone();
} else {
($x -> {value}, $rem -> {value}) =
$CALC -> _div($x -> {value}, $y -> {value});
$x -> {sign} = $xsign eq $ysign ? '+' : '-';
$x -> {sign} = '+' if $CALC -> _is_zero($x -> {value});
$x -> round(@r);
}
if (wantarray) {
$rem -> {sign} = $xsign;
$rem -> {sign} = '+' if $CALC -> _is_zero($rem -> {value});
$rem -> {_a} = $x -> {_a};
$rem -> {_p} = $x -> {_p};
$rem -> round(@r);
return ($x, $rem);
}
return $x;
}
sub bmod {
# This is the remainder after floored division.
# Set up parameters.
my ($class, $x, $y, @r) = (ref($_[0]), @_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
($class, $x, $y, @r) = objectify(2, @_);
}
return $x if $x -> modify('bmod');
$r[3] = $y; # no push!
# At least one argument is NaN.
if ($x -> is_nan() || $y -> is_nan()) {
return $x -> bnan();
}
# Modulo zero. See documentation for bdiv().
if ($y -> is_zero()) {
return $x;
}
# Numerator (dividend) is +/-inf.
if ($x -> is_inf()) {
return $x -> bnan();
}
# Denominator (divisor) is +/-inf.
if ($y -> is_inf()) {
if ($x -> is_zero() || $x -> bcmp(0) == $y -> bcmp(0)) {
return $x;
} else {
return $x -> binf($y -> sign());
}
}
# 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 -> {sign} = '+'; # do not leave -0
} else {
$x -> {value} = $CALC -> _sub($y -> {value}, $x -> {value}, 1) # $y-$x
if ($x -> {sign} ne $y -> {sign});
$x -> {sign} = $y -> {sign};
}
$x -> round(@r);
}
sub btmod {
# Remainder after truncated division.
# set up parameters
my ($class, $x, $y, @r) = (ref($_[0]), @_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
($class, $x, $y, @r) = objectify(2, @_);
}
return $x if $x -> modify('btmod');
# At least one argument is NaN.
if ($x -> is_nan() || $y -> is_nan()) {
return $x -> bnan();
}
# Modulo zero. See documentation for btdiv().
if ($y -> is_zero()) {
return $x;
}
# Numerator (dividend) is +/-inf.
if ($x -> is_inf()) {
return $x -> bnan();
}
# Denominator (divisor) is +/-inf.
if ($y -> is_inf()) {
return $x;
}
return $upgrade -> btmod($upgrade -> new($x), $upgrade -> new($y), @r)
if defined $upgrade;
$r[3] = $y; # no push!
my $xsign = $x -> {sign};
my $ysign = $y -> {sign};
$x -> {value} = $CALC -> _mod($x -> {value}, $y -> {value});
$x -> {sign} = $xsign;
$x -> {sign} = '+' if $CALC -> _is_zero($x -> {value});
$x -> round(@r);
return $x;
}
sub bmodinv {
# Return modular multiplicative inverse:
#
# z is the modular inverse of x (mod y) if and only if
#
# x*z ≡ 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 ($class, $x, $y, @r) = (undef, @_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
($class, $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 ($class, $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
# Modulo zero. See documentation for Math::BigInt's bmod() method.
if ($mod -> is_zero()) {
if ($num -> is_zero()) {
return $class -> bnan();
} else {
return $num -> copy();
}
}
# 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 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 ($class, $x, $y, @r) = (ref($_[0]), @_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
($class, $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($class) || $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 blog {
# Return the logarithm of the operand. If a second operand is defined, that
# value is used as the base, otherwise the base is assumed to be Euler's
# constant.
# Don't objectify the base, since an undefined base, as in $x->blog() or
# $x->blog(undef) signals that the base is Euler's number.
# set up parameters
my ($class, $x, $base, @r) = (undef, @_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
($class, $x, $base, @r) = objectify(2, @_);
}
return $x if $x->modify('blog');
# Handle all exception cases and all trivial cases. I have used Wolfram
# Alpha (http://www.wolframalpha.com) as the reference for these cases.
return $x -> bnan() if $x -> is_nan();
if (defined $base) {
$base = $class -> new($base) unless ref $base;
if ($base -> is_nan() || $base -> is_one()) {
return $x -> bnan();
} elsif ($base -> is_inf() || $base -> is_zero()) {
return $x -> bnan() if $x -> is_inf() || $x -> is_zero();
return $x -> bzero();
} elsif ($base -> is_negative()) { # -inf < base < 0
return $x -> bzero() if $x -> is_one(); # x = 1
return $x -> bone() if $x == $base; # x = base
return $x -> bnan(); # otherwise
}
return $x -> bone() if $x == $base; # 0 < base && 0 < x < inf
}
# We now know that the base is either undefined or >= 2 and finite.
return $x -> binf('+') if $x -> is_inf(); # x = +/-inf
return $x -> bnan() if $x -> is_neg(); # -inf < x < 0
return $x -> bzero() if $x -> is_one(); # x = 1
return $x -> binf('-') if $x -> is_zero(); # x = 0
# At this point we are done handling all exception cases and trivial cases.
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 bexp {
# Calculate e ** $x (Euler's number to the power of X), truncated to
# an integer value.
my ($class, $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) {
$x = $u;
} else {
$u = $u->as_int();
# modify $x in place
$x->{value} = $u->{value};
$x->round(@r);
}
}
sub bnok {
# Calculate n over k (binomial coefficient or "choose" function) as integer.
# set up parameters
my ($class, $x, $y, @r) = (ref($_[0]), @_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
($class, $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 || substr($y->{sign}, 0, 1) eq "-";
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
my $n = $x -> {value};
my $k = $y -> {value};
# If k > n/2, or, equivalently, 2*k > n, compute nok(n, k) as
# nok(n, n-k) to minimize the number if iterations in the loop.
{
my $twok = $CALC->_mul($CALC->_two(), $CALC->_copy($k));
if ($CALC->_acmp($twok, $n) > 0) {
$k = $CALC->_sub($CALC->_copy($n), $k);
}
}
if ($CALC->_is_zero($k)) {
$n = $CALC->_one();
} else {
# Make a copy of the original n, since we'll be modifying n
# in-place.
my $n_orig = $CALC->_copy($n);
$CALC->_sub($n, $k);
$CALC->_inc($n);
my $f = $CALC->_copy($n);
$CALC->_inc($f);
my $d = $CALC->_two();
# while f <= n (the original n, that is) ...
while ($CALC->_acmp($f, $n_orig) <= 0) {
$CALC->_mul($n, $f);
$CALC->_div($n, $d);
$CALC->_inc($f);
$CALC->_inc($d);
}
}
$x -> {value} = $n;
}
$x->round(@r);
}
sub bsin {
# Calculate sinus(x) to N digits. Unless upgrading is in effect, returns the
# result truncated to an integer.
my ($class, $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 bcos {
# Calculate cosinus(x) to N digits. Unless upgrading is in effect, returns the
# result truncated to an integer.
my ($class, $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 batan {
# Calculate arcus tangens of x to N digits. Unless upgrading is in effect, returns the
# result truncated to an integer.
my ($class, $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);
}
sub batan2 {
# calculate arcus tangens of ($y/$x)
# set up parameters
my ($class, $y, $x, @r) = (ref($_[0]), @_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
($class, $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 Math::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($class->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($class->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 bsqrt {
# calculate square root of $x
my ($class, $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 ($class, $x, $y, @r) = (ref($_[0]), @_);
$y = $class->new(2) unless defined $y;
# objectify is costly, so avoid it
if ((!ref($x)) || (ref($x) ne ref($y))) {
($class, $x, $y, @r) = objectify(2, $class || $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 bfac {
# (BINT or num_str, BINT or num_str) return BINT
# compute factorial number from $x, modify $x in place
my ($class, $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 blsft {
# (BINT or num_str, BINT or num_str) return BINT
# compute x << y, base n, y >= 0
# set up parameters
my ($class, $x, $y, $b, @r) = (ref($_[0]), @_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
($class, $x, $y, $b, @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();
$b = 2 if !defined $b;
return $x -> bnan() if $b <= 0 || $y -> {sign} eq '-';
$x -> {value} = $CALC -> _lsft($x -> {value}, $y -> {value}, $b);
$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 ($class, $x, $y, $b, @r) = (ref($_[0]), @_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
($class, $x, $y, $b, @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
$b = 2 if !defined $b;
return $x -> bnan() if $b <= 0 || $y -> {sign} eq '-';
# this only works for negative numbers when shifting in base 2
if (($x -> {sign} eq '-') && ($b == 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 = $class -> 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}, $b);
$x -> round(@r);
}
###############################################################################
# Bitwise methods
###############################################################################
sub band {
#(BINT or num_str, BINT or num_str) return BINT
# compute x & y
# set up parameters
my ($class, $x, $y, @r) = (ref($_[0]), @_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
($class, $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($class, $x, $y, $sx, $sy, @r);
}
sub bior {
#(BINT or num_str, BINT or num_str) return BINT
# compute x | y
# set up parameters
my ($class, $x, $y, @r) = (ref($_[0]), @_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
($class, $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($class, $x, $y, $sx, $sy, @r);
}
sub bxor {
#(BINT or num_str, BINT or num_str) return BINT
# compute x ^ y
# set up parameters
my ($class, $x, $y, @r) = (ref($_[0]), @_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
($class, $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($class, $x, $y, $sx, $sy, @r);
}
sub bnot {
# (num_str or BINT) return BINT
# represent ~x as twos-complement number
# we don't need $class, so undef instead of ref($_[0]) make it slightly faster
my ($class, $x, $a, $p, $r) = ref($_[0]) ? (undef, @_) : objectify(1, @_);
return $x if $x->modify('bnot');
$x->binc()->bneg(); # binc already does round
}
###############################################################################
# Rounding methods
###############################################################################
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 $class = 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 = ${"$class\::accuracy"} unless defined $a;
$p = ${"$class\::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 = ${"$class\::round_mode"} unless defined $r;
if ($r !~ /^(even|odd|[+-]inf|zero|trunc|common)$/) {
Carp::croak("Unknown round mode '$r'");
}
# now round, by calling either bround or bfround:
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 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 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 $class = ref($x) || $x;
$x = $class->new($x) unless ref $x;
my ($scale, $mode) = $x->_scale_p(@_);
return $x if !defined $scale || $x->modify('bfround'); # no-op
# no-op for Math::BigInt objects 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 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 bfloor {
# round towards minus infinity; no-op since it's already integer
my ($class, $x, @r) = ref($_[0]) ? (undef, @_) : objectify(1, @_);
$x->round(@r);
}
sub bceil {
# round towards plus infinity; no-op since it's already int
my ($class, $x, @r) = ref($_[0]) ? (undef, @_) : objectify(1, @_);
$x->round(@r);
}
sub bint {
# round towards zero; no-op since it's already integer
my ($class, $x, @r) = ref($_[0]) ? (undef, @_) : objectify(1, @_);
$x->round(@r);
}
###############################################################################
# Other mathematical methods
###############################################################################
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 ($class, @args) = objectify(0, @_);
my $x = shift @args;
$x = ref($x) && $x -> isa($class) ? $x -> copy() : $class -> new($x);
return $class->bnan() if $x->{sign} !~ /^[+-]$/; # x NaN?
while (@args) {
my $y = shift @args;
$y = $class->new($y) unless ref($y) && $y -> isa($class);
return $class->bnan() if $y->{sign} !~ /^[+-]$/; # y NaN?
$x->{value} = $CALC->_gcd($x->{value}, $y->{value});
last if $CALC->_is_one($x->{value});
}
return $x -> babs();
}
sub blcm {
# (BINT or num_str, BINT or num_str) return BINT
# does not modify arguments, but returns new object
# Least Common Multiple
my ($class, @args) = objectify(0, @_);
my $x = shift @args;
$x = ref($x) && $x -> isa($class) ? $x -> copy() : $class -> new($x);
return $class->bnan() if $x->{sign} !~ /^[+-]$/; # x NaN?
while (@args) {
my $y = shift @args;
$y = $class -> new($y) unless ref($y) && $y -> isa($class);
return $x->bnan() if $y->{sign} !~ /^[+-]$/; # y not integer
$x -> {value} = $CALC->_lcm($x -> {value}, $y -> {value});
}
return $x -> babs();
}
###############################################################################
# Object property methods
###############################################################################
sub sign {
# return the sign of the number: +/-/-inf/+inf/NaN
my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);
$x->{sign};
}
sub digit {
# return the nth decimal digit, negative values count backward, 0 is right
my ($class, $x, $n) = ref($_[0]) ? (undef, @_) : objectify(1, @_);
$n = $n->numify() if ref($n);
$CALC->_digit($x->{value}, $n || 0);
}
sub length {
my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);
my $e = $CALC->_len($x->{value});
wantarray ? ($e, 0) : $e;
}
sub exponent {
# return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
my ($class, $x) = ref($_[0]) ? (ref($_[0]), $_[0]) : objectify(1, @_);
if ($x->{sign} !~ /^[+-]$/) {
my $s = $x->{sign};
$s =~ s/^[+-]//; # NaN, -inf, +inf => NaN or inf
return $class->new($s);
}
return $class->bzero() if $x->is_zero();
# 12300 => 2 trailing zeros => exponent is 2
$class->new($CALC->_zeros($x->{value}));
}
sub mantissa {
# return the mantissa (compatible to Math::BigFloat, e.g. reduced)
my ($class, $x) = ref($_[0]) ? (ref($_[0]), $_[0]) : objectify(1, @_);
if ($x->{sign} !~ /^[+-]$/) {
# for NaN, +inf, -inf: keep the sign
return $class->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 ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);
($x->mantissa(), $x->exponent());
}
sub sparts {
my $self = shift;
my $class = ref $self;
Carp::croak("sparts() is an instance method, not a class method")
unless $class;
# Not-a-number.
if ($self -> is_nan()) {
my $mant = $self -> copy(); # mantissa
return $mant unless wantarray; # scalar context
my $expo = $class -> bnan(); # exponent
return ($mant, $expo); # list context
}
# Infinity.
if ($self -> is_inf()) {
my $mant = $self -> copy(); # mantissa
return $mant unless wantarray; # scalar context
my $expo = $class -> binf('+'); # exponent
return ($mant, $expo); # list context
}
# Finite number.
my $mant = $self -> copy();
my $nzeros = $CALC -> _zeros($mant -> {value});
$mant -> brsft($nzeros, 10) if $nzeros != 0;
return $mant unless wantarray;
my $expo = $class -> new($nzeros);
return ($mant, $expo);
}
sub nparts {
my $self = shift;
my $class = ref $self;
Carp::croak("nparts() is an instance method, not a class method")
unless $class;
# Not-a-number.
if ($self -> is_nan()) {
my $mant = $self -> copy(); # mantissa
return $mant unless wantarray; # scalar context
my $expo = $class -> bnan(); # exponent
return ($mant, $expo); # list context
}
# Infinity.
if ($self -> is_inf()) {
my $mant = $self -> copy(); # mantissa
return $mant unless wantarray; # scalar context
my $expo = $class -> binf('+'); # exponent
return ($mant, $expo); # list context
}
# Finite number.
my ($mant, $expo) = $self -> sparts();
if ($mant -> bcmp(0)) {
my ($ndigtot, $ndigfrac) = $mant -> length();
my $expo10adj = $ndigtot - $ndigfrac - 1;
if ($expo10adj != 0) {
return $upgrade -> new($self) -> nparts() if $upgrade;
$mant -> bnan();
return $mant unless wantarray;
$expo -> badd($expo10adj);
return ($mant, $expo);
}
}
return $mant unless wantarray;
return ($mant, $expo);
}
sub eparts {
my $self = shift;
my $class = ref $self;
Carp::croak("eparts() is an instance method, not a class method")
unless $class;
# Not-a-number and Infinity.
return $self -> sparts() if $self -> is_nan() || $self -> is_inf();
# Finite number.
my ($mant, $expo) = $self -> sparts();
if ($mant -> bcmp(0)) {
my $ndigmant = $mant -> length();
$expo -> badd($ndigmant);
# $c is the number of digits that will be in the integer part of the
# final mantissa.
my $c = $expo -> copy() -> bdec() -> bmod(3) -> binc();
$expo -> bsub($c);
if ($ndigmant > $c) {
return $upgrade -> new($self) -> eparts() if $upgrade;
$mant -> bnan();
return $mant unless wantarray;
return ($mant, $expo);
}
$mant -> blsft($c - $ndigmant, 10);
}
return $mant unless wantarray;
return ($mant, $expo);
}
sub dparts {
my $self = shift;
my $class = ref $self;
Carp::croak("dparts() is an instance method, not a class method")
unless $class;
my $int = $self -> copy();
return $int unless wantarray;
my $frc = $class -> bzero();
return ($int, $frc);
}
###############################################################################
# String conversion methods
###############################################################################
sub bstr {
my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);
if ($x->{sign} ne '+' && $x->{sign} ne '-') {
return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
return 'inf'; # +inf
}
my $str = $CALC->_str($x->{value});
return $x->{sign} eq '-' ? "-$str" : $str;
}
# Scientific notation with significand/mantissa as an integer, e.g., "12345" is
# written as "1.2345e+4".
sub bsstr {
my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);
if ($x->{sign} ne '+' && $x->{sign} ne '-') {
return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
return 'inf'; # +inf
}
my ($m, $e) = $x -> parts();
my $str = $CALC->_str($m->{value}) . 'e+' . $CALC->_str($e->{value});
return $x->{sign} eq '-' ? "-$str" : $str;
}
# Normalized notation, e.g., "12345" is written as "12345e+0".
sub bnstr {
my $x = shift;
if ($x->{sign} ne '+' && $x->{sign} ne '-') {
return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
return 'inf'; # +inf
}
return $x -> bstr() if $x -> is_nan() || $x -> is_inf();
my ($mant, $expo) = $x -> parts();
# The "fraction posision" is the position (offset) for the decimal point
# relative to the end of the digit string.
my $fracpos = $mant -> length() - 1;
if ($fracpos == 0) {
my $str = $CALC->_str($mant->{value}) . "e+" . $CALC->_str($expo->{value});
return $x->{sign} eq '-' ? "-$str" : $str;
}
$expo += $fracpos;
my $mantstr = $CALC->_str($mant -> {value});
substr($mantstr, -$fracpos, 0) = '.';
my $str = $mantstr . 'e+' . $CALC->_str($expo -> {value});
return $x->{sign} eq '-' ? "-$str" : $str;
}
# Engineering notation, e.g., "12345" is written as "12.345e+3".
sub bestr {
my $x = shift;
if ($x->{sign} ne '+' && $x->{sign} ne '-') {
return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
return 'inf'; # +inf
}
my ($mant, $expo) = $x -> parts();
my $sign = $mant -> sign();
$mant -> babs();
my $mantstr = $CALC->_str($mant -> {value});
my $mantlen = CORE::length($mantstr);
my $dotidx = 1;
$expo += $mantlen - 1;
my $c = $expo -> copy() -> bmod(3);
$expo -= $c;
$dotidx += $c;
if ($mantlen < $dotidx) {
$mantstr .= "0" x ($dotidx - $mantlen);
} elsif ($mantlen > $dotidx) {
substr($mantstr, $dotidx, 0) = ".";
}
my $str = $mantstr . 'e+' . $CALC->_str($expo -> {value});
return $sign eq "-" ? "-$str" : $str;
}
# Decimal notation, e.g., "12345".
sub bdstr {
my $x = shift;
if ($x->{sign} ne '+' && $x->{sign} ne '-') {
return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
return 'inf'; # +inf
}
my $str = $CALC->_str($x->{value});
return $x->{sign} eq '-' ? "-$str" : $str;
}
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_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 $oct = $CALC->_as_oct($x->{value});
return $x->{sign} eq '-' ? "-$oct" : $oct;
}
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_bytes {
# return a byte string
my $x = shift;
$x = $class->new($x) if !ref($x);
Carp::croak("as_bytes() requires a finite, non-negative integer")
if $x -> is_neg() || ! $x -> is_int();
return $CALC->_as_bytes($x->{value});
}
###############################################################################
# Other conversion methods
###############################################################################
sub numify {
# Make a Perl scalar number from a Math::BigInt object.
my $x = shift;
$x = $class->new($x) unless ref $x;
if ($x -> is_nan()) {
require Math::Complex;
my $inf = Math::Complex::Inf();
return $inf - $inf;
}
if ($x -> is_inf()) {
require Math::Complex;
my $inf = Math::Complex::Inf();
return $x -> is_negative() ? -$inf : $inf;
}
my $num = 0 + $CALC->_num($x->{value});
return $x->{sign} eq '-' ? -$num : $num;
}
###############################################################################
# Private methods and functions.
###############################################################################
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) {
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) {
# Don't do anything with undefs. This special treatment is necessary
# because blog() might have a second operand which is undef, to signify
# that the default Euler base should be used.
next unless defined $a[$i];
my $ref = ref $a[$i];
# Perl scalars are fed to the appropriate constructor.
unless ($ref) {
$a[$i] = $a[0] -> new($a[$i]);
next;
}
# If it is an object of the right class, all is fine.
next if $ref -> isa($a[0]);
# Upgrading is OK, so skip further tests if the argument is upgraded.
if (defined $up && $ref -> isa($up)) {
next;
}
# See if we can call one of the as_xxx() methods. We don't know whether
# the as_xxx() method returns an object or a scalar, so re-check
# afterwards.
my $recheck = 0;
if ($a[0] -> isa('Math::BigInt')) {
if ($a[$i] -> can('as_int')) {
$a[$i] = $a[$i] -> as_int();
$recheck = 1;
} elsif ($a[$i] -> can('as_number')) {
$a[$i] = $a[$i] -> as_number();
$recheck = 1;
}
}
elsif ($a[0] -> isa('Math::BigFloat')) {
if ($a[$i] -> can('as_float')) {
$a[$i] = $a[$i] -> as_float();
$recheck = $1;
}
}
# If we called one of the as_xxx() methods, recheck.
if ($recheck) {
$ref = ref($a[$i]);
# Perl scalars are fed to the appropriate constructor.
unless ($ref) {
$a[$i] = $a[0] -> new($a[$i]);
next;
}
# If it is an object of the right class, all is fine.
next if $ref -> isa($a[0]);
}
# Last resort.
$a[$i] = $a[0] -> new($a[$i]);
}
# Reset the downgrading.
${"$a[0]::downgrade"} = $down;
return @a;
}
sub import {
my $class = 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 { $class->new(shift) },
binary => sub { $class->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;
$class->SUPER::import(@a); # need it for subclasses
$class->export_to_level(1, $class, @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{$lib} || 0) < 2) {
Carp::carp("$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)) {
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';
Carp::carp("Cannot load outdated $lib v$ver, please upgrade");
$WARN{$lib} = 2; # never warn again
}
}
}
}
if ($CALC eq '') {
if ($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 _register_callback {
my ($class, $callback) = @_;
if (ref($callback) ne 'CODE') {
Carp::croak("$callback is not a coderef");
}
$CALLBACKS{$class} = $callback;
}
sub _split_dec_string {
my $str = shift;
if ($str =~ s/
^
# leading whitespace
( \s* )
# optional sign
( [+-]? )
# significand
(
\d+ (?: _ \d+ )*
(?:
\.
(?: \d+ (?: _ \d+ )* )?
)?
|
\.
\d+ (?: _ \d+ )*
)
# optional exponent
(?:
[Ee]
( [+-]? )
( \d+ (?: _ \d+ )* )
)?
# trailing stuff
( \D .*? )?
\z
//x) {
my $leading = $1;
my $significand_sgn = $2 || '+';
my $significand_abs = $3;
my $exponent_sgn = $4 || '+';
my $exponent_abs = $5 || '0';
my $trailing = $6;
# Remove underscores and leading zeros.
$significand_abs =~ tr/_//d;
$exponent_abs =~ tr/_//d;
$significand_abs =~ s/^0+(.)/$1/;
$exponent_abs =~ s/^0+(.)/$1/;
# If the significand contains a dot, remove it and adjust the exponent
# accordingly. E.g., "1234.56789e+3" -> "123456789e-2"
my $idx = index $significand_abs, '.';
if ($idx > -1) {
$significand_abs =~ s/0+\z//;
substr($significand_abs, $idx, 1) = '';
my $exponent = $exponent_sgn . $exponent_abs;
$exponent .= $idx - CORE::length($significand_abs);
$exponent_abs = abs $exponent;
$exponent_sgn = $exponent < 0 ? '-' : '+';
}
return($leading,
$significand_sgn, $significand_abs,
$exponent_sgn, $exponent_abs,
$trailing);
}
return undef;
}
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
}
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 _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, -$follow) =~ /[^0]/ ? 1 : 0;
}
sub _find_round_parameters {
# After any operation or when calling round(), the result is rounded by
# regarding the A & P from arguments, local parameters, or globals.
# !!!!!!! If you change this, remember to change round(), too! !!!!!!!!!!
# This procedure finds the round parameters, but it is for speed reasons
# duplicated in round. Otherwise, it is tested by the testsuite and used
# by bdiv().
# returns ($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 $class = ref($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 = ${"$class\::accuracy"} unless defined $a;
$p = ${"$class\::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 = ${"$class\::round_mode"} unless defined $r;
if ($r !~ /^(even|odd|[+-]inf|zero|trunc|common)$/) {
Carp::croak("Unknown round mode '$r'");
}
$a = int($a) if defined $a;
$p = int($p) if defined $p;
($self, $a, $p, $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 falls back to
# pure Perl if the GMP library is not installed):
# (See also the L<MATH LIBRARY> section!)
# warns if Math::BigInt::GMP cannot be found
use Math::BigInt lib => 'GMP';
# to suppress the warning use this:
# use Math::BigInt try => 'GMP';
# dies if GMP cannot be loaded:
# use Math::BigInt only => 'GMP';
my $str = '1234567890';
my @values = (64, 74, 18);
my $n = 1; my $sign = '-';
# Configuration methods (may be used as class methods and instance methods)
Math::BigInt->accuracy(); # get class accuracy
Math::BigInt->accuracy($n); # set class accuracy
Math::BigInt->precision(); # get class precision
Math::BigInt->precision($n); # set class precision
Math::BigInt->round_mode(); # get class rounding mode
Math::BigInt->round_mode($m); # set global round mode, must be one of
# 'even', 'odd', '+inf', '-inf', 'zero',
# 'trunc', or 'common'
Math::BigInt->config(); # return hash with configuration
# Constructor methods (when the class methods below are used as instance
# methods, the value is assigned the invocand)
$x = Math::BigInt->new($str); # defaults to 0
$x = Math::BigInt->new('0x123'); # from hexadecimal
$x = Math::BigInt->new('0b101'); # from binary
$x = Math::BigInt->from_hex('cafe'); # from hexadecimal
$x = Math::BigInt->from_oct('377'); # from octal
$x = Math::BigInt->from_bin('1101'); # from binary
$x = Math::BigInt->bzero(); # create a +0
$x = Math::BigInt->bone(); # create a +1
$x = Math::BigInt->bone('-'); # create a -1
$x = Math::BigInt->binf(); # create a +inf
$x = Math::BigInt->binf('-'); # create a -inf
$x = Math::BigInt->bnan(); # create a Not-A-Number
$x = Math::BigInt->bpi(); # returns pi
$y = $x->copy(); # make a copy (unlike $y = $x)
$y = $x->as_int(); # return as a Math::BigInt
# Boolean methods (these don't modify the invocand)
$x->is_zero(); # if $x is 0
$x->is_one(); # if $x is +1
$x->is_one("+"); # ditto
$x->is_one("-"); # if $x is -1
$x->is_inf(); # if $x is +inf or -inf
$x->is_inf("+"); # if $x is +inf
$x->is_inf("-"); # if $x is -inf
$x->is_nan(); # if $x is NaN
$x->is_positive(); # if $x > 0
$x->is_pos(); # ditto
$x->is_negative(); # if $x < 0
$x->is_neg(); # ditto
$x->is_odd(); # if $x is odd
$x->is_even(); # if $x is even
$x->is_int(); # if $x is an integer
# Comparison methods
$x->bcmp($y); # compare numbers (undef, < 0, == 0, > 0)
$x->bacmp($y); # compare absolutely (undef, < 0, == 0, > 0)
$x->beq($y); # true if and only if $x == $y
$x->bne($y); # true if and only if $x != $y
$x->blt($y); # true if and only if $x < $y
$x->ble($y); # true if and only if $x <= $y
$x->bgt($y); # true if and only if $x > $y
$x->bge($y); # true if and only if $x >= $y
# Arithmetic methods
$x->bneg(); # negation
$x->babs(); # absolute value
$x->bsgn(); # sign function (-1, 0, 1, or NaN)
$x->bnorm(); # normalize (no-op)
$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)
$x->bmuladd($y,$z); # $x = $x * $y + $z
$x->bdiv($y); # division (floored), set $x to quotient
# return (quo,rem) or quo if scalar
$x->btdiv($y); # division (truncated), set $x to quotient
# return (quo,rem) or quo if scalar
$x->bmod($y); # modulus (x % y)
$x->btmod($y); # modulus (truncated)
$x->bmodinv($mod); # modular multiplicative inverse
$x->bmodpow($y,$mod); # modular exponentiation (($x ** $y) % $mod)
$x->bpow($y); # power of arguments (x ** y)
$x->blog(); # logarithm of $x to base e (Euler's number)
$x->blog($base); # logarithm of $x to base $base (e.g., base 2)
$x->bexp(); # calculate e ** $x where e is Euler's number
$x->bnok($y); # x over y (binomial coefficient n over k)
$x->bsin(); # sine
$x->bcos(); # cosine
$x->batan(); # inverse tangent
$x->batan2($y); # two-argument inverse tangent
$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)
$x->blsft($n); # left shift $n places in base 2
$x->blsft($n,$b); # left shift $n places in base $b
# returns (quo,rem) or quo (scalar context)
$x->brsft($n); # right shift $n places in base 2
$x->brsft($n,$b); # right shift $n places in base $b
# returns (quo,rem) or quo (scalar context)
# Bitwise methods
$x->band($y); # bitwise and
$x->bior($y); # bitwise inclusive or
$x->bxor($y); # bitwise exclusive or
$x->bnot(); # bitwise not (two's complement)
# Rounding methods
$x->round($A,$P,$mode); # round to accuracy or precision using
# rounding mode $mode
$x->bround($n); # accuracy: preserve $n digits
$x->bfround($n); # $n > 0: round to $nth digit left of dec. point
# $n < 0: round to $nth digit right of dec. point
$x->bfloor(); # round towards minus infinity
$x->bceil(); # round towards plus infinity
$x->bint(); # round towards zero
# Other mathematical methods
$x->bgcd($y); # greatest common divisor
$x->blcm($y); # least common multiple
# Object property methods (do not modify the invocand)
$x->sign(); # the sign, either +, - or NaN
$x->digit($n); # the nth digit, counting from the right
$x->digit(-$n); # the nth digit, counting from the left
$x->length(); # return number of digits in number
($xl,$f) = $x->length(); # length of number and length of fraction
# part, latter is always 0 digits long
# for Math::BigInt objects
$x->mantissa(); # return (signed) mantissa as a Math::BigInt
$x->exponent(); # return exponent as a Math::BigInt
$x->parts(); # return (mantissa,exponent) as a Math::BigInt
$x->sparts(); # mantissa and exponent (as integers)
$x->nparts(); # mantissa and exponent (normalised)
$x->eparts(); # mantissa and exponent (engineering notation)
$x->dparts(); # integer and fraction part
# Conversion methods (do not modify the invocand)
$x->bstr(); # decimal notation, possibly zero padded
$x->bsstr(); # string in scientific notation with integers
$x->bnstr(); # string in normalized notation
$x->bestr(); # string in engineering notation
$x->bdstr(); # string in decimal notation
$x->as_hex(); # as signed hexadecimal string with prefixed 0x
$x->as_bin(); # as signed binary string with prefixed 0b
$x->as_oct(); # as signed octal string with prefixed 0
$x->as_bytes(); # as byte string
# Other conversion methods
$x->numify(); # return as scalar (might overflow or underflow)
=head1 DESCRIPTION
Math::BigInt provides support for arbitrary precision integers. Overloading is
also provided for Perl operators.
=head2 Input
Input values to these routines may be any scalar number or string that looks
like a number and represents an integer.
=over
=item *
Leading and trailing whitespace is ignored.
=item *
Leading and trailing zeros are ignored.
=item *
If the string has a "0x" prefix, it is interpreted as a hexadecimal number.
=item *
If the string has a "0b" prefix, it is interpreted as a binary number.
=item *
One underline is allowed between any two digits.
=item *
If the string can not be interpreted, NaN is returned.
=back
Octal numbers are typically prefixed by "0", but since leading zeros are
stripped, these methods can not automatically recognize octal numbers, so use
the constructor from_oct() to interpret octal strings.
Some examples of valid string input
Input string Resulting value
123 123
1.23e2 123
12300e-2 123
0xcafe 51966
0b1101 13
67_538_754 67538754
-4_5_6.7_8_9e+0_1_0 -4567890000000
Input given as scalar numbers might lose precision. Quote your input to ensure
that no digits are lost:
$x = Math::BigInt->new( 56789012345678901234 ); # bad
$x = Math::BigInt->new('56789012345678901234'); # good
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();
$nan = Math::BigInt->bnan();
=head2 Output
Output values are usually Math::BigInt objects.
Boolean operators C<is_zero()>, C<is_one()>, C<is_inf()>, etc. return true or
false.
Comparison operators C<bcmp()> and C<bacmp()>) return -1, 0, 1, or
undef.
=head1 METHODS
=head2 Configuration methods
Each of the methods below (except config(), accuracy() and precision()) accepts
three additional parameters. These arguments C<$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.
Setting a class variable effects all object instance that are created
afterwards.
=over
=item accuracy()
Math::BigInt->accuracy(5); # set class accuracy
$x->accuracy(5); # set instance accuracy
$A = Math::BigInt->accuracy(); # get class accuracy
$A = $x->accuracy(); # get instance accuracy
Set or get the accuracy, i.e., the number of significant digits. The accuracy
must be an integer. If the accuracy is set to C<undef>, no rounding is done.
Alternatively, one can round the results explicitly using one of L</round()>,
L</bround()> or L</bfround()> or by passing the desired accuracy to the method
as an additional parameter:
my $x = Math::BigInt->new(30000);
my $y = Math::BigInt->new(7);
print scalar $x->copy()->bdiv($y, 2); # prints 4300
print scalar $x->copy()->bdiv($y)->bround(2); # prints 4300
Please see the section about L</ACCURACY and PRECISION> for further details.
$y = Math::BigInt->new(1234567); # $y is not rounded
Math::BigInt->accuracy(4); # set class accuracy to 4
$x = Math::BigInt->new(1234567); # $x is rounded automatically
print "$x $y"; # prints "1235000 1234567"
print $x->accuracy(); # prints "4"
print $y->accuracy(); # also prints "4", since
# class accuracy is 4
Math::BigInt->accuracy(5); # set class accuracy to 5
print $x->accuracy(); # prints "4", since instance
# accuracy is 4
print $y->accuracy(); # prints "5", since no instance
# accuracy, and class accuracy is 5
Note: 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.
=item precision()
Math::BigInt->precision(-2); # set class precision
$x->precision(-2); # set instance precision
$P = Math::BigInt->precision(); # get class precision
$P = $x->precision(); # get instance precision
Set or get the precision, i.e., the place to round relative to the decimal
point. The precision must be a integer. Setting the precision to $P means that
each number is rounded up or down, depending on the rounding mode, to the
nearest multiple of 10**$P. If the precision is set to C<undef>, no rounding is
done.
You might 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.
Please see the section about L</ACCURACY and PRECISION> for further details.
$y = Math::BigInt->new(1234567); # $y is not rounded
Math::BigInt->precision(4); # set class precision to 4
$x = Math::BigInt->new(1234567); # $x is rounded automatically
print $x; # prints "1230000"
Note: 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.
=item div_scale()
Set/get the fallback accuracy. This is the accuracy used when neither accuracy
nor precision is set explicitly. It is used when a computation might otherwise
attempt to return an infinite number of digits.
=item round_mode()
Set/get the rounding mode.
=item upgrade()
Set/get the class for upgrading. When a computation might result in a
non-integer, the operands are upgraded to this class. This is used for instance
by L<bignum>. The default is C<undef>, thus the following operation creates
a Math::BigInt, not a Math::BigFloat:
my $i = Math::BigInt->new(123);
my $f = Math::BigFloat->new('123.1');
print $i + $f, "\n"; # prints 246
=item downgrade()
Set/get the class for downgrading. The default is C<undef>. Downgrading is not
done by Math::BigInt.
=item modify()
$x->modify('bpowd');
This method returns 0 if the object can be modified with the given operation,
or 1 if not.
This is used for instance by L<Math::BigInt::Constant>.
=item config()
use Data::Dumper;
print Dumper ( Math::BigInt->config() );
print Math::BigInt->config()->{lib},"\n";
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:
accuracy precision round_mode div_scale
upgrade downgrade trap_inf trap_nan
Example:
$new_cfg = Math::BigInt->config(
{ trap_inf => 1, precision => 5 }
);
=back
=head2 Constructor methods
=over
=item new()
$x = Math::BigInt->new($str,$A,$P,$R);
Creates a new Math::BigInt object from a scalar or another Math::BigInt object.
The input is accepted as decimal, hexadecimal (with leading '0x') or binary
(with leading '0b').
See L</Input> for more info on accepted input formats.
=item from_hex()
$x = Math::BigInt->from_hex("0xcafe"); # input is hexadecimal
Interpret input as a hexadecimal string. A "0x" or "x" prefix is optional. A
single underscore character may be placed right after the prefix, if present,
or between any two digits. If the input is invalid, a NaN is returned.
=item from_oct()
$x = Math::BigInt->from_oct("0775"); # input is octal
Interpret the input as an octal string and return the corresponding value. A
"0" (zero) prefix is optional. A single underscore character may be placed
right after the prefix, if present, or between any two digits. If the input is
invalid, a NaN is returned.
=item from_bin()
$x = Math::BigInt->from_bin("0b10011"); # input is binary
Interpret the input as a binary string. A "0b" or "b" prefix is optional. A
single underscore character may be placed right after the prefix, if present,
or between any two digits. If the input is invalid, a NaN is returned.
=item from_bytes()
$x = Math::BigInt->from_bytes("\xf3\x6b"); # $x = 62315
Interpret the input as a byte string, assuming big endian byte order. The
output is always a non-negative, finite integer.
In some special cases, from_bytes() matches the conversion done by unpack():
$b = "\x4e"; # one char byte string
$x = Math::BigInt->from_bytes($b); # = 78
$y = unpack "C", $b; # ditto, but scalar
$b = "\xf3\x6b"; # two char byte string
$x = Math::BigInt->from_bytes($b); # = 62315
$y = unpack "S>", $b; # ditto, but scalar
$b = "\x2d\xe0\x49\xad"; # four char byte string
$x = Math::BigInt->from_bytes($b); # = 769673645
$y = unpack "L>", $b; # ditto, but scalar
$b = "\x2d\xe0\x49\xad\x2d\xe0\x49\xad"; # eight char byte string
$x = Math::BigInt->from_bytes($b); # = 3305723134637787565
$y = unpack "Q>", $b; # ditto, but scalar
=item bzero()
$x = Math::BigInt->bzero();
$x->bzero();
Returns a new Math::BigInt object representing zero. If used as an instance
method, assigns the value to the invocand.
=item bone()
$x = Math::BigInt->bone(); # +1
$x = Math::BigInt->bone("+"); # +1
$x = Math::BigInt->bone("-"); # -1
$x->bone(); # +1
$x->bone("+"); # +1
$x->bone('-'); # -1
Creates a new Math::BigInt object representing one. The optional argument is
either '-' or '+', indicating whether you want plus one or minus one. If used
as an instance method, assigns the value to the invocand.
=item binf()
$x = Math::BigInt->binf($sign);
Creates a new Math::BigInt object representing infinity. The optional argument
is either '-' or '+', indicating whether you want infinity or minus infinity.
If used as an instance method, assigns the value to the invocand.
$x->binf();
$x->binf('-');
=item bnan()
$x = Math::BigInt->bnan();
Creates a new Math::BigInt object representing NaN (Not A Number). If used as
an instance method, assigns the value to the invocand.
$x->bnan();
=item bpi()
$x = Math::BigInt->bpi(100); # 3
$x->bpi(100); # 3
Creates a new Math::BigInt object representing PI. If used as an instance
method, assigns the value to the invocand. With Math::BigInt this always
returns 3.
If upgrading is in effect, returns PI, rounded to N digits with the current
rounding mode:
use Math::BigFloat;
use Math::BigInt upgrade => "Math::BigFloat";
print Math::BigInt->bpi(3), "\n"; # 3.14
print Math::BigInt->bpi(100), "\n"; # 3.1415....
=item copy()
$x->copy(); # make a true copy of $x (unlike $y = $x)
=item as_int()
=item as_number()
These methods are called when Math::BigInt encounters an object it doesn't know
how to handle. For instance, assume $x is a Math::BigInt, or subclass thereof,
and $y is defined, but not a Math::BigInt, or subclass thereof. If you do
$x -> badd($y);
$y needs to be converted into an object that $x can deal with. This is done by
first checking if $y is something that $x might be upgraded to. If that is the
case, no further attempts are made. The next is to see if $y supports the
method C<as_int()>. If it does, C<as_int()> is called, but if it doesn't, the
next thing is to see if $y supports the method C<as_number()>. If it does,
C<as_number()> is called. The method C<as_int()> (and C<as_number()>) is
expected to return either an object that has the same class as $x, a subclass
thereof, or a string that C<ref($x)-E<gt>new()> can parse to create an object.
C<as_number()> is an alias to C<as_int()>. C<as_number> was introduced in
v1.22, while C<as_int()> was introduced in v1.68.
In Math::BigInt, C<as_int()> has the same effect as C<copy()>.
=back
=head2 Boolean methods
None of these methods modify the invocand object.
=over
=item is_zero()
$x->is_zero(); # true if $x is 0
Returns true if the invocand is zero and false otherwise.
=item is_one( [ SIGN ])
$x->is_one(); # true if $x is +1
$x->is_one("+"); # ditto
$x->is_one("-"); # true if $x is -1
Returns true if the invocand is one and false otherwise.
=item is_finite()
$x->is_finite(); # true if $x is not +inf, -inf or NaN
Returns true if the invocand is a finite number, i.e., it is neither +inf,
-inf, nor NaN.
=item is_inf( [ SIGN ] )
$x->is_inf(); # true if $x is +inf
$x->is_inf("+"); # ditto
$x->is_inf("-"); # true if $x is -inf
Returns true if the invocand is infinite and false otherwise.
=item is_nan()
$x->is_nan(); # true if $x is NaN
=item is_positive()
=item is_pos()
$x->is_positive(); # true if > 0
$x->is_pos(); # ditto
Returns true if the invocand is positive and false otherwise. A C<NaN> is
neither positive nor negative.
=item is_negative()
=item is_neg()
$x->is_negative(); # true if < 0
$x->is_neg(); # ditto
Returns true if the invocand is negative and false otherwise. A C<NaN> is
neither positive nor negative.
=item is_odd()
$x->is_odd(); # true if odd, false for even
Returns true if the invocand is odd and false otherwise. C<NaN>, C<+inf>, and
C<-inf> are neither odd nor even.
=item is_even()
$x->is_even(); # true if $x is even
Returns true if the invocand is even and false otherwise. C<NaN>, C<+inf>,
C<-inf> are not integers and are neither odd nor even.
=item is_int()
$x->is_int(); # true if $x is an integer
Returns true if the invocand is an integer and false otherwise. C<NaN>,
C<+inf>, C<-inf> are not integers.
=back
=head2 Comparison methods
None of these methods modify the invocand object. Note that a C<NaN> is neither
less than, greater than, or equal to anything else, even a C<NaN>.
=over
=item bcmp()
$x->bcmp($y);
Returns -1, 0, 1 depending on whether $x is less than, equal to, or grater than
$y. Returns undef if any operand is a NaN.
=item bacmp()
$x->bacmp($y);
Returns -1, 0, 1 depending on whether the absolute value of $x is less than,
equal to, or grater than the absolute value of $y. Returns undef if any operand
is a NaN.
=item beq()
$x -> beq($y);
Returns true if and only if $x is equal to $y, and false otherwise.
=item bne()
$x -> bne($y);
Returns true if and only if $x is not equal to $y, and false otherwise.
=item blt()
$x -> blt($y);
Returns true if and only if $x is equal to $y, and false otherwise.
=item ble()
$x -> ble($y);
Returns true if and only if $x is less than or equal to $y, and false
otherwise.
=item bgt()
$x -> bgt($y);
Returns true if and only if $x is greater than $y, and false otherwise.
=item bge()
$x -> bge($y);
Returns true if and only if $x is greater than or equal to $y, and false
otherwise.
=back
=head2 Arithmetic methods
These methods modify the invocand object and returns it.
=over
=item 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.
=item babs()
$x->babs();
Set the number to its absolute value, e.g. change the sign from '-' to '+'
and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
numbers.
=item bsgn()
$x->bsgn();
Signum function. Set the number to -1, 0, or 1, depending on whether the
number is negative, zero, or positive, respectively. Does not modify NaNs.
=item bnorm()
$x->bnorm(); # normalize (no-op)
Normalize the number. This is a no-op and is provided only for backwards
compatibility.
=item binc()
$x->binc(); # increment x by 1
=item bdec()
$x->bdec(); # decrement x by 1
=item badd()
$x->badd($y); # addition (add $y to $x)
=item bsub()
$x->bsub($y); # subtraction (subtract $y from $x)
=item bmul()
$x->bmul($y); # multiplication (multiply $x by $y)
=item bmuladd()
$x->bmuladd($y,$z);
Multiply $x by $y, and then add $z to the result,
This method was added in v1.87 of Math::BigInt (June 2007).
=item bdiv()
$x->bdiv($y); # divide, set $x to quotient
Divides $x by $y by doing floored division (F-division), where the quotient is
the floored (rounded towards negative infinity) quotient of the two operands.
In list context, returns the quotient and the remainder. The remainder is
either zero or has the same sign as the second operand. In scalar context, only
the quotient is returned.
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
non-zero) 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)
The behavior of the overloaded 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 finite $x and finite, non-zero $y.
Perl's "use integer" might change the behaviour of % and / for scalars. This is
because under 'use integer' Perl does what the underlying C library thinks is
right, and this varies. However, "use integer" does not change the way things
are done with Math::BigInt objects.
=item btdiv()
$x->btdiv($y); # divide, set $x to quotient
Divides $x by $y by doing truncated division (T-division), where quotient is
the truncated (rouneded towards zero) quotient of the two operands. In list
context, returns the quotient and the remainder. The remainder is either zero
or has the same sign as the first operand. In scalar context, only the quotient
is returned.
=item bmod()
$x->bmod($y); # modulus (x % y)
Returns $x modulo $y, i.e., the remainder after floored division (F-division).
This method is like Perl's % operator. See L</bdiv()>.
=item btmod()
$x->btmod($y); # modulus
Returns the remainer after truncated division (T-division). See L</btdiv()>.
=item bmodinv()
$x->bmodinv($mod); # modular multiplicative inverse
Returns the multiplicative inverse of C<$x> modulo C<$mod>. If
$y = $x -> copy() -> bmodinv($mod)
then C<$y> is the number closest to zero, and with the same sign as C<$mod>,
satisfying
($x * $y) % $mod = 1 % $mod
If C<$x> and C<$y> are non-zero, they must be relative primes, i.e.,
C<bgcd($y, $mod)==1>. 'C<NaN>' is returned when no modular multiplicative
inverse exists.
=item bmodpow()
$num->bmodpow($exp,$mod); # modular exponentiation
# ($num**$exp % $mod)
Returns the value of C<$num> taken to the power C<$exp> in the modulus
C<$mod> using binary exponentiation. 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)
=item bpow()
$x->bpow($y); # power of arguments (x ** y)
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 modifies
$x, the last one won't:
print bpow($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 blog()
$x->blog($base, $accuracy); # logarithm of x to the base $base
If C<$base> is not defined, Euler's number (e) is used:
print $x->blog(undef, 100); # log(x) to 100 digits
=item bexp()
$x->bexp($accuracy); # calculate e ** X
Calculates the expression C<e ** $x> where C<e> is Euler's number.
This method was added in v1.82 of Math::BigInt (April 2007).
See also L</blog()>.
=item bnok()
$x->bnok($y); # x over y (binomial coefficient n over k)
Calculates the binomial coefficient n over k, also called the "choose"
function. The result is equivalent to:
( n ) n!
| - | = -------
( k ) k!(n-k)!
This method was added in v1.84 of Math::BigInt (April 2007).
=item bsin()
my $x = Math::BigInt->new(1);
print $x->bsin(100), "\n";
Calculate the sine of $x, modifying $x in place.
In Math::BigInt, unless upgrading is in effect, the result is truncated to an
integer.
This method was added in v1.87 of Math::BigInt (June 2007).
=item bcos()
my $x = Math::BigInt->new(1);
print $x->bcos(100), "\n";
Calculate the cosine of $x, modifying $x in place.
In Math::BigInt, unless upgrading is in effect, the result is truncated to an
integer.
This method was added in v1.87 of Math::BigInt (June 2007).
=item batan()
my $x = Math::BigFloat->new(0.5);
print $x->batan(100), "\n";
Calculate the arcus tangens of $x, modifying $x in place.
In Math::BigInt, unless upgrading is in effect, the result is truncated to an
integer.
This method was added in v1.87 of Math::BigInt (June 2007).
=item batan2()
my $x = Math::BigInt->new(1);
my $y = Math::BigInt->new(1);
print $y->batan2($x), "\n";
Calculate the arcus tangens of C<$y> divided by C<$x>, modifying $y in place.
In Math::BigInt, unless upgrading is in effect, the result is truncated to an
integer.
This method was added in v1.87 of Math::BigInt (June 2007).
=item bsqrt()
$x->bsqrt(); # calculate square-root
C<bsqrt()> returns the square root 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 broot()
$x->broot($N);
Calculates the N'th root of C<$x>.
=item bfac()
$x->bfac(); # factorial of $x (1*2*3*4*..*$x)
=item brsft()
$x->brsft($n); # right shift $n places in base 2
$x->brsft($n, $b); # right shift $n places in base $b
The latter is equivalent to
$x -> bdiv($b -> copy() -> bpow($n))
=item blsft()
$x->blsft($n); # left shift $n places in base 2
$x->blsft($n, $b); # left shift $n places in base $b
The latter is equivalent to
$x -> bmul($b -> copy() -> bpow($n))
=back
=head2 Bitwise methods
=over
=item band()
$x->band($y); # bitwise and
=item bior()
$x->bior($y); # bitwise inclusive or
=item bxor()
$x->bxor($y); # bitwise exclusive or
=item bnot()
$x->bnot(); # bitwise not (two's complement)
Two's complement (bitwise not). This is equivalent to, but faster than,
$x->binc()->bneg();
=back
=head2 Rounding methods
=over
=item round()
$x->round($A,$P,$round_mode);
Round $x to accuracy C<$A> or precision C<$P> using the round mode
C<$round_mode>.
=item bround()
$x->bround($N); # accuracy: preserve $N digits
Rounds $x to an accuracy of $N digits.
=item bfround()
$x->bfround($N);
Rounds to a multiple of 10**$N. Examples:
Input N Result
123456.123456 3 123500
123456.123456 2 123450
123456.123456 -2 123456.12
123456.123456 -3 123456.123
=item bfloor()
$x->bfloor();
Round $x towards minus infinity, i.e., set $x to the largest integer less than
or equal to $x.
=item bceil()
$x->bceil();
Round $x towards plus infinity, i.e., set $x to the smallest integer greater
than or equal to $x).
=item bint()
$x->bint();
Round $x towards zero.
=back
=head2 Other mathematical methods
=over
=item bgcd()
$x -> bgcd($y); # GCD of $x and $y
$x -> bgcd($y, $z, ...); # GCD of $x, $y, $z, ...
Returns the greatest common divisor (GCD).
=item blcm()
$x -> blcm($y); # LCM of $x and $y
$x -> blcm($y, $z, ...); # LCM of $x, $y, $z, ...
Returns the least common multiple (LCM).
=back
=head2 Object property methods
=over
=item 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'
=item digit()
$x->digit($n); # return the nth digit, counting from right
If C<$n> is negative, returns the digit counting from left.
=item 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
Math::BigInt objects, the length of the fraction part is always 0.
The following probably doesn't 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 mantissa()
$x->mantissa();
Return the signed mantissa of $x as a Math::BigInt.
=item exponent()
$x->exponent();
Return the exponent of $x as a Math::BigInt.
=item parts()
$x->parts();
Returns the significand (mantissa) and the exponent as integers. In
Math::BigFloat, both are returned as Math::BigInt objects.
=item sparts()
Returns the significand (mantissa) and the exponent as integers. In scalar
context, only the significand is returned. The significand is the integer with
the smallest absolute value. The output of C<sparts()> corresponds to the
output from C<bsstr()>.
In Math::BigInt, this method is identical to C<parts()>.
=item nparts()
Returns the significand (mantissa) and exponent corresponding to normalized
notation. In scalar context, only the significand is returned. For finite
non-zero numbers, the significand's absolute value is greater than or equal to
1 and less than 10. The output of C<nparts()> corresponds to the output from
C<bnstr()>. In Math::BigInt, if the significand can not be represented as an
integer, upgrading is performed or NaN is returned.
=item eparts()
Returns the significand (mantissa) and exponent corresponding to engineering
notation. In scalar context, only the significand is returned. For finite
non-zero numbers, the significand's absolute value is greater than or equal to
1 and less than 1000, and the exponent is a multiple of 3. The output of
C<eparts()> corresponds to the output from C<bestr()>. In Math::BigInt, if the
significand can not be represented as an integer, upgrading is performed or NaN
is returned.
=item dparts()
Returns the integer part and the fraction part. If the fraction part can not be
represented as an integer, upgrading is performed or NaN is returned. The
output of C<dparts()> corresponds to the output from C<bdstr()>.
=back
=head2 String conversion methods
=over
=item bstr()
Returns a string representing the number using decimal notation. In
Math::BigFloat, the output is zero padded according to the current accuracy or
precision, if any of those are defined.
=item bsstr()
Returns a string representing the number using scientific notation where both
the significand (mantissa) and the exponent are integers. The output
corresponds to the output from C<sparts()>.
123 is returned as "123e+0"
1230 is returned as "123e+1"
12300 is returned as "123e+2"
12000 is returned as "12e+3"
10000 is returned as "1e+4"
=item bnstr()
Returns a string representing the number using normalized notation, the most
common variant of scientific notation. For finite non-zero numbers, the
absolute value of the significand is less than or equal to 1 and less than 10.
The output corresponds to the output from C<nparts()>.
123 is returned as "1.23e+2"
1230 is returned as "1.23e+3"
12300 is returned as "1.23e+4"
12000 is returned as "1.2e+4"
10000 is returned as "1e+4"
=item bestr()
Returns a string representing the number using engineering notation. For finite
non-zero numbers, the absolute value of the significand is less than or equal
to 1 and less than 1000, and the exponent is a multiple of 3. The output
corresponds to the output from C<eparts()>.
123 is returned as "123e+0"
1230 is returned as "1.23e+3"
12300 is returned as "12.3e+3"
12000 is returned as "12e+3"
10000 is returned as "10e+3"
=item bdstr()
Returns a string representing the number using decimal notation. The output
corresponds to the output from C<dparts()>.
123 is returned as "123"
1230 is returned as "1230"
12300 is returned as "12300"
12000 is returned as "12000"
10000 is returned as "10000"
=item as_hex()
$x->as_hex();
Returns a string representing the number using hexadecimal notation. The output
is prefixed by "0x".
=item as_bin()
$x->as_bin();
Returns a string representing the number using binary notation. The output is
prefixed by "0b".
=item as_oct()
$x->as_oct();
Returns a string representing the number using octal notation. The output is
prefixed by "0".
=item as_bytes()
$x = Math::BigInt->new("1667327589");
$s = $x->as_bytes(); # $s = "cafe"
Returns a byte string representing the number using big endian byte order. The
invocand must be a non-negative, finite integer.
=back
=head2 Other conversion methods
=over
=item numify()
print $x->numify();
Returns a Perl scalar from $x. It is used automatically whenever a scalar is
needed, for instance in array index operations.
=back
=head1 ACCURACY and PRECISION
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::BigInt and
Math::BigFloat 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) are be used.
=head2 Precision P
Precision is 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 at least 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 initial value was). It could also
have p < 0, when the digits after the decimal point are zero.
The string output (of floating point numbers) is 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 Math::BigInt objects, 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,
123.45000 has 8 and 0.000123 has 3.
The string output (of floating point numbers) is 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 Math::BigInt objects, 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
=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.
=item 'common'
round up if the digit immediately to the right of the rounding place is 5 or
greater, otherwise round down. E.g., 0.15 becomes 0.2 and 0.149 becomes 0.1.
=back
The handling of A & P in MBI/MBF (the old core code shipped with Perl versions
<= 5.7.2) is like this:
=over
=item Precision
* bfround($p) is able to round to $p number of digits after the decimal
point
* otherwise P is unused
=item Accuracy (significant digits)
* bround($a) rounds to $a significant digits
* only bdiv() and bsqrt() take A as (optional) parameter
+ other operations simply create the same number (bneg etc), or
more (bmul) of digits
+ rounding/truncating is only done when explicitly calling one
of bround or bfround, and never for Math::BigInt (not implemented)
* bsqrt() simply hands its accuracy argument over to bdiv.
* the documentation and the comment in the code indicate two
different ways on how bdiv() determines the maximum number
of digits it should calculate, and the actual code does yet
another thing
POD:
max($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
So for lx = 3, ly = 9, scale = 10, scale will actually be 16
(10+9-3). Actually, the 'difference' added to the scale is cal-
culated from the number of "significant digits" in dividend and
divisor, which is derived by looking at the length of the man-
tissa. 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 signif-
icant digits. The rounding after the division then uses the
remainder and $y to determine whether 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
=item Setting/Accessing
* You can set the A global via Math::BigInt->accuracy() or
Math::BigFloat->accuracy() or whatever class you are using.
* You can also set P globally by using Math::SomeClass->precision()
likewise.
* Globals are classwide, and not inherited by subclasses.
* to undefine A, use Math::SomeCLass->accuracy(undef);
* to undefine P, use Math::SomeClass->precision(undef);
* Setting Math::SomeClass->accuracy() clears automatically
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 Math::SomeClass->accuracy()
* to find out the current global P, use Math::SomeClass->precision()
* use $x->accuracy() respective $x->precision() for the local
setting of $x.
* Please note that $x->accuracy() respective $x->precision()
return eventually defined global A or P, when $x's A or P is not
set.
=item Creating numbers
* When you create a number, you can give the 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, NO rounding will occur, and the globals will
NOT be used. This is used by subclasses to create numbers without
suffering rounding in the parent. Thus a subclass is able to have its own
globals enforced upon creation of a number by using
$x = Math::BigInt->new($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 Math::BigInt objects 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 Math::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 bdiv/bsqrt) and will not
be rounded.
* There is another setting for bdiv() (and thus for bsqrt()). If neither of
A or P is defined, bdiv() 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 :-)
* bdiv 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 bdiv() and bsqrt() 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()->bround(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
* bsqrt() will hand its arguments to bdiv(), 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 $x->accuracy() or
$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.
* $x->accuracy() clears $x->precision(), and vice versa.
=item Rounding
* the rounding routines will use the respective global or local settings.
bround() is for accuracy rounding, while 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', 'common'
* you can set/get the global R by using Math::SomeClass->round_mode()
or by setting $Math::SomeClass::round_mode
* after each operation, $result->round() is called, and the result may
eventually be rounded (that is, if A or P were set either locally,
globally or as parameter to the operation)
* to manually round a number, call $x->round($A,$P,$round_mode);
this will round the number by using the appropriate rounding function
and then normalize it.
* rounding modifies the local settings of the number:
$x = Math::BigFloat->new(123.456);
$x->accuracy(5);
$x->bround(4);
Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
will be 4 from now on.
=item Default values
* R: 'even'
* F: 40
* A: undef
* P: undef
=item Remarks
* The defaults are set up so that the new code gives the same results as
the old code (except in a few cases on bdiv):
+ 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 Math::BigInt has extensive handling of inf and NaN, certain quirks
remain.
=over
=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
Math::BigInt.
=back
=head1 INTERNALS
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 try => 'Calc';
You can change this backend library by using:
use Math::BigInt try => 'GMP';
B<Note>: General purpose packages should not be explicit about the library to
use; let the script author decide which is best.
If your script works with huge numbers and Calc is too slow for them, you can
also for the loading of one of these libraries and if none of them can be used,
the code dies:
use Math::BigInt only => 'GMP,Pari';
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 try => 'Foo,Math::BigInt::Bar';
The library that is loaded last is used. Note that this can be overwritten at
any time by loading a different library, and numbers constructed with different
libraries cannot be used in math operations together.
=head3 What library to use?
B<Note>: General purpose packages should not be explicit about the library to
use; let the script author decide which is best.
L<Math::BigInt::GMP> and L<Math::BigInt::Pari> are in cases involving big
numbers much faster than Calc, however it is slower when dealing with very
small numbers (less than about 20 digits) and when converting very large
numbers to decimal (for instance for printing, rounding, calculating their
length in decimal etc).
So please select carefully what library you want to use.
Different low-level libraries use different formats to store the numbers.
However, 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 get '+inf' when dividing a positive number by 0, and '-inf'
when dividing any negative number by 0.
=head1 EXAMPLES
use Math::BigInt;
sub bigint { Math::BigInt->new(shift); }
$x = Math::BigInt->bstr("1234") # string "1234"
$x = "$x"; # same as bstr()
$x = Math::BigInt->bneg("1234"); # Math::BigInt "-1234"
$x = Math::BigInt->babs("-12345"); # Math::BigInt "12345"
$x = Math::BigInt->bnorm("-0.00"); # Math::BigInt "0"
$x = bigint(1) + bigint(2); # Math::BigInt "3"
$x = bigint(1) + "2"; # ditto (auto-Math::BigIntify of "2")
$x = bigint(1); # Math::BigInt "1"
$x = $x + 5 / 2; # Math::BigInt "3"
$x = $x ** 3; # Math::BigInt "27"
$x *= 2; # Math::BigInt "54"
$x = Math::BigInt->new(0); # Math::BigInt "0"
$x--; # Math::BigInt "-1"
$x = Math::BigInt->badd(4,5) # Math::BigInt "9"
print $x->bsstr(); # 9e+0
Examples for rounding:
use Math::BigFloat;
use Test::More;
$x = Math::BigFloat->new(123.4567);
$y = Math::BigFloat->new(123.456789);
Math::BigFloat->accuracy(4); # no more A than 4
is ($x->copy()->bround(),123.4); # even rounding
print $x->copy()->bround(),"\n"; # 123.4
Math::BigFloat->round_mode('odd'); # round to odd
print $x->copy()->bround(),"\n"; # 123.5
Math::BigFloat->accuracy(5); # no more A than 5
Math::BigFloat->round_mode('odd'); # round to odd
print $x->copy()->bround(),"\n"; # 123.46
$y = $x->copy()->bround(4),"\n"; # A = 4: 123.4
print "$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()->bround(),"\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 is calculated using Perl scalars.
Please note that strings and floating point constants are not affected, so that
use Math::BigInt qw/:constant/;
$x = 1234567890123456789012345678901234567890
+ 123456789123456789;
$y = '1234567890123456789012345678901234567890'
+ '123456789123456789';
does not give you what you expect. 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 Math::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";
prints 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. See the section
L</MATH LIBRARY> for more information.
For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
=head1 SUBCLASSING
=head2 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
=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 needs to provide its 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 '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 L<bignum>:
use bignum;
Also good for one-liners:
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 always hands up their work:
div bsqrt blog bexp bpi bsin bcos batan batan2
All other methods upgrade themselves only when one (or all) of their arguments
are of the class mentioned in $upgrade.
=head1 EXPORTS
C<Math::BigInt> exports nothing by default, but can export the following
methods:
bgcd
blcm
=head1 CAVEATS
Some things might not work as you expect them. Below is documented what is
known to be troublesome:
=over
=item Comparing numbers as strings
Both C<bstr()> and C<bsstr()> as well as stringify via overload drop the
leading '+'. 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> and L<Test::More>, which stringify arguments before comparing them.
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 now works as expected:
use Test::More tests => 1;
use Math::BigInt;
my $x = Math::BigInt -> new(3*3);
my $y = Math::BigInt -> new(3*3);
is($x,3*3, 'multiplication');
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 represents 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::More tests => 3;
use Math::BigInt;
$x = Math::BigInt->new('1e56'); $y = 1e56;
is($x,$y); # fails
is($x->bsstr(),$y); # okay
$y = Math::BigInt->new($y);
is($x,$y); # okay
Alternatively, simply use C<< <=> >> for comparisons, this always gets it
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()> returns (at least for Perl v5.7.1 and up) another Math::BigInt, not a
Perl scalar:
$x = Math::BigInt->new(123);
$y = int($x); # 123 as a Math::BigInt
$x = Math::BigFloat->new(123.45);
$y = int($x); # 123 as a Math::BigFloat
If you want a real Perl scalar, use C<numify()>:
$y = $x->numify(); # 123 as a scalar
This is seldom necessary, though, because this is done automatically, like when
you access an array:
$z = $array[$x]; # does work automatically
=item Modifying and =
Beware of:
$x = Math::BigFloat->new(5);
$y = $x;
This makes a second reference to the B<same> object and stores it in $y. Thus
anything that modifies $x (except overloaded operators) also modifies $y, and
vice versa. Or in other words, C<=> is only safe if you modify your
Math::BigInt objects 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 first makes a copy and then
multiply it by 2:
$y = $x->copy()->bmul(2);
See also the documentation for overload.pm regarding C<=>.
=item Overloading -$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 gets overwritten.
This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
=item Mixing different object types
With overloaded operators, it is the first (dominating) operand that determines
which method is called. Here are some examples showing what actually gets
called in various cases.
use Math::BigInt;
use Math::BigFloat;
$mbf = Math::BigFloat->new(5);
$mbi2 = Math::BigInt->new(5);
$mbi = Math::BigInt->new(2);
# what actually gets called:
$float = $mbf + $mbi; # $mbf->badd($mbi)
$float = $mbf / $mbi; # $mbf->bdiv($mbi)
$integer = $mbi + $mbf; # $mbi->badd($mbf)
$integer = $mbi2 / $mbi; # $mbi2->bdiv($mbi)
$integer = $mbi2 / $mbf; # $mbi2->bdiv($mbf)
For instance, Math::BigInt->bdiv() always returns a Math::BigInt, regardless of
whether the second operant is a Math::BigFloat. To get a Math::BigFloat you
either need to call the operation manually, make sure each operand already is a
Math::BigFloat, or cast to that type via Math::BigFloat->new():
$float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
Beware of casting the entire expression, as this would cast the
result, at which point it is too late:
$float = Math::BigFloat->new($mbi2 / $mbi); # = 2
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;
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.
=back
=head1 BUGS
Please report any bugs or feature requests to
C<bug-math-bigint at rt.cpan.org>, or through the web interface at
L<https://rt.cpan.org/Ticket/Create.html?Queue=Math-BigInt> (requires login).
We will be notified, and then you'll automatically be notified of progress on
your bug as I make changes.
=head1 SUPPORT
You can find documentation for this module with the perldoc command.
perldoc Math::BigInt
You can also look for information at:
=over 4
=item * RT: CPAN's request tracker
L<https://rt.cpan.org/Public/Dist/Display.html?Name=Math-BigInt>
=item * AnnoCPAN: Annotated CPAN documentation
L<http://annocpan.org/dist/Math-BigInt>
=item * CPAN Ratings
L<http://cpanratings.perl.org/dist/Math-BigInt>
=item * Search CPAN
L<http://search.cpan.org/dist/Math-BigInt/>
=item * CPAN Testers Matrix
L<http://matrix.cpantesters.org/?dist=Math-BigInt>
=item * The Bignum mailing list
=over 4
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=head1 LICENSE
This program is free software; you may redistribute it and/or modify it under
the same terms as Perl itself.
=head1 SEE ALSO
L<Math::BigFloat> and L<Math::BigRat> as well as the backends
L<Math::BigInt::FastCalc>, L<Math::BigInt::GMP>, and L<Math::BigInt::Pari>.
The pragmas L<bignum>, L<bigint> and L<bigrat> also might be of interest
because they solve the autoupgrading/downgrading issue, at least partly.
=head1 AUTHORS
=over 4
=item *
Mark Biggar, overloaded interface by Ilya Zakharevich, 1996-2001.
=item *
Completely rewritten by Tels L<http://bloodgate.com>, 2001-2008.
=item *
Florian Ragwitz E<lt>flora@cpan.orgE<gt>, 2010.
=item *
Peter John Acklam E<lt>pjacklam@online.noE<gt>, 2011-.
=back
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