# "Mike had an infinite amount to do and a negative amount of time in which
# to do it." - Before and After
# The following hash values are used:
# value: unsigned int with actual value (as a Math::BigInt::Calc or similiar)
# sign : +,-,NaN,+inf,-inf
# _f : flags, used by MBF to flag parts of a float as untouchable
# Remember not to take shortcuts ala $xs = $x->{value}; $CALC->foo($xs); since
# underlying lib might change the reference!
my $class = "Math::BigInt";
# This is a patched v1.60, containing a fix for the "1234567890\n" bug
@EXPORT_OK = qw( objectify _swap bgcd blcm);
use vars qw
/$round_mode $accuracy $precision $div_scale $rnd_mode/;
use vars qw
/$upgrade $downgrade/;
# Inside overload, the first arg is always an object. If the original code had
# it reversed (like $x = 2 * $y), then the third paramater indicates this
# swapping. To make it work, we use a helper routine which not only reswaps the
# params, but also makes a new object in this case. See _swap() for details,
# especially the cases of operators with different classes.
# For overloaded ops with only one argument we simple use $_[0]->copy() to
# Thus inheritance of overload operators becomes possible and transparent for
# our subclasses without the need to repeat the entire overload section there.
'=' => sub { $_[0]->copy(); },
# '+' and '-' do not use _swap, since it is a triffle slower. If you want to
# override _swap (if ever), then override overload of '+' and '-', too!
# for sub it is a bit tricky to keep b: b-a => -a+b
'-' => sub { my $c = $_[0]->copy; $_[2] ?
$c->bneg()->badd($_[1]) :
'+' => sub { $_[0]->copy()->badd($_[1]); },
# some shortcuts for speed (assumes that reversed order of arguments is routed
# to normal '+' and we thus can always modify first arg. If this is changed,
# this breaks and must be adjusted.)
'+=' => sub { $_[0]->badd($_[1]); },
'-=' => sub { $_[0]->bsub($_[1]); },
'*=' => sub { $_[0]->bmul($_[1]); },
'/=' => sub { scalar $_[0]->bdiv($_[1]); },
'%=' => sub { $_[0]->bmod($_[1]); },
'^=' => sub { $_[0]->bxor($_[1]); },
'&=' => sub { $_[0]->band($_[1]); },
'|=' => sub { $_[0]->bior($_[1]); },
'**=' => sub { $_[0]->bpow($_[1]); },
# not supported by Perl yet
ref($_[0])->bcmp($_[1],$_[0]) :
"$_[1]" cmp $_[0]->bstr() :
$_[0]->bstr() cmp "$_[1]" },
'log' => sub { $_[0]->copy()->blog(); },
'int' => sub { $_[0]->copy(); },
'neg' => sub { $_[0]->copy()->bneg(); },
'abs' => sub { $_[0]->copy()->babs(); },
'sqrt' => sub { $_[0]->copy()->bsqrt(); },
'~' => sub { $_[0]->copy()->bnot(); },
'*' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bmul($a[1]); },
'/' => sub { my @a = ref($_[0])->_swap(@_);scalar $a[0]->bdiv($a[1]);},
'%' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bmod($a[1]); },
'**' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bpow($a[1]); },
'<<' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->blsft($a[1]); },
'>>' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->brsft($a[1]); },
'&' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->band($a[1]); },
'|' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bior($a[1]); },
'^' => sub { my @a = ref($_[0])->_swap(@_); $a[0]->bxor($a[1]); },
# can modify arg of ++ and --, so avoid a new-copy for speed, but don't
# use $_[0]->__one(), it modifies $_[0] to be 1!
'++' => sub { $_[0]->binc() },
'--' => sub { $_[0]->bdec() },
# if overloaded, O(1) instead of O(N) and twice as fast for small numbers
# this kludge is needed for perl prior 5.6.0 since returning 0 here fails :-/
# v5.6.1 dumps on that: return !$_[0]->is_zero() || undef; :-(
my $t = !$_[0]->is_zero();
# the original qw() does not work with the TIESCALAR below, why?
# Order of arguments unsignificant
'""' => sub { $_[0]->bstr(); },
'0+' => sub { $_[0]->numify(); }
##############################################################################
# global constants, flags and accessory
use constant MB_NEVER_ROUND
=> 0x0001;
my $NaNOK=1; # are NaNs ok?
my $nan = 'NaN'; # constants for easier life
my $CALC = 'Math::BigInt::Calc'; # module to do low level math
my $IMPORT = 0; # did import() yet?
$round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
$upgrade = undef; # default is no upgrade
$downgrade = undef; # default is no downgrade
##############################################################################
# the old code had $rnd_mode, so we need to support it, too
sub TIESCALAR
{ my ($class) = @_; bless \
$round_mode, $class; }
sub FETCH
{ return $round_mode; }
sub STORE
{ $rnd_mode = $_[0]->round_mode($_[1]); }
BEGIN { tie
$rnd_mode, 'Math::BigInt'; }
##############################################################################
# make Class->round_mode() work
my $class = ref($self) || $self || __PACKAGE__
;
die "Unknown round mode $m"
if $m !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/;
return ${"${class}::round_mode"} = $m;
return ${"${class}::round_mode"};
# make Class->upgrade() work
my $class = ref($self) || $self || __PACKAGE__
;
return ${"${class}::upgrade"} = $u;
return ${"${class}::upgrade"};
# make Class->downgrade() work
my $class = ref($self) || $self || __PACKAGE__
;
return ${"${class}::downgrade"} = $u;
return ${"${class}::downgrade"};
# make Class->round_mode() work
my $class = ref($self) || $self || __PACKAGE__
;
die ('div_scale must be greater than zero') if $_[0] < 0;
${"${class}::div_scale"} = shift;
return ${"${class}::div_scale"};
# $x->accuracy($a); ref($x) $a
# $x->accuracy(); ref($x)
# Class->accuracy(); class
# Class->accuracy($a); class $a
my $class = ref($x) || $x || __PACKAGE__
;
die ('accuracy must not be zero') if defined $a && $a == 0;
# $object->accuracy() or fallback to global
$x->bround($a) if defined $a;
$x->{_a
} = $a; # set/overwrite, even if not rounded
$x->{_p
} = undef; # clear P
${"${class}::accuracy"} = $a;
${"${class}::precision"} = undef; # clear P
# $object->accuracy() or fallback to global
$r = $x->{_a
} if ref($x);
# but don't return global undef, when $x's accuracy is 0!
$r = ${"${class}::accuracy"} if !defined $r;
# $x->precision($p); ref($x) $p
# $x->precision(); ref($x)
# Class->precision(); class
# Class->precision($p); class $p
my $class = ref($x) || $x || __PACKAGE__
;
# $object->precision() or fallback to global
$x->bfround($p) if defined $p;
$x->{_p
} = $p; # set/overwrite, even if not rounded
$x->{_a
} = undef; # clear A
${"${class}::precision"} = $p;
${"${class}::accuracy"} = undef; # clear A
# $object->precision() or fallback to global
$r = $x->{_p
} if ref($x);
# but don't return global undef, when $x's precision is 0!
$r = ${"${class}::precision"} if !defined $r;
# return (later set?) configuration data as hash ref
my $class = shift || 'Math::BigInt';
lib_version
=> ${"${lib}::VERSION"},
qw
/upgrade downgrade precision accuracy round_mode VERSION div_scale/)
$cfg->{lc($_)} = ${"${class}::$_"};
# select accuracy parameter based on precedence,
# used by bround() and bfround(), may return undef for scale (means no op)
my ($x,$s,$m,$scale,$mode) = @_;
$scale = $x->{_a
} if !defined $scale;
$scale = $s if (!defined $scale);
$mode = $m if !defined $mode;
# select precision parameter based on precedence,
# used by bround() and bfround(), may return undef for scale (means no op)
my ($x,$s,$m,$scale,$mode) = @_;
$scale = $x->{_p
} if !defined $scale;
$scale = $s if (!defined $scale);
$mode = $m if !defined $mode;
##############################################################################
# if two arguments, the first one is the class to "swallow" subclasses
return unless ref($x); # only for objects
my $self = {}; bless $self,$c;
$self->{value
} = $CALC->_copy($x->{value
}); next;
if (!($r = ref($x->{$k})))
$self->{$k} = $x->{$k}; next;
$self->{$k} = \
${$x->{$k}};
$self->{$k} = [ @
{$x->{$k}} ];
foreach my $h (keys %{$x->{$k}})
$self->{$k}->{$h} = $x->{$k}->{$h};
$self->{$k} = $xk->copy();
$self->{$k} = $xk->new($xk);
# create a new BigInt object from a string or another BigInt object.
# see hash keys documented at top
# the argument could be an object, so avoid ||, && etc on it, this would
# cause costly overloaded code to be called. The only allowed ops are
my ($class,$wanted,$a,$p,$r) = @_;
# avoid numify-calls by not using || on $wanted!
return $class->bzero($a,$p) if !defined $wanted; # default to 0
return $class->copy($wanted,$a,$p,$r)
if ref($wanted) && $wanted->isa($class); # MBI or subclass
$class->import() if $IMPORT == 0; # make require work
my $self = bless {}, $class;
# shortcut for "normal" numbers
if ((!ref $wanted) && ($wanted =~ /^([+-]?)[1-9][0-9]*\z/))
$self->{sign
} = $1 || '+';
# remove sign without touching wanted
my $t = $wanted; $t =~ s/^[+-]//; $ref = \
$t;
$self->{value
} = $CALC->_new($ref);
if ( (defined $a) || (defined $p)
|| (defined ${"${class}::precision"})
|| (defined ${"${class}::accuracy"})
$self->round($a,$p,$r) unless (@_ == 4 && !defined $a && !defined $p);
# handle '+inf', '-inf' first
if ($wanted =~ /^[+-]?inf$/)
$self->{value
} = $CALC->_zero();
$self->{sign
} = $wanted; $self->{sign
} = '+inf' if $self->{sign
} eq 'inf';
# split str in m mantissa, e exponent, i integer, f fraction, v value, s sign
my ($mis,$miv,$mfv,$es,$ev) = _split
(\
$wanted);
die "$wanted is not a number initialized to $class" if !$NaNOK;
$self->{value
} = $CALC->_zero();
$self->{value
} = $mis->{value
};
$self->{sign
} = $mis->{sign
};
return $self; # throw away $mis
# make integer from mantissa by adjusting exp, then convert to bigint
$self->{sign
} = $$mis; # store sign
$self->{value
} = $CALC->_zero(); # for all the NaN cases
my $e = int("$$es$$ev"); # exponent (avoid recursion)
my $diff = $e - CORE
::length($$mfv);
if ($diff < 0) # Not integer
return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
# adjust fraction and add it to value
# print "diff > 0 $$miv\n";
$$miv = $$miv . ($$mfv . '0' x
$diff);
if ($$mfv ne '') # e <= 0
# fraction and negative/zero E => NOI
#print "NOI 2 \$\$mfv '$$mfv'\n";
return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
if ($$miv !~ s/0{$e}$//) # can strip so many zero's?
return $upgrade->new($wanted,$a,$p,$r) if defined $upgrade;
$self->{sign
} = '+' if $$miv eq '0'; # normalize -0 => +0
$self->{value
} = $CALC->_new($miv) if $self->{sign
} =~ /^[+-]$/;
# if any of the globals is set, use them to round and store them inside $self
# do not round for new($x,undef,undef) since that is used by MBF to signal
$self->round($a,$p,$r) unless @_ == 4 && !defined $a && !defined $p;
# create a bigint 'NaN', if given a BigInt, set it to 'NaN'
$self = $class if !defined $self;
my $c = $self; $self = {}; bless $self, $c;
$self->import() if $IMPORT == 0; # make require work
return if $self->modify('bnan');
# use subclass to initialize
# otherwise do our own thing
$self->{value
} = $CALC->_zero();
delete $self->{_a
}; delete $self->{_p
}; # rounding NaN is silly
# create a bigint '+-inf', if given a BigInt, set it to '+-inf'
# the sign is either '+', or if given, used from there
my $sign = shift; $sign = '+' if !defined $sign || $sign !~ /^-(inf)?$/;
$self = $class if !defined $self;
my $c = $self; $self = {}; bless $self, $c;
$self->import() if $IMPORT == 0; # make require work
return if $self->modify('binf');
# use subclass to initialize
# otherwise do our own thing
$self->{value
} = $CALC->_zero();
$sign = $sign . 'inf' if $sign !~ /inf$/; # - => -inf
($self->{_a
},$self->{_p
}) = @_; # take over requested rounding
# create a bigint '+0', if given a BigInt, set it to 0
$self = $class if !defined $self;
my $c = $self; $self = {}; bless $self, $c;
$self->import() if $IMPORT == 0; # make require work
return if $self->modify('bzero');
if ($self->can('_bzero'))
# use subclass to initialize
# otherwise do our own thing
$self->{value
} = $CALC->_zero();
# call like: $x->bzero($a,$p,$r,$y);
($self,$self->{_a
},$self->{_p
}) = $self->_find_round_parameters(@_);
if ( (!defined $self->{_a
}) || (defined $_[0] && $_[0] > $self->{_a
}));
if ( (!defined $self->{_p
}) || (defined $_[1] && $_[1] > $self->{_p
}));
# create a bigint '+1' (or -1 if given sign '-'),
# if given a BigInt, set it to +1 or -1, respecively
my $sign = shift; $sign = '+' if !defined $sign || $sign ne '-';
$self = $class if !defined $self;
my $c = $self; $self = {}; bless $self, $c;
$self->import() if $IMPORT == 0; # make require work
return if $self->modify('bone');
# use subclass to initialize
# otherwise do our own thing
$self->{value
} = $CALC->_one();
# call like: $x->bone($sign,$a,$p,$r,$y);
($self,$self->{_a
},$self->{_p
}) = $self->_find_round_parameters(@_);
if ( (!defined $self->{_a
}) || (defined $_[0] && $_[0] > $self->{_a
}));
if ( (!defined $self->{_p
}) || (defined $_[1] && $_[1] > $self->{_p
}));
##############################################################################
# (ref to BFLOAT or num_str ) return num_str
# Convert number from internal format to scientific string format.
# internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
# my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
if ($x->{sign
} !~ /^[+-]$/)
return $x->{sign
} unless $x->{sign
} eq '+inf'; # -inf, NaN
my ($m,$e) = $x->parts();
# MBF: my $s = $e->{sign}; $s = '' if $s eq '-'; my $sep = 'e'.$s;
return $m->bstr().$sign.$e->bstr();
# make a string from bigint object
my $x = shift; $class = ref($x) || $x; $x = $class->new(shift) if !ref($x);
# my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
if ($x->{sign
} !~ /^[+-]$/)
return $x->{sign
} unless $x->{sign
} eq '+inf'; # -inf, NaN
my $es = ''; $es = $x->{sign
} if $x->{sign
} eq '-';
return $es.${$CALC->_str($x->{value
})};
# Make a "normal" scalar from a BigInt object
my $x = shift; $x = $class->new($x) unless ref $x;
return $x->{sign
} if $x->{sign
} !~ /^[+-]$/;
my $num = $CALC->_num($x->{value
});
return -$num if $x->{sign
} eq '-';
##############################################################################
# public stuff (usually prefixed with "b")
# return the sign of the number: +/-/-inf/+inf/NaN
my ($self,$x) = ref($_[0]) ?
(ref($_[0]),$_[0]) : objectify
(1,@_);
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.
# This procedure finds the round parameters, but it is for speed reasons
# duplicated in round. Otherwise, it is tested by the testsuite and used
my ($self,$a,$p,$r,@args) = @_;
# $a accuracy, if given by caller
# $p precision, if given by caller
# $r round_mode, if given by caller
# @args all 'other' arguments (0 for unary, 1 for binary ops)
# leave bigfloat parts alone
return ($self) if exists $self->{_f
} && $self->{_f
} & MB_NEVER_ROUND
!= 0;
my $c = ref($self); # find out class of argument(s)
# now pick $a or $p, but only if we have got "arguments"
# take the defined one, or if both defined, the one that is smaller
$a = $_->{_a
} if (defined $_->{_a
}) && (!defined $a || $_->{_a
} < $a);
# even if $a is defined, take $p, to signal error for both defined
# take the defined one, or if both defined, the one that is bigger
$p = $_->{_p
} if (defined $_->{_p
}) && (!defined $p || $_->{_p
} > $p);
# if still none defined, use globals (#2)
$a = ${"$c\::accuracy"} unless defined $a;
$p = ${"$c\::precision"} unless defined $p;
return ($self) unless defined $a || defined $p; # early out
# set A and set P is an fatal error
return ($self->bnan()) if defined $a && defined $p;
$r = ${"$c\::round_mode"} unless defined $r;
die "Unknown round mode '$r'" if $r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/;
# Round $self according to given parameters, or given second argument's
# parameters or global defaults
# for speed reasons, _find_round_parameters is embeded here:
my ($self,$a,$p,$r,@args) = @_;
# $a accuracy, if given by caller
# $p precision, if given by caller
# $r round_mode, if given by caller
# @args all 'other' arguments (0 for unary, 1 for binary ops)
# leave bigfloat parts alone
return ($self) if exists $self->{_f
} && $self->{_f
} & MB_NEVER_ROUND
!= 0;
my $c = ref($self); # find out class of argument(s)
# now pick $a or $p, but only if we have got "arguments"
# take the defined one, or if both defined, the one that is smaller
$a = $_->{_a
} if (defined $_->{_a
}) && (!defined $a || $_->{_a
} < $a);
# even if $a is defined, take $p, to signal error for both defined
# take the defined one, or if both defined, the one that is bigger
$p = $_->{_p
} if (defined $_->{_p
}) && (!defined $p || $_->{_p
} > $p);
# if still none defined, use globals (#2)
$a = ${"$c\::accuracy"} unless defined $a;
$p = ${"$c\::precision"} unless defined $p;
return $self unless defined $a || defined $p; # early out
# set A and set P is an fatal error
return $self->bnan() if defined $a && defined $p;
$r = ${"$c\::round_mode"} unless defined $r;
die "Unknown round mode '$r'" if $r !~ /^(even|odd|\+inf|\-inf|zero|trunc)$/;
# now round, by calling either fround or ffround:
$self->bround($a,$r) if !defined $self->{_a
} || $self->{_a
} >= $a;
else # both can't be undefined due to early out
$self->bfround($p,$r) if !defined $self->{_p
} || $self->{_p
} <= $p;
$self->bnorm(); # after round, normalize
# (numstr or BINT) return BINT
# Normalize number -- no-op here
my ($self,$x) = ref($_[0]) ?
(ref($_[0]),$_[0]) : objectify
(1,@_);
# (BINT or num_str) return BINT
# make number absolute, or return absolute BINT from string
my ($self,$x) = ref($_[0]) ?
(ref($_[0]),$_[0]) : objectify
(1,@_);
return $x if $x->modify('babs');
# post-normalized abs for internal use (does nothing for NaN)
# (BINT or num_str) return BINT
# negate number or make a negated number from string
my ($self,$x) = ref($_[0]) ?
(ref($_[0]),$_[0]) : objectify
(1,@_);
return $x if $x->modify('bneg');
# for +0 dont negate (to have always normalized)
$x->{sign
} =~ tr/+-/-+/ if !$x->is_zero(); # does nothing for NaN
# Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
# (BINT or num_str, BINT or num_str) return cond_code
my ($self,$x,$y) = (ref($_[0]),@_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
($self,$x,$y) = objectify
(2,@_);
if (($x->{sign
} !~ /^[+-]$/) || ($y->{sign
} !~ /^[+-]$/))
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';
# check sign for speed first
return 1 if $x->{sign
} eq '+' && $y->{sign
} eq '-'; # does also 0 <=> -y
return -1 if $x->{sign
} eq '-' && $y->{sign
} eq '+'; # does also -x <=> 0
# have same sign, so compare absolute values. Don't make tests for zero here
# because it's actually slower than testin in Calc (especially w/ Pari et al)
# post-normalized compare for internal use (honors signs)
return $CALC->_acmp($x->{value
},$y->{value
});
$CALC->_acmp($y->{value
},$x->{value
}); # swaped (lib returns 0,1,-1)
# Compares 2 values, ignoring their signs.
# Returns one of undef, <0, =0, >0. (suitable for sort)
# (BINT, BINT) return cond_code
my ($self,$x,$y) = (ref($_[0]),@_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
($self,$x,$y) = objectify
(2,@_);
if (($x->{sign
} !~ /^[+-]$/) || ($y->{sign
} !~ /^[+-]$/))
return undef if (($x->{sign
} eq $nan) || ($y->{sign
} eq $nan));
return 0 if $x->{sign
} =~ /^[+-]inf$/ && $y->{sign
} =~ /^[+-]inf$/;
return +1; # inf is always bigger
$CALC->_acmp($x->{value
},$y->{value
}); # lib does only 0,1,-1
# add second arg (BINT or string) to first (BINT) (modifies first)
my ($self,$x,$y,@r) = (ref($_[0]),@_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
($self,$x,$y,@r) = objectify
(2,@_);
return $x if $x->modify('badd');
return $upgrade->badd($x,$y,@r) if defined $upgrade &&
((!$x->isa($self)) || (!$y->isa($self)));
if (($x->{sign
} !~ /^[+-]$/) || ($y->{sign
} !~ /^[+-]$/))
return $x->bnan() if (($x->{sign
} eq $nan) || ($y->{sign
} eq $nan));
if (($x->{sign
} =~ /^[+-]inf$/) && ($y->{sign
} =~ /^[+-]inf$/))
# +inf++inf or -inf+-inf => same, rest is NaN
return $x if $x->{sign
} eq $y->{sign
};
# +-inf + something => +inf
# something +-inf => +-inf
$x->{sign
} = $y->{sign
}, return $x if $y->{sign
} =~ /^[+-]inf$/;
my ($sx, $sy) = ( $x->{sign
}, $y->{sign
} ); # get signs
$x->{value
} = $CALC->_add($x->{value
},$y->{value
}); # same sign, abs add
my $a = $CALC->_acmp ($y->{value
},$x->{value
}); # absolute compare
#print "swapped sub (a=$a)\n";
$x->{value
} = $CALC->_sub($y->{value
},$x->{value
},1); # abs sub w/ swap
# speedup, if equal, set result to 0
#print "equal sub, result = 0\n";
$x->{value
} = $CALC->_zero();
#print "unswapped sub (a=$a)\n";
$x->{value
} = $CALC->_sub($x->{value
}, $y->{value
}); # abs sub
$x->round(@r) if !exists $x->{_f
} || $x->{_f
} & MB_NEVER_ROUND
== 0;
# (BINT or num_str, BINT or num_str) return num_str
# subtract second arg from first, modify first
my ($self,$x,$y,@r) = (ref($_[0]),@_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
($self,$x,$y,@r) = objectify
(2,@_);
return $x if $x->modify('bsub');
# upgrade done by badd():
# return $upgrade->badd($x,$y,@r) if defined $upgrade &&
# ((!$x->isa($self)) || (!$y->isa($self)));
$x->round(@r) if !exists $x->{_f
} || $x->{_f
} & MB_NEVER_ROUND
== 0;
$y->{sign
} =~ tr/+\-/-+/; # does nothing for NaN
$x->badd($y,@r); # badd does not leave internal zeros
$y->{sign
} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
$x; # already rounded by badd() or no round necc.
my ($self,$x,$a,$p,$r) = ref($_[0]) ?
(ref($_[0]),@_) : objectify
(1,@_);
return $x if $x->modify('binc');
$x->{value
} = $CALC->_inc($x->{value
});
$x->round($a,$p,$r) if !exists $x->{_f
} || $x->{_f
} & MB_NEVER_ROUND
== 0;
elsif ($x->{sign
} eq '-')
$x->{value
} = $CALC->_dec($x->{value
});
$x->{sign
} = '+' if $CALC->_is_zero($x->{value
}); # -1 +1 => -0 => +0
$x->round($a,$p,$r) if !exists $x->{_f
} || $x->{_f
} & MB_NEVER_ROUND
== 0;
$x->badd($self->__one(),$a,$p,$r); # badd does round
my ($self,$x,$a,$p,$r) = ref($_[0]) ?
(ref($_[0]),@_) : objectify
(1,@_);
return $x if $x->modify('bdec');
my $zero = $CALC->_is_zero($x->{value
}) && $x->{sign
} eq '+';
if (($x->{sign
} eq '-') || $zero)
$x->{value
} = $CALC->_inc($x->{value
});
$x->{sign
} = '-' if $zero; # 0 => 1 => -1
$x->{sign
} = '+' if $CALC->_is_zero($x->{value
}); # -1 +1 => -0 => +0
$x->round($a,$p,$r) if !exists $x->{_f
} || $x->{_f
} & MB_NEVER_ROUND
== 0;
elsif ($x->{sign
} eq '+')
$x->{value
} = $CALC->_dec($x->{value
});
$x->round($a,$p,$r) if !exists $x->{_f
} || $x->{_f
} & MB_NEVER_ROUND
== 0;
$x->badd($self->__one('-'),$a,$p,$r); # badd does round
my ($self,$x,$base,$a,$p,$r) = ref($_[0]) ?
(ref($_[0]),@_) : objectify
(1,@_);
return $upgrade->blog($x,$base,$a,$p,$r) if defined $upgrade;
# (BINT or num_str, BINT or num_str) return BINT
# does not modify arguments, but returns new object
# Lowest Common Multiplicator
while (@_) { $x = __lcm
($x,shift); }
# (BINT or num_str, BINT or num_str) return BINT
# does not modify arguments, but returns new object
# GCD -- Euclids algorithm, variant C (Knuth Vol 3, pg 341 ff)
$y = __PACKAGE__
->new($y) if !ref($y);
my $x = $y->copy(); # keep arguments
$y = shift; $y = $self->new($y) if !ref($y);
return $x->bnan() if $y->{sign
} !~ /^[+-]$/; # y NaN?
$x->{value
} = $CALC->_gcd($x->{value
},$y->{value
}); last if $x->is_one();
$y = shift; $y = $self->new($y) if !ref($y);
$x = __gcd
($x,$y->copy()); last if $x->is_one(); # _gcd handles NaN
# (num_str or BINT) return BINT
# represent ~x as twos-complement number
# we don't need $self, so undef instead of ref($_[0]) make it slightly faster
my ($self,$x,$a,$p,$r) = ref($_[0]) ?
(undef,@_) : objectify
(1,@_);
return $x if $x->modify('bnot');
$x->bneg()->bdec(); # bdec already does round
# return true if arg (BINT or num_str) is zero (array '+', '0')
# we don't need $self, so undef instead of ref($_[0]) make it slightly faster
my ($self,$x) = ref($_[0]) ?
(undef,$_[0]) : objectify
(1,@_);
return 0 if $x->{sign
} !~ /^\+$/; # -, NaN & +-inf aren't
$CALC->_is_zero($x->{value
});
# return true if arg (BINT or num_str) is NaN
my ($self,$x) = ref($_[0]) ?
(ref($_[0]),$_[0]) : objectify
(1,@_);
return 1 if $x->{sign
} eq $nan;
# return true if arg (BINT or num_str) is +-inf
my ($self,$x,$sign) = ref($_[0]) ?
(ref($_[0]),@_) : objectify
(1,@_);
$sign = '' if !defined $sign;
return 1 if $sign eq $x->{sign
}; # match ("+inf" eq "+inf")
return 0 if $sign !~ /^([+-]|)$/;
return 1 if ($x->{sign
} =~ /^[+-]inf$/);
$sign = quotemeta($sign.'inf');
return 1 if ($x->{sign
} =~ /^$sign$/);
# return true if arg (BINT or num_str) is +1
# we don't need $self, so undef instead of ref($_[0]) make it slightly faster
my ($self,$x,$sign) = ref($_[0]) ?
(undef,@_) : objectify
(1,@_);
$sign = '' if !defined $sign; $sign = '+' if $sign ne '-';
return 0 if $x->{sign
} ne $sign; # -1 != +1, NaN, +-inf aren't either
$CALC->_is_one($x->{value
});
# return true when arg (BINT or num_str) is odd, false for even
# we don't need $self, so undef instead of ref($_[0]) make it slightly faster
my ($self,$x) = ref($_[0]) ?
(undef,$_[0]) : objectify
(1,@_);
return 0 if $x->{sign
} !~ /^[+-]$/; # NaN & +-inf aren't
$CALC->_is_odd($x->{value
});
# return true when arg (BINT or num_str) is even, false for odd
# we don't need $self, so undef instead of ref($_[0]) make it slightly faster
my ($self,$x) = ref($_[0]) ?
(undef,$_[0]) : objectify
(1,@_);
return 0 if $x->{sign
} !~ /^[+-]$/; # NaN & +-inf aren't
$CALC->_is_even($x->{value
});
# return true when arg (BINT or num_str) is positive (>= 0)
# we don't need $self, so undef instead of ref($_[0]) make it slightly faster
my ($self,$x) = ref($_[0]) ?
(undef,$_[0]) : objectify
(1,@_);
return 1 if $x->{sign
} =~ /^\+/;
# return true when arg (BINT or num_str) is negative (< 0)
# we don't need $self, so undef instead of ref($_[0]) make it slightly faster
my ($self,$x) = ref($_[0]) ?
(undef,$_[0]) : objectify
(1,@_);
return 1 if ($x->{sign
} =~ /^-/);
# return true when arg (BINT or num_str) is an integer
# always true for BigInt, but different for Floats
# we don't need $self, so undef instead of ref($_[0]) make it slightly faster
my ($self,$x) = ref($_[0]) ?
(undef,$_[0]) : objectify
(1,@_);
$x->{sign
} =~ /^[+-]$/ ?
1 : 0; # inf/-inf/NaN aren't
###############################################################################
# multiply two numbers -- stolen from Knuth Vol 2 pg 233
# (BINT or num_str, BINT or num_str) return BINT
my ($self,$x,$y,@r) = (ref($_[0]),@_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
($self,$x,$y,@r) = objectify
(2,@_);
return $x if $x->modify('bmul');
return $x->bnan() if (($x->{sign
} eq $nan) || ($y->{sign
} eq $nan));
if (($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 $upgrade->bmul($x,$y,@r)
if defined $upgrade && $y->isa($upgrade);
$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) if !exists $x->{_f
} || $x->{_f
} & MB_NEVER_ROUND
== 0;
# helper function that handles +-inf cases for bdiv()/bmod() to reuse code
# NaN if x == NaN or y == NaN or x==y==0
return wantarray ?
($x->bnan(),$self->bnan()) : $x->bnan()
if (($x->is_nan() || $y->is_nan()) ||
($x->is_zero() && $y->is_zero()));
# +-inf / +-inf == NaN, reminder also NaN
if (($x->{sign
} =~ /^[+-]inf$/) && ($y->{sign
} =~ /^[+-]inf$/))
return wantarray ?
($x->bnan(),$self->bnan()) : $x->bnan();
# x / +-inf => 0, remainder x (works even if x == 0)
if ($y->{sign
} =~ /^[+-]inf$/)
my $t = $x->copy(); # bzero clobbers up $x
return wantarray ?
($x->bzero(),$t) : $x->bzero()
# 5 / 0 => +inf, -6 / 0 => -inf
# +inf / 0 = inf, inf, and -inf / 0 => -inf, -inf
# exception: -8 / 0 has remainder -8, not 8
# exception: -inf / 0 has remainder -inf, not inf
# +-inf / 0 => special case for -inf
return wantarray ?
($x,$x->copy()) : $x if $x->is_inf();
if (!$x->is_zero() && !$x->is_inf())
my $t = $x->copy(); # binf clobbers up $x
($x->binf($x->{sign
}),$t) : $x->binf($x->{sign
})
# last case: +-inf / ordinary number
$sign = '-inf' if substr($x->{sign
},0,1) ne $y->{sign
};
return wantarray ?
($x,$self->bzero()) : $x;
# (dividend: BINT or num_str, divisor: BINT or num_str) return
# (BINT,BINT) (quo,rem) or BINT (only rem)
my ($self,$x,$y,@r) = (ref($_[0]),@_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
($self,$x,$y,@r) = objectify
(2,@_);
return $x if $x->modify('bdiv');
return $self->_div_inf($x,$y)
if (($x->{sign
} !~ /^[+-]$/) || ($y->{sign
} !~ /^[+-]$/) || $y->is_zero());
return $upgrade->bdiv($upgrade->new($x),$y,@r)
if defined $upgrade && !$y->isa($self);
wantarray ?
($x->round(@r),$self->bzero(@r)):$x->round(@r) if $x->is_zero();
# Is $x in the interval [0, $y) (aka $x <= $y) ?
my $cmp = $CALC->_acmp($x->{value
},$y->{value
});
if (($cmp < 0) and (($x->{sign
} eq $y->{sign
}) or !wantarray))
return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
return $x->bzero()->round(@r) unless wantarray;
my $t = $x->copy(); # make copy first, because $x->bzero() clobbers $x
return ($x->bzero()->round(@r),$t);
# shortcut, both are the same, so set to +/- 1
$x->__one( ($x->{sign
} ne $y->{sign
} ?
'-' : '+') );
return $x unless wantarray;
return ($x->round(@r),$self->bzero(@r));
return $upgrade->bdiv($upgrade->new($x),$upgrade->new($y),@r)
# calc new sign and in case $y == +/- 1, return $x
my $xsign = $x->{sign
}; # keep
$x->{sign
} = ($x->{sign
} ne $y->{sign
} ?
'-' : '+');
# check for / +-1 (cant use $y->is_one due to '-'
if ($CALC->_is_one($y->{value
}))
return wantarray ?
($x->round(@r),$self->bzero(@r)) : $x->round(@r);
my $rem = $self->bzero();
($x->{value
},$rem->{value
}) = $CALC->_div($x->{value
},$y->{value
});
$x->{sign
} = '+' if $CALC->_is_zero($x->{value
});
if (! $CALC->_is_zero($rem->{value
}))
$rem->{sign
} = $y->{sign
};
$rem = $y-$rem if $xsign ne $y->{sign
}; # one of them '-'
$rem->{sign
} = '+'; # dont leave -0
return ($x,$rem->round(@r));
$x->{value
} = $CALC->_div($x->{value
},$y->{value
});
$x->{sign
} = '+' if $CALC->_is_zero($x->{value
});
$x->round(@r) if !exists $x->{_f
} || $x->{_f
} & MB_NEVER_ROUND
== 0;
###############################################################################
# (BINT or num_str, BINT or num_str) return BINT
my ($self,$x,$y,@r) = (ref($_[0]),@_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
($self,$x,$y,@r) = objectify
(2,@_);
return $x if $x->modify('bmod');
if (($x->{sign
} !~ /^[+-]$/) || ($y->{sign
} !~ /^[+-]$/) || $y->is_zero())
my ($d,$r) = $self->_div_inf($x,$y);
$x->{value
} = $r->{value
};
# calc new sign and in case $y == +/- 1, return $x
$x->{value
} = $CALC->_mod($x->{value
},$y->{value
});
if (!$CALC->_is_zero($x->{value
}))
if ($xsign ne $y->{sign
})
my $t = $CALC->_copy($x->{value
}); # copy $x
$x->{value
} = $CALC->_copy($y->{value
}); # copy $y to $x
$x->{value
} = $CALC->_sub($y->{value
},$t,1); # $y-$x
$x->{sign
} = '+'; # dont leave -0
$x->round(@r) if !exists $x->{_f
} || $x->{_f
} & MB_NEVER_ROUND
== 0;
my ($t,$rem) = $self->bdiv($x->copy(),$y,@r); # slow way (also rounds)
foreach (qw
/value sign _a _p/)
# modular inverse. given a number which is (hopefully) relatively
# prime to the modulus, calculate its inverse using Euclid's
# alogrithm. if the number is not relatively prime to the modulus
# (i.e. their gcd is not one) then NaN is returned.
my ($self,$x,$y,@r) = (ref($_[0]),@_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
($self,$x,$y,@r) = objectify
(2,@_);
return $x if $x->modify('bmodinv');
if ($y->{sign
} ne '+' # -, NaN, +inf, -inf
|| $x->is_zero() # or num == 0
|| $x->{sign
} !~ /^[+-]$/ # or num NaN, inf, -inf
# put least residue into $x if $x was negative, and thus make it positive
$x->bmod($y) if $x->{sign
} eq '-';
if ($CALC->can('_modinv'))
$x->{value
} = $CALC->_modinv($x->{value
},$y->{value
});
$x->bnan() if !defined $x->{value
} ; # in case there was none
my ($u, $u1) = ($self->bzero(), $self->bone());
my ($a, $b) = ($y->copy(), $x->copy());
# first step need always be done since $num (and thus $b) is never 0
# Note that the loop is aligned so that the check occurs between #2 and #1
# thus saving us one step #2 at the loop end. Typical loop count is 1. Even
# a case with 28 loops still gains about 3% with this layout.
($a, $q, $b) = ($b, $a->bdiv($b)); # step #1
($u, $u1) = ($u1, $u->bsub($u1->copy()->bmul($q))); # step #2
($a, $q, $b) = ($b, $a->bdiv($b)); # step #1 again
# if the gcd is not 1, then return NaN! It would be pointless to
# have called bgcd to check this first, because we would then be performing
# the same Euclidean Algorithm *twice*
return $x->bnan() unless $a->is_one();
$x->{value
} = $u1->{value
};
$x->{sign
} = $u1->{sign
};
# takes a very large number to a very large exponent in a given very
# large modulus, quickly, thanks to binary exponentation. supports
my ($self,$num,$exp,$mod,@r) = objectify
(3,@_);
return $num if $num->modify('bmodpow');
# check modulus for valid values
return $num->bnan() if ($mod->{sign
} ne '+' # NaN, - , -inf, +inf
# check exponent for valid values
if ($exp->{sign
} =~ /\w/)
# i.e., if it's NaN, +inf, or -inf...
$num->bmodinv ($mod) if ($exp->{sign
} eq '-');
# check num for valid values (also NaN if there was no inverse but $exp < 0)
return $num->bnan() if $num->{sign
} !~ /^[+-]$/;
if ($CALC->can('_modpow'))
# $mod is positive, sign on $exp is ignored, result also positive
$num->{value
} = $CALC->_modpow($num->{value
},$exp->{value
},$mod->{value
});
return $num->bzero(@r) if $mod->is_one();
return $num->bone('+',@r) if $num->is_zero() or $num->is_one();
# $num->bmod($mod); # if $x is large, make it smaller first
my $acc = $num->copy(); # but this is not really faster...
$num->bone(); # keep ref to $num
my $expbin = $exp->as_bin(); $expbin =~ s/^[-]?0b//; # ignore sign and prefix
my $len = length($expbin);
if( substr($expbin,$len,1) eq '1')
$num->bmul($acc)->bmod($mod);
$acc->bmul($acc)->bmod($mod);
###############################################################################
# (BINT or num_str, BINT or num_str) return BINT
# compute factorial numbers
# modifies first argument
my ($self,$x,@r) = ref($_[0]) ?
(ref($_[0]),@_) : objectify
(1,@_);
return $x if $x->modify('bfac');
return $x->bnan() if $x->{sign
} ne '+'; # inf, NnN, <0 etc => NaN
return $x->bone('+',@r) if $x->is_zero() || $x->is_one(); # 0 or 1 => 1
$x->{value
} = $CALC->_fac($x->{value
});
# seems we need not to temp. clear A/P of $x since the result is the same
while ($f->bacmp($n) < 0)
$x->bmul($f); $f->binc();
$x->bmul($f,@r); # last step and also round
# (BINT or num_str, BINT or num_str) return BINT
# compute power of two numbers -- stolen from Knuth Vol 2 pg 233
# modifies first argument
my ($self,$x,$y,@r) = (ref($_[0]),@_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
($self,$x,$y,@r) = objectify
(2,@_);
return $x if $x->modify('bpow');
return $upgrade->bpow($upgrade->new($x),$y,@r)
if defined $upgrade && !$y->isa($self);
return $x if $x->{sign
} =~ /^[+-]inf$/; # -inf/+inf ** x
return $x->bnan() if $x->{sign
} eq $nan || $y->{sign
} eq $nan;
return $x->bone('+',@r) if $y->is_zero();
return $x->round(@r) if $x->is_one() || $y->is_one();
if ($x->{sign
} eq '-' && $CALC->_is_one($x->{value
}))
# if $x == -1 and odd/even y => +1/-1
return $y->is_odd() ?
$x->round(@r) : $x->babs()->round(@r);
# my Casio FX-5500L has a bug here: -1 ** 2 is -1, but -1 * -1 is 1;
# 1 ** -y => 1 / (1 ** |y|)
# so do test for negative $y after above's clause
return $x->bnan() if $y->{sign
} eq '-';
return $x->round(@r) if $x->is_zero(); # 0**y => 0 (if not y <= 0)
$x->{value
} = $CALC->_pow($x->{value
},$y->{value
});
$x->round(@r) if !exists $x->{_f
} || $x->{_f
} & MB_NEVER_ROUND
== 0;
# based on the assumption that shifting in base 10 is fast, and that mul
# works faster if numbers are small: we count trailing zeros (this step is
# O(1)..O(N), but in case of O(N) we save much more time due to this),
# stripping them out of the multiplication, and add $count * $y zeros
# 300 ** 3 == 300*300*300 == 3*3*3 . '0' x 2 * 3 == 27 . '0' x 6
# creates deep recursion since brsft/blsft use bpow sometimes.
# my $zeros = $x->_trailing_zeros();
# $x->brsft($zeros,10); # remove zeros
# $x->bpow($y); # recursion (will not branch into here again)
# $zeros = $y * $zeros; # real number of zeros to add
my $pow2 = $self->__one();
my $y_bin = $y->as_bin(); $y_bin =~ s/^0b//;
my $len = length($y_bin);
$pow2->bmul($x) if substr($y_bin,$len,1) eq '1'; # is odd?
$x->round(@r) if !exists $x->{_f
} || $x->{_f
} & MB_NEVER_ROUND
== 0;
# (BINT or num_str, BINT or num_str) return BINT
# compute x << y, base n, y >= 0
my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
($self,$x,$y,$n,@r) = objectify
(2,@_);
return $x if $x->modify('blsft');
return $x->bnan() if ($x->{sign
} !~ /^[+-]$/ || $y->{sign
} !~ /^[+-]$/);
return $x->round(@r) if $y->is_zero();
$n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign
} eq '-';
my $t; $t = $CALC->_lsft($x->{value
},$y->{value
},$n) if $CALC->can('_lsft');
$x->{value
} = $t; return $x->round(@r);
return $x->bmul( $self->bpow($n, $y, @r), @r );
# (BINT or num_str, BINT or num_str) return BINT
# compute x >> y, base n, y >= 0
my ($self,$x,$y,$n,@r) = (ref($_[0]),@_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
($self,$x,$y,$n,@r) = objectify
(2,@_);
return $x if $x->modify('brsft');
return $x->bnan() if ($x->{sign
} !~ /^[+-]$/ || $y->{sign
} !~ /^[+-]$/);
return $x->round(@r) if $y->is_zero();
return $x->bzero(@r) if $x->is_zero(); # 0 => 0
$n = 2 if !defined $n; return $x->bnan() if $n <= 0 || $y->{sign
} eq '-';
# this only works for negative numbers when shifting in base 2
if (($x->{sign
} eq '-') && ($n == 2))
return $x->round(@r) if $x->is_one('-'); # -1 => -1
# 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
$bin =~ s/^-0b//; # strip '-0b' prefix
$bin =~ tr/10/01/; # flip bits
if (CORE
::length($bin) <= $y)
$bin = '0'; # shifting to far right creates -1
# 0, because later increment makes
# that 1, attached '-' makes it '-1'
# because -1 >> x == -1 !
$bin =~ s/.{$y}$//; # cut off at the right side
$bin = '1' . $bin; # extend left side by one dummy '1'
$bin =~ tr/10/01/; # flip bits back
my $res = $self->new('0b'.$bin); # add prefix and convert back
$res->binc(); # remember to increment
$x->{value
} = $res->{value
}; # take over value
return $x->round(@r); # we are done now, magic, isn't?
$x->bdec(); # n == 2, but $y == 1: this fixes it
my $t; $t = $CALC->_rsft($x->{value
},$y->{value
},$n) if $CALC->can('_rsft');
$x->bdiv($self->bpow($n,$y, @r), @r);
#(BINT or num_str, BINT or num_str) return BINT
my ($self,$x,$y,@r) = (ref($_[0]),@_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
($self,$x,$y,@r) = objectify
(2,@_);
return $x if $x->modify('band');
local $Math::BigInt
::upgrade
= undef;
return $x->bnan() if ($x->{sign
} !~ /^[+-]$/ || $y->{sign
} !~ /^[+-]$/);
return $x->bzero(@r) if $y->is_zero() || $x->is_zero();
my $sign = 0; # sign of result
$sign = 1 if ($x->{sign
} eq '-') && ($y->{sign
} eq '-');
my $sx = 1; $sx = -1 if $x->{sign
} eq '-';
my $sy = 1; $sy = -1 if $y->{sign
} eq '-';
if ($CALC->can('_and') && $sx == 1 && $sy == 1)
$x->{value
} = $CALC->_and($x->{value
},$y->{value
});
my $m = $self->bone(); my ($xr,$yr);
my $x10000 = $self->new (0x1000);
my $y1 = copy
(ref($x),$y); # make copy
$y1->babs(); # and positive
my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
use integer
; # need this for negative bools
while (!$x1->is_zero() && !$y1->is_zero())
($x1, $xr) = bdiv
($x1, $x10000);
($y1, $yr) = bdiv
($y1, $x10000);
# make both op's numbers!
$x->badd( bmul
( $class->new(
abs($sx*int($xr->numify()) & $sy*int($yr->numify()))),
#(BINT or num_str, BINT or num_str) return BINT
my ($self,$x,$y,@r) = (ref($_[0]),@_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
($self,$x,$y,@r) = objectify
(2,@_);
return $x if $x->modify('bior');
local $Math::BigInt
::upgrade
= undef;
return $x->bnan() if ($x->{sign
} !~ /^[+-]$/ || $y->{sign
} !~ /^[+-]$/);
return $x->round(@r) if $y->is_zero();
my $sign = 0; # sign of result
$sign = 1 if ($x->{sign
} eq '-') || ($y->{sign
} eq '-');
my $sx = 1; $sx = -1 if $x->{sign
} eq '-';
my $sy = 1; $sy = -1 if $y->{sign
} eq '-';
# don't use lib for negative values
if ($CALC->can('_or') && $sx == 1 && $sy == 1)
$x->{value
} = $CALC->_or($x->{value
},$y->{value
});
my $m = $self->bone(); my ($xr,$yr);
my $x10000 = $self->new(0x10000);
my $y1 = copy
(ref($x),$y); # make copy
$y1->babs(); # and positive
my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
use integer
; # need this for negative bools
while (!$x1->is_zero() || !$y1->is_zero())
($x1, $xr) = bdiv
($x1,$x10000);
($y1, $yr) = bdiv
($y1,$x10000);
# make both op's numbers!
$x->badd( bmul
( $class->new(
abs($sx*int($xr->numify()) | $sy*int($yr->numify()))),
#(BINT or num_str, BINT or num_str) return BINT
my ($self,$x,$y,@r) = (ref($_[0]),@_);
# objectify is costly, so avoid it
if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
($self,$x,$y,@r) = objectify
(2,@_);
return $x if $x->modify('bxor');
local $Math::BigInt
::upgrade
= undef;
return $x->bnan() if ($x->{sign
} !~ /^[+-]$/ || $y->{sign
} !~ /^[+-]$/);
return $x->round(@r) if $y->is_zero();
my $sign = 0; # sign of result
$sign = 1 if $x->{sign
} ne $y->{sign
};
my $sx = 1; $sx = -1 if $x->{sign
} eq '-';
my $sy = 1; $sy = -1 if $y->{sign
} eq '-';
# don't use lib for negative values
if ($CALC->can('_xor') && $sx == 1 && $sy == 1)
$x->{value
} = $CALC->_xor($x->{value
},$y->{value
});
my $m = $self->bone(); my ($xr,$yr);
my $x10000 = $self->new(0x10000);
my $y1 = copy
(ref($x),$y); # make copy
$y1->babs(); # and positive
my $x1 = $x->copy()->babs(); $x->bzero(); # modify x in place!
use integer
; # need this for negative bools
while (!$x1->is_zero() || !$y1->is_zero())
($x1, $xr) = bdiv
($x1, $x10000);
($y1, $yr) = bdiv
($y1, $x10000);
# make both op's numbers!
$x->badd( bmul
( $class->new(
abs($sx*int($xr->numify()) ^ $sy*int($yr->numify()))),
my ($self,$x) = ref($_[0]) ?
(ref($_[0]),$_[0]) : objectify
(1,@_);
my $e = $CALC->_len($x->{value
});
return wantarray ?
($e,0) : $e;
# return the nth decimal digit, negative values count backward, 0 is right
my ($self,$x,$n) = ref($_[0]) ?
(ref($_[0]),@_) : objectify
(1,@_);
$CALC->_digit($x->{value
},$n||0);
# return the amount of trailing zeros in $x
$x = $class->new($x) unless ref $x;
return 0 if $x->is_zero() || $x->is_odd() || $x->{sign
} !~ /^[+-]$/;
return $CALC->_zeros($x->{value
}) if $CALC->can('_zeros');
# if not: since we do not know underlying internal representation:
my $es = "$x"; $es =~ /([0]*)$/;
return 0 if !defined $1; # no zeros
CORE
::length("$1"); # as string, not as +0!
my ($self,$x,@r) = ref($_[0]) ?
(ref($_[0]),@_) : objectify
(1,@_);
return $x if $x->modify('bsqrt');
return $x->bnan() if $x->{sign
} ne '+'; # -x or inf or NaN => NaN
return $x->bzero(@r) if $x->is_zero(); # 0 => 0
return $x->round(@r) if $x->is_one(); # 1 => 1
return $upgrade->bsqrt($x,@r) if defined $upgrade;
$x->{value
} = $CALC->_sqrt($x->{value
});
return $x->bone('+',@r) if $x < 4; # 2,3 => 1
my $l = int($x->length()/2);
$x->bone(); # keep ref($x), but modify it
my $last = $self->bzero();
while ($last != $x && $lastlast != $x)
$lastlast = $last; $last = $x;
$x-- if $x * $x > $y; # overshot?
# return a copy of the exponent (here always 0, NaN or 1 for $m == 0)
my ($self,$x) = ref($_[0]) ?
(ref($_[0]),$_[0]) : objectify
(1,@_);
if ($x->{sign
} !~ /^[+-]$/)
my $s = $x->{sign
}; $s =~ s/^[+-]//;
return $self->new($s); # -inf,+inf => inf
return $e->binc() if $x->is_zero();
$e += $x->_trailing_zeros();
# return the mantissa (compatible to Math::BigFloat, e.g. reduced)
my ($self,$x) = ref($_[0]) ?
(ref($_[0]),$_[0]) : objectify
(1,@_);
if ($x->{sign
} !~ /^[+-]$/)
return $self->new($x->{sign
}); # keep + or - sign
my $zeros = $m->_trailing_zeros();
$m->brsft($zeros,10) if $zeros != 0;
# return a copy of both the exponent and the mantissa
my ($self,$x) = ref($_[0]) ?
(ref($_[0]),$_[0]) : objectify
(1,@_);
return ($x->mantissa(),$x->exponent());
##############################################################################
# precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
# $n == 0 || $n == 1 => round to integer
my $x = shift; $x = $class->new($x) unless ref $x;
my ($scale,$mode) = $x->_scale_p($x->precision(),$x->round_mode(),@_);
return $x if !defined $scale; # no-op
return $x if $x->modify('bfround');
# no-op for BigInts if $n <= 0
$x->{_a
} = undef; # clear an eventual set A
$x->{_p
} = $scale; return $x;
$x->bround( $x->length()-$scale, $mode);
$x->{_a
} = undef; # bround sets {_a}
$x->{_p
} = $scale; # so correct it
return 0 if $len == 1; # '5' is trailed by invisible zeros
return 0 if $follow > $len || $follow < 1;
# since we do not know underlying represention of $x, use decimal string
#my $r = substr ($$xs,-$follow);
my $r = substr ("$x",-$follow);
return 1 if $r =~ /[^0]/;
# to make life easier for switch between MBF and MBI (autoload fxxx()
# like MBF does for bxxx()?)
# accuracy: +$n preserve $n digits from left,
# -$n preserve $n digits from right (f.i. for 0.1234 style in MBF)
# and overwrite the rest with 0's, return normalized number
# do not return $x->bnorm(), but $x
my $x = shift; $x = $class->new($x) unless ref $x;
my ($scale,$mode) = $x->_scale_a($x->accuracy(),$x->round_mode(),@_);
return $x if !defined $scale; # no-op
return $x if $x->modify('bround');
if ($x->is_zero() || $scale == 0)
$x->{_a
} = $scale if !defined $x->{_a
} || $x->{_a
} > $scale; # 3 > 2
return $x if $x->{sign
} !~ /^[+-]$/; # inf, NaN
# we have fewer digits than we want to scale to
# scale < 0, but > -len (not >=!)
if (($scale < 0 && $scale < -$len-1) || ($scale >= $len))
$x->{_a
} = $scale if !defined $x->{_a
} || $x->{_a
} > $scale; # 3 > 2
# count of 0's to pad, from left (+) or right (-): 9 - +6 => 3, or |-6| => 6
my ($pad,$digit_round,$digit_after);
$pad = abs($scale-1) if $scale < 0;
# do not use digit(), it is costly for binary => decimal
my $xs = $CALC->_str($x->{value
});
# pad: 123: 0 => -1, at 1 => -2, at 2 => -3, at 3 => -4
# pad+1: 123: 0 => 0, at 1 => -1, at 2 => -2, at 3 => -3
$digit_round = '0'; $digit_round = substr($$xs,$pl,1) if $pad <= $len;
$pl++; $pl ++ if $pad >= $len;
$digit_after = '0'; $digit_after = substr($$xs,$pl,1) if $pad > 0;
# in case of 01234 we round down, for 6789 up, and only in case 5 we look
# closer at the remaining digits of the original $x, remember decision
my $round_up = 1; # default round up
($mode eq 'trunc') || # trunc by round down
($digit_after =~ /[01234]/) || # round down anyway,
($digit_after eq '5') && # not 5000...0000
($x->_scan_for_nonzero($pad,$xs) == 0) &&
($mode eq 'even') && ($digit_round =~ /[24680]/) ||
($mode eq 'odd') && ($digit_round =~ /[13579]/) ||
($mode eq '+inf') && ($x->{sign
} eq '-') ||
($mode eq '-inf') && ($x->{sign
} eq '+') ||
($mode eq 'zero') # round down if zero, sign adjusted below
my $put_back = 0; # not yet modified
if (($pad > 0) && ($pad <= $len))
substr($$xs,-$pad,$pad) = '0' x
$pad;
$x->bzero(); # round to '0'
if ($round_up) # what gave test above?
$pad = $len, $$xs = '0'x
$pad if $scale < 0; # tlr: whack 0.51=>1.0
# we modify directly the string variant instead of creating a number and
# adding it, since that is faster (we already have the string)
my $c = 0; $pad ++; # for $pad == $len case
$c = substr($$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 in if needed
$x->{_a
} = $scale if $scale >= 0;
$x->{_a
} = 0 if $scale < -$len;
# return integer less or equal then number, since it is already integer,
my ($self,$x,@r) = ref($_[0]) ?
(ref($_[0]),@_) : objectify
(1,@_);
# return integer greater or equal then number, since it is already integer,
my ($self,$x,@r) = ref($_[0]) ?
(ref($_[0]),@_) : objectify
(1,@_);
##############################################################################
# private stuff (internal use only)
# internal speedup, set argument to 1, or create a +/- 1
my $x = $self->bone(); # $x->{value} = $CALC->_one();
$x->{sign
} = shift || '+';
# Overload will swap params if first one is no object ref so that the first
# one is always an object ref. In this case, third param is true.
# This routine is to overcome the effect of scalar,$object creating an object
# of the class of this package, instead of the second param $object. This
# happens inside overload, when the overload section of this package is
# inherited by sub classes.
# For overload cases (and this is used only there), we need to preserve the
# args, hence the copy().
# You can override this method in a subclass, the overload section will call
# $object->_swap() to make sure it arrives at the proper subclass, with some
# exceptions like '+' and '-'. To make '+' and '-' work, you also need to
# specify your own overload for them.
# object, (object|scalar) => preserve first and make copy
# scalar, object => swapped, re-swap and create new from first
# (using class of second object, not $class!!)
my $self = shift; # for override in subclass
my $c = ref ($_[0]) || $class; # fallback $class should not happen
return ( $c->new($_[1]), $_[0] );
return ( $_[0]->copy(), $_[1] );
# check for strings, if yes, return objects instead
# the first argument is number of args objectify() should look at it will
# return $count+1 elements, the first will be a classname. This is because
# overloaded '""' calls bstr($object,undef,undef) and this would result in
# useless objects beeing created and thrown away. So we cannot simple loop
# over @_. If the given count is 0, all arguments will be used.
# If the second arg is a ref, use it as class.
# If not, try to use it as classname, unless undef, then use $class
# (aka Math::BigInt). The latter shouldn't happen,though.
# $x->badd(1); => ref x, scalar y
# Class->badd(1,2); => classname x (scalar), scalar x, scalar y
# Class->badd( Class->(1),2); => classname x (scalar), ref x, scalar y
# Math::BigInt::badd(1,2); => scalar x, scalar y
# In the last case we check number of arguments to turn it silently into
# $class,1,2. (We can not take '1' as class ;o)
# badd($class,1) is not supported (it should, eventually, try to add undef)
# currently it tries 'Math::BigInt' + 1, which will not work.
# some shortcut for the common cases
return (ref($_[1]),$_[1]) if (@_ == 2) && ($_[0]||0 == 1) && ref($_[1]);
my $count = abs(shift || 0);
my (@a,$k,$d); # resulting array, temp, and downgrade
# okay, got object as first
# nope, got 1,2 (Class->xxx(1) => Class,1 and not supported)
$a[0] = shift if $_[0] =~ /^[A-Z].*::/; # classname as first?
# disable downgrading, because Math::BigFLoat->foo('1.0','2.0') needs floats
if (defined ${"$a[0]::downgrade"})
$d = ${"$a[0]::downgrade"};
${"$a[0]::downgrade"} = undef;
my $up = ${"$a[0]::upgrade"};
# print "Now in objectify, my class is today $a[0]\n";
elsif (!defined $up && ref($k) ne $a[0])
# foreign object, try to convert to integer
$k->can('as_number') ?
$k = $k->as_number() : $k = $a[0]->new($k);
elsif (!defined $up && ref($k) ne $a[0])
# foreign object, try to convert to integer
$k->can('as_number') ?
$k = $k->as_number() : $k = $a[0]->new($k);
push @a,@_; # return other params, too
die "$class objectify needs list context" unless wantarray;
${"$a[0]::downgrade"} = $d;
my @a; my $l = scalar @_;
for ( my $i = 0; $i < $l ; $i++ )
if ($_[$i] eq ':constant')
# this causes overlord er load to step in
overload
::constant integer
=> sub { $self->new(shift) };
overload
::constant binary
=> sub { $self->new(shift) };
elsif ($_[$i] eq 'upgrade')
$upgrade = $_[$i+1]; # or undef to disable
elsif ($_[$i] =~ /^lib$/i)
# this causes a different low lib to take care...
# any non :constant stuff is handled by our parent, Exporter
# even if @_ is empty, to give it a chance
$self->SUPER::import
(@a); # need it for subclasses
$self->export_to_level(1,$self,@a); # need it for MBF
# try to load core math lib
my @c = split /\s*,\s*/,$CALC;
push @c,'Calc'; # if all fail, try this
$CALC = ''; # signal error
next if ($lib || '') eq '';
$lib = 'Math::BigInt::'.$lib if $lib !~ /^Math::BigInt/i;
# 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
$file = File
::Spec
->catfile (@parts, $file);
eval { require "$file"; $lib->import( @c ); }
$CALC = $lib, last if $@
eq ''; # no error in loading lib?
die "Couldn't load any math lib, not even the default" if $CALC eq '';
# convert a (ref to) big hex string to BigInt, return undef for error
my $x = Math
::BigInt
->bzero();
$$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
$$hs =~ s/([0-9a-fA-F])_([0-9a-fA-F])/$1$2/g;
return $x->bnan() if $$hs !~ /^[\-\+]?0x[0-9A-Fa-f]+$/;
my $sign = '+'; $sign = '-' if ($$hs =~ /^-/);
$$hs =~ s/^[+-]//; # strip sign
if ($CALC->can('_from_hex'))
$x->{value
} = $CALC->_from_hex($hs);
my $mul = Math
::BigInt
->bzero(); $mul++;
my $x65536 = Math
::BigInt
->new(65536);
my $len = CORE
::length($$hs)-2;
$len = int($len/4); # 4-digit parts, w/o '0x'
$val = substr($$hs,$i,4);
$val =~ s/^[+-]?0x// if $len == 0; # for last part only because
$val = hex($val); # hex does not like wrong chars
$x += $mul * $val if $val != 0;
$mul *= $x65536 if $len >= 0; # skip last mul
$x->{sign
} = $sign unless $CALC->_is_zero($x->{value
}); # no '-0'
# convert a (ref to) big binary string to BigInt, return undef for error
my $x = Math
::BigInt
->bzero();
$$bs =~ s/([01])_([01])/$1$2/g;
$$bs =~ s/([01])_([01])/$1$2/g;
return $x->bnan() if $$bs !~ /^[+-]?0b[01]+$/;
my $sign = '+'; $sign = '-' if ($$bs =~ /^\-/);
$$bs =~ s/^[+-]//; # strip sign
if ($CALC->can('_from_bin'))
$x->{value
} = $CALC->_from_bin($bs);
my $mul = Math
::BigInt
->bzero(); $mul++;
my $x256 = Math
::BigInt
->new(256);
my $len = CORE
::length($$bs)-2;
$len = int($len/8); # 8-digit parts, w/o '0b'
$val = substr($$bs,$i,8);
$val =~ s/^[+-]?0b// if $len == 0; # for last part only
#$val = oct('0b'.$val); # does not work on Perl prior to 5.6.0
# $val = ('0' x (8-CORE::length($val))).$val if CORE::length($val) < 8;
$val = ord(pack('B8',substr('00000000'.$val,-8,8)));
$x += $mul * $val if $val != 0;
$mul *= $x256 if $len >= 0; # skip last mul
$x->{sign
} = $sign unless $CALC->_is_zero($x->{value
}); # no '-0'
# (ref to num_str) return num_str
# internal, take apart a string and return the pieces
# strip leading/trailing whitespace, leading zeros, underscore and reject
# strip white space at front, also extranous leading zeros
$$x =~ s/^\s*([-]?)0*([0-9])/$1$2/g; # will not strip ' .2'
$$x =~ s/^\s+//; # but this will
$$x =~ s/\s+$//g; # strip white space at end
# shortcut, if nothing to split, return early
if ($$x =~ /^[+-]?\d+\z/)
$$x =~ s/^([+-])0*([0-9])/$2/; my $sign = $1 || '+';
return (\
$sign, $x, \'', \'', \
0);
return if $$x !~ /^[+-]?(\.?[0-9]|0b[0-1]|0x[0-9a-fA-F])/;
return __from_hex
($x) if $$x =~ /^[\-\+]?0x/; # hex string
return __from_bin
($x) if $$x =~ /^[\-\+]?0b/; # binary string
# strip underscores between digits
$$x =~ s/(\d)_(\d)/$1$2/g;
$$x =~ s/(\d)_(\d)/$1$2/g; # do twice for 1_2_3
# 2.1234 # 0.12 # 1 # 1E1 # 2.134E1 # 434E-10 # 1.02009E-2
# .2 # 1_2_3.4_5_6 # 1.4E1_2_3 # 1e3 # +.2
return if $$x =~ /[Ee].*[Ee]/; # more than one E => error
my ($m,$e) = split /[Ee]/,$$x;
$e = '0' if !defined $e || $e eq "";
# sign,value for exponent,mantint,mantfrac
my ($es,$ev,$mis,$miv,$mfv);
if ($e =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
return if $m eq '.' || $m eq '';
my ($mi,$mf,$last) = split /\./,$m;
return if defined $last; # last defined => 1.2.3 or others
$mi = '0' if !defined $mi;
$mi .= '0' if $mi =~ /^[\-\+]?$/;
$mf = '0' if !defined $mf || $mf eq '';
if ($mi =~ /^([+-]?)0*(\d+)$/) # strip leading zeros
$mis = $1||'+'; $miv = $2;
return unless ($mf =~ /^(\d*?)0*$/); # strip trailing zeros
return (\
$mis,\
$miv,\
$mfv,\
$es,\
$ev);
return; # NaN, not a 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
# return as hex string, with prefixed 0x
my $x = shift; $x = $class->new($x) if !ref($x);
return $x->bstr() if $x->{sign
} !~ /^[+-]$/; # inf, nan etc
return '0x0' if $x->is_zero();
$s = $x->{sign
} if $x->{sign
} eq '-';
if ($CALC->can('_as_hex'))
$es = ${$CALC->_as_hex($x->{value
})};
my $x1 = $x->copy()->babs(); my ($xr,$x10000,$h);
$x10000 = Math
::BigInt
->new (0x10000); $h = 'h4';
$x10000 = Math
::BigInt
->new (0x1000); $h = 'h3';
($x1, $xr) = bdiv
($x1,$x10000);
$es .= unpack($h,pack('v',$xr->numify()));
$es =~ s/^[0]+//; # strip leading zeros
# return as binary string, with prefixed 0b
my $x = shift; $x = $class->new($x) if !ref($x);
return $x->bstr() if $x->{sign
} !~ /^[+-]$/; # inf, nan etc
return '0b0' if $x->is_zero();
$s = $x->{sign
} if $x->{sign
} eq '-';
if ($CALC->can('_as_bin'))
$es = ${$CALC->_as_bin($x->{value
})};
my $x1 = $x->copy()->babs(); my ($xr,$x10000,$b);
$x10000 = Math
::BigInt
->new (0x10000); $b = 'b16';
$x10000 = Math
::BigInt
->new (0x1000); $b = 'b12';
($x1, $xr) = bdiv
($x1,$x10000);
$es .= unpack($b,pack('v',$xr->numify()));
$es =~ s/^[0]+//; # strip leading zeros
##############################################################################
# internal calculation routines (others are in Math::BigInt::Calc etc)
# (BINT or num_str, BINT or num_str) return BINT
# does modify first argument
my $x = shift; my $ty = shift;
return $x->bnan() if ($x->{sign
} eq $nan) || ($ty->{sign
} eq $nan);
return $x * $ty / bgcd
($x,$ty);
# (BINT or num_str, BINT or num_str) return BINT
# does modify both arguments
# GCD -- Euclids algorithm E, Knuth Vol 2 pg 296
return $x->bnan() if $x->{sign
} !~ /^[+-]$/ || $ty->{sign
} !~ /^[+-]$/;
($x, $ty) = ($ty,bmod
($x,$ty));
###############################################################################
# this method return 0 if the object can be modified, or 1 for not
# We use a fast use constant statement here, to avoid costly calls. Subclasses
# may override it with special code (f.i. Math::BigInt::Constant does so)
Math::BigInt - Arbitrary size integer math package
$x = Math::BigInt->new($str); # defaults to 0
$nan = Math::BigInt->bnan(); # create a NotANumber
$zero = Math::BigInt->bzero(); # create a +0
$inf = Math::BigInt->binf(); # create a +inf
$inf = Math::BigInt->binf('-'); # create a -inf
$one = Math::BigInt->bone(); # create a +1
$one = Math::BigInt->bone('-'); # create a -1
$x->is_zero(); # true if arg is +0
$x->is_nan(); # true if arg is NaN
$x->is_one(); # true if arg is +1
$x->is_one('-'); # true if arg is -1
$x->is_odd(); # true if odd, false for even
$x->is_even(); # true if even, false for odd
$x->is_positive(); # true if >= 0
$x->is_negative(); # true if < 0
$x->is_inf(sign); # true if +inf, or -inf (sign is default '+')
$x->is_int(); # true if $x is an integer (not a float)
$x->bcmp($y); # compare numbers (undef,<0,=0,>0)
$x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
$x->sign(); # return the sign, either +,- or NaN
$x->digit($n); # return the nth digit, counting from right
$x->digit(-$n); # return the nth digit, counting from left
# The following all modify their first argument:
$x->bzero(); # set $x to 0
$x->bnan(); # set $x to NaN
$x->bone(); # set $x to +1
$x->bone('-'); # set $x to -1
$x->binf(); # set $x to inf
$x->binf('-'); # set $x to -inf
$x->babs(); # absolute value
$x->bnorm(); # normalize (no-op)
$x->bnot(); # two's complement (bit wise not)
$x->binc(); # increment x by 1
$x->bdec(); # decrement x by 1
$x->badd($y); # addition (add $y to $x)
$x->bsub($y); # subtraction (subtract $y from $x)
$x->bmul($y); # multiplication (multiply $x by $y)
$x->bdiv($y); # divide, set $x to quotient
# return (quo,rem) or quo if scalar
$x->bmod($y); # modulus (x % y)
$x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
$x->bmodinv($mod); # the inverse of $x in the given modulus $mod
$x->bpow($y); # power of arguments (x ** y)
$x->blsft($y); # left shift
$x->brsft($y); # right shift
$x->blsft($y,$n); # left shift, by base $n (like 10)
$x->brsft($y,$n); # right shift, by base $n (like 10)
$x->band($y); # bitwise and
$x->bior($y); # bitwise inclusive or
$x->bxor($y); # bitwise exclusive or
$x->bnot(); # bitwise not (two's complement)
$x->bsqrt(); # calculate square-root
$x->bfac(); # factorial of $x (1*2*3*4*..$x)
$x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r
$x->bround($N); # accuracy: preserve $N digits
$x->bfround($N); # round to $Nth digit, no-op for BigInts
# The following do not modify their arguments in BigInt, but do in BigFloat:
$x->bfloor(); # return integer less or equal than $x
$x->bceil(); # return integer greater or equal than $x
# The following do not modify their arguments:
bgcd(@values); # greatest common divisor (no OO style)
blcm(@values); # lowest common multiplicator (no OO style)
$x->length(); # return number of digits in number
($x,$f) = $x->length(); # length of number and length of fraction part,
# latter is always 0 digits long for BigInt's
$x->exponent(); # return exponent as BigInt
$x->mantissa(); # return (signed) mantissa as BigInt
$x->parts(); # return (mantissa,exponent) as BigInt
$x->copy(); # make a true copy of $x (unlike $y = $x;)
$x->as_number(); # return as BigInt (in BigInt: same as copy())
$x->bstr(); # normalized string
$x->bsstr(); # normalized string in scientific notation
$x->as_hex(); # as signed hexadecimal string with prefixed 0x
$x->as_bin(); # as signed binary string with prefixed 0b
Math::BigInt->config(); # return hash containing configuration/version
# precision and accuracy (see section about rounding for more)
$x->precision(); # return P of $x (or global, if P of $x undef)
$x->precision($n); # set P of $x to $n
$x->accuracy(); # return A of $x (or global, if A of $x undef)
$x->accuracy($n); # set A $x to $n
Math::BigInt->precision(); # get/set global P for all BigInt objects
Math::BigInt->accuracy(); # get/set global A for all BigInt objects
All operators (inlcuding basic math operations) are overloaded if you
declare your big integers as
$i = new Math::BigInt '123_456_789_123_456_789';
Operations with overloaded operators preserve the arguments which is
Big integer values are strings of the form C</^[+-]\d+$/> with leading
'-0' canonical value '-0', normalized '0'
' -123_123_123' canonical value '-123123123'
'1_23_456_7890' canonical value '1234567890'
Input values to these routines may be either Math::BigInt objects or
strings of the form C</^[+-]?[\d]+\.?[\d]*E?[+-]?[\d]*$/>.
You can include one underscore between any two digits. The input string may
have leading and trailing whitespace, which will be ignored. In later
versions, a more strict (no whitespace at all) or more lax (whitespace
allowed everywhere) input checking will also be possible.
This means integer values like 1.01E2 or even 1000E-2 are also accepted.
Non integer values result in NaN.
Math::BigInt::new() defaults to 0, while Math::BigInt::new('') results
bnorm() on a BigInt object is now effectively a no-op, since the numbers
are always stored in normalized form. On a string, it creates a BigInt
Output values are BigInt objects (normalized), except for bstr(), which
returns a string in normalized form.
Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
return either undef, <0, 0 or >0 and are suited for sort.
Each of the methods below accepts three additional parameters. These arguments
$A, $P and $R are accuracy, precision and round_mode. Please see more in the
section about ACCURACY and ROUNDIND.
print Dumper ( Math::BigInt->config() );
Returns a hash containing the configuration, e.g. the version number, lib
$x->accuracy(5); # local for $x
$class->accuracy(5); # global for all members of $class
Set or get the global or local accuracy, aka how many significant digits the
results have. Please see the section about L<ACCURACY AND PRECISION> for
Value must be greater than zero. Pass an undef value to disable it:
Math::BigInt->accuracy(undef);
Returns the current accuracy. For C<$x->accuracy()> it will return either the
local accuracy, or if not defined, the global. This means the return value
represents the accuracy that will be in effect for $x:
$y = Math::BigInt->new(1234567); # unrounded
print Math::BigInt->accuracy(4),"\n"; # set 4, print 4
$x = Math::BigInt->new(123456); # will be automatically rounded
print "$x $y\n"; # '123500 1234567'
print $x->accuracy(),"\n"; # will be 4
print $y->accuracy(),"\n"; # also 4, since global is 4
print Math::BigInt->accuracy(5),"\n"; # set to 5, print 5
print $x->accuracy(),"\n"; # still 4
print $y->accuracy(),"\n"; # 5, since global is 5
Shifts $x right by $y in base $n. Default is base 2, used are usually 10 and
Right shifting usually amounts to dividing $x by $n ** $y and truncating the
$x = Math::BigInt->new(10);
$x->brsft(1); # same as $x >> 1: 5
$x = Math::BigInt->new(1234);
$x->brsft(2,10); # result 12
There is one exception, and that is base 2 with negative $x:
$x = Math::BigInt->new(-5);
This will print -3, not -2 (as it would if you divide -5 by 2 and truncate the
$x = Math::BigInt->new($str,$A,$P,$R);
Creates a new BigInt object from a string or another BigInt object. The
input is accepted as decimal, hex (with leading '0x') or binary (with leading
$x = Math::BigInt->bnan();
Creates a new BigInt object representing NaN (Not A Number).
If used on an object, it will set it to NaN:
$x = Math::BigInt->bzero();
Creates a new BigInt object representing zero.
If used on an object, it will set it to zero:
$x = Math::BigInt->binf($sign);
Creates a new BigInt object representing infinity. The optional argument is
either '-' or '+', indicating whether you want infinity or minus infinity.
If used on an object, it will set it to infinity:
$x = Math::BigInt->binf($sign);
Creates a new BigInt object representing one. The optional argument is
either '-' or '+', indicating whether you want one or minus one.
If used on an object, it will set it to one:
=head2 is_one()/is_zero()/is_nan()/is_inf()
$x->is_zero(); # true if arg is +0
$x->is_nan(); # true if arg is NaN
$x->is_one(); # true if arg is +1
$x->is_one('-'); # true if arg is -1
$x->is_inf(); # true if +inf
$x->is_inf('-'); # true if -inf (sign is default '+')
These methods all test the BigInt for beeing one specific value and return
true or false depending on the input. These are faster than doing something
=head2 is_positive()/is_negative()
$x->is_positive(); # true if >= 0
$x->is_negative(); # true if < 0
The methods return true if the argument is positive or negative, respectively.
C<NaN> is neither positive nor negative, while C<+inf> counts as positive, and
C<-inf> is negative. A C<zero> is positive.
These methods are only testing the sign, and not the value.
=head2 is_odd()/is_even()/is_int()
$x->is_odd(); # true if odd, false for even
$x->is_even(); # true if even, false for odd
$x->is_int(); # true if $x is an integer
The return true when the argument satisfies the condition. C<NaN>, C<+inf>,
C<-inf> are not integers and are neither odd nor even.
Compares $x with $y and takes the sign into account.
Returns -1, 0, 1 or undef.
Compares $x with $y while ignoring their. Returns -1, 0, 1 or undef.
Return the sign, of $x, meaning either C<+>, C<->, C<-inf>, C<+inf> or NaN.
$x->digit($n); # return the nth digit, counting from right
Negate the number, e.g. change the sign between '+' and '-', or between '+inf'
and '-inf', respectively. Does nothing for NaN or zero.
Set the number to it's absolute value, e.g. change the sign from '-' to '+'
and from '-inf' to '+inf', respectively. Does nothing for NaN or positive
$x->bnorm(); # normalize (no-op)
$x->bnot(); # two's complement (bit wise not)
$x->binc(); # increment x by 1
$x->bdec(); # decrement x by 1
$x->badd($y); # addition (add $y to $x)
$x->bsub($y); # subtraction (subtract $y from $x)
$x->bmul($y); # multiplication (multiply $x by $y)
$x->bdiv($y); # divide, set $x to quotient
# return (quo,rem) or quo if scalar
$x->bmod($y); # modulus (x % y)
$num->bmodinv($mod); # modular inverse
Returns the inverse of C<$num> in the given modulus C<$mod>. 'C<NaN>' is
returned unless C<$num> is relatively prime to C<$mod>, i.e. unless
$num->bmodpow($exp,$mod); # modular exponentation ($num**$exp % $mod)
Returns the value of C<$num> taken to the power C<$exp> in the modulus
C<$mod> using binary exponentation. C<bmodpow> is far superior to
because C<bmodpow> is much faster--it reduces internal variables into
the modulus whenever possible, so it operates on smaller numbers.
C<bmodpow> also supports negative exponents.
$x->bpow($y); # power of arguments (x ** y)
$x->blsft($y); # left shift
$x->blsft($y,$n); # left shift, by base $n (like 10)
$x->brsft($y); # right shift
$x->brsft($y,$n); # right shift, by base $n (like 10)
$x->band($y); # bitwise and
$x->bior($y); # bitwise inclusive or
$x->bxor($y); # bitwise exclusive or
$x->bnot(); # bitwise not (two's complement)
$x->bsqrt(); # calculate square-root
$x->bfac(); # factorial of $x (1*2*3*4*..$x)
$x->round($A,$P,$round_mode); # round to accuracy or precision using mode $r
$x->bround($N); # accuracy: preserve $N digits
$x->bfround($N); # round to $Nth digit, no-op for BigInts
Set $x to the integer less or equal than $x. This is a no-op in BigInt, but
does change $x in BigFloat.
Set $x to the integer greater or equal than $x. This is a no-op in BigInt, but
does change $x in BigFloat.
bgcd(@values); # greatest common divisor (no OO style)
blcm(@values); # lowest common multiplicator (no OO style)
($xl,$fl) = $x->length();
Returns the number of digits in the decimal representation of the number.
In list context, returns the length of the integer and fraction part. For
BigInt's, the length of the fraction part will always be 0.
Return the exponent of $x as BigInt.
Return the signed mantissa of $x as BigInt.
$x->parts(); # return (mantissa,exponent) as BigInt
$x->copy(); # make a true copy of $x (unlike $y = $x;)
$x->as_number(); # return as BigInt (in BigInt: same as copy())
$x->bstr(); # normalized string
$x->bsstr(); # normalized string in scientific notation
$x->as_hex(); # as signed hexadecimal string with prefixed 0x
$x->as_bin(); # as signed binary string with prefixed 0b
=head1 ACCURACY and PRECISION
Since version v1.33, Math::BigInt and Math::BigFloat have full support for
accuracy and precision based rounding, both automatically after every
operation as well as manually.
This section describes the accuracy/precision handling in Math::Big* as it
used to be and as it is now, complete with an explanation of all terms and
Not yet implemented things (but with correct description) are marked with '!',
things that need to be answered are marked with '?'.
In the next paragraph follows a short description of terms used here (because
these may differ from terms used by others people or documentation).
During the rest of this document, the shortcuts A (for accuracy), P (for
precision), F (fallback) and R (rounding mode) will be used.
A fixed number of digits before (positive) or after (negative)
the decimal point. For example, 123.45 has a precision of -2. 0 means an
integer like 123 (or 120). A precision of 2 means two digits to the left
of the decimal point are zero, so 123 with P = 1 becomes 120. Note that
numbers with zeros before the decimal point may have different precisions,
because 1200 can have p = 0, 1 or 2 (depending on what the inital value
was). It could also have p < 0, when the digits after the decimal point
The string output (of floating point numbers) will be padded with zeros:
Initial value P A Result String
------------------------------------------------------------
1234.01 2 1234.01 1234.01
1234.01 5 1234.01 1234.01000
For BigInts, no padding occurs.
Number of significant digits. Leading zeros are not counted. A
number may have an accuracy greater than the non-zero digits
when there are zeros in it or trailing zeros. For example, 123.456 has
A of 6, 10203 has 5, 123.0506 has 7, 123.450000 has 8 and 0.000123 has 3.
The string output (of floating point numbers) will be padded with zeros:
Initial value P A Result String
------------------------------------------------------------
1234.01 6 1234.01 1234.01
1234.1 8 1234.1 1234.1000
For BigInts, no padding occurs.
When both A and P are undefined, this is used as a fallback accuracy when
When rounding a number, different 'styles' or 'kinds'
of rounding are possible. (Note that random rounding, as in
Math::Round, is not implemented.)
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:
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.
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.
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.
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,
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.
The handling of A & P in MBI/MBF (the old core code shipped with Perl
versions <= 5.7.2) is like this:
* ffround($p) is able to round to $p number of digits after the decimal
=item Accuracy (significant digits)
* fround($a) rounds to $a significant digits
* only fdiv() and fsqrt() take A as (optional) paramater
+ other operations simply create the same number (fneg etc), or more (fmul)
+ rounding/truncating is only done when explicitly calling one of fround
or ffround, and never for BigInt (not implemented)
* fsqrt() simply hands its accuracy argument over to fdiv.
* the documentation and the comment in the code indicate two different ways
on how fdiv() determines the maximum number of digits it should calculate,
and the actual code does yet another thing
max($Math::BigFloat::div_scale,length(dividend)+length(divisor))
result has at most max(scale, length(dividend), length(divisor)) digits
scale = max(scale, length(dividend)-1,length(divisor)-1);
scale += length(divisior) - length(dividend);
So for lx = 3, ly = 9, scale = 10, scale will actually be 16 (10+9-3).
Actually, the 'difference' added to the scale is calculated from the
number of "significant digits" in dividend and divisor, which is derived
by looking at the length of the mantissa. Which is wrong, since it includes
the + sign (oups) and actually gets 2 for '+100' and 4 for '+101'. Oups
again. Thus 124/3 with div_scale=1 will get you '41.3' based on the strange
assumption that 124 has 3 significant digits, while 120/7 will get you
'17', not '17.1' since 120 is thought to have 2 significant digits.
The rounding after the division then uses the remainder and $y to determine
wether it must round up or down.
? I have no idea which is the right way. That's why I used a slightly more
? simple scheme and tweaked the few failing testcases to match it.
This is how it works now:
* 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, take Math::SomeClass->accuracy()
* to find out the current global P, take Math::SomeClass->precision()
* use $x->accuracy() respective $x->precision() for the local setting of $x.
* Please note that $x->accuracy() respecive $x->precision() fall back to the
defined globals, when $x's A or P is not set.
* When you create a number, you can give it's desired A or P via:
$x = Math::BigInt->new($number,$A,$P);
* Only one of A or P can be defined, otherwise the result is NaN
* If no A or P is give ($x = Math::BigInt->new($number) form), then the
globals (if set) will be used. Thus changing the global defaults later on
will not change the A or P of previously created numbers (i.e., A and P of
$x will be what was in effect when $x was created)
* If given undef for A and P, B<no> rounding will occur, and the globals will
B<not> be used. This is used by subclasses to create numbers without
suffering rounding in the parent. Thus a subclass is able to have it's own
globals enforced upon creation of a number by using
$x = Math::BigInt->new($number,undef,undef):
use Math::Bigint::SomeSubclass;
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.
* If A or P are enabled/defined, they are used to round the result of each
operation according to the rules below
* Negative P is ignored in Math::BigInt, since BigInts never have digits
* Math::BigFloat uses Math::BigInts internally, but setting A or P inside
Math::BigInt as globals should not tamper with the parts of a BigFloat.
Thus a flag is used to mark all Math::BigFloat numbers as 'never round'
* It only makes sense that a number has only one of A or P at a time.
Since you can set/get both A and P, there is a rule that will practically
enforce only A or P to be in effect at a time, even if both are set.
This is called precedence.
* If two objects are involved in an operation, and one of them has A in
effect, and the other P, this results in an error (NaN).
* A takes precendence over P (Hint: A comes before P). If A is defined, it
is used, otherwise P is used. If neither of them is defined, nothing is
used, i.e. the result will have as many digits as it can (with an
exception for fdiv/fsqrt) and will not be rounded.
* There is another setting for fdiv() (and thus for fsqrt()). If neither of
A or P is defined, fdiv() will use a fallback (F) of $div_scale digits.
If either the dividend's or the divisor's mantissa has more digits than
the value of F, the higher value will be used instead of F.
This is to limit the digits (A) of the result (just consider what would
happen with unlimited A and P in the case of 1/3 :-)
* fdiv will calculate (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 three modes:
+ never round (this is the default):
This is done by setting A and P to undef. No math operation
will round the result, with fdiv() and fsqrt() as exceptions to guard
against overflows. You must explicitely call bround(), bfround() or
round() (the latter with parameters).
Note: Once you have rounded a number, the settings will 'stick' on it
and 'infect' all other numbers engaged in math operations with it, since
local settings have the highest precedence. So, to get SaferRound[tm],
use a copy() before rounding like this:
$x = Math::BigFloat->new(12.34);
$y = Math::BigFloat->new(98.76);
$z = $x * $y; # 1218.6984
print $x->copy()->fround(3); # 12.3 (but A is now 3!)
$z = $x * $y; # still 1218.6984, without
# copy would have been 1210!
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,
?Maybe an extra option that forbids local A & P settings would be in order,
?so that intermediate rounding does not 'poison' further math?
* 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
(local: per object, global: global default, parameter: argument to sub)
+ 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)
* fsqrt() will hand its arguments to fdiv(), as it used to, only now for two
arguments (A and P) instead of one
* You can set A and P locally by using $x->accuracy() and $x->precision()
and thus force different A and P for different objects/numbers.
* Setting A or P this way immediately rounds $x to the new value.
* $x->accuracy() clears $x->precision(), and vice versa.
* the rounding routines will use the respective global or local settings.
fround()/bround() is for accuracy rounding, while ffround()/bfround()
* the two rounding functions take as the second parameter one of the
following rounding modes (R):
'even', 'odd', '+inf', '-inf', 'zero', 'trunc'
* you can set and get the global R by using Math::SomeClass->round_mode()
or by setting $Math::SomeClass::round_mode
* after each operation, $result->round() is called, and the result may
eventually be rounded (that is, if A or P were set either locally,
globally or as parameter to the operation)
* to manually round a number, call $x->round($A,$P,$round_mode);
this will round the number by using the appropriate rounding function
* rounding modifies the local settings of the number:
$x = Math::BigFloat->new(123.456);
Here 4 takes precedence over 5, so 123.5 is the result and $x->accuracy()
* The defaults are set up so that the new code gives the same results as
the old code (except in a few cases on fdiv):
+ Both A and P are undefined and thus will not be used for rounding
+ round() is thus a no-op, unless given extra parameters A and P
The actual numbers are stored as unsigned big integers (with seperate sign).
You should neither care about nor depend on the internal representation; it
might change without notice. Use only method calls like C<< $x->sign(); >>
instead relying on the internal hash keys like in C<< $x->{sign}; >>.
Math with the numbers is done (by default) by a module called
Math::BigInt::Calc. This is equivalent to saying:
use Math::BigInt lib => 'Calc';
You can change this by using:
use Math::BigInt lib => 'BitVect';
The following would first try to find Math::BigInt::Foo, then
Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
use Math::BigInt lib => 'Foo,Math::BigInt::Bar';
Calc.pm uses as internal format an array of elements of some decimal base
(usually 1e5 or 1e7) with the least significant digit first, while BitVect.pm
uses a bit vector of base 2, most significant bit first. Other modules might
use even different means of representing the numbers. See the respective
module documentation for further details.
The sign is either '+', '-', 'NaN', '+inf' or '-inf' and stored seperately.
A sign of 'NaN' is used to represent the result when input arguments are not
numbers or as a result of 0/0. '+inf' and '-inf' represent plus respectively
minus infinity. You will get '+inf' when dividing a positive number by 0, and
'-inf' when dividing any negative number by 0.
=head2 mantissa(), exponent() and parts()
C<mantissa()> and C<exponent()> return the said parts of the BigInt such
print "ok\n" if $x == $y;
C<< ($m,$e) = $x->parts() >> is just a shortcut that gives you both of them
in one go. Both the returned mantissa and exponent have a sign.
Currently, for BigInts C<$e> will be always 0, except for NaN, +inf and -inf,
where it will be NaN; and for $x == 0, where it will be 1
(to be compatible with Math::BigFloat's internal representation of a zero as
C<$m> will always be a copy of the original number. The relation between $e
and $m might change in the future, but will always be equivalent in a
numerical sense, e.g. $m might get minimized.
sub bint { Math::BigInt->new(shift); }
$x = Math::BigInt->bstr("1234") # string "1234"
$x = "$x"; # same as bstr()
$x = Math::BigInt->bneg("1234"); # Bigint "-1234"
$x = Math::BigInt->babs("-12345"); # Bigint "12345"
$x = Math::BigInt->bnorm("-0 00"); # BigInt "0"
$x = bint(1) + bint(2); # BigInt "3"
$x = bint(1) + "2"; # ditto (auto-BigIntify of "2")
$x = bint(1); # BigInt "1"
$x = $x + 5 / 2; # BigInt "3"
$x = $x ** 3; # BigInt "27"
$x = Math::BigInt->new(0); # BigInt "0"
$x = Math::BigInt->badd(4,5) # BigInt "9"
print $x->bsstr(); # 9e+0
$x = Math::BigFloat->new(123.4567);
$y = Math::BigFloat->new(123.456789);
Math::BigFloat->accuracy(4); # no more A than 4
ok ($x->copy()->fround(),123.4); # even rounding
print $x->copy()->fround(),"\n"; # 123.4
Math::BigFloat->round_mode('odd'); # round to odd
print $x->copy()->fround(),"\n"; # 123.5
Math::BigFloat->accuracy(5); # no more A than 5
Math::BigFloat->round_mode('odd'); # round to odd
print $x->copy()->fround(),"\n"; # 123.46
$y = $x->copy()->fround(4),"\n"; # A = 4: 123.4
print "$y, ",$y->accuracy(),"\n"; # 123.4, 4
Math::BigFloat->accuracy(undef); # A not important now
Math::BigFloat->precision(2); # P important
print $x->copy()->bnorm(),"\n"; # 123.46
print $x->copy()->fround(),"\n"; # 123.46
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.
perl -MMath::BigInt=:constant -e 'print 2**100,"\n"'
prints the integer value of C<2**100>. Note that without conversion of
constants the expression 2**100 will be calculated as perl scalar.
Please note that strings and floating point constants are not affected,
use Math::BigInt qw/:constant/;
$x = 1234567890123456789012345678901234567890
$y = '1234567890123456789012345678901234567890'
do not work. You need an explicit Math::BigInt->new() around one of the
operands. You should also quote large constants to protect loss of precision:
$x = Math::BigInt->new('1234567889123456789123456789123456789');
Without the quotes Perl would convert the large number to a floating point
constant at compile time and then hand the result to BigInt, which results in
an truncated result or a NaN.
This also applies to integers that look like floating point constants:
use Math::BigInt ':constant';
will print nothing but newlines. Use either L<bignum> or L<Math::BigFloat>
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 $x takes
more time then the actual addition.
With a technique called copy-on-write, the cost of copying with overload could
be minimized or even completely avoided. A test implementation of COW did show
performance gains for overloaded math, but introduced a performance loss due
to a constant overhead for all other operatons.
The rewritten version of this module is slower on certain operations, like
new(), bstr() and numify(). The reason are that it does now more work and
handles more cases. The time spent in these operations is usually gained in
the other operations so that programs on the average should get faster. If
they don't, please contect the author.
Some operations may be slower for small numbers, but are significantly faster
for big numbers. Other operations are now constant (O(1), like bneg(), babs()
etc), instead of O(N) and thus nearly always take much less time. These
optimizations were done on purpose.
If you find the Calc module to slow, try to install any of the replacement
modules and see if they help you.
=head2 Alternative math libraries
You can use an alternative library to drive Math::BigInt via:
use Math::BigInt lib => 'Module';
See L<MATH LIBRARY> for more information.
For more benchmark results see L<http://bloodgate.com/perl/benchmarks.html>.
=head1 Subclassing Math::BigInt
The basic design of Math::BigInt allows simple subclasses with very little
work, as long as a few simple rules are followed:
The public API must remain consistent, i.e. if a sub-class is overloading
addition, the sub-class must use the same name, in this case badd(). The
reason for this is that Math::BigInt is optimized to call the object methods
The private object hash keys like C<$x->{sign}> may not be changed, but
additional keys can be added, like C<$x->{_custom}>.
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.
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
At the very minimum, any subclass will need to provide it's own C<new()> and can
store additional hash keys in the object. There are also some package globals
that must be defined, e.g.:
$precision = -2; # round to 2 decimal places
Additionally, you might want to provide the following two globals to allow
auto-upgrading and auto-downgrading to work correctly:
This allows Math::BigInt to correctly retrieve package globals from the
subclass, like C<$SubClass::precision>. See t/Math/BigInt/Subclass.pm or
t/Math/BigFloat/SubClass.pm completely functional subclass examples.
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
use Math::BigInt upgrade => 'Foo::Bar';
certain operations will 'upgrade' their calculation and thus the result to
the class Foo::Bar. Usually this is used in conjunction with Math::BigFloat:
use Math::BigInt upgrade => 'Math::BigFloat';
As a shortcut, you can use the module C<bignum>:
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.
The following methods upgrade themselves unconditionally; that is if upgrade
is in effect, they will always hand up their work:
Beware: This list is not complete.
All other methods upgrade themselves only when one (or all) of their
arguments are of the class mentioned in $upgrade (This might change in later
versions to a more sophisticated scheme):
Under Perl prior to 5.6.0 having an C<use Math::BigInt ':constant';> and
C<eval()> in your code will crash with "Out of memory". This is probably an
overload/exporter bug. You can workaround by not having C<eval()>
and ':constant' at the same time or upgrade your Perl to a newer version.
=item Fails to load Calc on Perl prior 5.6.0
Since eval(' use ...') can not be used in conjunction with ':constant', BigInt
will fall back to eval { require ... } when loading the math lib on Perls
prior to 5.6.0. This simple replaces '::' with '/' and thus might fail on
filesystems using a different seperator.
Some things might not work as you expect them. Below is documented what is
=item stringify, bstr(), bsstr() and 'cmp'
Both stringify and bstr() now drop the leading '+'. The old code would return
'+3', the new returns '3'. This is to be consistent with Perl and to make
cmp (especially with overloading) to work as you expect. It also solves
problems with Test.pm, it's ok() uses 'eq' internally.
Mark said, when asked about to drop the '+' altogether, or make only cmp work:
I agree (with the first alternative), don't add the '+' on positive
numbers. It's not as important anymore with the new internal
form for numbers. It made doing things like abs and neg easier,
but those have to be done differently now anyway.
So, the following examples will now work all as expected:
BEGIN { plan tests => 1 }
my $x = new Math::BigInt 3*3;
my $y = new Math::BigInt 3*3;
print "$x eq 9" if $x eq $y;
print "$x eq 9" if $x eq '9';
print "$x eq 9" if $x eq 3*3;
Additionally, the following still works:
print "$x == 9" if $x == $y;
print "$x == 9" if $x == 9;
print "$x == 9" if $x == 3*3;
There is now a C<bsstr()> method to get the string in scientific notation aka
C<1e+2> instead of C<100>. Be advised that overloaded 'eq' always uses bstr()
for comparisation, but Perl will represent some numbers as 100 and others
as 1e+308. If in doubt, convert both arguments to Math::BigInt before doing eq:
BEGIN { plan tests => 3 }
$x = Math::BigInt->new('1e56'); $y = 1e56;
ok ($x->bsstr(),$y); # okay
$y = Math::BigInt->new($y);
Alternatively, simple use <=> for comparisations, that will get it always
right. There is not yet a way to get a number automatically represented as
a string that matches exactly the way Perl represents it.
C<int()> will return (at least for Perl v5.7.1 and up) another BigInt, not a
$x = Math::BigInt->new(123);
$y = int($x); # BigInt 123
$x = Math::BigFloat->new(123.45);
$y = int($x); # BigInt 123
In all Perl versions you can use C<as_number()> for the same effect:
$x = Math::BigFloat->new(123.45);
$y = $x->as_number(); # BigInt 123
This also works for other subclasses, like Math::String.
It is yet unlcear whether overloaded int() should return a scalar or a BigInt.
The following will probably not do what you expect:
$c = Math::BigInt->new(123);
print $c->length(),"\n"; # prints 30
It prints both the number of digits in the number and in the fraction part
since print calls C<length()> in list context. Use something like:
print scalar $c->length(),"\n"; # prints 3
The following will probably not do what you expect:
print $c->bdiv(10000),"\n";
It prints both quotient and remainder since print calls C<bdiv()> in list
context. Also, C<bdiv()> will modify $c, so be carefull. You probably want
print scalar $c->bdiv(10000),"\n"; # or if you want to modify $c
The quotient is always the greatest integer less than or equal to the
real-valued quotient of the two operands, and the remainder (when it is
nonzero) always has the same sign as the second operand; so, for
As a consequence, the behavior of the operator % agrees with the
behavior of Perl's built-in % operator (as documented in the perlop
manpage), and the equation
$x == ($x / $y) * $y + ($x % $y)
holds true for any $x and $y, which justifies calling the two return
values of bdiv() the quotient and remainder. The only exception to this rule
are when $y == 0 and $x is negative, then the remainder will also be
negative. See below under "infinity handling" for the reasoning behing this.
Perl's 'use integer;' changes the behaviour of % and / for scalars, but will
not change BigInt's way to do things. This is because under 'use integer' Perl
will do what the underlying C thinks is right and this is different for each
system. If you need BigInt's behaving exactly like Perl's 'use integer', bug
the author to implement it ;)
Here are some examples that explain the reasons why certain results occur while
The following table shows the result of the division and the remainder, so that
the equation above holds true. Some "ordinary" cases are strewn in to show more
A / B = C, R so that C * B + R = A
=========================================================
5 / 8 = 0, 5 0 * 8 + 5 = 5
0 / 8 = 0, 0 0 * 8 + 0 = 0
0 / inf = 0, 0 0 * inf + 0 = 0
0 /-inf = 0, 0 0 * -inf + 0 = 0
5 / inf = 0, 5 0 * inf + 5 = 5
5 /-inf = 0, 5 0 * -inf + 5 = 5
-5/ inf = 0, -5 0 * inf + -5 = -5
-5/-inf = 0, -5 0 * -inf + -5 = -5
inf/ 5 = inf, 0 inf * 5 + 0 = inf
-inf/ 5 = -inf, 0 -inf * 5 + 0 = -inf
inf/ -5 = -inf, 0 -inf * -5 + 0 = inf
-inf/ -5 = inf, 0 inf * -5 + 0 = -inf
5/ 5 = 1, 0 1 * 5 + 0 = 5
-5/ -5 = 1, 0 1 * -5 + 0 = -5
inf/ inf = 1, 0 1 * inf + 0 = inf
-inf/-inf = 1, 0 1 * -inf + 0 = -inf
inf/-inf = -1, 0 -1 * -inf + 0 = inf
-inf/ inf = -1, 0 1 * -inf + 0 = -inf
8/ 0 = inf, 8 inf * 0 + 8 = 8
inf/ 0 = inf, inf inf * 0 + inf = inf
These cases below violate the "remainder has the sign of the second of the two
arguments", since they wouldn't match up otherwise.
A / B = C, R so that C * B + R = A
========================================================
-inf/ 0 = -inf, -inf -inf * 0 + inf = -inf
-8/ 0 = -inf, -8 -inf * 0 + 8 = -8
$x = Math::BigFloat->new(5);
It will not do what you think, e.g. making a copy of $x. Instead it just makes
a second reference to the B<same> object and stores it in $y. Thus anything
that modifies $x (except overloaded operators) will modify $y, and vice versa.
Or in other words, C<=> is only safe if you modify your BigInts only via
overloaded math. As soon as you use a method call it breaks:
print "$x, $y\n"; # prints '10, 10'
If you want a true copy of $x, use:
You can also chain the calls like this, this will make first a copy and then
$y = $x->copy()->bmul(2);
See also the documentation for overload.pm regarding C<=>.
C<bpow()> (and the rounding functions) now modifies the first argument and
returns it, unlike the old code which left it alone and only returned the
result. This is to be consistent with C<badd()> etc. The first three will
modify $x, the last one won't:
print bpow($x,$i),"\n"; # modify $x
print $x->bpow($i),"\n"; # ditto
print $x **= $i,"\n"; # the same
print $x ** $i,"\n"; # leave $x alone
The form C<$x **= $y> is faster than C<$x = $x ** $y;>, though.
since overload calls C<sub($x,0,1);> instead of C<neg($x)>. The first variant
needs to preserve $x since it does not know that it later will get overwritten.
This makes a copy of $x and takes O(N), but $x->bneg() is O(1).
With Copy-On-Write, this issue would be gone, but C-o-W is not implemented
since it is slower for all other things.
=item Mixing different object types
In Perl you will get a floating point value if you do one of the following:
With overloaded math, only the first two variants will result in a BigFloat:
$mbf = Math::BigFloat->new(5);
$mbi2 = Math::BigInteger->new(5);
$mbi = Math::BigInteger->new(2);
# what actually gets called:
$float = $mbf + $mbi; # $mbf->badd()
$float = $mbf / $mbi; # $mbf->bdiv()
$integer = $mbi + $mbf; # $mbi->badd()
$integer = $mbi2 / $mbi; # $mbi2->bdiv()
$integer = $mbi2 / $mbf; # $mbi2->bdiv()
This is because math with overloaded operators follows the first (dominating)
operand, and the operation of that is called and returns thus the result. So,
Math::BigInt::bdiv() will always return a Math::BigInt, regardless whether
the result should be a Math::BigFloat or the second operant is one.
To get a Math::BigFloat you either need to call the operation manually,
make sure the operands are already of the proper type or casted to that type
via Math::BigFloat->new():
$float = Math::BigFloat->new($mbi2) / $mbi; # = 2.5
Beware of simple "casting" the entire expression, this would only convert
the already computed result:
$float = Math::BigFloat->new($mbi2 / $mbi); # = 2.0 thus wrong!
Beware also of the order of more complicated expressions like:
$integer = ($mbi2 + $mbi) / $mbf; # int / float => int
$integer = $mbi2 / Math::BigFloat->new($mbi); # ditto
If in doubt, break the expression into simpler terms, or cast all operands
to the desired resulting type.
Scalar values are a bit different, since:
will both result in the proper type due to the way the overloaded math works.
This section also applies to other overloaded math packages, like Math::String.
One solution to you problem might be L<autoupgrading|upgrading>.
C<bsqrt()> works only good if the result is a big integer, e.g. the square
root of 144 is 12, but from 12 the square root is 3, regardless of rounding
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
For negative numbers in base see also L<brsft|brsft>.
This program is free software; you may redistribute it and/or modify it under
the same terms as Perl itself.
L<Math::BigFloat> and L<Math::Big> as well as L<Math::BigInt::BitVect>,
L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
more documentation including a full version history, testcases, empty
subclass files and benchmarks.
Original code by Mark Biggar, overloaded interface by Ilya Zakharevich.
Completely rewritten by Tels http://bloodgate.com in late 2000, 2001.