package Math::BigFloat;
#
-# Mike grinned. 'Two down, infinity to go' - Mike Nostrus in Before and After
+# Mike grinned. 'Two down, infinity to go' - Mike Nostrus in 'Before and After'
#
# The following hash values are internally used:
-# _e: exponent (BigInt)
-# _m: mantissa (absolute BigInt)
-# sign: +,-,"NaN" if not a number
-# _a: accuracy
-# _p: precision
-# _f: flags, used to signal MBI not to touch our private parts
-
-$VERSION = '1.31';
+# _e : exponent (ref to $CALC object)
+# _m : mantissa (ref to $CALC object)
+# _es : sign of _e
+# sign : +,-,+inf,-inf, or "NaN" if not a number
+# _a : accuracy
+# _p : precision
+
+$VERSION = '1.51';
require 5.005;
-use Exporter;
-use File::Spec;
-# use Math::BigInt;
-@ISA = qw( Exporter Math::BigInt);
+
+require Exporter;
+@ISA = qw(Exporter Math::BigInt);
use strict;
-use vars qw/$AUTOLOAD $accuracy $precision $div_scale $round_mode $rnd_mode/;
-use vars qw/$upgrade $downgrade $MBI/;
+# $_trap_inf/$_trap_nan are internal and should never be accessed from outside
+use vars qw/$AUTOLOAD $accuracy $precision $div_scale $round_mode $rnd_mode
+ $upgrade $downgrade $_trap_nan $_trap_inf/;
my $class = "Math::BigFloat";
use overload
;
##############################################################################
-# global constants, flags and accessory
-
-use constant MB_NEVER_ROUND => 0x0001;
+# global constants, flags and assorted stuff
-# are NaNs ok?
-my $NaNOK=1;
-# constant for easier life
-my $nan = 'NaN';
+# the following are public, but their usage is not recommended. Use the
+# accessor methods instead.
# class constants, use Class->constant_name() to access
$round_mode = 'even'; # one of 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'
$upgrade = undef;
$downgrade = undef;
-$MBI = 'Math::BigInt'; # the package we are using for our private parts
- # changable by use Math::BigFloat with => 'package'
+# the package we are using for our private parts, defaults to:
+# Math::BigInt->config()->{lib}
+my $MBI = 'Math::BigInt::FastCalc';
+
+# are NaNs ok? (otherwise it dies when encountering an NaN) set w/ config()
+$_trap_nan = 0;
+# the same for infinity
+$_trap_inf = 0;
+
+# constant for easier life
+my $nan = 'NaN';
+
+my $IMPORT = 0; # was import() called yet? used to make require work
+
+# some digits of accuracy for blog(undef,10); which we use in blog() for speed
+my $LOG_10 =
+ '2.3025850929940456840179914546843642076011014886287729760333279009675726097';
+my $LOG_10_A = length($LOG_10)-1;
+# ditto for log(2)
+my $LOG_2 =
+ '0.6931471805599453094172321214581765680755001343602552541206800094933936220';
+my $LOG_2_A = length($LOG_2)-1;
+my $HALF = '0.5'; # made into an object if necc.
##############################################################################
# the old code had $rnd_mode, so we need to support it, too
sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
BEGIN
- {
- $rnd_mode = 'even';
- tie $rnd_mode, 'Math::BigFloat';
+ {
+ # when someone set's $rnd_mode, we catch this and check the value to see
+ # whether it is valid or not.
+ $rnd_mode = 'even'; tie $rnd_mode, 'Math::BigFloat';
}
##############################################################################
-# in case we call SUPER::->foo() and this wants to call modify()
-# sub modify () { 0; }
-
{
# valid method aliases for AUTOLOAD
my %methods = map { $_ => 1 }
qw / fadd fsub fmul fdiv fround ffround fsqrt fmod fstr fsstr fpow fnorm
- fint facmp fcmp fzero fnan finf finc fdec flog ffac
- fceil ffloor frsft flsft fone flog
+ fint facmp fcmp fzero fnan finf finc fdec flog ffac fneg
+ fceil ffloor frsft flsft fone flog froot
/;
# valid method's that can be hand-ed up (for AUTOLOAD)
my %hand_ups = map { $_ => 1 }
- qw / is_nan is_inf is_negative is_positive
- accuracy precision div_scale round_mode fneg fabs babs fnot
+ qw / is_nan is_inf is_negative is_positive is_pos is_neg
+ accuracy precision div_scale round_mode fabs fnot
objectify upgrade downgrade
bone binf bnan bzero
/;
- sub method_alias { return exists $methods{$_[0]||''}; }
- sub method_hand_up { return exists $hand_ups{$_[0]||''}; }
+ sub method_alias { exists $methods{$_[0]||''}; }
+ sub method_hand_up { exists $hand_ups{$_[0]||''}; }
}
##############################################################################
return $class->bzero() if !defined $wanted; # default to 0
return $wanted->copy() if UNIVERSAL::isa($wanted,'Math::BigFloat');
+ $class->import() if $IMPORT == 0; # make require work
+
my $self = {}; bless $self, $class;
# shortcut for bigints and its subclasses
if ((ref($wanted)) && (ref($wanted) ne $class))
{
- $self->{_m} = $wanted->as_number(); # get us a bigint copy
- $self->{_e} = $MBI->bzero();
- $self->{_m}->babs();
+ $self->{_m} = $wanted->as_number()->{value}; # get us a bigint copy
+ $self->{_e} = $MBI->_zero();
+ $self->{_es} = '+';
$self->{sign} = $wanted->sign();
return $self->bnorm();
}
- # got string
+ # else: got a string
+
# handle '+inf', '-inf' first
- if ($wanted =~ /^[+-]?inf$/)
+ if ($wanted =~ /^[+-]?inf\z/)
{
return $downgrade->new($wanted) if $downgrade;
- $self->{_e} = $MBI->bzero();
- $self->{_m} = $MBI->bzero();
- $self->{sign} = $wanted;
- $self->{sign} = '+inf' if $self->{sign} eq 'inf';
- return $self->bnorm();
+ $self->{sign} = $wanted; # set a default sign for bstr()
+ return $self->binf($wanted);
+ }
+
+ # shortcut for simple forms like '12' that neither have trailing nor leading
+ # zeros
+ if ($wanted =~ /^([+-]?)([1-9][0-9]*[1-9])$/)
+ {
+ $self->{_e} = $MBI->_zero();
+ $self->{_es} = '+';
+ $self->{sign} = $1 || '+';
+ $self->{_m} = $MBI->_new($2);
+ return $self->round(@r) if !$downgrade;
}
- #print "new string '$wanted'\n";
- my ($mis,$miv,$mfv,$es,$ev) = Math::BigInt::_split(\$wanted);
+
+ my ($mis,$miv,$mfv,$es,$ev) = Math::BigInt::_split($wanted);
if (!ref $mis)
{
- die "$wanted is not a number initialized to $class" if !$NaNOK;
+ if ($_trap_nan)
+ {
+ require Carp;
+ Carp::croak ("$wanted is not a number initialized to $class");
+ }
return $downgrade->bnan() if $downgrade;
- $self->{_e} = $MBI->bzero();
- $self->{_m} = $MBI->bzero();
+ $self->{_e} = $MBI->_zero();
+ $self->{_es} = '+';
+ $self->{_m} = $MBI->_zero();
$self->{sign} = $nan;
}
else
{
- # make integer from mantissa by adjusting exp, then convert to bigint
- # undef,undef to signal MBI that we don't need no bloody rounding
- $self->{_e} = $MBI->new("$$es$$ev",undef,undef); # exponent
- $self->{_m} = $MBI->new("$$miv$$mfv",undef,undef); # create mant.
+ # make integer from mantissa by adjusting exp, then convert to int
+ $self->{_e} = $MBI->_new($$ev); # exponent
+ $self->{_es} = $$es || '+';
+ my $mantissa = "$$miv$$mfv"; # create mant.
+ $mantissa =~ s/^0+(\d)/$1/; # strip leading zeros
+ $self->{_m} = $MBI->_new($mantissa); # create mant.
+
# 3.123E0 = 3123E-3, and 3.123E-2 => 3123E-5
- $self->{_e} -= CORE::length($$mfv) if CORE::length($$mfv) != 0;
+ if (CORE::length($$mfv) != 0)
+ {
+ my $len = $MBI->_new( CORE::length($$mfv));
+ ($self->{_e}, $self->{_es}) =
+ _e_sub ($self->{_e}, $len, $self->{_es}, '+');
+ }
+ # we can only have trailing zeros on the mantissa if $$mfv eq ''
+ else
+ {
+ # Use a regexp to count the trailing zeros in $$miv instead of _zeros()
+ # because that is faster, especially when _m is not stored in base 10.
+ my $zeros = 0; $zeros = CORE::length($1) if $$miv =~ /[1-9](0*)$/;
+ if ($zeros != 0)
+ {
+ my $z = $MBI->_new($zeros);
+ # turn '120e2' into '12e3'
+ $MBI->_rsft ( $self->{_m}, $z, 10);
+ ($self->{_e}, $self->{_es}) =
+ _e_add ( $self->{_e}, $z, $self->{_es}, '+');
+ }
+ }
$self->{sign} = $$mis;
+
+ # for something like 0Ey, set y to 1, and -0 => +0
+ # Check $$miv for being '0' and $$mfv eq '', because otherwise _m could not
+ # have become 0. That's faster than to call $MBI->_is_zero().
+ $self->{sign} = '+', $self->{_e} = $MBI->_one()
+ if $$miv eq '0' and $$mfv eq '';
+
+ return $self->round(@r) if !$downgrade;
}
# if downgrade, inf, NaN or integers go down
- if ($downgrade && $self->{_e}->{sign} eq '+')
+ if ($downgrade && $self->{_es} eq '+')
{
-# print "downgrading $$miv$$mfv"."E$$es$$ev";
- if ($self->{_e}->is_zero())
+ if ($MBI->_is_zero( $self->{_e} ))
{
- $self->{_m}->{sign} = $$mis; # negative if wanted
- return $downgrade->new($self->{_m});
+ return $downgrade->new($$mis . $MBI->_str( $self->{_m} ));
}
- return $downgrade->new("$$mis$$miv$$mfv"."E$$es$$ev");
+ return $downgrade->new($self->bsstr());
+ }
+ $self->bnorm()->round(@r); # first normalize, then round
+ }
+
+sub copy
+ {
+ my ($c,$x);
+ if (@_ > 1)
+ {
+ # if two arguments, the first one is the class to "swallow" subclasses
+ ($c,$x) = @_;
+ }
+ else
+ {
+ $x = shift;
+ $c = ref($x);
}
- # print "mbf new $self->{sign} $self->{_m} e $self->{_e} ",ref($self),"\n";
- $self->bnorm()->round(@r); # first normalize, then round
+ return unless ref($x); # only for objects
+
+ my $self = {}; bless $self,$c;
+
+ $self->{sign} = $x->{sign};
+ $self->{_es} = $x->{_es};
+ $self->{_m} = $MBI->_copy($x->{_m});
+ $self->{_e} = $MBI->_copy($x->{_e});
+ $self->{_a} = $x->{_a} if defined $x->{_a};
+ $self->{_p} = $x->{_p} if defined $x->{_p};
+ $self;
}
sub _bnan
{
- # used by parent class bone() to initialize number to 1
+ # used by parent class bone() to initialize number to NaN
my $self = shift;
- $self->{_m} = $MBI->bzero();
- $self->{_e} = $MBI->bzero();
+
+ if ($_trap_nan)
+ {
+ require Carp;
+ my $class = ref($self);
+ Carp::croak ("Tried to set $self to NaN in $class\::_bnan()");
+ }
+
+ $IMPORT=1; # call our import only once
+ $self->{_m} = $MBI->_zero();
+ $self->{_e} = $MBI->_zero();
+ $self->{_es} = '+';
}
sub _binf
{
- # used by parent class bone() to initialize number to 1
+ # used by parent class bone() to initialize number to +-inf
my $self = shift;
- $self->{_m} = $MBI->bzero();
- $self->{_e} = $MBI->bzero();
+
+ if ($_trap_inf)
+ {
+ require Carp;
+ my $class = ref($self);
+ Carp::croak ("Tried to set $self to +-inf in $class\::_binf()");
+ }
+
+ $IMPORT=1; # call our import only once
+ $self->{_m} = $MBI->_zero();
+ $self->{_e} = $MBI->_zero();
+ $self->{_es} = '+';
}
sub _bone
{
# used by parent class bone() to initialize number to 1
my $self = shift;
- $self->{_m} = $MBI->bone();
- $self->{_e} = $MBI->bzero();
+ $IMPORT=1; # call our import only once
+ $self->{_m} = $MBI->_one();
+ $self->{_e} = $MBI->_zero();
+ $self->{_es} = '+';
}
sub _bzero
{
- # used by parent class bone() to initialize number to 1
+ # used by parent class bone() to initialize number to 0
my $self = shift;
- $self->{_m} = $MBI->bzero();
- $self->{_e} = $MBI->bone();
+ $IMPORT=1; # call our import only once
+ $self->{_m} = $MBI->_zero();
+ $self->{_e} = $MBI->_one();
+ $self->{_es} = '+';
}
sub isa
UNIVERSAL::isa($self,$class);
}
+sub config
+ {
+ # return (later set?) configuration data as hash ref
+ my $class = shift || 'Math::BigFloat';
+
+ my $cfg = $class->SUPER::config(@_);
+
+ # now we need only to override the ones that are different from our parent
+ $cfg->{class} = $class;
+ $cfg->{with} = $MBI;
+ $cfg;
+ }
+
##############################################################################
# string conversation
# (ref to BFLOAT or num_str ) return num_str
# Convert number from internal format to (non-scientific) string format.
# internal format is always normalized (no leading zeros, "-0" => "+0")
- my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
- #my $x = shift; my $class = ref($x) || $x;
- #$x = $class->new(shift) unless ref($x);
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
- #die "Oups! e was $nan" if $x->{_e}->{sign} eq $nan;
- #die "Oups! m was $nan" if $x->{_m}->{sign} eq $nan;
if ($x->{sign} !~ /^[+-]$/)
{
return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
return 'inf'; # +inf
}
-
+
my $es = '0'; my $len = 1; my $cad = 0; my $dot = '.';
- my $not_zero = ! $x->is_zero();
+ # $x is zero?
+ my $not_zero = !($x->{sign} eq '+' && $MBI->_is_zero($x->{_m}));
if ($not_zero)
{
- $es = $x->{_m}->bstr();
+ $es = $MBI->_str($x->{_m});
$len = CORE::length($es);
- if (!$x->{_e}->is_zero())
+ my $e = $MBI->_num($x->{_e});
+ $e = -$e if $x->{_es} eq '-';
+ if ($e < 0)
{
- if ($x->{_e}->sign() eq '-')
+ $dot = '';
+ # if _e is bigger than a scalar, the following will blow your memory
+ if ($e <= -$len)
{
- $dot = '';
- if ($x->{_e} <= -$len)
- {
- # print "style: 0.xxxx\n";
- my $r = $x->{_e}->copy(); $r->babs()->bsub( CORE::length($es) );
- $es = '0.'. ('0' x $r) . $es; $cad = -($len+$r);
- }
- else
- {
- # print "insert '.' at $x->{_e} in '$es'\n";
- substr($es,$x->{_e},0) = '.'; $cad = $x->{_e};
- }
+ my $r = abs($e) - $len;
+ $es = '0.'. ('0' x $r) . $es; $cad = -($len+$r);
}
else
{
- # expand with zeros
- $es .= '0' x $x->{_e}; $len += $x->{_e}; $cad = 0;
+ substr($es,$e,0) = '.'; $cad = $MBI->_num($x->{_e});
+ $cad = -$cad if $x->{_es} eq '-';
}
}
+ elsif ($e > 0)
+ {
+ # expand with zeros
+ $es .= '0' x $e; $len += $e; $cad = 0;
+ }
} # if not zero
- $es = $x->{sign}.$es if $x->{sign} eq '-';
- # if set accuracy or precision, pad with zeros
+
+ $es = '-'.$es if $x->{sign} eq '-';
+ # if set accuracy or precision, pad with zeros on the right side
if ((defined $x->{_a}) && ($not_zero))
{
# 123400 => 6, 0.1234 => 4, 0.001234 => 4
$zeros = $x->{_a} - $len if $cad != $len;
$es .= $dot.'0' x $zeros if $zeros > 0;
}
- elsif ($x->{_p} || 0 < 0)
+ elsif ((($x->{_p} || 0) < 0))
{
# 123400 => 6, 0.1234 => 4, 0.001234 => 6
my $zeros = -$x->{_p} + $cad;
# (ref to BFLOAT or num_str ) return num_str
# Convert number from internal format to scientific string format.
# internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
- my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
- #my $x = shift; my $class = ref($x) || $x;
- #$x = $class->new(shift) unless ref($x);
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
- #die "Oups! e was $nan" if $x->{_e}->{sign} eq $nan;
- #die "Oups! m was $nan" if $x->{_m}->{sign} eq $nan;
if ($x->{sign} !~ /^[+-]$/)
{
return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
return 'inf'; # +inf
}
- my $sign = $x->{_e}->{sign}; $sign = '' if $sign eq '-';
- my $sep = 'e'.$sign;
- $x->{_m}->bstr().$sep.$x->{_e}->bstr();
+ my $sep = 'e'.$x->{_es};
+ my $sign = $x->{sign}; $sign = '' if $sign eq '+';
+ $sign . $MBI->_str($x->{_m}) . $sep . $MBI->_str($x->{_e});
}
sub numify
{
# Make a number from a BigFloat object
- # simple return string and let Perl's atoi()/atof() handle the rest
- my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+ # simple return a string and let Perl's atoi()/atof() handle the rest
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
$x->bsstr();
}
##############################################################################
# public stuff (usually prefixed with "b")
+sub bneg
+ {
+ # (BINT or num_str) return BINT
+ # negate number or make a negated number from string
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ return $x if $x->modify('bneg');
+
+ # for +0 dont negate (to have always normalized +0). Does nothing for 'NaN'
+ $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $MBI->_is_zero($x->{_m}));
+ $x;
+ }
+
# tels 2001-08-04
-# todo: this must be overwritten and return NaN for non-integer values
+# XXX TODO this must be overwritten and return NaN for non-integer values
# band(), bior(), bxor(), too
#sub bnot
# {
sub bcmp
{
# Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
- # (BFLOAT or num_str, BFLOAT or num_str) return cond_code
- my ($self,$x,$y) = objectify(2,@_);
+
+ # set up parameters
+ my ($self,$x,$y) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y) = objectify(2,@_);
+ }
+
+ return $upgrade->bcmp($x,$y) if defined $upgrade &&
+ ((!$x->isa($self)) || (!$y->isa($self)));
if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
{
return 1 if $yz && $x->{sign} eq '+'; # +x <=> 0
# adjust so that exponents are equal
- my $lxm = $x->{_m}->length();
- my $lym = $y->{_m}->length();
+ my $lxm = $MBI->_len($x->{_m});
+ my $lym = $MBI->_len($y->{_m});
# the numify somewhat limits our length, but makes it much faster
- my $lx = $lxm + $x->{_e}->numify();
- my $ly = $lym + $y->{_e}->numify();
+ my ($xes,$yes) = (1,1);
+ $xes = -1 if $x->{_es} ne '+';
+ $yes = -1 if $y->{_es} ne '+';
+ my $lx = $lxm + $xes * $MBI->_num($x->{_e});
+ my $ly = $lym + $yes * $MBI->_num($y->{_e});
my $l = $lx - $ly; $l = -$l if $x->{sign} eq '-';
return $l <=> 0 if $l != 0;
my $ym = $y->{_m};
if ($diff > 0)
{
- $ym = $y->{_m}->copy()->blsft($diff,10);
+ $ym = $MBI->_copy($y->{_m});
+ $ym = $MBI->_lsft($ym, $MBI->_new($diff), 10);
}
elsif ($diff < 0)
{
- $xm = $x->{_m}->copy()->blsft(-$diff,10);
+ $xm = $MBI->_copy($x->{_m});
+ $xm = $MBI->_lsft($xm, $MBI->_new(-$diff), 10);
}
- my $rc = $xm->bacmp($ym);
+ my $rc = $MBI->_acmp($xm,$ym);
$rc = -$rc if $x->{sign} eq '-'; # -124 < -123
$rc <=> 0;
}
{
# Compares 2 values, ignoring their signs.
# Returns one of undef, <0, =0, >0. (suitable for sort)
- # (BFLOAT or num_str, BFLOAT or num_str) return cond_code
- my ($self,$x,$y) = objectify(2,@_);
+
+ # set up parameters
+ my ($self,$x,$y) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y) = objectify(2,@_);
+ }
+
+ return $upgrade->bacmp($x,$y) if defined $upgrade &&
+ ((!$x->isa($self)) || (!$y->isa($self)));
# handle +-inf and NaN's
if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/)
return 1 if $yz && !$xz; # +x <=> 0
# adjust so that exponents are equal
- my $lxm = $x->{_m}->length();
- my $lym = $y->{_m}->length();
+ my $lxm = $MBI->_len($x->{_m});
+ my $lym = $MBI->_len($y->{_m});
+ my ($xes,$yes) = (1,1);
+ $xes = -1 if $x->{_es} ne '+';
+ $yes = -1 if $y->{_es} ne '+';
# the numify somewhat limits our length, but makes it much faster
- my $lx = $lxm + $x->{_e}->numify();
- my $ly = $lym + $y->{_e}->numify();
+ my $lx = $lxm + $xes * $MBI->_num($x->{_e});
+ my $ly = $lym + $yes * $MBI->_num($y->{_e});
my $l = $lx - $ly;
return $l <=> 0 if $l != 0;
my $ym = $y->{_m};
if ($diff > 0)
{
- $ym = $y->{_m}->copy()->blsft($diff,10);
+ $ym = $MBI->_copy($y->{_m});
+ $ym = $MBI->_lsft($ym, $MBI->_new($diff), 10);
}
elsif ($diff < 0)
{
- $xm = $x->{_m}->copy()->blsft(-$diff,10);
+ $xm = $MBI->_copy($x->{_m});
+ $xm = $MBI->_lsft($xm, $MBI->_new(-$diff), 10);
}
- $xm->bacmp($ym) <=> 0;
+ $MBI->_acmp($xm,$ym);
}
sub badd
{
# add second arg (BFLOAT or string) to first (BFLOAT) (modifies first)
# return result as BFLOAT
- my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
- #print "mbf badd $x $y\n";
+ # set up parameters
+ my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+ }
+
# inf and NaN handling
if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
{
return $x;
}
+ return $upgrade->badd($x,$y,$a,$p,$r) if defined $upgrade &&
+ ((!$x->isa($self)) || (!$y->isa($self)));
+
# speed: no add for 0+y or x+0
return $x->bround($a,$p,$r) if $y->is_zero(); # x+0
if ($x->is_zero()) # 0+y
{
# make copy, clobbering up x (modify in place!)
- $x->{_e} = $y->{_e}->copy();
- $x->{_m} = $y->{_m}->copy();
+ $x->{_e} = $MBI->_copy($y->{_e});
+ $x->{_es} = $y->{_es};
+ $x->{_m} = $MBI->_copy($y->{_m});
$x->{sign} = $y->{sign} || $nan;
return $x->round($a,$p,$r,$y);
}
# take lower of the two e's and adapt m1 to it to match m2
my $e = $y->{_e};
- $e = $MBI->bzero() if !defined $e; # if no BFLOAT ?
- $e = $e->copy(); # make copy (didn't do it yet)
- $e->bsub($x->{_e});
- my $add = $y->{_m}->copy();
- if ($e->{sign} eq '-') # < 0
+ $e = $MBI->_zero() if !defined $e; # if no BFLOAT?
+ $e = $MBI->_copy($e); # make copy (didn't do it yet)
+
+ my $es;
+
+ ($e,$es) = _e_sub($e, $x->{_e}, $y->{_es} || '+', $x->{_es});
+
+ my $add = $MBI->_copy($y->{_m});
+
+ if ($es eq '-') # < 0
{
- my $e1 = $e->copy()->babs();
- #$x->{_m} *= (10 ** $e1);
- $x->{_m}->blsft($e1,10);
- $x->{_e} += $e; # need the sign of e
+ $MBI->_lsft( $x->{_m}, $e, 10);
+ ($x->{_e},$x->{_es}) = _e_add($x->{_e}, $e, $x->{_es}, $es);
}
- elsif (!$e->is_zero()) # > 0
+ elsif (!$MBI->_is_zero($e)) # > 0
{
- #$add *= (10 ** $e);
- $add->blsft($e,10);
+ $MBI->_lsft($add, $e, 10);
}
# else: both e are the same, so just leave them
- $x->{_m}->{sign} = $x->{sign}; # fiddle with signs
- $add->{sign} = $y->{sign};
- $x->{_m} += $add; # finally do add/sub
- $x->{sign} = $x->{_m}->{sign}; # re-adjust signs
- $x->{_m}->{sign} = '+'; # mantissa always positiv
- # delete trailing zeros, then round
- return $x->bnorm()->round($a,$p,$r,$y);
- }
-
-sub bsub
- {
- # (BigFloat or num_str, BigFloat or num_str) return BigFloat
- # subtract second arg from first, modify first
- my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
- if ($y->is_zero()) # still round for not adding zero
+ if ($x->{sign} eq $y->{sign})
{
- return $x->round($a,$p,$r);
+ # add
+ $x->{_m} = $MBI->_add($x->{_m}, $add);
}
-
- $y->{sign} =~ tr/+\-/-+/; # does nothing for NaN
- $x->badd($y,$a,$p,$r); # badd does not leave internal zeros
- $y->{sign} =~ tr/+\-/-+/; # refix $y (does nothing for NaN)
- $x; # already rounded by badd()
+ else
+ {
+ ($x->{_m}, $x->{sign}) =
+ _e_add($x->{_m}, $add, $x->{sign}, $y->{sign});
+ }
+
+ # delete trailing zeros, then round
+ $x->bnorm()->round($a,$p,$r,$y);
}
+# sub bsub is inherited from Math::BigInt!
+
sub binc
{
# increment arg by one
- my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+ my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
- if ($x->{_e}->sign() eq '-')
+ if ($x->{_es} eq '-')
{
- return $x->badd($self->bone(),$a,$p,$r); # digits after dot
+ return $x->badd($self->bone(),@r); # digits after dot
}
- if (!$x->{_e}->is_zero())
+ if (!$MBI->_is_zero($x->{_e})) # _e == 0 for NaN, inf, -inf
{
- $x->{_m}->blsft($x->{_e},10); # 1e2 => 100
- $x->{_e}->bzero();
+ # 1e2 => 100, so after the shift below _m has a '0' as last digit
+ $x->{_m} = $MBI->_lsft($x->{_m}, $x->{_e},10); # 1e2 => 100
+ $x->{_e} = $MBI->_zero(); # normalize
+ $x->{_es} = '+';
+ # we know that the last digit of $x will be '1' or '9', depending on the
+ # sign
}
# now $x->{_e} == 0
if ($x->{sign} eq '+')
{
- $x->{_m}->binc();
- return $x->bnorm()->bround($a,$p,$r);
+ $MBI->_inc($x->{_m});
+ return $x->bnorm()->bround(@r);
}
elsif ($x->{sign} eq '-')
{
- $x->{_m}->bdec();
- $x->{sign} = '+' if $x->{_m}->is_zero(); # -1 +1 => -0 => +0
- return $x->bnorm()->bround($a,$p,$r);
+ $MBI->_dec($x->{_m});
+ $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # -1 +1 => -0 => +0
+ return $x->bnorm()->bround(@r);
}
# inf, nan handling etc
- $x->badd($self->__one(),$a,$p,$r); # does round
+ $x->badd($self->bone(),@r); # badd() does round
}
sub bdec
{
# decrement arg by one
- my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+ my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
- if ($x->{_e}->sign() eq '-')
+ if ($x->{_es} eq '-')
{
- return $x->badd($self->bone('-'),$a,$p,$r); # digits after dot
+ return $x->badd($self->bone('-'),@r); # digits after dot
}
- if (!$x->{_e}->is_zero())
+ if (!$MBI->_is_zero($x->{_e}))
{
- $x->{_m}->blsft($x->{_e},10); # 1e2 => 100
- $x->{_e}->bzero();
+ $x->{_m} = $MBI->_lsft($x->{_m}, $x->{_e},10); # 1e2 => 100
+ $x->{_e} = $MBI->_zero(); # normalize
+ $x->{_es} = '+';
}
# now $x->{_e} == 0
my $zero = $x->is_zero();
# <= 0
if (($x->{sign} eq '-') || $zero)
{
- $x->{_m}->binc();
- $x->{sign} = '-' if $zero; # 0 => 1 => -1
- $x->{sign} = '+' if $x->{_m}->is_zero(); # -1 +1 => -0 => +0
- return $x->bnorm()->round($a,$p,$r);
+ $MBI->_inc($x->{_m});
+ $x->{sign} = '-' if $zero; # 0 => 1 => -1
+ $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # -1 +1 => -0 => +0
+ return $x->bnorm()->round(@r);
}
# > 0
elsif ($x->{sign} eq '+')
{
- $x->{_m}->bdec();
- return $x->bnorm()->round($a,$p,$r);
+ $MBI->_dec($x->{_m});
+ return $x->bnorm()->round(@r);
}
# inf, nan handling etc
- $x->badd($self->bone('-'),$a,$p,$r); # does round
+ $x->badd($self->bone('-'),@r); # does round
}
+sub DEBUG () { 0; }
+
sub blog
{
- my ($self,$x,$base,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(2,@_);
-
- # http://www.efunda.com/math/taylor_series/logarithmic.cfm?search_string=log
-
- # u = x-1, v = x+1
- # _ _
- # Taylor: | u 1 u^3 1 u^5 |
- # ln (x) = 2 | --- + - * --- + - * --- + ... | x > 0
- # |_ v 3 v^3 5 v^5 _|
+ my ($self,$x,$base,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
- # This takes much more steps to calculate the result:
- # u = x-1
- # _ _
- # Taylor: | u 1 u^2 1 u^3 |
- # ln (x) = 2 | --- + - * --- + - * --- + ... | x > 1/2
- # |_ x 2 x^2 3 x^3 _|
+ # $base > 0, $base != 1; if $base == undef default to $base == e
+ # $x >= 0
# we need to limit the accuracy to protect against overflow
my $fallback = 0;
- my $scale = 0;
- my @params = $x->_find_round_parameters($a,$p,$r);
+ my ($scale,@params);
+ ($x,@params) = $x->_find_round_parameters($a,$p,$r);
+
+ # also takes care of the "error in _find_round_parameters?" case
+ return $x->bnan() if $x->{sign} ne '+' || $x->is_zero();
+
# no rounding at all, so must use fallback
- if (scalar @params == 1)
+ if (scalar @params == 0)
{
# simulate old behaviour
- $params[1] = $self->div_scale(); # and round to it as accuracy
- $scale = $params[1]+4; # at least four more for proper round
- $params[3] = $r; # round mode by caller or undef
+ $params[0] = $self->div_scale(); # and round to it as accuracy
+ $params[1] = undef; # P = undef
+ $scale = $params[0]+4; # at least four more for proper round
+ $params[2] = $r; # round mode by caller or undef
$fallback = 1; # to clear a/p afterwards
}
else
{
# the 4 below is empirical, and there might be cases where it is not
# enough...
- $scale = abs($params[1] || $params[2]) + 4; # take whatever is defined
+ $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
}
return $x->bzero(@params) if $x->is_one();
- return $x->bnan() if $x->{sign} ne '+' || $x->is_zero();
- #return $x->bone('+',@params) if $x->bcmp($base) == 0;
+ # base not defined => base == Euler's constant e
+ if (defined $base)
+ {
+ # make object, since we don't feed it through objectify() to still get the
+ # case of $base == undef
+ $base = $self->new($base) unless ref($base);
+ # $base > 0; $base != 1
+ return $x->bnan() if $base->is_zero() || $base->is_one() ||
+ $base->{sign} ne '+';
+ # if $x == $base, we know the result must be 1.0
+ if ($x->bcmp($base) == 0)
+ {
+ $x->bone('+',@params);
+ if ($fallback)
+ {
+ # clear a/p after round, since user did not request it
+ delete $x->{_a}; delete $x->{_p};
+ }
+ return $x;
+ }
+ }
# when user set globals, they would interfere with our calculation, so
- # disable then and later re-enable them
+ # disable them and later re-enable them
no strict 'refs';
my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
delete $x->{_a}; delete $x->{_p};
# need to disable $upgrade in BigInt, to avoid deep recursion
local $Math::BigInt::upgrade = undef;
+ local $Math::BigFloat::downgrade = undef;
+
+ # upgrade $x if $x is not a BigFloat (handle BigInt input)
+ if (!$x->isa('Math::BigFloat'))
+ {
+ $x = Math::BigFloat->new($x);
+ $self = ref($x);
+ }
+
+ my $done = 0;
+
+ # If the base is defined and an integer, try to calculate integer result
+ # first. This is very fast, and in case the real result was found, we can
+ # stop right here.
+ if (defined $base && $base->is_int() && $x->is_int())
+ {
+ my $i = $MBI->_copy( $x->{_m} );
+ $MBI->_lsft( $i, $x->{_e}, 10 ) unless $MBI->_is_zero($x->{_e});
+ my $int = Math::BigInt->bzero();
+ $int->{value} = $i;
+ $int->blog($base->as_number());
+ # if ($exact)
+ if ($base->as_number()->bpow($int) == $x)
+ {
+ # found result, return it
+ $x->{_m} = $int->{value};
+ $x->{_e} = $MBI->_zero();
+ $x->{_es} = '+';
+ $x->bnorm();
+ $done = 1;
+ }
+ }
+
+ if ($done == 0)
+ {
+ # first calculate the log to base e (using reduction by 10 (and probably 2))
+ $self->_log_10($x,$scale);
+
+ # and if a different base was requested, convert it
+ if (defined $base)
+ {
+ $base = Math::BigFloat->new($base) unless $base->isa('Math::BigFloat');
+ # not ln, but some other base (don't modify $base)
+ $x->bdiv( $base->copy()->blog(undef,$scale), $scale );
+ }
+ }
- my ($case,$limit,$v,$u,$below,$factor,$two,$next,$over,$f);
-
- if (3 < 5)
- #if ($x <= Math::BigFloat->new("0.5"))
- {
- $case = 0;
- # print "case $case $x < 0.5\n";
- $v = $x->copy(); $v->binc(); # v = x+1
- $x->bdec(); $u = $x->copy(); # u = x-1; x = x-1
- $x->bdiv($v,$scale); # first term: u/v
- $below = $v->copy();
- $over = $u->copy();
- $u *= $u; $v *= $v; # u^2, v^2
- $below->bmul($v); # u^3, v^3
- $over->bmul($u);
- $factor = $self->new(3); $f = $self->new(2);
- }
- #else
- # {
- # $case = 1;
- # print "case 1 $x > 0.5\n";
- # $v = $x->copy(); # v = x
- # $u = $x->copy(); $u->bdec(); # u = x-1;
- # $x->bdec(); $x->bdiv($v,$scale); # first term: x-1/x
- # $below = $v->copy();
- # $over = $u->copy();
- # $below->bmul($v); # u^2, v^2
- # $over->bmul($u);
- # $factor = $self->new(2); $f = $self->bone();
- # }
+ # shortcut to not run through _find_round_parameters again
+ if (defined $params[0])
+ {
+ $x->bround($params[0],$params[2]); # then round accordingly
+ }
+ else
+ {
+ $x->bfround($params[1],$params[2]); # then round accordingly
+ }
+ if ($fallback)
+ {
+ # clear a/p after round, since user did not request it
+ delete $x->{_a}; delete $x->{_p};
+ }
+ # restore globals
+ $$abr = $ab; $$pbr = $pb;
+
+ $x;
+ }
+
+sub _log
+ {
+ # internal log function to calculate ln() based on Taylor series.
+ # Modifies $x in place.
+ my ($self,$x,$scale) = @_;
+
+ # in case of $x == 1, result is 0
+ return $x->bzero() if $x->is_one();
+
+ # http://www.efunda.com/math/taylor_series/logarithmic.cfm?search_string=log
+
+ # u = x-1, v = x+1
+ # _ _
+ # Taylor: | u 1 u^3 1 u^5 |
+ # ln (x) = 2 | --- + - * --- + - * --- + ... | x > 0
+ # |_ v 3 v^3 5 v^5 _|
+
+ # This takes much more steps to calculate the result and is thus not used
+ # u = x-1
+ # _ _
+ # Taylor: | u 1 u^2 1 u^3 |
+ # ln (x) = 2 | --- + - * --- + - * --- + ... | x > 1/2
+ # |_ x 2 x^2 3 x^3 _|
+
+ my ($limit,$v,$u,$below,$factor,$two,$next,$over,$f);
+
+ $v = $x->copy(); $v->binc(); # v = x+1
+ $x->bdec(); $u = $x->copy(); # u = x-1; x = x-1
+ $x->bdiv($v,$scale); # first term: u/v
+ $below = $v->copy();
+ $over = $u->copy();
+ $u *= $u; $v *= $v; # u^2, v^2
+ $below->bmul($v); # u^3, v^3
+ $over->bmul($u);
+ $factor = $self->new(3); $f = $self->new(2);
+
+ my $steps = 0 if DEBUG;
$limit = $self->new("1E-". ($scale-1));
- #my $steps = 0;
while (3 < 5)
{
# we calculate the next term, and add it to the last
# when the next term is below our limit, it won't affect the outcome
# anymore, so we stop
- $next = $over->copy()->bdiv($below->copy()->bmul($factor),$scale);
- last if $next->bcmp($limit) <= 0;
+
+ # calculating the next term simple from over/below will result in quite
+ # a time hog if the input has many digits, since over and below will
+ # accumulate more and more digits, and the result will also have many
+ # digits, but in the end it is rounded to $scale digits anyway. So if we
+ # round $over and $below first, we save a lot of time for the division
+ # (not with log(1.2345), but try log (123**123) to see what I mean. This
+ # can introduce a rounding error if the division result would be f.i.
+ # 0.1234500000001 and we round it to 5 digits it would become 0.12346, but
+ # if we truncated $over and $below we might get 0.12345. Does this matter
+ # for the end result? So we give $over and $below 4 more digits to be
+ # on the safe side (unscientific error handling as usual... :+D
+
+ $next = $over->copy->bround($scale+4)->bdiv(
+ $below->copy->bmul($factor)->bround($scale+4),
+ $scale);
+
+## old version:
+## $next = $over->copy()->bdiv($below->copy()->bmul($factor),$scale);
+
+ last if $next->bacmp($limit) <= 0;
+
+ delete $next->{_a}; delete $next->{_p};
$x->badd($next);
- # print "step $steps $x\n";
# calculate things for the next term
$over *= $u; $below *= $v; $factor->badd($f);
- #$steps++;
+ if (DEBUG)
+ {
+ $steps++; print "step $steps = $x\n" if $steps % 10 == 0;
+ }
}
- $x->bmul(2) if $case == 0;
- #print "took $steps steps\n";
-
- # shortcut to not run trough _find_round_parameters again
- if (defined $params[1])
- {
- $x->bround($params[1],$params[3]); # then round accordingly
+ $x->bmul($f); # $x *= 2
+ print "took $steps steps\n" if DEBUG;
+ }
+
+sub _log_10
+ {
+ # Internal log function based on reducing input to the range of 0.1 .. 9.99
+ # and then "correcting" the result to the proper one. Modifies $x in place.
+ my ($self,$x,$scale) = @_;
+
+ # taking blog() from numbers greater than 10 takes a *very long* time, so we
+ # break the computation down into parts based on the observation that:
+ # blog(x*y) = blog(x) + blog(y)
+ # We set $y here to multiples of 10 so that $x is below 1 (the smaller $x is
+ # the faster it get's, especially because 2*$x takes about 10 times as long,
+ # so by dividing $x by 10 we make it at least factor 100 faster...)
+
+ # The same observation is valid for numbers smaller than 0.1 (e.g. computing
+ # log(1) is fastest, and the farther away we get from 1, the longer it takes)
+ # so we also 'break' this down by multiplying $x with 10 and subtract the
+ # log(10) afterwards to get the correct result.
+
+ # calculate nr of digits before dot
+ my $dbd = $MBI->_num($x->{_e});
+ $dbd = -$dbd if $x->{_es} eq '-';
+ $dbd += $MBI->_len($x->{_m});
+
+ # more than one digit (e.g. at least 10), but *not* exactly 10 to avoid
+ # infinite recursion
+
+ my $calc = 1; # do some calculation?
+
+ # disable the shortcut for 10, since we need log(10) and this would recurse
+ # infinitely deep
+ if ($x->{_es} eq '+' && $MBI->_is_one($x->{_e}) && $MBI->_is_one($x->{_m}))
+ {
+ $dbd = 0; # disable shortcut
+ # we can use the cached value in these cases
+ if ($scale <= $LOG_10_A)
+ {
+ $x->bzero(); $x->badd($LOG_10);
+ $calc = 0; # no need to calc, but round
+ }
}
else
{
- $x->bfround($params[2],$params[3]); # then round accordingly
+ # disable the shortcut for 2, since we maybe have it cached
+ if (($MBI->_is_zero($x->{_e}) && $MBI->_is_two($x->{_m})))
+ {
+ $dbd = 0; # disable shortcut
+ # we can use the cached value in these cases
+ if ($scale <= $LOG_2_A)
+ {
+ $x->bzero(); $x->badd($LOG_2);
+ $calc = 0; # no need to calc, but round
+ }
+ }
}
- if ($fallback)
+
+ # if $x = 0.1, we know the result must be 0-log(10)
+ if ($calc != 0 && $x->{_es} eq '-' && $MBI->_is_one($x->{_e}) &&
+ $MBI->_is_one($x->{_m}))
{
- # clear a/p after round, since user did not request it
- $x->{_a} = undef; $x->{_p} = undef;
+ $dbd = 0; # disable shortcut
+ # we can use the cached value in these cases
+ if ($scale <= $LOG_10_A)
+ {
+ $x->bzero(); $x->bsub($LOG_10);
+ $calc = 0; # no need to calc, but round
+ }
}
- # restore globals
- $$abr = $ab; $$pbr = $pb;
- $x;
+ return if $calc == 0; # already have the result
+
+ # default: these correction factors are undef and thus not used
+ my $l_10; # value of ln(10) to A of $scale
+ my $l_2; # value of ln(2) to A of $scale
+
+ # $x == 2 => 1, $x == 13 => 2, $x == 0.1 => 0, $x == 0.01 => -1
+ # so don't do this shortcut for 1 or 0
+ if (($dbd > 1) || ($dbd < 0))
+ {
+ # convert our cached value to an object if not already (avoid doing this
+ # at import() time, since not everybody needs this)
+ $LOG_10 = $self->new($LOG_10,undef,undef) unless ref $LOG_10;
+
+ #print "x = $x, dbd = $dbd, calc = $calc\n";
+ # got more than one digit before the dot, or more than one zero after the
+ # dot, so do:
+ # log(123) == log(1.23) + log(10) * 2
+ # log(0.0123) == log(1.23) - log(10) * 2
+
+ if ($scale <= $LOG_10_A)
+ {
+ # use cached value
+ $l_10 = $LOG_10->copy(); # copy for mul
+ }
+ else
+ {
+ # else: slower, compute it (but don't cache it, because it could be big)
+ # also disable downgrade for this code path
+ local $Math::BigFloat::downgrade = undef;
+ $l_10 = $self->new(10)->blog(undef,$scale); # scale+4, actually
+ }
+ $dbd-- if ($dbd > 1); # 20 => dbd=2, so make it dbd=1
+ $l_10->bmul( $self->new($dbd)); # log(10) * (digits_before_dot-1)
+ my $dbd_sign = '+';
+ if ($dbd < 0)
+ {
+ $dbd = -$dbd;
+ $dbd_sign = '-';
+ }
+ ($x->{_e}, $x->{_es}) =
+ _e_sub( $x->{_e}, $MBI->_new($dbd), $x->{_es}, $dbd_sign); # 123 => 1.23
+
+ }
+
+ # Now: 0.1 <= $x < 10 (and possible correction in l_10)
+
+ ### Since $x in the range 0.5 .. 1.5 is MUCH faster, we do a repeated div
+ ### or mul by 2 (maximum times 3, since x < 10 and x > 0.1)
+
+ $HALF = $self->new($HALF) unless ref($HALF);
+
+ my $twos = 0; # default: none (0 times)
+ my $two = $self->new(2);
+ while ($x->bacmp($HALF) <= 0)
+ {
+ $twos--; $x->bmul($two);
+ }
+ while ($x->bacmp($two) >= 0)
+ {
+ $twos++; $x->bdiv($two,$scale+4); # keep all digits
+ }
+ # $twos > 0 => did mul 2, < 0 => did div 2 (never both)
+ # calculate correction factor based on ln(2)
+ if ($twos != 0)
+ {
+ $LOG_2 = $self->new($LOG_2,undef,undef) unless ref $LOG_2;
+ if ($scale <= $LOG_2_A)
+ {
+ # use cached value
+ $l_2 = $LOG_2->copy(); # copy for mul
+ }
+ else
+ {
+ # else: slower, compute it (but don't cache it, because it could be big)
+ # also disable downgrade for this code path
+ local $Math::BigFloat::downgrade = undef;
+ $l_2 = $two->blog(undef,$scale); # scale+4, actually
+ }
+ $l_2->bmul($twos); # * -2 => subtract, * 2 => add
+ }
+
+ $self->_log($x,$scale); # need to do the "normal" way
+ $x->badd($l_10) if defined $l_10; # correct it by ln(10)
+ $x->badd($l_2) if defined $l_2; # and maybe by ln(2)
+ # all done, $x contains now the result
}
sub blcm
my ($self,@arg) = objectify(0,@_);
my $x = $self->new(shift @arg);
- while (@arg) { $x = _lcm($x,shift @arg); }
+ while (@arg) { $x = Math::BigInt::__lcm($x,shift @arg); }
$x;
}
-sub bgcd
- {
- # (BFLOAT or num_str, BFLOAT or num_str) return BINT
+sub bgcd
+ {
+ # (BINT or num_str, BINT or num_str) return BINT
# does not modify arguments, but returns new object
- # GCD -- Euclids algorithm Knuth Vol 2 pg 296
-
- my ($self,@arg) = objectify(0,@_);
- my $x = $self->new(shift @arg);
- while (@arg) { $x = _gcd($x,shift @arg); }
+
+ my $y = shift;
+ $y = __PACKAGE__->new($y) if !ref($y);
+ my $self = ref($y);
+ my $x = $y->copy()->babs(); # keep arguments
+
+ return $x->bnan() if $x->{sign} !~ /^[+-]$/ # x NaN?
+ || !$x->is_int(); # only for integers now
+
+ while (@_)
+ {
+ my $t = shift; $t = $self->new($t) if !ref($t);
+ $y = $t->copy()->babs();
+
+ return $x->bnan() if $y->{sign} !~ /^[+-]$/ # y NaN?
+ || !$y->is_int(); # only for integers now
+
+ # greatest common divisor
+ while (! $y->is_zero())
+ {
+ ($x,$y) = ($y->copy(), $x->copy()->bmod($y));
+ }
+
+ last if $x->is_one();
+ }
$x;
}
+##############################################################################
+
+sub _e_add
+ {
+ # Internal helper sub to take two positive integers and their signs and
+ # then add them. Input ($CALC,$CALC,('+'|'-'),('+'|'-')),
+ # output ($CALC,('+'|'-'))
+ my ($x,$y,$xs,$ys) = @_;
+
+ # if the signs are equal we can add them (-5 + -3 => -(5 + 3) => -8)
+ if ($xs eq $ys)
+ {
+ $x = $MBI->_add ($x, $y ); # a+b
+ # the sign follows $xs
+ return ($x, $xs);
+ }
+
+ my $a = $MBI->_acmp($x,$y);
+ if ($a > 0)
+ {
+ $x = $MBI->_sub ($x , $y); # abs sub
+ }
+ elsif ($a == 0)
+ {
+ $x = $MBI->_zero(); # result is 0
+ $xs = '+';
+ }
+ else # a < 0
+ {
+ $x = $MBI->_sub ( $y, $x, 1 ); # abs sub
+ $xs = $ys;
+ }
+ ($x,$xs);
+ }
+
+sub _e_sub
+ {
+ # Internal helper sub to take two positive integers and their signs and
+ # then subtract them. Input ($CALC,$CALC,('+'|'-'),('+'|'-')),
+ # output ($CALC,('+'|'-'))
+ my ($x,$y,$xs,$ys) = @_;
+
+ # flip sign
+ $ys =~ tr/+-/-+/;
+ _e_add($x,$y,$xs,$ys); # call add (does subtract now)
+ }
+
###############################################################################
# is_foo methods (is_negative, is_positive are inherited from BigInt)
sub is_int
{
# return true if arg (BFLOAT or num_str) is an integer
- my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
return 1 if ($x->{sign} =~ /^[+-]$/) && # NaN and +-inf aren't
- $x->{_e}->{sign} eq '+'; # 1e-1 => no integer
+ $x->{_es} eq '+'; # 1e-1 => no integer
0;
}
sub is_zero
{
# return true if arg (BFLOAT or num_str) is zero
- my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
- return 1 if $x->{sign} eq '+' && $x->{_m}->is_zero();
+ return 1 if $x->{sign} eq '+' && $MBI->_is_zero($x->{_m});
0;
}
sub is_one
{
# return true if arg (BFLOAT or num_str) is +1 or -1 if signis given
- my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+ my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
- my $sign = shift || ''; $sign = '+' if $sign ne '-';
+ $sign = '+' if !defined $sign || $sign ne '-';
return 1
- if ($x->{sign} eq $sign && $x->{_e}->is_zero() && $x->{_m}->is_one());
+ if ($x->{sign} eq $sign &&
+ $MBI->_is_zero($x->{_e}) && $MBI->_is_one($x->{_m}));
0;
}
sub is_odd
{
# return true if arg (BFLOAT or num_str) is odd or false if even
- my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
return 1 if ($x->{sign} =~ /^[+-]$/) && # NaN & +-inf aren't
- ($x->{_e}->is_zero() && $x->{_m}->is_odd());
+ ($MBI->_is_zero($x->{_e}) && $MBI->_is_odd($x->{_m}));
0;
}
sub is_even
{
# return true if arg (BINT or num_str) is even or false if odd
- my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
- return 1 if ($x->{_e}->{sign} eq '+' # 123.45 is never
- && $x->{_m}->is_even()); # but 1200 is
+ return 1 if ($x->{_es} eq '+' # 123.45 is never
+ && $MBI->_is_even($x->{_m})); # but 1200 is
0;
}
{
# multiply two numbers -- stolen from Knuth Vol 2 pg 233
# (BINT or num_str, BINT or num_str) return BINT
- my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+
+ # set up parameters
+ my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+ }
return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
}
# handle result = 0
return $x->bzero() if $x->is_zero() || $y->is_zero();
+
+ return $upgrade->bmul($x,$y,$a,$p,$r) if defined $upgrade &&
+ ((!$x->isa($self)) || (!$y->isa($self)));
# aEb * cEd = (a*c)E(b+d)
- $x->{_m}->bmul($y->{_m});
- $x->{_e}->badd($y->{_e});
+ $MBI->_mul($x->{_m},$y->{_m});
+ ($x->{_e}, $x->{_es}) = _e_add($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es});
+
# adjust sign:
$x->{sign} = $x->{sign} ne $y->{sign} ? '-' : '+';
return $x->bnorm()->round($a,$p,$r,$y);
{
# (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return
# (BFLOAT,BFLOAT) (quo,rem) or BFLOAT (only rem)
- my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+
+ # set up parameters
+ my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+ }
return $self->_div_inf($x,$y)
if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
# we need to limit the accuracy to protect against overflow
my $fallback = 0;
- my $scale = 0;
- my @params = $x->_find_round_parameters($a,$p,$r,$y);
+ my (@params,$scale);
+ ($x,@params) = $x->_find_round_parameters($a,$p,$r,$y);
+
+ return $x if $x->is_nan(); # error in _find_round_parameters?
# no rounding at all, so must use fallback
- if (scalar @params == 1)
+ if (scalar @params == 0)
{
# simulate old behaviour
- $params[1] = $self->div_scale(); # and round to it as accuracy
- $scale = $params[1]+4; # at least four more for proper round
- $params[3] = $r; # round mode by caller or undef
+ $params[0] = $self->div_scale(); # and round to it as accuracy
+ $scale = $params[0]+4; # at least four more for proper round
+ $params[2] = $r; # round mode by caller or undef
$fallback = 1; # to clear a/p afterwards
}
else
{
# the 4 below is empirical, and there might be cases where it is not
# enough...
- $scale = abs($params[1] || $params[2]) + 4; # take whatever is defined
+ $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
}
- my $lx = $x->{_m}->length(); my $ly = $y->{_m}->length();
+
+ my $rem; $rem = $self->bzero() if wantarray;
+
+ $y = $self->new($y) unless $y->isa('Math::BigFloat');
+
+ my $lx = $MBI->_len($x->{_m}); my $ly = $MBI->_len($y->{_m});
$scale = $lx if $lx > $scale;
$scale = $ly if $ly > $scale;
my $diff = $ly - $lx;
$scale += $diff if $diff > 0; # if lx << ly, but not if ly << lx!
-
- # make copy of $x in case of list context for later reminder calculation
- my $rem;
- if (wantarray && !$y->is_one())
- {
- $rem = $x->copy();
- }
- $x->{sign} = $x->{sign} ne $y->sign() ? '-' : '+';
+ # already handled inf/NaN/-inf above:
- # check for / +-1 ( +/- 1E0)
- if (!$y->is_one())
- {
- # promote BigInts and it's subclasses (except when already a BigFloat)
- $y = $self->new($y) unless $y->isa('Math::BigFloat');
+ # check that $y is not 1 nor -1 and cache the result:
+ my $y_not_one = !($MBI->_is_zero($y->{_e}) && $MBI->_is_one($y->{_m}));
- #print "bdiv $y ",ref($y),"\n";
- # need to disable $upgrade in BigInt, to avoid deep recursion
- local $Math::BigInt::upgrade = undef; # should be parent class vs MBI
+ # flipping the sign of $y will also flip the sign of $x for the special
+ # case of $x->bsub($x); so we can catch it below:
+ my $xsign = $x->{sign};
+ $y->{sign} =~ tr/+-/-+/;
- # calculate the result to $scale digits and then round it
- # a * 10 ** b / c * 10 ** d => a/c * 10 ** (b-d)
- $x->{_m}->blsft($scale,10);
- $x->{_m}->bdiv( $y->{_m} ); # a/c
- $x->{_e}->bsub( $y->{_e} ); # b-d
- $x->{_e}->bsub($scale); # correct for 10**scale
- $x->bnorm(); # remove trailing 0's
+ if ($xsign ne $x->{sign})
+ {
+ # special case of $x /= $x results in 1
+ $x->bone(); # "fixes" also sign of $y, since $x is $y
}
+ else
+ {
+ # correct $y's sign again
+ $y->{sign} =~ tr/+-/-+/;
+ # continue with normal div code:
+
+ # make copy of $x in case of list context for later reminder calculation
+ if (wantarray && $y_not_one)
+ {
+ $rem = $x->copy();
+ }
+
+ $x->{sign} = $x->{sign} ne $y->sign() ? '-' : '+';
+
+ # check for / +-1 ( +/- 1E0)
+ if ($y_not_one)
+ {
+ # promote BigInts and it's subclasses (except when already a BigFloat)
+ $y = $self->new($y) unless $y->isa('Math::BigFloat');
+
+ # calculate the result to $scale digits and then round it
+ # a * 10 ** b / c * 10 ** d => a/c * 10 ** (b-d)
+ $MBI->_lsft($x->{_m},$MBI->_new($scale),10);
+ $MBI->_div ($x->{_m},$y->{_m}); # a/c
+
+ # correct exponent of $x
+ ($x->{_e},$x->{_es}) = _e_sub($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es});
+ # correct for 10**scale
+ ($x->{_e},$x->{_es}) = _e_sub($x->{_e}, $MBI->_new($scale), $x->{_es}, '+');
+ $x->bnorm(); # remove trailing 0's
+ }
+ } # ende else $x != $y
- # shortcut to not run trough _find_round_parameters again
- if (defined $params[1])
+ # shortcut to not run through _find_round_parameters again
+ if (defined $params[0])
{
- $x->bround($params[1],$params[3]); # then round accordingly
+ delete $x->{_a}; # clear before round
+ $x->bround($params[0],$params[2]); # then round accordingly
}
else
{
- $x->bfround($params[2],$params[3]); # then round accordingly
+ delete $x->{_p}; # clear before round
+ $x->bfround($params[1],$params[2]); # then round accordingly
}
if ($fallback)
{
# clear a/p after round, since user did not request it
- $x->{_a} = undef; $x->{_p} = undef;
+ delete $x->{_a}; delete $x->{_p};
}
-
+
if (wantarray)
{
- if (!$y->is_one())
- {
- $rem->bmod($y,$params[1],$params[2],$params[3]); # copy already done
- }
- else
+ if ($y_not_one)
{
- $rem = $self->bzero();
+ $rem->bmod($y,@params); # copy already done
}
if ($fallback)
{
# clear a/p after round, since user did not request it
- $rem->{_a} = undef; $rem->{_p} = undef;
+ delete $rem->{_a}; delete $rem->{_p};
}
return ($x,$rem);
}
sub bmod
{
# (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return reminder
- my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+ # set up parameters
+ my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+ }
+
+ # handle NaN, inf, -inf
if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
{
my ($d,$re) = $self->SUPER::_div_inf($x,$y);
- return $re->round($a,$p,$r,$y);
+ $x->{sign} = $re->{sign};
+ $x->{_e} = $re->{_e};
+ $x->{_m} = $re->{_m};
+ return $x->round($a,$p,$r,$y);
}
- return $x->bnan() if $x->is_zero() && $y->is_zero();
- return $x if $y->is_zero();
- return $x->bnan() if $x->is_nan() || $y->is_nan();
- return $x->bzero() if $y->is_one() || $x->is_zero();
+ if ($y->is_zero())
+ {
+ return $x->bnan() if $x->is_zero();
+ return $x;
+ }
+
+ return $x->bzero() if $x->is_zero()
+ || ($x->is_int() &&
+ # check that $y == +1 or $y == -1:
+ ($MBI->_is_zero($y->{_e}) && $MBI->_is_one($y->{_m})));
- # inf handling is missing here
-
my $cmp = $x->bacmp($y); # equal or $x < $y?
return $x->bzero($a,$p) if $cmp == 0; # $x == $y => result 0
$x->{sign} = $y->{sign}; # calc sign first
return $x->round($a,$p,$r) if $cmp < 0 && $neg == 0; # $x < $y => result $x
- my $ym = $y->{_m}->copy();
+ my $ym = $MBI->_copy($y->{_m});
# 2e1 => 20
- $ym->blsft($y->{_e},10) if $y->{_e}->{sign} eq '+' && !$y->{_e}->is_zero();
+ $MBI->_lsft( $ym, $y->{_e}, 10)
+ if $y->{_es} eq '+' && !$MBI->_is_zero($y->{_e});
# if $y has digits after dot
my $shifty = 0; # correct _e of $x by this
- if ($y->{_e}->{sign} eq '-') # has digits after dot
+ if ($y->{_es} eq '-') # has digits after dot
{
# 123 % 2.5 => 1230 % 25 => 5 => 0.5
- $shifty = $y->{_e}->copy()->babs(); # no more digits after dot
- $x->blsft($shifty,10); # 123 => 1230, $y->{_m} is already 25
+ $shifty = $MBI->_num($y->{_e}); # no more digits after dot
+ $MBI->_lsft($x->{_m}, $y->{_e}, 10);# 123 => 1230, $y->{_m} is already 25
}
# $ym is now mantissa of $y based on exponent 0
my $shiftx = 0; # correct _e of $x by this
- if ($x->{_e}->{sign} eq '-') # has digits after dot
+ if ($x->{_es} eq '-') # has digits after dot
{
# 123.4 % 20 => 1234 % 200
- $shiftx = $x->{_e}->copy()->babs(); # no more digits after dot
- $ym->blsft($shiftx,10);
+ $shiftx = $MBI->_num($x->{_e}); # no more digits after dot
+ $MBI->_lsft($ym, $x->{_e}, 10); # 123 => 1230
}
# 123e1 % 20 => 1230 % 20
- if ($x->{_e}->{sign} eq '+' && !$x->{_e}->is_zero())
+ if ($x->{_es} eq '+' && !$MBI->_is_zero($x->{_e}))
{
- $x->{_m}->blsft($x->{_e},10);
+ $MBI->_lsft( $x->{_m}, $x->{_e},10); # es => '+' here
}
- $x->{_e} = $MBI->bzero() unless $x->{_e}->is_zero();
-
- $x->{_e}->bsub($shiftx) if $shiftx != 0;
- $x->{_e}->bsub($shifty) if $shifty != 0;
+
+ $x->{_e} = $MBI->_new($shiftx);
+ $x->{_es} = '+';
+ $x->{_es} = '-' if $shiftx != 0 || $shifty != 0;
+ $MBI->_add( $x->{_e}, $MBI->_new($shifty)) if $shifty != 0;
# now mantissas are equalized, exponent of $x is adjusted, so calc result
- $x->{_m}->bmod($ym);
+ $x->{_m} = $MBI->_mod( $x->{_m}, $ym);
- $x->{sign} = '+' if $x->{_m}->is_zero(); # fix sign for -0
+ $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # fix sign for -0
$x->bnorm();
if ($neg != 0) # one of them negative => correct in place
{
- my $r = $y - $x;
- $x->{_m} = $r->{_m};
- $x->{_e} = $r->{_e};
- $x->{sign} = '+' if $x->{_m}->is_zero(); # fix sign for -0
- $x->bnorm();
+ my $r = $y - $x;
+ $x->{_m} = $r->{_m};
+ $x->{_e} = $r->{_e};
+ $x->{_es} = $r->{_es};
+ $x->{sign} = '+' if $MBI->_is_zero($x->{_m}); # fix sign for -0
+ $x->bnorm();
+ }
+
+ $x->round($a,$p,$r,$y); # round and return
+ }
+
+sub broot
+ {
+ # calculate $y'th root of $x
+
+ # set up parameters
+ my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+ }
+
+ # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
+ return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
+ $y->{sign} !~ /^\+$/;
+
+ return $x if $x->is_zero() || $x->is_one() || $x->is_inf() || $y->is_one();
+
+ # we need to limit the accuracy to protect against overflow
+ my $fallback = 0;
+ my (@params,$scale);
+ ($x,@params) = $x->_find_round_parameters($a,$p,$r);
+
+ return $x if $x->is_nan(); # error in _find_round_parameters?
+
+ # no rounding at all, so must use fallback
+ if (scalar @params == 0)
+ {
+ # simulate old behaviour
+ $params[0] = $self->div_scale(); # and round to it as accuracy
+ $scale = $params[0]+4; # at least four more for proper round
+ $params[2] = $r; # iound mode by caller or undef
+ $fallback = 1; # to clear a/p afterwards
+ }
+ else
+ {
+ # the 4 below is empirical, and there might be cases where it is not
+ # enough...
+ $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
+ }
+
+ # when user set globals, they would interfere with our calculation, so
+ # disable them and later re-enable them
+ no strict 'refs';
+ my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
+ my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
+ # we also need to disable any set A or P on $x (_find_round_parameters took
+ # them already into account), since these would interfere, too
+ delete $x->{_a}; delete $x->{_p};
+ # need to disable $upgrade in BigInt, to avoid deep recursion
+ local $Math::BigInt::upgrade = undef; # should be really parent class vs MBI
+
+ # remember sign and make $x positive, since -4 ** (1/2) => -2
+ my $sign = 0; $sign = 1 if $x->{sign} eq '-'; $x->{sign} = '+';
+
+ my $is_two = 0;
+ if ($y->isa('Math::BigFloat'))
+ {
+ $is_two = ($y->{sign} eq '+' && $MBI->_is_two($y->{_m}) && $MBI->_is_zero($y->{_e}));
+ }
+ else
+ {
+ $is_two = ($y == 2);
+ }
+
+ # normal square root if $y == 2:
+ if ($is_two)
+ {
+ $x->bsqrt($scale+4);
+ }
+ elsif ($y->is_one('-'))
+ {
+ # $x ** -1 => 1/$x
+ my $u = $self->bone()->bdiv($x,$scale);
+ # copy private parts over
+ $x->{_m} = $u->{_m};
+ $x->{_e} = $u->{_e};
+ $x->{_es} = $u->{_es};
+ }
+ else
+ {
+ # calculate the broot() as integer result first, and if it fits, return
+ # it rightaway (but only if $x and $y are integer):
+
+ my $done = 0; # not yet
+ if ($y->is_int() && $x->is_int())
+ {
+ my $i = $MBI->_copy( $x->{_m} );
+ $MBI->_lsft( $i, $x->{_e}, 10 ) unless $MBI->_is_zero($x->{_e});
+ my $int = Math::BigInt->bzero();
+ $int->{value} = $i;
+ $int->broot($y->as_number());
+ # if ($exact)
+ if ($int->copy()->bpow($y) == $x)
+ {
+ # found result, return it
+ $x->{_m} = $int->{value};
+ $x->{_e} = $MBI->_zero();
+ $x->{_es} = '+';
+ $x->bnorm();
+ $done = 1;
+ }
+ }
+ if ($done == 0)
+ {
+ my $u = $self->bone()->bdiv($y,$scale+4);
+ delete $u->{_a}; delete $u->{_p}; # otherwise it conflicts
+ $x->bpow($u,$scale+4); # el cheapo
+ }
+ }
+ $x->bneg() if $sign == 1;
+
+ # shortcut to not run through _find_round_parameters again
+ if (defined $params[0])
+ {
+ $x->bround($params[0],$params[2]); # then round accordingly
+ }
+ else
+ {
+ $x->bfround($params[1],$params[2]); # then round accordingly
}
-
- $x->round($a,$p,$r,$y); # round and return
+ if ($fallback)
+ {
+ # clear a/p after round, since user did not request it
+ delete $x->{_a}; delete $x->{_p};
+ }
+ # restore globals
+ $$abr = $ab; $$pbr = $pb;
+ $x;
}
sub bsqrt
{
- # calculate square root; this should probably
- # use a different test to see whether the accuracy we want is...
+ # calculate square root
my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
- return $x->bnan() if $x->{sign} eq 'NaN' || $x->{sign} =~ /^-/; # <0, NaN
- return $x if $x->{sign} eq '+inf'; # +inf
- return $x if $x->is_zero() || $x->is_one();
+ return $x->bnan() if $x->{sign} !~ /^[+]/; # NaN, -inf or < 0
+ return $x if $x->{sign} eq '+inf'; # sqrt(inf) == inf
+ return $x->round($a,$p,$r) if $x->is_zero() || $x->is_one();
# we need to limit the accuracy to protect against overflow
my $fallback = 0;
- my $scale = 0;
- my @params = $x->_find_round_parameters($a,$p,$r);
+ my (@params,$scale);
+ ($x,@params) = $x->_find_round_parameters($a,$p,$r);
+
+ return $x if $x->is_nan(); # error in _find_round_parameters?
# no rounding at all, so must use fallback
- if (scalar @params == 1)
+ if (scalar @params == 0)
{
# simulate old behaviour
- $params[1] = $self->div_scale(); # and round to it as accuracy
- $scale = $params[1]+4; # at least four more for proper round
- $params[3] = $r; # round mode by caller or undef
+ $params[0] = $self->div_scale(); # and round to it as accuracy
+ $scale = $params[0]+4; # at least four more for proper round
+ $params[2] = $r; # round mode by caller or undef
$fallback = 1; # to clear a/p afterwards
}
else
{
# the 4 below is empirical, and there might be cases where it is not
# enough...
- $scale = abs($params[1] || $params[2]) + 4; # take whatever is defined
+ $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
}
# when user set globals, they would interfere with our calculation, so
# need to disable $upgrade in BigInt, to avoid deep recursion
local $Math::BigInt::upgrade = undef; # should be really parent class vs MBI
- my $xas = $x->as_number();
+ my $i = $MBI->_copy( $x->{_m} );
+ $MBI->_lsft( $i, $x->{_e}, 10 ) unless $MBI->_is_zero($x->{_e});
+ my $xas = Math::BigInt->bzero();
+ $xas->{value} = $i;
+
my $gs = $xas->copy()->bsqrt(); # some guess
-# print "guess $gs\n";
- if (($x->{_e}->{sign} ne '-') # guess can't be accurate if there are
+ if (($x->{_es} ne '-') # guess can't be accurate if there are
# digits after the dot
&& ($xas->bacmp($gs * $gs) == 0)) # guess hit the nail on the head?
{
- # exact result
- $x->{_m} = $gs; $x->{_e} = $MBI->bzero(); $x->bnorm();
- # shortcut to not run trough _find_round_parameters again
- if (defined $params[1])
+ # exact result, copy result over to keep $x
+ $x->{_m} = $gs->{value}; $x->{_e} = $MBI->_zero(); $x->{_es} = '+';
+ $x->bnorm();
+ # shortcut to not run through _find_round_parameters again
+ if (defined $params[0])
{
- $x->bround($params[1],$params[3]); # then round accordingly
+ $x->bround($params[0],$params[2]); # then round accordingly
}
else
{
- $x->bfround($params[2],$params[3]); # then round accordingly
+ $x->bfround($params[1],$params[2]); # then round accordingly
}
if ($fallback)
{
# clear a/p after round, since user did not request it
- $x->{_a} = undef; $x->{_p} = undef;
+ delete $x->{_a}; delete $x->{_p};
}
# re-enable A and P, upgrade is taken care of by "local"
${"$self\::accuracy"} = $ab; ${"$self\::precision"} = $pb;
return $x;
}
- $gs = $self->new( $gs ); # BigInt to BigFloat
-
- my $lx = $x->{_m}->length();
- $scale = $lx if $scale < $lx;
- my $e = $self->new("1E-$scale"); # make test variable
+
+ # sqrt(2) = 1.4 because sqrt(2*100) = 1.4*10; so we can increase the accuracy
+ # of the result by multipyling the input by 100 and then divide the integer
+ # result of sqrt(input) by 10. Rounding afterwards returns the real result.
- my $y = $x->copy();
- my $two = $self->new(2);
- my $diff = $e;
- # promote BigInts and it's subclasses (except when already a BigFloat)
- $y = $self->new($y) unless $y->isa('Math::BigFloat');
+ # The following steps will transform 123.456 (in $x) into 123456 (in $y1)
+ my $y1 = $MBI->_copy($x->{_m});
- my $rem;
- while ($diff->bacmp($e) >= 0)
- {
- $rem = $y->copy()->bdiv($gs,$scale);
- $rem = $y->copy()->bdiv($gs,$scale)->badd($gs)->bdiv($two,$scale);
- $diff = $rem->copy()->bsub($gs);
- $gs = $rem->copy();
- }
- # copy over to modify $x
- $x->{_m} = $rem->{_m}; $x->{_e} = $rem->{_e};
+ my $length = $MBI->_len($y1);
- # shortcut to not run trough _find_round_parameters again
- if (defined $params[1])
- {
- $x->bround($params[1],$params[3]); # then round accordingly
- }
- else
- {
- $x->bfround($params[2],$params[3]); # then round accordingly
- }
- if ($fallback)
- {
- # clear a/p after round, since user did not request it
- $x->{_a} = undef; $x->{_p} = undef;
- }
- # restore globals
- $$abr = $ab; $$pbr = $pb;
- $x;
- }
+ # Now calculate how many digits the result of sqrt(y1) would have
+ my $digits = int($length / 2);
-sub bfac
- {
- # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT
- # compute factorial numbers
- # modifies first argument
- my ($self,$x,@r) = objectify(1,@_);
+ # But we need at least $scale digits, so calculate how many are missing
+ my $shift = $scale - $digits;
- return $x->bnan()
- if (($x->{sign} ne '+') || # inf, NaN, <0 etc => NaN
- ($x->{_e}->{sign} ne '+')); # digits after dot?
+ # That should never happen (we take care of integer guesses above)
+ # $shift = 0 if $shift < 0;
- return $x->bone(@r) if $x->is_zero() || $x->is_one(); # 0 or 1 => 1
-
- # use BigInt's bfac() for faster calc
- $x->{_m}->blsft($x->{_e},10); # un-norm m
- $x->{_e}->bzero(); # norm $x again
- $x->{_m}->bfac(); # factorial
- $x->bnorm()->round(@r);
- }
+ # Multiply in steps of 100, by shifting left two times the "missing" digits
+ my $s2 = $shift * 2;
-sub _pow2
- {
- # Calculate a power where $y is a non-integer, like 2 ** 0.5
- my ($x,$y,$a,$p,$r) = @_;
- my $self = ref($x);
-
- # we need to limit the accuracy to protect against overflow
- my $fallback = 0;
- my $scale = 0;
- my @params = $x->_find_round_parameters($a,$p,$r);
+ # We now make sure that $y1 has the same odd or even number of digits than
+ # $x had. So when _e of $x is odd, we must shift $y1 by one digit left,
+ # because we always must multiply by steps of 100 (sqrt(100) is 10) and not
+ # steps of 10. The length of $x does not count, since an even or odd number
+ # of digits before the dot is not changed by adding an even number of digits
+ # after the dot (the result is still odd or even digits long).
+ $s2++ if $MBI->_is_odd($x->{_e});
- # no rounding at all, so must use fallback
- if (scalar @params == 1)
+ $MBI->_lsft( $y1, $MBI->_new($s2), 10);
+
+ # now take the square root and truncate to integer
+ $y1 = $MBI->_sqrt($y1);
+
+ # By "shifting" $y1 right (by creating a negative _e) we calculate the final
+ # result, which is than later rounded to the desired scale.
+
+ # calculate how many zeros $x had after the '.' (or before it, depending
+ # on sign of $dat, the result should have half as many:
+ my $dat = $MBI->_num($x->{_e});
+ $dat = -$dat if $x->{_es} eq '-';
+ $dat += $length;
+
+ if ($dat > 0)
{
- # simulate old behaviour
- $params[1] = $self->div_scale(); # and round to it as accuracy
- $scale = $params[1]+4; # at least four more for proper round
- $params[3] = $r; # round mode by caller or undef
- $fallback = 1; # to clear a/p afterwards
+ # no zeros after the dot (e.g. 1.23, 0.49 etc)
+ # preserve half as many digits before the dot than the input had
+ # (but round this "up")
+ $dat = int(($dat+1)/2);
}
else
{
- # the 4 below is empirical, and there might be cases where it is not
- # enough...
- $scale = abs($params[1] || $params[2]) + 4; # take whatever is defined
- }
-
- # when user set globals, they would interfere with our calculation, so
- # disable then and later re-enable them
- no strict 'refs';
- my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
- my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
- # we also need to disable any set A or P on $x (_find_round_parameters took
- # them already into account), since these would interfere, too
- delete $x->{_a}; delete $x->{_p};
- # need to disable $upgrade in BigInt, to avoid deep recursion
- local $Math::BigInt::upgrade = undef;
-
- # split the second argument into its integer and fraction part
- # we calculate the result then from these two parts, like in
- # 2 ** 2.4 == (2 ** 2) * (2 ** 0.4)
- my $c = $self->new($y->as_number()); # integer part
- my $d = $y-$c; # fractional part
- my $xc = $x->copy(); # a temp. copy
-
- # now calculate binary fraction from the decimal fraction on the fly
- # f.i. 0.654:
- # 0.654 * 2 = 1.308 > 1 => 0.1 ( 1.308 - 1 = 0.308)
- # 0.308 * 2 = 0.616 < 1 => 0.10
- # 0.616 * 2 = 1.232 > 1 => 0.101 ( 1.232 - 1 = 0.232)
- # and so on...
- # The process stops when the result is exactly one, or when we have
- # enough accuracy
-
- # From the binary fraction we calculate the result as follows:
- # we assume the fraction ends in 1, and we remove this one first.
- # For each digit after the dot, assume 1 eq R and 0 eq XR, where R means
- # take square root and X multiply with the original X.
-
- my $i = 0;
- while ($i++ < 50)
- {
- $d->badd($d); # * 2
- last if $d->is_one(); # == 1
- $x->bsqrt(); # 0
- if ($d > 1)
- {
- $x->bsqrt(); $x->bmul($xc); $d->bdec(); # 1
- }
- print "at $x\n";
+ $dat = int(($dat)/2);
}
- # assume fraction ends in 1
- $x->bsqrt(); # 1
- if (!$c->is_one())
+ $dat -= $MBI->_len($y1);
+ if ($dat < 0)
{
- $x->bmul( $xc->bpow($c) );
+ $dat = abs($dat);
+ $x->{_e} = $MBI->_new( $dat );
+ $x->{_es} = '-';
}
- elsif (!$c->is_zero())
- {
- $x->bmul( $xc );
+ else
+ {
+ $x->{_e} = $MBI->_new( $dat );
+ $x->{_es} = '+';
}
- # done
+ $x->{_m} = $y1;
+ $x->bnorm();
- # shortcut to not run trough _find_round_parameters again
- if (defined $params[1])
+ # shortcut to not run through _find_round_parameters again
+ if (defined $params[0])
{
- $x->bround($params[1],$params[3]); # then round accordingly
+ $x->bround($params[0],$params[2]); # then round accordingly
}
else
{
- $x->bfround($params[2],$params[3]); # then round accordingly
+ $x->bfround($params[1],$params[2]); # then round accordingly
}
if ($fallback)
{
# clear a/p after round, since user did not request it
- $x->{_a} = undef; $x->{_p} = undef;
+ delete $x->{_a}; delete $x->{_p};
}
# restore globals
$$abr = $ab; $$pbr = $pb;
$x;
}
+sub bfac
+ {
+ # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT
+ # compute factorial number, modifies first argument
+
+ # set up parameters
+ my ($self,$x,@r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ ($self,$x,@r) = objectify(1,@_) if !ref($x);
+
+ return $x if $x->{sign} eq '+inf'; # inf => inf
+ return $x->bnan()
+ if (($x->{sign} ne '+') || # inf, NaN, <0 etc => NaN
+ ($x->{_es} ne '+')); # digits after dot?
+
+ # use BigInt's bfac() for faster calc
+ if (! $MBI->_is_zero($x->{_e}))
+ {
+ $MBI->_lsft($x->{_m}, $x->{_e},10); # change 12e1 to 120e0
+ $x->{_e} = $MBI->_zero(); # normalize
+ $x->{_es} = '+';
+ }
+ $MBI->_fac($x->{_m}); # calculate factorial
+ $x->bnorm()->round(@r); # norm again and round result
+ }
+
sub _pow
{
# Calculate a power where $y is a non-integer, like 2 ** 0.5
my $self = ref($x);
# if $y == 0.5, it is sqrt($x)
- return $x->bsqrt($a,$p,$r,$y) if $y->bcmp('0.5') == 0;
+ $HALF = $self->new($HALF) unless ref($HALF);
+ return $x->bsqrt($a,$p,$r,$y) if $y->bcmp($HALF) == 0;
+
+ # Using:
+ # a ** x == e ** (x * ln a)
# u = y * ln x
- # _ _
- # Taylor: | u u^2 u^3 |
- # x ** y = 1 + | --- + --- + * ----- + ... |
- # |_ 1 1*2 1*2*3 _|
+ # _ _
+ # Taylor: | u u^2 u^3 |
+ # x ** y = 1 + | --- + --- + ----- + ... |
+ # |_ 1 1*2 1*2*3 _|
# we need to limit the accuracy to protect against overflow
my $fallback = 0;
- my $scale = 0;
- my @params = $x->_find_round_parameters($a,$p,$r);
+ my ($scale,@params);
+ ($x,@params) = $x->_find_round_parameters($a,$p,$r);
+
+ return $x if $x->is_nan(); # error in _find_round_parameters?
# no rounding at all, so must use fallback
- if (scalar @params == 1)
+ if (scalar @params == 0)
{
# simulate old behaviour
- $params[1] = $self->div_scale(); # and round to it as accuracy
- $scale = $params[1]+4; # at least four more for proper round
- $params[3] = $r; # round mode by caller or undef
+ $params[0] = $self->div_scale(); # and round to it as accuracy
+ $params[1] = undef; # disable P
+ $scale = $params[0]+4; # at least four more for proper round
+ $params[2] = $r; # round mode by caller or undef
$fallback = 1; # to clear a/p afterwards
}
else
{
# the 4 below is empirical, and there might be cases where it is not
# enough...
- $scale = abs($params[1] || $params[2]) + 4; # take whatever is defined
+ $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
}
# when user set globals, they would interfere with our calculation, so
- # disable then and later re-enable them
+ # disable them and later re-enable them
no strict 'refs';
my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
my ($limit,$v,$u,$below,$factor,$next,$over);
- $u = $x->copy()->blog($scale)->bmul($y);
+ $u = $x->copy()->blog(undef,$scale)->bmul($y);
$v = $self->bone(); # 1
$factor = $self->new(2); # 2
$x->bone(); # first term: 1
$below = $v->copy();
$over = $u->copy();
-
+
$limit = $self->new("1E-". ($scale-1));
#my $steps = 0;
while (3 < 5)
# when the next term is below our limit, it won't affect the outcome
# anymore, so we stop
$next = $over->copy()->bdiv($below,$scale);
- last if $next->bcmp($limit) <= 0;
+ last if $next->bacmp($limit) <= 0;
$x->badd($next);
-# print "at $x\n";
# calculate things for the next term
$over *= $u; $below *= $factor; $factor->binc();
+
+ last if $x->{sign} !~ /^[-+]$/;
+
#$steps++;
}
- # shortcut to not run trough _find_round_parameters again
- if (defined $params[1])
+ # shortcut to not run through _find_round_parameters again
+ if (defined $params[0])
{
- $x->bround($params[1],$params[3]); # then round accordingly
+ $x->bround($params[0],$params[2]); # then round accordingly
}
else
{
- $x->bfround($params[2],$params[3]); # then round accordingly
+ $x->bfround($params[1],$params[2]); # then round accordingly
}
if ($fallback)
{
# clear a/p after round, since user did not request it
- $x->{_a} = undef; $x->{_p} = undef;
+ delete $x->{_a}; delete $x->{_p};
}
# restore globals
$$abr = $ab; $$pbr = $pb;
# compute power of two numbers, second arg is used as integer
# modifies first argument
- my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+ # set up parameters
+ my ($self,$x,$y,$a,$p,$r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+ }
- return $x if $x->{sign} =~ /^[+-]inf$/;
return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
- return $x->bone() if $y->is_zero();
+ return $x if $x->{sign} =~ /^[+-]inf$/;
+
+ # -2 ** -2 => NaN
+ return $x->bnan() if $x->{sign} eq '-' && $y->{sign} eq '-';
+
+ # cache the result of is_zero
+ my $y_is_zero = $y->is_zero();
+ return $x->bone() if $y_is_zero;
return $x if $x->is_one() || $y->is_one();
- return $x->_pow2($y,$a,$p,$r) if !$y->is_int(); # non-integer power
+ my $x_is_zero = $x->is_zero();
+ return $x->_pow($y,$a,$p,$r) if !$x_is_zero && !$y->is_int(); # non-integer power
+
+ my $y1 = $y->as_number()->{value}; # make MBI part
- my $y1 = $y->as_number(); # make bigint
# if ($x == -1)
- if ($x->{sign} eq '-' && $x->{_m}->is_one() && $x->{_e}->is_zero())
+ if ($x->{sign} eq '-' && $MBI->_is_one($x->{_m}) && $MBI->_is_zero($x->{_e}))
{
# if $x == -1 and odd/even y => +1/-1 because +-1 ^ (+-1) => +-1
- return $y1->is_odd() ? $x : $x->babs(1);
+ return $MBI->_is_odd($y1) ? $x : $x->babs(1);
}
- if ($x->is_zero())
+ if ($x_is_zero)
{
+ return $x->bone() if $y_is_zero;
return $x if $y->{sign} eq '+'; # 0**y => 0 (if not y <= 0)
- # 0 ** -y => 1 / (0 ** y) => / 0! (1 / 0 => +inf)
- $x->binf();
+ # 0 ** -y => 1 / (0 ** y) => 1 / 0! (1 / 0 => +inf)
+ return $x->binf();
}
+ my $new_sign = '+';
+ $new_sign = $MBI->_is_odd($y1) ? '-' : '+' if $x->{sign} ne '+';
+
# calculate $x->{_m} ** $y and $x->{_e} * $y separately (faster)
- $y1->babs();
- $x->{_m}->bpow($y1);
- $x->{_e}->bmul($y1);
- $x->{sign} = $nan if $x->{_m}->{sign} eq $nan || $x->{_e}->{sign} eq $nan;
+ $x->{_m} = $MBI->_pow( $x->{_m}, $y1);
+ $x->{_e} = $MBI->_mul ($x->{_e}, $y1);
+
+ $x->{sign} = $new_sign;
$x->bnorm();
if ($y->{sign} eq '-')
{
# modify $x in place!
- my $z = $x->copy(); $x->bzero()->binc();
+ my $z = $x->copy(); $x->bone();
return $x->bdiv($z,$a,$p,$r); # round in one go (might ignore y's A!)
}
$x->round($a,$p,$r,$y);
# expects and returns normalized numbers!
my $x = shift; my $self = ref($x) || $x; $x = $self->new(shift) if !ref($x);
- return $x if $x->modify('bfround');
-
- my ($scale,$mode) = $x->_scale_p($self->precision(),$self->round_mode(),@_);
- return $x if !defined $scale; # no-op
+ my ($scale,$mode) = $x->_scale_p(@_);
+ return $x if !defined $scale || $x->modify('bfround'); # no-op
# never round a 0, +-inf, NaN
if ($x->is_zero())
return $x;
}
return $x if $x->{sign} !~ /^[+-]$/;
- # print "MBF bfround $x to scale $scale mode $mode\n";
# don't round if x already has lower precision
return $x if (defined $x->{_p} && $x->{_p} < 0 && $scale < $x->{_p});
$x->{_p} = $scale; # remember round in any case
- $x->{_a} = undef; # and clear A
+ delete $x->{_a}; # and clear A
if ($scale < 0)
{
- # print "bfround scale $scale e $x->{_e}\n";
# round right from the '.'
- return $x if $x->{_e} >= 0; # nothing to round
+
+ return $x if $x->{_es} eq '+'; # e >= 0 => nothing to round
+
$scale = -$scale; # positive for simplicity
- my $len = $x->{_m}->length(); # length of mantissa
- my $dad = -$x->{_e}; # digits after dot
- my $zad = 0; # zeros after dot
- $zad = -$len-$x->{_e} if ($x->{_e} < -$len);# for 0.00..00xxx style
- #print "scale $scale dad $dad zad $zad len $len\n";
+ my $len = $MBI->_len($x->{_m}); # length of mantissa
+ # the following poses a restriction on _e, but if _e is bigger than a
+ # scalar, you got other problems (memory etc) anyway
+ my $dad = -(0+ ($x->{_es}.$MBI->_num($x->{_e}))); # digits after dot
+ my $zad = 0; # zeros after dot
+ $zad = $dad - $len if (-$dad < -$len); # for 0.00..00xxx style
+
+ # p rint "scale $scale dad $dad zad $zad len $len\n";
# number bsstr len zad dad
# 0.123 123e-3 3 0 3
# 0.0123 123e-4 3 1 4
$scale = $dbd+$scale;
}
}
- # print "round to $x->{_m} to $scale\n";
}
else
{
+ # round left from the '.'
+
# 123 => 100 means length(123) = 3 - $scale (2) => 1
- my $dbt = $x->{_m}->length();
+ my $dbt = $MBI->_len($x->{_m});
# digits before dot
- my $dbd = $dbt + $x->{_e};
+ my $dbd = $dbt + ($x->{_es} . $MBI->_num($x->{_e}));
# should be the same, so treat it as this
$scale = 1 if $scale == 0;
# shortcut if already integer
{
$scale = $dbd - $scale;
}
-
}
- # print "using $scale for $x->{_m} with '$mode'\n";
# pass sign to bround for rounding modes '+inf' and '-inf'
- $x->{_m}->{sign} = $x->{sign};
- $x->{_m}->bround($scale,$mode);
- $x->{_m}->{sign} = '+'; # fix sign back
+ my $m = bless { sign => $x->{sign}, value => $x->{_m} }, 'Math::BigInt';
+ $m->bround($scale,$mode);
+ $x->{_m} = $m->{value}; # get our mantissa back
$x->bnorm();
}
{
# accuracy: preserve $N digits, and overwrite the rest with 0's
my $x = shift; my $self = ref($x) || $x; $x = $self->new(shift) if !ref($x);
-
- die ('bround() needs positive accuracy') if ($_[0] || 0) < 0;
- my ($scale,$mode) = $x->_scale_a($self->accuracy(),$self->round_mode(),@_);
- return $x if !defined $scale; # no-op
+ if (($_[0] || 0) < 0)
+ {
+ require Carp; Carp::croak ('bround() needs positive accuracy');
+ }
- return $x if $x->modify('bround');
+ my ($scale,$mode) = $x->_scale_a(@_);
+ return $x if !defined $scale || $x->modify('bround'); # no-op
# scale is now either $x->{_a}, $accuracy, or the user parameter
# test whether $x already has lower accuracy, do nothing in this case
# but do round if the accuracy is the same, since a math operation might
# want to round a number with A=5 to 5 digits afterwards again
- return $x if defined $_[0] && defined $x->{_a} && $x->{_a} < $_[0];
+ return $x if defined $x->{_a} && $x->{_a} < $scale;
# scale < 0 makes no sense
+ # scale == 0 => keep all digits
# never round a +-inf, NaN
- return $x if ($scale < 0) || $x->{sign} !~ /^[+-]$/;
+ return $x if ($scale <= 0) || $x->{sign} !~ /^[+-]$/;
- # 1: $scale == 0 => keep all digits
- # 2: never round a 0
- # 3: if we should keep more digits than the mantissa has, do nothing
- if ($scale == 0 || $x->is_zero() || $x->{_m}->length() <= $scale)
+ # 1: never round a 0
+ # 2: if we should keep more digits than the mantissa has, do nothing
+ if ($x->is_zero() || $MBI->_len($x->{_m}) <= $scale)
{
$x->{_a} = $scale if !defined $x->{_a} || $x->{_a} > $scale;
return $x;
}
# pass sign to bround for '+inf' and '-inf' rounding modes
- $x->{_m}->{sign} = $x->{sign};
- $x->{_m}->bround($scale,$mode); # round mantissa
- $x->{_m}->{sign} = '+'; # fix sign back
- # $x->{_m}->{_a} = undef; $x->{_m}->{_p} = undef;
+ my $m = bless { sign => $x->{sign}, value => $x->{_m} }, 'Math::BigInt';
+
+ $m->bround($scale,$mode); # round mantissa
+ $x->{_m} = $m->{value}; # get our mantissa back
$x->{_a} = $scale; # remember rounding
- $x->{_p} = undef; # and clear P
+ delete $x->{_p}; # and clear P
$x->bnorm(); # del trailing zeros gen. by bround()
}
return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
# if $x has digits after dot
- if ($x->{_e}->{sign} eq '-')
+ if ($x->{_es} eq '-')
{
- #$x->{_m}->brsft(-$x->{_e},10);
- #$x->{_e}->bzero();
- #$x-- if $x->{sign} eq '-';
-
- $x->{_e}->{sign} = '+'; # negate e
- $x->{_m}->brsft($x->{_e},10); # cut off digits after dot
- $x->{_e}->bzero(); # trunc/norm
- $x->{_m}->binc() if $x->{sign} eq '-'; # decrement if negative
+ $x->{_m} = $MBI->_rsft($x->{_m},$x->{_e},10); # cut off digits after dot
+ $x->{_e} = $MBI->_zero(); # trunc/norm
+ $x->{_es} = '+'; # abs e
+ $MBI->_inc($x->{_m}) if $x->{sign} eq '-'; # increment if negative
}
$x->round($a,$p,$r);
}
return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
# if $x has digits after dot
- if ($x->{_e}->{sign} eq '-')
+ if ($x->{_es} eq '-')
{
- #$x->{_m}->brsft(-$x->{_e},10);
- #$x->{_e}->bzero();
- #$x++ if $x->{sign} eq '+';
-
- $x->{_e}->{sign} = '+'; # negate e
- $x->{_m}->brsft($x->{_e},10); # cut off digits after dot
- $x->{_e}->bzero(); # trunc/norm
- $x->{_m}->binc() if $x->{sign} eq '+'; # decrement if negative
+ $x->{_m} = $MBI->_rsft($x->{_m},$x->{_e},10); # cut off digits after dot
+ $x->{_e} = $MBI->_zero(); # trunc/norm
+ $x->{_es} = '+'; # abs e
+ $MBI->_inc($x->{_m}) if $x->{sign} eq '+'; # increment if positive
}
$x->round($a,$p,$r);
}
sub brsft
{
- # shift right by $y (divide by power of 2)
- my ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_);
+ # shift right by $y (divide by power of $n)
+
+ # set up parameters
+ my ($self,$x,$y,$n,$a,$p,$r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_);
+ }
return $x if $x->modify('brsft');
return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
- $n = 2 if !defined $n; $n = Math::BigFloat->new($n);
- $x->bdiv($n ** $y,$a,$p,$r,$y);
+ $n = 2 if !defined $n; $n = $self->new($n);
+ $x->bdiv($n->bpow($y),$a,$p,$r,$y);
}
sub blsft
{
- # shift right by $y (divide by power of 2)
- my ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_);
+ # shift left by $y (multiply by power of $n)
+
+ # set up parameters
+ my ($self,$x,$y,$n,$a,$p,$r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,$n,$a,$p,$r) = objectify(2,@_);
+ }
- return $x if $x->modify('brsft');
+ return $x if $x->modify('blsft');
return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
- $n = 2 if !defined $n; $n = Math::BigFloat->new($n);
- $x->bmul($n ** $y,$a,$p,$r,$y);
+ $n = 2 if !defined $n; $n = $self->new($n);
+ $x->bmul($n->bpow($y),$a,$p,$r,$y);
}
###############################################################################
sub DESTROY
{
- # going through AUTOLOAD for every DESTROY is costly, so avoid it by empty sub
+ # going through AUTOLOAD for every DESTROY is costly, avoid it by empty sub
}
sub AUTOLOAD
# or falling back to MBI::bxxx()
my $name = $AUTOLOAD;
- $name =~ s/.*:://; # split package
+ $name =~ s/(.*):://; # split package
+ my $c = $1 || $class;
no strict 'refs';
+ $c->import() if $IMPORT == 0;
if (!method_alias($name))
{
if (!defined $name)
{
# delayed load of Carp and avoid recursion
require Carp;
- Carp::croak ("Can't call a method without name");
+ Carp::croak ("$c: Can't call a method without name");
}
if (!method_hand_up($name))
{
# delayed load of Carp and avoid recursion
require Carp;
- Carp::croak ("Can't call $class\-\>$name, not a valid method");
+ Carp::croak ("Can't call $c\-\>$name, not a valid method");
}
# try one level up, but subst. bxxx() for fxxx() since MBI only got bxxx()
$name =~ s/^f/b/;
- return &{"$MBI"."::$name"}(@_);
+ return &{"Math::BigInt"."::$name"}(@_);
}
my $bname = $name; $bname =~ s/^f/b/;
- *{$class."::$name"} = \&$bname;
- &$bname; # uses @_
+ $c .= "::$name";
+ *{$c} = \&{$bname};
+ &{$c}; # uses @_
}
sub exponent
if ($x->{sign} !~ /^[+-]$/)
{
my $s = $x->{sign}; $s =~ s/^[+-]//;
- return $self->new($s); # -inf, +inf => +inf
+ return Math::BigInt->new($s); # -inf, +inf => +inf
}
- return $x->{_e}->copy();
+ Math::BigInt->new( $x->{_es} . $MBI->_str($x->{_e}));
}
sub mantissa
if ($x->{sign} !~ /^[+-]$/)
{
my $s = $x->{sign}; $s =~ s/^[+]//;
- return $self->new($s); # -inf, +inf => +inf
+ return Math::BigInt->new($s); # -inf, +inf => +inf
}
- my $m = $x->{_m}->copy(); # faster than going via bstr()
+ my $m = Math::BigInt->new( $MBI->_str($x->{_m}));
$m->bneg() if $x->{sign} eq '-';
$m;
my $s = $x->{sign}; $s =~ s/^[+]//; my $se = $s; $se =~ s/^[-]//;
return ($self->new($s),$self->new($se)); # +inf => inf and -inf,+inf => inf
}
- my $m = $x->{_m}->copy(); # faster than going via bstr()
+ my $m = Math::BigInt->bzero();
+ $m->{value} = $MBI->_copy($x->{_m});
$m->bneg() if $x->{sign} eq '-';
- return ($m,$x->{_e}->copy());
+ ($m, Math::BigInt->new( $x->{_es} . $MBI->_num($x->{_e}) ));
}
##############################################################################
sub import
{
my $self = shift;
- my $l = scalar @_; my $j = 0; my @a = @_;
- my $lib = '';
- for ( my $i = 0; $i < $l ; $i++, $j++)
+ my $l = scalar @_;
+ my $lib = ''; my @a;
+ $IMPORT=1;
+ for ( my $i = 0; $i < $l ; $i++)
{
if ( $_[$i] eq ':constant' )
{
- # this rest causes overlord er load to step in
- # print "overload @_\n";
+ # This causes overlord er load to step in. 'binary' and 'integer'
+ # are handled by BigInt.
overload::constant float => sub { $self->new(shift); };
- splice @a, $j, 1; $j--;
}
elsif ($_[$i] eq 'upgrade')
{
# this causes upgrading
$upgrade = $_[$i+1]; # or undef to disable
- my $s = 2; $s = 1 if @a-$j < 2; # avoid "can not modify non-existant..."
- splice @a, $j, $s; $j -= $s;
+ $i++;
}
elsif ($_[$i] eq 'downgrade')
{
# this causes downgrading
$downgrade = $_[$i+1]; # or undef to disable
- my $s = 2; $s = 1 if @a-$j < 2; # avoid "can not modify non-existant..."
- splice @a, $j, $s; $j -= $s;
+ $i++;
}
elsif ($_[$i] eq 'lib')
{
+ # alternative library
$lib = $_[$i+1] || ''; # default Calc
- my $s = 2; $s = 1 if @a-$j < 2; # avoid "can not modify non-existant..."
- splice @a, $j, $s; $j -= $s;
+ $i++;
}
elsif ($_[$i] eq 'with')
{
- $MBI = $_[$i+1] || 'Math::BigInt'; # default Math::BigInt
- my $s = 2; $s = 1 if @a-$j < 2; # avoid "can not modify non-existant..."
- splice @a, $j, $s; $j -= $s;
+ # alternative class for our private parts()
+ # XXX: no longer supported
+ # $MBI = $_[$i+1] || 'Math::BigInt';
+ $i++;
+ }
+ else
+ {
+ push @a, $_[$i];
}
}
- my @parts = split /::/, $MBI; # Math::BigInt => Math BigInt
- my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
- $file = File::Spec->catdir (@parts, $file);
+
+ $lib =~ tr/a-zA-Z0-9,://cd; # restrict to sane characters
# let use Math::BigInt lib => 'GMP'; use Math::BigFloat; still work
my $mbilib = eval { Math::BigInt->config()->{lib} };
- $lib .= ",$mbilib" if defined $mbilib;
- require $file;
- $MBI->import ( lib => $lib, 'objectify' );
+ if ((defined $mbilib) && ($MBI eq 'Math::BigInt::Calc'))
+ {
+ # MBI already loaded
+ Math::BigInt->import('lib',"$lib,$mbilib", 'objectify');
+ }
+ else
+ {
+ # MBI not loaded, or with ne "Math::BigInt::Calc"
+ $lib .= ",$mbilib" if defined $mbilib;
+ $lib =~ s/^,//; # don't leave empty
+
+ # replacement library can handle lib statement, but also could ignore it
+
+ # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
+ # used in the same script, or eval inside import(). So we require MBI:
+ require Math::BigInt;
+ Math::BigInt->import( lib => $lib, 'objectify' );
+ }
+ if ($@)
+ {
+ require Carp; Carp::croak ("Couldn't load $lib: $! $@");
+ }
+ # find out which one was actually loaded
+ $MBI = Math::BigInt->config()->{lib};
+ # register us with MBI to get notified of future lib changes
+ Math::BigInt::_register_callback( $self, sub { $MBI = $_[0]; } );
+
# any non :constant stuff is handled by our parent, Exporter
# even if @_ is empty, to give it a chance
$self->SUPER::import(@a); # for subclasses
{
# adjust m and e so that m is smallest possible
# round number according to accuracy and precision settings
- my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
return $x if $x->{sign} !~ /^[+-]$/; # inf, nan etc
-# if (!$x->{_m}->is_odd())
-# {
- my $zeros = $x->{_m}->_trailing_zeros(); # correct for trailing zeros
- if ($zeros != 0)
+ my $zeros = $MBI->_zeros($x->{_m}); # correct for trailing zeros
+ if ($zeros != 0)
+ {
+ my $z = $MBI->_new($zeros);
+ $x->{_m} = $MBI->_rsft ($x->{_m}, $z, 10);
+ if ($x->{_es} eq '-')
{
- $x->{_m}->brsft($zeros,10); $x->{_e}->badd($zeros);
+ if ($MBI->_acmp($x->{_e},$z) >= 0)
+ {
+ $x->{_e} = $MBI->_sub ($x->{_e}, $z);
+ $x->{_es} = '+' if $MBI->_is_zero($x->{_e});
+ }
+ else
+ {
+ $x->{_e} = $MBI->_sub ( $MBI->_copy($z), $x->{_e});
+ $x->{_es} = '+';
+ }
}
- # for something like 0Ey, set y to 1, and -0 => +0
- $x->{sign} = '+', $x->{_e}->bone() if $x->{_m}->is_zero();
-# }
- # this is to prevent automatically rounding when MBI's globals are set
- $x->{_m}->{_f} = MB_NEVER_ROUND;
- $x->{_e}->{_f} = MB_NEVER_ROUND;
- # 'forget' that mantissa was rounded via MBI::bround() in MBF's bfround()
- $x->{_m}->{_a} = undef; $x->{_e}->{_a} = undef;
- $x->{_m}->{_p} = undef; $x->{_e}->{_p} = undef;
+ else
+ {
+ $x->{_e} = $MBI->_add ($x->{_e}, $z);
+ }
+ }
+ else
+ {
+ # $x can only be 0Ey if there are no trailing zeros ('0' has 0 trailing
+ # zeros). So, for something like 0Ey, set y to 1, and -0 => +0
+ $x->{sign} = '+', $x->{_es} = '+', $x->{_e} = $MBI->_one()
+ if $MBI->_is_zero($x->{_m});
+ }
+
$x; # MBI bnorm is no-op, so dont call it
}
##############################################################################
-# internal calculation routines
+
+sub as_hex
+ {
+ # return number as hexadecimal string (only for integers defined)
+ my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+
+ return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
+ return '0x0' if $x->is_zero();
+
+ return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex!?
+
+ my $z = $MBI->_copy($x->{_m});
+ if (! $MBI->_is_zero($x->{_e})) # > 0
+ {
+ $MBI->_lsft( $z, $x->{_e},10);
+ }
+ $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z));
+ $z->as_hex();
+ }
+
+sub as_bin
+ {
+ # return number as binary digit string (only for integers defined)
+ my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+
+ return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
+ return '0b0' if $x->is_zero();
+
+ return $nan if $x->{_es} ne '+'; # how to do 1e-1 in hex!?
+
+ my $z = $MBI->_copy($x->{_m});
+ if (! $MBI->_is_zero($x->{_e})) # > 0
+ {
+ $MBI->_lsft( $z, $x->{_e},10);
+ }
+ $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z));
+ $z->as_bin();
+ }
sub as_number
{
# return copy as a bigint representation of this BigFloat number
my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
- my $z = $x->{_m}->copy();
- if ($x->{_e}->{sign} eq '-') # < 0
+ my $z = $MBI->_copy($x->{_m});
+ if ($x->{_es} eq '-') # < 0
{
- $x->{_e}->{sign} = '+'; # flip
- $z->brsft($x->{_e},10);
- $x->{_e}->{sign} = '-'; # flip back
+ $MBI->_rsft( $z, $x->{_e},10);
}
- elsif (!$x->{_e}->is_zero()) # > 0
+ elsif (! $MBI->_is_zero($x->{_e})) # > 0
{
- $z->blsft($x->{_e},10);
+ $MBI->_lsft( $z, $x->{_e},10);
}
- $z->{sign} = $x->{sign};
+ $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z));
$z;
}
my $class = ref($x) || $x;
$x = $class->new(shift) unless ref($x);
- return 1 if $x->{_m}->is_zero();
- my $len = $x->{_m}->length();
- $len += $x->{_e} if $x->{_e}->sign() eq '+';
+ return 1 if $MBI->_is_zero($x->{_m});
+
+ my $len = $MBI->_len($x->{_m});
+ $len += $MBI->_num($x->{_e}) if $x->{_es} eq '+';
if (wantarray())
{
- my $t = $MBI->bzero();
- $t = $x->{_e}->copy()->babs() if $x->{_e}->sign() eq '-';
- return ($len,$t);
+ my $t = 0;
+ $t = $MBI->_num($x->{_e}) if $x->{_es} eq '-';
+ return ($len, $t);
}
$len;
}
$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_pos(); # true if >= 0
+ $x->is_neg(); # true if < 0
$x->is_inf(sign); # true if +inf, or -inf (default is '+')
$x->bcmp($y); # compare numbers (undef,<0,=0,>0)
$x->digit($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:
-
+ # The following all modify their first argument. If you want to preserve
+ # $x, use $z = $x->copy()->bXXX($y); See under L<CAVEATS> for why this is
+ # necessary when mixing $a = $b assignments with non-overloaded math.
+
# set
$x->bzero(); # set $i to 0
$x->bnan(); # set $i to NaN
$x->badd($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 $i to quotient
+ $x->bdiv($y); # divide, set $x to quotient
# return (quo,rem) or quo if scalar
- $x->bmod($y); # modulus
- $x->bpow($y); # power of arguments (a**b)
+ $x->bmod($y); # modulus ($x % $y)
+ $x->bpow($y); # power of arguments ($x ** $y)
$x->blsft($y); # left shift
$x->brsft($y); # right shift
# return (quo,rem) or quo if scalar
- $x->blog($base); # logarithm of $x, base defaults to e
- # (other bases than e not supported yet)
+ $x->blog(); # logarithm of $x to base e (Euler's number)
+ $x->blog($base); # logarithm of $x to base $base (f.i. 2)
$x->band($y); # bit-wise and
$x->bior($y); # bit-wise inclusive or
$x->bnot(); # bit-wise not (two's complement)
$x->bsqrt(); # calculate square-root
+ $x->broot($y); # $y'th root of $x (e.g. $y == 3 => cubic root)
$x->bfac(); # factorial of $x (1*2*3*4*..$x)
- $x->bround($N); # accuracy: preserver $N digits
+ $x->bround($N); # accuracy: preserve $N digits
$x->bfround($N); # precision: round to the $Nth digit
+ $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
blcm(@values); # lowest common multiplicator
$x->bstr(); # return string
$x->bsstr(); # return string in scientific notation
-
- $x->bfloor(); # return integer less or equal than $x
- $x->bceil(); # return integer greater or equal than $x
-
+
+ $x->as_int(); # return $x as BigInt
$x->exponent(); # return exponent as BigInt
$x->mantissa(); # return mantissa as BigInt
$x->parts(); # return (mantissa,exponent) as BigInt
$x->length(); # number of digits (w/o sign and '.')
($l,$f) = $x->length(); # number of digits, and length of fraction
+ $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
+
+ # these get/set the appropriate global value for all BigFloat objects
+ Math::BigFloat->precision(); # Precision
+ Math::BigFloat->accuracy(); # Accuracy
+ Math::BigFloat->round_mode(); # rounding mode
+
=head1 DESCRIPTION
-All operators (inlcuding basic math operations) are overloaded if you
+All operators (including basic math operations) are overloaded if you
declare your big floating point numbers as
$i = new Math::BigFloat '12_3.456_789_123_456_789E-2';
=back
-all with optional leading and trailing zeros and/or spaces. Additonally,
+all with optional leading and trailing zeros and/or spaces. Additionally,
numbers are allowed to have an underscore between any two digits.
Empty strings as well as other illegal numbers results in 'NaN'.
The string output will always have leading and trailing zeros stripped and drop
a plus sign. C<bstr()> will give you always the form with a decimal point,
-while C<bsstr()> (for scientific) gives you the scientific notation.
+while C<bsstr()> (s for scientific) gives you the scientific notation.
Input bstr() bsstr()
'-0' '0' '0E1'
C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
return either undef, <0, 0 or >0 and are suited for sort.
-Actual math is done by using BigInts to represent the mantissa and exponent.
+Actual math is done by using the class defined with C<with => Class;> (which
+defaults to BigInts) to represent the mantissa and exponent.
+
The sign C</^[+-]$/> is stored separately. The string 'NaN' is used to
represent the result when input arguments are not numbers, as well as
the result of dividing by zero.
See also: L<Rounding|Rounding>.
-Math::BigFloat supports both precision and accuracy. For a full documentation,
-examples and tips on these topics please see the large section in
-L<Math::BigInt>.
+Math::BigFloat supports both precision (rounding to a certain place before or
+after the dot) and accuracy (rounding to a certain number of digits). For a
+full documentation, examples and tips on these topics please see the large
+section about rounding in L<Math::BigInt>.
+
+Since things like C<sqrt(2)> or C<1 / 3> must presented with a limited
+accuracy lest a operation consumes all resources, each operation produces
+no more than the requested number of digits.
+
+If there is no gloabl precision or accuracy set, B<and> the operation in
+question was not called with a requested precision or accuracy, B<and> the
+input $x has no accuracy or precision set, then a fallback parameter will
+be used. For historical reasons, it is called C<div_scale> and can be accessed
+via:
+
+ $d = Math::BigFloat->div_scale(); # query
+ Math::BigFloat->div_scale($n); # set to $n digits
-Since things like sqrt(2) or 1/3 must presented with a limited precision lest
-a operation consumes all resources, each operation produces no more than
-C<Math::BigFloat::precision()> digits.
+The default value for C<div_scale> is 40.
-In case the result of one operation has more precision than specified,
+In case the result of one operation has more digits than specified,
it is rounded. The rounding mode taken is either the default mode, or the one
supplied to the operation after the I<scale>:
$x = Math::BigFloat->new(2);
- Math::BigFloat::precision(5); # 5 digits max
- $y = $x->copy()->bdiv(3); # will give 0.66666
- $y = $x->copy()->bdiv(3,6); # will give 0.666666
- $y = $x->copy()->bdiv(3,6,'odd'); # will give 0.666667
- Math::BigFloat::round_mode('zero');
- $y = $x->copy()->bdiv(3,6); # will give 0.666666
+ Math::BigFloat->accuracy(5); # 5 digits max
+ $y = $x->copy()->bdiv(3); # will give 0.66667
+ $y = $x->copy()->bdiv(3,6); # will give 0.666667
+ $y = $x->copy()->bdiv(3,6,undef,'odd'); # will give 0.666667
+ Math::BigFloat->round_mode('zero');
+ $y = $x->copy()->bdiv(3,6); # will also give 0.666667
+
+Note that C<< Math::BigFloat->accuracy() >> and C<< Math::BigFloat->precision() >>
+set the global variables, and thus B<any> newly created number will be subject
+to the global rounding B<immediately>. This means that in the examples above, the
+C<3> as argument to C<bdiv()> will also get an accuracy of B<5>.
+
+It is less confusing to either calculate the result fully, and afterwards
+round it explicitly, or use the additional parameters to the math
+functions like so:
+
+ use Math::BigFloat;
+ $x = Math::BigFloat->new(2);
+ $y = $x->copy()->bdiv(3);
+ print $y->bround(5),"\n"; # will give 0.66667
+
+ or
+
+ use Math::BigFloat;
+ $x = Math::BigFloat->new(2);
+ $y = $x->copy()->bdiv(3,5); # will give 0.66667
+ print "$y\n";
=head2 Rounding
=item fround ( -$scale ) and fround ( 0 )
-These are effetively no-ops.
+These are effectively no-ops.
=back
the following: 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'.
The default rounding mode is 'even'. By using
-C<< Math::BigFloat::round_mode($round_mode); >> you can get and set the default
+C<< Math::BigFloat->round_mode($round_mode); >> you can get and set the default
mode for subsequent rounding. The usage of C<$Math::BigFloat::$round_mode> is
no longer supported.
The second parameter to the round functions then overrides the default
temporarily.
-The C<< as_number() >> function returns a BigInt from a Math::BigFloat. It uses
+The C<as_number()> function returns a BigInt from a Math::BigFloat. It uses
'trunc' as rounding mode to make it equivalent to:
$x = 2.5;
$x = Math::BigFloat->new(2.5);
$y = $x->as_number('odd'); # $y = 3
-=head1 EXAMPLES
-
- # not ready yet
+=head1 METHODS
+
+=head2 accuracy
+
+ $x->accuracy(5); # local for $x
+ CLASS->accuracy(5); # global for all members of CLASS
+ # Note: This also applies to new()!
+
+ $A = $x->accuracy(); # read out accuracy that affects $x
+ $A = CLASS->accuracy(); # read out global accuracy
+
+Set or get the global or local accuracy, aka how many significant digits the
+results have. If you set a global accuracy, then this also applies to new()!
+
+Warning! The accuracy I<sticks>, e.g. once you created a number under the
+influence of C<< CLASS->accuracy($A) >>, all results from math operations with
+that number will also be rounded.
+
+In most cases, you should probably round the results explicitly using one of
+L<round()>, L<bround()> or L<bfround()> or by passing the desired accuracy
+to the math operation as additional parameter:
+
+ my $x = Math::BigInt->new(30000);
+ my $y = Math::BigInt->new(7);
+ print scalar $x->copy()->bdiv($y, 2); # print 4300
+ print scalar $x->copy()->bdiv($y)->bround(2); # print 4300
+
+=head2 precision()
+
+ $x->precision(-2); # local for $x, round at the second digit right of the dot
+ $x->precision(2); # ditto, round at the second digit left of the dot
+
+ CLASS->precision(5); # Global for all members of CLASS
+ # This also applies to new()!
+ CLASS->precision(-5); # ditto
+
+ $P = CLASS->precision(); # read out global precision
+ $P = $x->precision(); # read out precision that affects $x
+
+Note: You probably want to use L<accuracy()> instead. With L<accuracy> you
+set the number of digits each result should have, with L<precision> you
+set the place where to round!
=head1 Autocreating constants
use Math::BigFloat lib => 'Foo,Math::BigInt::Bar';
Calc.pm uses as internal format an array of elements of some decimal base
-(usually 1e7, but this might be differen for some systems) with the least
+(usually 1e7, but this might be different for some systems) 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
use Math::BigFloat lib => 'GMP';
-Use the lib, Luke! And see L<Using Math::BigInt::Lite> for more details.
+It is also possible to just require Math::BigFloat:
+
+ require Math::BigFloat;
+
+This will load the necessary things (like BigInt) when they are needed, and
+automatically.
+
+Use the lib, Luke! And see L<Using Math::BigInt::Lite> for more details than
+you ever wanted to know about loading a different library.
=head2 Using Math::BigInt::Lite
# 3
use Math::BigFloat with => 'Math::BigInt::Lite', lib => 'GMP,Pari';
-If you want to use Math::BigInt's, too, simple add a Math::BigInt B<before>:
+There is no need for a "use Math::BigInt;" statement, even if you want to
+use Math::BigInt's, since Math::BigFloat will needs Math::BigInt and thus
+always loads it. But if you add it, add it B<before>:
# 4
use Math::BigInt;
That would try to load Foo, Bar, Baz and Calc (in that order). Or in other
words, Math::BigFloat will try to retain previously loaded libs when you
-don't specify it one.
+don't specify it onem but if you specify one, it will try to load them.
Actually, the lib loading order would be "Bar,Baz,Calc", and then
-"Foo,Bar,Baz,Calc", but independend of which lib exists, the result is the
-same as trying the latter load alone, except for the fact that Bar or Baz
-might be loaded needlessly in an intermidiate step
+"Foo,Bar,Baz,Calc", but independent of which lib exists, the result is the
+same as trying the latter load alone, except for the fact that one of Bar or
+Baz might be loaded needlessly in an intermediate step (and thus hang around
+and waste memory). If neither Bar nor Baz exist (or don't work/compile), they
+will still be tried to be loaded, but this is not as time/memory consuming as
+actually loading one of them. Still, this type of usage is not recommended due
+to these issues.
-The old way still works though:
+The old way (loading the lib only in BigInt) still works though:
# 6
use Math::BigInt lib => 'Bar,Baz';
use Math::BigFloat;
-But B<examples #3 and #4 are recommended> for usage.
+You can even load Math::BigInt afterwards:
-=head1 BUGS
+ # 7
+ use Math::BigFloat;
+ use Math::BigInt lib => 'Bar,Baz';
-=over 2
+But this has the same problems like #5, it will first load Calc
+(Math::BigFloat needs Math::BigInt and thus loads it) and then later Bar or
+Baz, depending on which of them works and is usable/loadable. Since this
+loads Calc unnecc., it is not recommended.
-=item *
+Since it also possible to just require Math::BigFloat, this poses the question
+about what libary this will use:
-The following does not work yet:
+ require Math::BigFloat;
+ my $x = Math::BigFloat->new(123); $x += 123;
- $m = $x->mantissa();
- $e = $x->exponent();
- $y = $m * ( 10 ** $e );
- print "ok\n" if $x == $y;
+It will use Calc. Please note that the call to import() is still done, but
+only when you use for the first time some Math::BigFloat math (it is triggered
+via any constructor, so the first time you create a Math::BigFloat, the load
+will happen in the background). This means:
-=item *
+ require Math::BigFloat;
+ Math::BigFloat->import ( lib => 'Foo,Bar' );
-There is no fmod() function yet.
+would be the same as:
-=back
+ use Math::BigFloat lib => 'Foo, Bar';
+
+But don't try to be clever to insert some operations in between:
+
+ require Math::BigFloat;
+ my $x = Math::BigFloat->bone() + 4; # load BigInt and Calc
+ Math::BigFloat->import( lib => 'Pari' ); # load Pari, too
+ $x = Math::BigFloat->bone()+4; # now use Pari
+
+While this works, it loads Calc needlessly. But maybe you just wanted that?
+
+B<Examples #3 is highly recommended> for daily usage.
+
+=head1 BUGS
+
+Please see the file BUGS in the CPAN distribution Math::BigInt for known bugs.
-=head1 CAVEAT
+=head1 CAVEATS
=over 1
print $c->bdiv(123.456),"\n";
It prints both quotient and reminder since print works in list context. Also,
-bdiv() will modify $c, so be carefull. You probably want to use
+bdiv() will modify $c, so be careful. You probably want to use
print $c / 123.456,"\n";
print scalar $c->bdiv(123.456),"\n"; # or if you want to modify $c
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 will modify $y, and vice versa.
-
- $x->bmul(2);
- print "$x, $y\n"; # prints '10, 10'
-
-If you want a true copy of $x, use:
-
- $y = $x->copy();
-
-See also the documentation in L<overload> regarding C<=>.
+that modifies $x will modify $y (except overloaded math operators), and vice
+versa. See L<Math::BigInt> for details and how to avoid that.
=item bpow
print $x->bpow($i),"\n"; # ditto
print $x ** $i,"\n"; # leave $x alone
+=item precision() vs. accuracy()
+
+A common pitfall is to use L<precision()> when you want to round a result to
+a certain number of digits:
+
+ use Math::BigFloat;
+
+ Math::BigFloat->precision(4); # does not do what you think it does
+ my $x = Math::BigFloat->new(12345); # rounds $x to "12000"!
+ print "$x\n"; # print "12000"
+ my $y = Math::BigFloat->new(3); # rounds $y to "0"!
+ print "$y\n"; # print "0"
+ $z = $x / $y; # 12000 / 0 => NaN!
+ print "$z\n";
+ print $z->precision(),"\n"; # 4
+
+Replacing L<precision> with L<accuracy> is probably not what you want, either:
+
+ use Math::BigFloat;
+
+ Math::BigFloat->accuracy(4); # enables global rounding:
+ my $x = Math::BigFloat->new(123456); # rounded immediately to "12350"
+ print "$x\n"; # print "123500"
+ my $y = Math::BigFloat->new(3); # rounded to "3
+ print "$y\n"; # print "3"
+ print $z = $x->copy()->bdiv($y),"\n"; # 41170
+ print $z->accuracy(),"\n"; # 4
+
+What you want to use instead is:
+
+ use Math::BigFloat;
+
+ my $x = Math::BigFloat->new(123456); # no rounding
+ print "$x\n"; # print "123456"
+ my $y = Math::BigFloat->new(3); # no rounding
+ print "$y\n"; # print "3"
+ print $z = $x->copy()->bdiv($y,4),"\n"; # 41150
+ print $z->accuracy(),"\n"; # undef
+
+In addition to computing what you expected, the last example also does B<not>
+"taint" the result with an accuracy or precision setting, which would
+influence any further operation.
+
=back
+=head1 SEE ALSO
+
+L<Math::BigInt>, L<Math::BigRat> and L<Math::Big> as well as
+L<Math::BigInt::BitVect>, L<Math::BigInt::Pari> and L<Math::BigInt::GMP>.
+
+The pragmas L<bignum>, L<bigint> and L<bigrat> might also be of interest
+because they solve the autoupgrading/downgrading issue, at least partly.
+
+The package at
+L<http://search.cpan.org/search?mode=module&query=Math%3A%3ABigInt> contains
+more documentation including a full version history, testcases, empty
+subclass files and benchmarks.
+
=head1 LICENSE
This program is free software; you may redistribute it and/or modify it under
=head1 AUTHORS
Mark Biggar, overloaded interface by Ilya Zakharevich.
-Completely rewritten by Tels http://bloodgate.com in 2001.
+Completely rewritten by Tels L<http://bloodgate.com> in 2001 - 2004, and still
+at it in 2005.
=cut
projects / p5sagit/p5-mst-13.2.git / blobdiff