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';
-require 5.005;
-use Exporter;
-use File::Spec;
-# use Math::BigInt;
-@ISA = qw( Exporter Math::BigInt);
+# _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 = '1.59';
+require 5.006;
+
+require Exporter;
+@ISA = qw/Math::BigInt/;
+@EXPORT_OK = qw/bpi/;
use strict;
-use vars qw/$AUTOLOAD $accuracy $precision $div_scale $round_mode $rnd_mode/;
-use vars qw/$upgrade $downgrade/;
+# $_trap_inf/$_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
-'<=>' => sub { $_[2] ?
+'<=>' => sub { my $rc = $_[2] ?
ref($_[0])->bcmp($_[1],$_[0]) :
- ref($_[0])->bcmp($_[0],$_[1])},
+ ref($_[0])->bcmp($_[0],$_[1]);
+ $rc = 1 unless defined $rc;
+ $rc <=> 0;
+ },
+# we need '>=' to get things like "1 >= NaN" right:
+'>=' => sub { my $rc = $_[2] ?
+ ref($_[0])->bcmp($_[1],$_[0]) :
+ ref($_[0])->bcmp($_[0],$_[1]);
+ # if there was a NaN involved, return false
+ return '' unless defined $rc;
+ $rc >= 0;
+ },
'int' => sub { $_[0]->as_number() }, # 'trunc' to bigint
;
##############################################################################
-# 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'
+# one of 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'
+$round_mode = 'even';
$accuracy = undef;
$precision = undef;
$div_scale = 40;
$upgrade = undef;
$downgrade = undef;
-my $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 nec.
##############################################################################
# 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 sets $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';
+
+ # we need both of them in this package:
+ *as_int = \&as_number;
}
##############################################################################
-# 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 ffac fneg
+ fceil ffloor frsft flsft fone flog froot fexp
/;
- # valid method's that can be hand-ed up (for AUTOLOAD)
+ # valid methods that can be handed 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
+ bsub
/;
- 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 $class))
+ if ((ref($wanted)) && UNIVERSAL::can( $wanted, "as_number"))
{
- $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 $self->bnorm();
}
- # got string
+ # else: got a string or something maskerading as number (with overload)
+
# handle '+inf', '-inf' first
- if ($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
+ {
+ # if two arguments, the first one is the class to "swallow" subclasses
+ if (@_ > 1)
+ {
+ my $self = bless {
+ sign => $_[1]->{sign},
+ _es => $_[1]->{_es},
+ _m => $MBI->_copy($_[1]->{_m}),
+ _e => $MBI->_copy($_[1]->{_e}),
+ }, $_[0] if @_ > 1;
+
+ $self->{_a} = $_[1]->{_a} if defined $_[1]->{_a};
+ $self->{_p} = $_[1]->{_p} if defined $_[1]->{_p};
+ return $self;
}
- # print "mbf new $self->{sign} $self->{_m} e $self->{_e} ",ref($self),"\n";
- $self->bnorm()->round(@r); # first normalize, then round
+
+ my $self = bless {
+ sign => $_[0]->{sign},
+ _es => $_[0]->{_es},
+ _m => $MBI->_copy($_[0]->{_m}),
+ _e => $MBI->_copy($_[0]->{_e}),
+ }, ref($_[0]);
+
+ $self->{_a} = $_[0]->{_a} if defined $_[0]->{_a};
+ $self->{_p} = $_[0]->{_p} if defined $_[0]->{_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
# return (later set?) configuration data as hash ref
my $class = shift || 'Math::BigFloat';
- my $cfg = $MBI->config();
+ if (@_ == 1 && ref($_[0]) ne 'HASH')
+ {
+ my $cfg = $class->SUPER::config();
+ return $cfg->{$_[0]};
+ }
+
+ my $cfg = $class->SUPER::config(@_);
- no strict 'refs';
+ # now we need only to override the ones that are different from our parent
$cfg->{class} = $class;
$cfg->{with} = $MBI;
- foreach (
- qw/upgrade downgrade precision accuracy round_mode VERSION div_scale/)
- {
- $cfg->{lc($_)} = ${"${class}::$_"};
- };
$cfg;
}
# (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,@r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,@r) = objectify(2,@_);
+ }
+
+ return $x if $x->modify('badd');
+
# inf and NaN handling
if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
{
return $x;
}
- return $upgrade->badd($x,$y,$a,$p,$r) if defined $upgrade &&
+ return $upgrade->badd($x,$y,@r) if defined $upgrade &&
((!$x->isa($self)) || (!$y->isa($self)));
+ $r[3] = $y; # no push!
+
# speed: no add for 0+y or x+0
- return $x->bround($a,$p,$r) if $y->is_zero(); # x+0
+ return $x->bround(@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);
+ return $x->round(@r);
}
# 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(@r);
}
+# 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,@_);
+
+ return $x if $x->modify('binc');
- 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,@_);
+
+ return $x if $x->modify('bdec');
- 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
+ my ($self,$x,$base,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
- # 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 _|
+ return $x if $x->modify('blog');
- # 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 number 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;
-
- 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();
- # }
- $limit = $self->new("1E-". ($scale-1));
- #my $steps = 0;
- while (3 < 5)
+ local $Math::BigFloat::downgrade = undef;
+
+ # upgrade $x if $x is not a BigFloat (handle BigInt input)
+ # XXX TODO: rebless!
+ if (!$x->isa('Math::BigFloat'))
{
- # 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;
- $x->badd($next);
- # print "step $steps $x\n";
- # calculate things for the next term
- $over *= $u; $below *= $v; $factor->badd($f);
- #$steps++;
+ $x = Math::BigFloat->new($x);
+ $self = ref($x);
}
- $x->bmul(2) if $case == 0;
- #print "took $steps steps\n";
- # shortcut to not run trough _find_round_parameters again
- if (defined $params[1])
+ 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())
{
- $x->bround($params[1],$params[3]); # then round accordingly
+ 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)
+ {
+ # base is undef, so base should be e (Euler's number), so 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 );
+ }
+ }
+
+ # 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[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 blcm
- {
- # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT
- # does not modify arguments, but returns new object
- # Lowest Common Multiplicator
-
- my ($self,@arg) = objectify(0,@_);
- my $x = $self->new(shift @arg);
- while (@arg) { $x = _lcm($x,shift @arg); }
- $x;
- }
-
-sub bgcd
- {
- # (BFLOAT or num_str, BFLOAT 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); }
- $x;
- }
-
-###############################################################################
-# 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,@_);
-
- return 1 if ($x->{sign} =~ /^[+-]$/) && # NaN and +-inf aren't
- $x->{_e}->{sign} eq '+'; # 1e-1 => no integer
- 0;
- }
-
-sub is_zero
+sub _len_to_steps
{
- # return true if arg (BFLOAT or num_str) is zero
- my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+ # Given D (digits in decimal), compute N so that N! (N factorial) is
+ # at least D digits long. D should be at least 50.
+ my $d = shift;
- return 1 if $x->{sign} eq '+' && $x->{_m}->is_zero();
- 0;
- }
+ # two constants for the Ramanujan estimate of ln(N!)
+ my $lg2 = log(2 * 3.14159265) / 2;
+ my $lg10 = log(10);
-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,@_);
+ # D = 50 => N => 42, so L = 40 and R = 50
+ my $l = 40; my $r = $d;
- my $sign = shift || ''; $sign = '+' if $sign ne '-';
- return 1
- if ($x->{sign} eq $sign && $x->{_e}->is_zero() && $x->{_m}->is_one());
- 0;
- }
+ # Otherwise this does not work under -Mbignum and we do not yet have "no bignum;" :(
+ $l = $l->numify if ref($l);
+ $r = $r->numify if ref($r);
+ $lg2 = $lg2->numify if ref($lg2);
+ $lg10 = $lg10->numify if ref($lg10);
-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,@_);
-
- return 1 if ($x->{sign} =~ /^[+-]$/) && # NaN & +-inf aren't
- ($x->{_e}->is_zero() && $x->{_m}->is_odd());
- 0;
+ # binary search for the right value (could this be written as the reverse of lg(n!)?)
+ while ($r - $l > 1)
+ {
+ my $n = int(($r - $l) / 2) + $l;
+ my $ramanujan =
+ int(($n * log($n) - $n + log( $n * (1 + 4*$n*(1+2*$n)) ) / 6 + $lg2) / $lg10);
+ $ramanujan > $d ? $r = $n : $l = $n;
+ }
+ $l;
}
-sub is_even
+sub bnok
{
- # return true if arg (BINT or num_str) is even or false if odd
- my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+ # Calculate n over k (binomial coefficient or "choose" function) as integer.
+ # set up parameters
+ my ($self,$x,$y,@r) = (ref($_[0]),@_);
- 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
- 0;
- }
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,@r) = objectify(2,@_);
+ }
-sub bmul
- {
- # 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,@_);
+ return $x if $x->modify('bnok');
- return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
+ return $x->bnan() if $x->is_nan() || $y->is_nan();
+ return $x->binf() if $x->is_inf();
- # inf handling
- if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
- {
- return $x->bnan() if $x->is_zero() || $y->is_zero();
- # result will always be +-inf:
- # +inf * +/+inf => +inf, -inf * -/-inf => +inf
- # +inf * -/-inf => -inf, -inf * +/+inf => -inf
- return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
- return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
- return $x->binf('-');
- }
- # 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)));
+ my $u = $x->as_int();
+ $u->bnok($y->as_int());
- # aEb * cEd = (a*c)E(b+d)
- $x->{_m}->bmul($y->{_m});
- $x->{_e}->badd($y->{_e});
- # adjust sign:
- $x->{sign} = $x->{sign} ne $y->{sign} ? '-' : '+';
- return $x->bnorm()->round($a,$p,$r,$y);
+ $x->{_m} = $u->{value};
+ $x->{_e} = $MBI->_zero();
+ $x->{_es} = '+';
+ $x->{sign} = '+';
+ $x->bnorm(@r);
}
-sub bdiv
+sub bexp
{
- # (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,@_);
-
- return $self->_div_inf($x,$y)
- if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
+ # Calculate e ** X (Euler's number to the power of X)
+ my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
- # x== 0 # also: or y == 1 or y == -1
- return wantarray ? ($x,$self->bzero()) : $x if $x->is_zero();
+ return $x if $x->modify('bexp');
- # upgrade ?
- return $upgrade->bdiv($upgrade->new($x),$y,$a,$p,$r) if defined $upgrade;
+ return $x->binf() if $x->{sign} eq '+inf';
+ return $x->bzero() if $x->{sign} eq '-inf';
# 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 ($scale,@params);
+ ($x,@params) = $x->_find_round_parameters($a,$p,$r);
+
+ # also takes care of the "error in _find_round_parameters?" case
+ return $x if $x->{sign} eq 'NaN';
# 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
- }
- my $lx = $x->{_m}->length(); my $ly = $y->{_m}->length();
- $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();
+ # the 4 below is empirical, and there might be cases where it's not enough...
+ $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
}
- $x->{sign} = $x->{sign} ne $y->sign() ? '-' : '+';
-
- # 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');
-
- #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
-
- # 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
- }
+ return $x->bone(@params) if $x->is_zero();
- # shortcut to not run trough _find_round_parameters again
- if (defined $params[1])
+ if (!$x->isa('Math::BigFloat'))
{
- $x->bround($params[1],$params[3]); # then round accordingly
+ $x = Math::BigFloat->new($x);
+ $self = ref($x);
}
- else
- {
- $x->bfround($params[2],$params[3]); # then round accordingly
- }
- if ($fallback)
+
+ # 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;
+ local $Math::BigFloat::downgrade = undef;
+
+ my $x_org = $x->copy();
+
+ # We use the following Taylor series:
+
+ # x x^2 x^3 x^4
+ # e = 1 + --- + --- + --- + --- ...
+ # 1! 2! 3! 4!
+
+ # The difference for each term is X and N, which would result in:
+ # 2 copy, 2 mul, 2 add, 1 inc, 1 div operations per term
+
+ # But it is faster to compute exp(1) and then raising it to the
+ # given power, esp. if $x is really big and an integer because:
+
+ # * The numerator is always 1, making the computation faster
+ # * the series converges faster in the case of x == 1
+ # * We can also easily check when we have reached our limit: when the
+ # term to be added is smaller than "1E$scale", we can stop - f.i.
+ # scale == 5, and we have 1/40320, then we stop since 1/40320 < 1E-5.
+ # * we can compute the *exact* result by simulating bigrat math:
+
+ # 1 1 gcd(3,4) = 1 1*24 + 1*6 5
+ # - + - = ---------- = --
+ # 6 24 6*24 24
+
+ # We do not compute the gcd() here, but simple do:
+ # 1 1 1*24 + 1*6 30
+ # - + - = --------- = --
+ # 6 24 6*24 144
+
+ # In general:
+ # a c a*d + c*b and note that c is always 1 and d = (b*f)
+ # - + - = ---------
+ # b d b*d
+
+ # This leads to: which can be reduced by b to:
+ # a 1 a*b*f + b a*f + 1
+ # - + - = --------- = -------
+ # b b*f b*b*f b*f
+
+ # The first terms in the series are:
+
+ # 1 1 1 1 1 1 1 1 13700
+ # -- + -- + -- + -- + -- + --- + --- + ---- = -----
+ # 1 1 2 6 24 120 720 5040 5040
+
+ # Note that we cannot simple reduce 13700/5040 to 685/252, but must keep A and B!
+
+ if ($scale <= 75)
+ {
+ # set $x directly from a cached string form
+ $x->{_m} = $MBI->_new(
+ "27182818284590452353602874713526624977572470936999595749669676277240766303535476");
+ $x->{sign} = '+';
+ $x->{_es} = '-';
+ $x->{_e} = $MBI->_new(79);
+ }
+ else
+ {
+ # compute A and B so that e = A / B.
+
+ # After some terms we end up with this, so we use it as a starting point:
+ my $A = $MBI->_new("90933395208605785401971970164779391644753259799242");
+ my $F = $MBI->_new(42); my $step = 42;
+
+ # Compute how many steps we need to take to get $A and $B sufficiently big
+ my $steps = _len_to_steps($scale - 4);
+# print STDERR "# Doing $steps steps for ", $scale-4, " digits\n";
+ while ($step++ <= $steps)
+ {
+ # calculate $a * $f + 1
+ $A = $MBI->_mul($A, $F);
+ $A = $MBI->_inc($A);
+ # increment f
+ $F = $MBI->_inc($F);
+ }
+ # compute $B as factorial of $steps (this is faster than doing it manually)
+ my $B = $MBI->_fac($MBI->_new($steps));
+
+# print "A ", $MBI->_str($A), "\nB ", $MBI->_str($B), "\n";
+
+ # compute A/B with $scale digits in the result (truncate, not round)
+ $A = $MBI->_lsft( $A, $MBI->_new($scale), 10);
+ $A = $MBI->_div( $A, $B );
+
+ $x->{_m} = $A;
+ $x->{sign} = '+';
+ $x->{_es} = '-';
+ $x->{_e} = $MBI->_new($scale);
+ }
+
+ # $x contains now an estimate of e, with some surplus digits, so we can round
+ if (!$x_org->is_one())
+ {
+ # raise $x to the wanted power and round it in one step:
+ $x->bpow($x_org, @params);
+ }
+ else
+ {
+ # else just round the already computed result
+ delete $x->{_a}; delete $x->{_p};
+ # 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; # return modified $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();
+
+ # XXX TODO: rewrite this in a similiar manner to bexp()
+
+ # 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));
+ 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
+
+ # 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);
+ # calculate things for the next term
+ $over *= $u; $below *= $v; $factor->badd($f);
+ if (DEBUG)
+ {
+ $steps++; print "step $steps = $x\n" if $steps % 10 == 0;
+ }
+ }
+ print "took $steps steps\n" if DEBUG;
+ $x->bmul($f); # $x *= 2
+ }
+
+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 becomes below 1 - the smaller
+ # $x is the faster it gets. Since 2*$x takes about 10 times as
+ # long, we make it faster by about a factor of 100 by dividing $x by 10.
+
+ # The same observation is valid for numbers smaller than 0.1, e.g. computing
+ # log(1) is fastest, and the further 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.
+
+ # To get $x even closer to 1, we also divide by 2 and then use log(2) to
+ # correct for this. For instance if $x is 2.4, we use the formula:
+ # blog(2.4 * 2) == blog (1.2) + blog(2)
+ # and thus calculate only blog(1.2) and blog(2), which is faster in total
+ # than calculating blog(2.4).
+
+ # In addition, the values for blog(2) and blog(10) are cached.
+
+ # 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); # modify $x in place
+ $calc = 0; # no need to calc, but round
+ }
+ # if we can't use the shortcut, we continue normally
+ }
+ else
+ {
+ # 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); # modify $x in place
+ $calc = 0; # no need to calc, but round
+ }
+ # if we can't use the shortcut, we continue normally
+ }
+ }
+
+ # 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}))
+ {
+ $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
+ }
+ }
+
+ 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
+
+ my $two = $self->new(2);
+
+ # $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 and cache result
+ # also disable downgrade for this code path
+ local $Math::BigFloat::downgrade = undef;
+
+ # shorten the time to calculate log(10) based on the following:
+ # log(1.25 * 8) = log(1.25) + log(8)
+ # = log(1.25) + log(2) + log(2) + log(2)
+
+ # first get $l_2 (and possible compute and cache log(2))
+ $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 the mul below
+ }
+ else
+ {
+ # else: slower, compute and cache result
+ $l_2 = $two->copy(); $self->_log($l_2, $scale); # scale+4, actually
+ $LOG_2 = $l_2->copy(); # cache the result for later
+ # the copy() is for mul below
+ $LOG_2_A = $scale;
+ }
+
+ # now calculate log(1.25):
+ $l_10 = $self->new('1.25'); $self->_log($l_10, $scale); # scale+4, actually
+
+ # log(1.25) + log(2) + log(2) + log(2):
+ $l_10->badd($l_2);
+ $l_10->badd($l_2);
+ $l_10->badd($l_2);
+ $LOG_10 = $l_10->copy(); # cache the result for later
+ # the copy() is for mul below
+ $LOG_10_A = $scale;
+ }
+ $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)
+ while ($x->bacmp($HALF) <= 0) # X <= 0.5
+ {
+ $twos--; $x->bmul($two);
+ }
+ while ($x->bacmp($two) >= 0) # X >= 2
+ {
+ $twos++; $x->bdiv($two,$scale+4); # keep all digits
+ }
+ # $twos > 0 => did mul 2, < 0 => did div 2 (but we never did both)
+ # So 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 the mul below
+ }
+ else
+ {
+ # else: slower, compute and cache result
+ # also disable downgrade for this code path
+ local $Math::BigFloat::downgrade = undef;
+ $l_2 = $two->copy(); $self->_log($l_2, $scale); # scale+4, actually
+ $LOG_2 = $l_2->copy(); # cache the result for later
+ # the copy() is for mul below
+ $LOG_2_A = $scale;
+ }
+ $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
+ $x;
+ }
+
+sub blcm
+ {
+ # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT
+ # does not modify arguments, but returns new object
+ # Lowest Common Multiplicator
+
+ my ($self,@arg) = objectify(0,@_);
+ my $x = $self->new(shift @arg);
+ while (@arg) { $x = Math::BigInt::__lcm($x,shift @arg); }
+ $x;
+ }
+
+sub bgcd
+ {
+ # (BINT or num_str, BINT or num_str) return BINT
+ # does not modify arguments, but returns new object
+
+ 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]) ? (undef,$_[0]) : objectify(1,@_);
+
+ (($x->{sign} =~ /^[+-]$/) && # NaN and +-inf aren't
+ ($x->{_es} eq '+')) ? 1 : 0; # 1e-1 => no integer
+ }
+
+sub is_zero
+ {
+ # return true if arg (BFLOAT or num_str) is zero
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ ($x->{sign} eq '+' && $MBI->_is_zero($x->{_m})) ? 1 : 0;
+ }
+
+sub is_one
+ {
+ # return true if arg (BFLOAT or num_str) is +1 or -1 if signis given
+ my ($self,$x,$sign) = ref($_[0]) ? (undef,@_) : objectify(1,@_);
+
+ $sign = '+' if !defined $sign || $sign ne '-';
+
+ ($x->{sign} eq $sign &&
+ $MBI->_is_zero($x->{_e}) &&
+ $MBI->_is_one($x->{_m}) ) ? 1 : 0;
+ }
+
+sub is_odd
+ {
+ # return true if arg (BFLOAT or num_str) is odd or false if even
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ (($x->{sign} =~ /^[+-]$/) && # NaN & +-inf aren't
+ ($MBI->_is_zero($x->{_e})) &&
+ ($MBI->_is_odd($x->{_m}))) ? 1 : 0;
+ }
+
+sub is_even
+ {
+ # return true if arg (BINT or num_str) is even or false if odd
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ (($x->{sign} =~ /^[+-]$/) && # NaN & +-inf aren't
+ ($x->{_es} eq '+') && # 123.45 isn't
+ ($MBI->_is_even($x->{_m}))) ? 1 : 0; # but 1200 is
+ }
+
+sub bmul
+ {
+ # multiply two numbers
+
+ # set up parameters
+ my ($self,$x,$y,@r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,@r) = objectify(2,@_);
+ }
+
+ return $x if $x->modify('bmul');
+
+ return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
+
+ # inf handling
+ if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
+ {
+ return $x->bnan() if $x->is_zero() || $y->is_zero();
+ # result will always be +-inf:
+ # +inf * +/+inf => +inf, -inf * -/-inf => +inf
+ # +inf * -/-inf => -inf, -inf * +/+inf => -inf
+ return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
+ return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
+ return $x->binf('-');
+ }
+
+ return $upgrade->bmul($x,$y,@r) if defined $upgrade &&
+ ((!$x->isa($self)) || (!$y->isa($self)));
+
+ # aEb * cEd = (a*c)E(b+d)
+ $MBI->_mul($x->{_m},$y->{_m});
+ ($x->{_e}, $x->{_es}) = _e_add($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es});
+
+ $r[3] = $y; # no push!
+
+ # adjust sign:
+ $x->{sign} = $x->{sign} ne $y->{sign} ? '-' : '+';
+ $x->bnorm->round(@r);
+ }
+
+sub bmuladd
+ {
+ # multiply two numbers and add the third to the result
+
+ # set up parameters
+ my ($self,$x,$y,$z,@r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,$z,@r) = objectify(3,@_);
+ }
+
+ return $x if $x->modify('bmuladd');
+
+ return $x->bnan() if (($x->{sign} eq $nan) ||
+ ($y->{sign} eq $nan) ||
+ ($z->{sign} eq $nan));
+
+ # inf handling
+ if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/))
+ {
+ return $x->bnan() if $x->is_zero() || $y->is_zero();
+ # result will always be +-inf:
+ # +inf * +/+inf => +inf, -inf * -/-inf => +inf
+ # +inf * -/-inf => -inf, -inf * +/+inf => -inf
+ return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
+ return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
+ return $x->binf('-');
+ }
+
+ return $upgrade->bmul($x,$y,@r) if defined $upgrade &&
+ ((!$x->isa($self)) || (!$y->isa($self)));
+
+ # aEb * cEd = (a*c)E(b+d)
+ $MBI->_mul($x->{_m},$y->{_m});
+ ($x->{_e}, $x->{_es}) = _e_add($x->{_e}, $y->{_e}, $x->{_es}, $y->{_es});
+
+ $r[3] = $y; # no push!
+
+ # adjust sign:
+ $x->{sign} = $x->{sign} ne $y->{sign} ? '-' : '+';
+
+ # z=inf handling (z=NaN handled above)
+ $x->{sign} = $z->{sign}, return $x if $z->{sign} =~ /^[+-]inf$/;
+
+ # take lower of the two e's and adapt m1 to it to match m2
+ my $e = $z->{_e};
+ $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}, $z->{_es} || '+', $x->{_es});
+
+ my $add = $MBI->_copy($z->{_m});
+
+ if ($es eq '-') # < 0
+ {
+ $MBI->_lsft( $x->{_m}, $e, 10);
+ ($x->{_e},$x->{_es}) = _e_add($x->{_e}, $e, $x->{_es}, $es);
+ }
+ elsif (!$MBI->_is_zero($e)) # > 0
+ {
+ $MBI->_lsft($add, $e, 10);
+ }
+ # else: both e are the same, so just leave them
+
+ if ($x->{sign} eq $z->{sign})
+ {
+ # add
+ $x->{_m} = $MBI->_add($x->{_m}, $add);
+ }
+ else
+ {
+ ($x->{_m}, $x->{sign}) =
+ _e_add($x->{_m}, $add, $x->{sign}, $z->{sign});
+ }
+
+ # delete trailing zeros, then round
+ $x->bnorm()->round(@r);
+ }
+
+sub bdiv
+ {
+ # (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return
+ # (BFLOAT,BFLOAT) (quo,rem) or BFLOAT (only rem)
+
+ # 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->modify('bdiv');
+
+ return $self->_div_inf($x,$y)
+ if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/) || $y->is_zero());
+
+ # x== 0 # also: or y == 1 or y == -1
+ return wantarray ? ($x,$self->bzero()) : $x if $x->is_zero();
+
+ # upgrade ?
+ return $upgrade->bdiv($upgrade->new($x),$y,$a,$p,$r) if defined $upgrade;
+
+ # 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,$y);
+
+ 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; # 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[0] || $params[1]) + 4; # take whatever is defined
+ }
+
+ 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!
+
+ # already handled inf/NaN/-inf above:
+
+ # 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}));
+
+ # 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/+-/-+/;
+
+ 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 through _find_round_parameters again
+ if (defined $params[0])
+ {
+ delete $x->{_a}; # clear before round
+ $x->bround($params[0],$params[2]); # then round accordingly
+ }
+ else
+ {
+ 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
+ delete $x->{_a}; delete $x->{_p};
+ }
+
+ if (wantarray)
+ {
+ if ($y_not_one)
+ {
+ $rem->bmod($y,@params); # copy already done
+ }
+ if ($fallback)
+ {
+ # clear a/p after round, since user did not request it
+ delete $rem->{_a}; delete $rem->{_p};
+ }
+ return ($x,$rem);
+ }
+ $x;
+ }
+
+sub bmod
+ {
+ # (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return reminder
+
+ # 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->modify('bmod');
+
+ # handle NaN, inf, -inf
+ if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
+ {
+ my ($d,$re) = $self->SUPER::_div_inf($x,$y);
+ $x->{sign} = $re->{sign};
+ $x->{_e} = $re->{_e};
+ $x->{_m} = $re->{_m};
+ return $x->round($a,$p,$r,$y);
+ }
+ 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})));
+
+ my $cmp = $x->bacmp($y); # equal or $x < $y?
+ return $x->bzero($a,$p) if $cmp == 0; # $x == $y => result 0
+
+ # only $y of the operands negative?
+ my $neg = 0; $neg = 1 if $x->{sign} ne $y->{sign};
+
+ $x->{sign} = $y->{sign}; # calc sign first
+ return $x->round($a,$p,$r) if $cmp < 0 && $neg == 0; # $x < $y => result $x
+
+ my $ym = $MBI->_copy($y->{_m});
+
+ # 2e1 => 20
+ $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->{_es} eq '-') # has digits after dot
+ {
+ # 123 % 2.5 => 1230 % 25 => 5 => 0.5
+ $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->{_es} eq '-') # has digits after dot
+ {
+ # 123.4 % 20 => 1234 % 200
+ $shiftx = $MBI->_num($x->{_e}); # no more digits after dot
+ $MBI->_lsft($ym, $x->{_e}, 10); # 123 => 1230
+ }
+ # 123e1 % 20 => 1230 % 20
+ if ($x->{_es} eq '+' && !$MBI->_is_zero($x->{_e}))
+ {
+ $MBI->_lsft( $x->{_m}, $x->{_e},10); # es => '+' here
+ }
+
+ $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} = $MBI->_mod( $x->{_m}, $ym);
+
+ $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->{_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,@_);
+ }
+
+ return $x if $x->modify('broot');
+
+ # NaN handling: $x ** 1/0, x or y NaN, or y inf/-inf or y == 0
+ return $x->bnan() if $x->{sign} !~ /^\+/ || $y->is_zero() ||
+ $y->{sign} !~ /^\+$/;
+
+ return $x 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
+ }
+ if ($fallback)
+ {
+ # clear a/p after round, since user did not request it
+ delete $x->{_a}; delete $x->{_p};
+ }
+ # restore globals
+ $$abr = $ab; $$pbr = $pb;
+ $x;
+ }
+
+sub bsqrt
+ {
+ # calculate square root
+ my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+
+ return $x if $x->modify('bsqrt');
+
+ return $x->bnan() if $x->{sign} !~ /^[+]/; # NaN, -inf or < 0
+ return $x if $x->{sign} eq '+inf'; # sqrt(inf) == inf
+ return $x->round($a,$p,$r) if $x->is_zero() || $x->is_one();
+
+ # we need to limit the accuracy to protect against overflow
+ my $fallback = 0;
+ my (@params,$scale);
+ ($x,@params) = $x->_find_round_parameters($a,$p,$r);
+
+ return $x if $x->is_nan(); # error in _find_round_parameters?
+
+ # no rounding at all, so must use fallback
+ if (scalar @params == 0)
+ {
+ # simulate old behaviour
+ $params[0] = $self->div_scale(); # and round to it as accuracy
+ $scale = $params[0]+4; # at least four more for proper round
+ $params[2] = $r; # round mode by caller or undef
+ $fallback = 1; # to clear a/p afterwards
+ }
+ else
+ {
+ # the 4 below is empirical, and there might be cases where it is not
+ # enough...
+ $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
+ }
+
+ # when user set globals, they would interfere with our calculation, so
+ # disable them and later re-enable them
+ no strict 'refs';
+ my $abr = "$self\::accuracy"; my $ab = $$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
+
+ 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
+
+ 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, 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[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};
+ }
+ # re-enable A and P, upgrade is taken care of by "local"
+ ${"$self\::accuracy"} = $ab; ${"$self\::precision"} = $pb;
+ return $x;
+ }
+
+ # 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.
+
+ # The following steps will transform 123.456 (in $x) into 123456 (in $y1)
+ my $y1 = $MBI->_copy($x->{_m});
+
+ my $length = $MBI->_len($y1);
+
+ # Now calculate how many digits the result of sqrt(y1) would have
+ my $digits = int($length / 2);
+
+ # But we need at least $scale digits, so calculate how many are missing
+ my $shift = $scale - $digits;
+
+ # That should never happen (we take care of integer guesses above)
+ # $shift = 0 if $shift < 0;
+
+ # Multiply in steps of 100, by shifting left two times the "missing" digits
+ my $s2 = $shift * 2;
+
+ # 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});
+
+ $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)
+ {
+ # 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
+ {
+ $dat = int(($dat)/2);
+ }
+ $dat -= $MBI->_len($y1);
+ if ($dat < 0)
+ {
+ $dat = abs($dat);
+ $x->{_e} = $MBI->_new( $dat );
+ $x->{_es} = '-';
+ }
+ else
+ {
+ $x->{_e} = $MBI->_new( $dat );
+ $x->{_es} = '+';
+ }
+ $x->{_m} = $y1;
+ $x->bnorm();
+
+ # 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
- $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);
+
+ # inf => inf
+ return $x if $x->modify('bfac') || $x->{sign} eq '+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 ** 0.3
+ my ($x,$y,@r) = @_;
+ my $self = ref($x);
+
+ # if $y == 0.5, it is sqrt($x)
+ $HALF = $self->new($HALF) unless ref($HALF);
+ return $x->bsqrt(@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 _|
+
+ # we need to limit the accuracy to protect against overflow
+ my $fallback = 0;
+ my ($scale,@params);
+ ($x,@params) = $x->_find_round_parameters(@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
+ $params[1] = undef; # disable P
+ $scale = $params[0]+4; # at least four more for proper round
+ $params[2] = $r[2]; # 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[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; $$pbr = undef;
+ # we also need to disable any set A or P on $x (_find_round_parameters took
+ # them already into account), since these would interfere, too
+ delete $x->{_a}; delete $x->{_p};
+ # need to disable $upgrade in BigInt, to avoid deep recursion
+ local $Math::BigInt::upgrade = undef;
+
+ my ($limit,$v,$u,$below,$factor,$next,$over);
+
+ $u = $x->copy()->blog(undef,$scale)->bmul($y);
+ $v = $self->bone(); # 1
+ $factor = $self->new(2); # 2
+ $x->bone(); # first term: 1
+
+ $below = $v->copy();
+ $over = $u->copy();
+
+ $limit = $self->new("1E-". ($scale-1));
+ #my $steps = 0;
+ while (3 < 5)
+ {
+ # we calculate the next term, and add it to the last
+ # when the next term is below our limit, it won't affect the outcome
+ # anymore, so we stop:
+ $next = $over->copy()->bdiv($below,$scale);
+ last if $next->bacmp($limit) <= 0;
+ $x->badd($next);
+ # calculate things for the next term
+ $over *= $u; $below *= $factor; $factor->binc();
+
+ last if $x->{sign} !~ /^[-+]$/;
+
+ #$steps++;
}
- if (wantarray)
+ # shortcut to not run through _find_round_parameters again
+ if (defined $params[0])
{
- if (!$y->is_one())
- {
- $rem->bmod($y,$params[1],$params[2],$params[3]); # copy already done
- }
- else
- {
- $rem = $self->bzero();
- }
- if ($fallback)
- {
- # clear a/p after round, since user did not request it
- $rem->{_a} = undef; $rem->{_p} = undef;
- }
- return ($x,$rem);
+ $x->bround($params[0],$params[2]); # then round accordingly
+ }
+ else
+ {
+ $x->bfround($params[1],$params[2]); # then round accordingly
+ }
+ if ($fallback)
+ {
+ # clear a/p after round, since user did not request it
+ delete $x->{_a}; delete $x->{_p};
}
+ # restore globals
+ $$abr = $ab; $$pbr = $pb;
$x;
}
-sub bmod
+sub bpow
{
- # (dividend: BFLOAT or num_str, divisor: BFLOAT or num_str) return reminder
- my ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+ # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT
+ # compute power of two numbers, second arg is used as integer
+ # modifies first argument
- if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
+ # 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])))
{
- my ($d,$re) = $self->SUPER::_div_inf($x,$y);
- return $re->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();
-
- # 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
+ ($self,$x,$y,$a,$p,$r) = objectify(2,@_);
+ }
- # only $y of the operands negative?
- my $neg = 0; $neg = 1 if $x->{sign} ne $y->{sign};
+ return $x if $x->modify('bpow');
- $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();
+ return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
+ return $x if $x->{sign} =~ /^[+-]inf$/;
- # 2e1 => 20
- $ym->blsft($y->{_e},10) if $y->{_e}->{sign} eq '+' && !$y->{_e}->is_zero();
-
- # if $y has digits after dot
- my $shifty = 0; # correct _e of $x by this
- if ($y->{_e}->{sign} eq '-') # has digits after dot
+ # 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();
+
+ 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
+
+ # if ($x == -1)
+ if ($x->{sign} eq '-' && $MBI->_is_one($x->{_m}) && $MBI->_is_zero($x->{_e}))
{
- # 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
+ # if $x == -1 and odd/even y => +1/-1 because +-1 ^ (+-1) => +-1
+ return $MBI->_is_odd($y1) ? $x : $x->babs(1);
+ }
+ if ($x_is_zero)
+ {
+ return $x if $y->{sign} eq '+'; # 0**y => 0 (if not y <= 0)
+ # 0 ** -y => 1 / (0 ** y) => 1 / 0! (1 / 0 => +inf)
+ return $x->binf();
}
- # $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
+ my $new_sign = '+';
+ $new_sign = $MBI->_is_odd($y1) ? '-' : '+' if $x->{sign} ne '+';
+
+ # calculate $x->{_m} ** $y and $x->{_e} * $y separately (faster)
+ $x->{_m} = $MBI->_pow( $x->{_m}, $y1);
+ $x->{_e} = $MBI->_mul ($x->{_e}, $y1);
+
+ $x->{sign} = $new_sign;
+ $x->bnorm();
+ if ($y->{sign} eq '-')
{
- # 123.4 % 20 => 1234 % 200
- $shiftx = $x->{_e}->copy()->babs(); # no more digits after dot
- $ym->blsft($shiftx,10);
+ # modify $x in place!
+ my $z = $x->copy(); $x->bone();
+ return scalar $x->bdiv($z,$a,$p,$r); # round in one go (might ignore y's A!)
}
- # 123e1 % 20 => 1230 % 20
- if ($x->{_e}->{sign} eq '+' && !$x->{_e}->is_zero())
+ $x->round($a,$p,$r,$y);
+ }
+
+sub bmodpow
+ {
+ # takes a very large number to a very large exponent in a given very
+ # large modulus, quickly, thanks to binary exponentation. Supports
+ # negative exponents.
+ my ($self,$num,$exp,$mod,@r) = objectify(3,@_);
+
+ return $num if $num->modify('bmodpow');
+
+ # check modulus for valid values
+ return $num->bnan() if ($mod->{sign} ne '+' # NaN, - , -inf, +inf
+ || $mod->is_zero());
+
+ # check exponent for valid values
+ if ($exp->{sign} =~ /\w/)
{
- $x->{_m}->blsft($x->{_e},10);
+ # i.e., if it's NaN, +inf, or -inf...
+ return $num->bnan();
}
- $x->{_e} = $MBI->bzero() unless $x->{_e}->is_zero();
-
- $x->{_e}->bsub($shiftx) if $shiftx != 0;
- $x->{_e}->bsub($shifty) if $shifty != 0;
-
- # now mantissas are equalized, exponent of $x is adjusted, so calc result
- $x->{_m}->bmod($ym);
+ $num->bmodinv ($mod) if ($exp->{sign} eq '-');
- $x->{sign} = '+' if $x->{_m}->is_zero(); # fix sign for -0
- $x->bnorm();
+ # check num for valid values (also NaN if there was no inverse but $exp < 0)
+ return $num->bnan() if $num->{sign} !~ /^[+-]$/;
- if ($neg != 0) # one of them negative => correct in place
+ # $mod is positive, sign on $exp is ignored, result also positive
+
+ # XXX TODO: speed it up when all three numbers are integers
+ $num->bpow($exp)->bmod($mod);
+ }
+
+###############################################################################
+# trigonometric functions
+
+# helper function for bpi() and batan2(), calculates arcus tanges (1/x)
+
+sub _atan_inv
+ {
+ # return a/b so that a/b approximates atan(1/x) to at least limit digits
+ my ($self, $x, $limit) = @_;
+
+ # Taylor: x^3 x^5 x^7 x^9
+ # atan = x - --- + --- - --- + --- - ...
+ # 3 5 7 9
+
+ # 1 1 1 1
+ # atan 1/x = - - ------- + ------- - ------- + ...
+ # x x^3 * 3 x^5 * 5 x^7 * 7
+
+ # 1 1 1 1
+ # atan 1/x = - - --------- + ---------- - ----------- + ...
+ # 5 3 * 125 5 * 3125 7 * 78125
+
+ # Subtraction/addition of a rational:
+
+ # 5 7 5*3 +- 7*4
+ # - +- - = ----------
+ # 4 3 4*3
+
+ # Term: N N+1
+ #
+ # a 1 a * d * c +- b
+ # ----- +- ------------------ = ----------------
+ # b d * c b * d * c
+
+ # since b1 = b0 * (d-2) * c
+
+ # a 1 a * d +- b / c
+ # ----- +- ------------------ = ----------------
+ # b d * c b * d
+
+ # and d = d + 2
+ # and c = c * x * x
+
+ # u = d * c
+ # stop if length($u) > limit
+ # a = a * u +- b
+ # b = b * u
+ # d = d + 2
+ # c = c * x * x
+ # sign = 1 - sign
+
+ my $a = $MBI->_one();
+ my $b = $MBI->_copy($x);
+
+ my $x2 = $MBI->_mul( $MBI->_copy($x), $b); # x2 = x * x
+ my $d = $MBI->_new( 3 ); # d = 3
+ my $c = $MBI->_mul( $MBI->_copy($x), $x2); # c = x ^ 3
+ my $two = $MBI->_new( 2 );
+
+ # run the first step unconditionally
+ my $u = $MBI->_mul( $MBI->_copy($d), $c);
+ $a = $MBI->_mul($a, $u);
+ $a = $MBI->_sub($a, $b);
+ $b = $MBI->_mul($b, $u);
+ $d = $MBI->_add($d, $two);
+ $c = $MBI->_mul($c, $x2);
+
+ # a is now a * (d-3) * c
+ # b is now b * (d-2) * c
+
+ # run the second step unconditionally
+ $u = $MBI->_mul( $MBI->_copy($d), $c);
+ $a = $MBI->_mul($a, $u);
+ $a = $MBI->_add($a, $b);
+ $b = $MBI->_mul($b, $u);
+ $d = $MBI->_add($d, $two);
+ $c = $MBI->_mul($c, $x2);
+
+ # a is now a * (d-3) * (d-5) * c * c
+ # b is now b * (d-2) * (d-4) * c * c
+
+ # so we can remove c * c from both a and b to shorten the numbers involved:
+ $a = $MBI->_div($a, $x2);
+ $b = $MBI->_div($b, $x2);
+ $a = $MBI->_div($a, $x2);
+ $b = $MBI->_div($b, $x2);
+
+# my $step = 0;
+ my $sign = 0; # 0 => -, 1 => +
+ while (3 < 5)
{
- 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();
+# $step++;
+# if (($i++ % 100) == 0)
+# {
+# print "a=",$MBI->_str($a),"\n";
+# print "b=",$MBI->_str($b),"\n";
+# }
+# print "d=",$MBI->_str($d),"\n";
+# print "x2=",$MBI->_str($x2),"\n";
+# print "c=",$MBI->_str($c),"\n";
+
+ my $u = $MBI->_mul( $MBI->_copy($d), $c);
+ # use _alen() for libs like GMP where _len() would be O(N^2)
+ last if $MBI->_alen($u) > $limit;
+ my ($bc,$r) = $MBI->_div( $MBI->_copy($b), $c);
+ if ($MBI->_is_zero($r))
+ {
+ # b / c is an integer, so we can remove c from all terms
+ # this happens almost every time:
+ $a = $MBI->_mul($a, $d);
+ $a = $MBI->_sub($a, $bc) if $sign == 0;
+ $a = $MBI->_add($a, $bc) if $sign == 1;
+ $b = $MBI->_mul($b, $d);
+ }
+ else
+ {
+ # b / c is not an integer, so we keep c in the terms
+ # this happens very rarely, for instance for x = 5, this happens only
+ # at the following steps:
+ # 1, 5, 14, 32, 72, 157, 340, ...
+ $a = $MBI->_mul($a, $u);
+ $a = $MBI->_sub($a, $b) if $sign == 0;
+ $a = $MBI->_add($a, $b) if $sign == 1;
+ $b = $MBI->_mul($b, $u);
+ }
+ $d = $MBI->_add($d, $two);
+ $c = $MBI->_mul($c, $x2);
+ $sign = 1 - $sign;
+
}
- $x->round($a,$p,$r,$y); # round and return
+# print "Took $step steps for ", $MBI->_str($x),"\n";
+# print "a=",$MBI->_str($a),"\n"; print "b=",$MBI->_str($b),"\n";
+ # return a/b so that a/b approximates atan(1/x)
+ ($a,$b);
}
-sub bsqrt
- {
- # calculate square root; this should probably
- # use a different test to see whether the accuracy we want is...
- my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+sub bpi
+ {
+ my ($self,$n) = @_;
+ if (@_ == 0)
+ {
+ $self = $class;
+ }
+ if (@_ == 1)
+ {
+ # called like Math::BigFloat::bpi(10);
+ $n = $self; $self = $class;
+ # called like Math::BigFloat->bpi();
+ $n = undef if $n eq 'Math::BigFloat';
+ }
+ $self = ref($self) if ref($self);
+ my $fallback = defined $n ? 0 : 1;
+ $n = 40 if !defined $n || $n < 1;
+
+ # after 黃見利 (Hwang Chien-Lih) (1997)
+ # pi/4 = 183 * atan(1/239) + 32 * atan(1/1023) – 68 * atan(1/5832)
+ # + 12 * atan(1/110443) - 12 * atan(1/4841182) - 100 * atan(1/6826318)
+
+ # a few more to prevent rounding errors
+ $n += 4;
+
+ my ($a,$b) = $self->_atan_inv( $MBI->_new(239),$n);
+ my ($c,$d) = $self->_atan_inv( $MBI->_new(1023),$n);
+ my ($e,$f) = $self->_atan_inv( $MBI->_new(5832),$n);
+ my ($g,$h) = $self->_atan_inv( $MBI->_new(110443),$n);
+ my ($i,$j) = $self->_atan_inv( $MBI->_new(4841182),$n);
+ my ($k,$l) = $self->_atan_inv( $MBI->_new(6826318),$n);
+
+ $MBI->_mul($a, $MBI->_new(732));
+ $MBI->_mul($c, $MBI->_new(128));
+ $MBI->_mul($e, $MBI->_new(272));
+ $MBI->_mul($g, $MBI->_new(48));
+ $MBI->_mul($i, $MBI->_new(48));
+ $MBI->_mul($k, $MBI->_new(400));
+
+ my $x = $self->bone(); $x->{_m} = $a; my $x_d = $self->bone(); $x_d->{_m} = $b;
+ my $y = $self->bone(); $y->{_m} = $c; my $y_d = $self->bone(); $y_d->{_m} = $d;
+ my $z = $self->bone(); $z->{_m} = $e; my $z_d = $self->bone(); $z_d->{_m} = $f;
+ my $u = $self->bone(); $u->{_m} = $g; my $u_d = $self->bone(); $u_d->{_m} = $h;
+ my $v = $self->bone(); $v->{_m} = $i; my $v_d = $self->bone(); $v_d->{_m} = $j;
+ my $w = $self->bone(); $w->{_m} = $k; my $w_d = $self->bone(); $w_d->{_m} = $l;
+ $x->bdiv($x_d, $n);
+ $y->bdiv($y_d, $n);
+ $z->bdiv($z_d, $n);
+ $u->bdiv($u_d, $n);
+ $v->bdiv($v_d, $n);
+ $w->bdiv($w_d, $n);
+
+ delete $x->{_a}; delete $y->{_a}; delete $z->{_a};
+ delete $u->{_a}; delete $v->{_a}; delete $w->{_a};
+ $x->badd($y)->bsub($z)->badd($u)->bsub($v)->bsub($w);
+
+ $x->bround($n-4);
+ delete $x->{_a} if $fallback == 1;
+ $x;
+ }
- 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();
+sub bcos
+ {
+ # Calculate a cosinus of x.
+ my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+
+ # Taylor: x^2 x^4 x^6 x^8
+ # cos = 1 - --- + --- - --- + --- ...
+ # 2! 4! 6! 8!
# 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(@r);
+
+ # constant object or error in _find_round_parameters?
+ return $x if $x->modify('bcos') || $x->is_nan();
+
+ return $x->bone(@r) if $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; # disable P
+ $scale = $params[0]+4; # at least four more for proper round
+ $params[2] = $r[2]; # 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
# 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
-
- my $xas = $x->as_number();
- my $gs = $xas->copy()->bsqrt(); # some guess
-
-# print "guess $gs\n";
- if (($x->{_e}->{sign} 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?
+ local $Math::BigInt::upgrade = undef;
+
+ my $last = 0;
+ my $over = $x * $x; # X ^ 2
+ my $x2 = $over->copy(); # X ^ 2; difference between terms
+ my $sign = 1; # start with -=
+ my $below = $self->new(2); my $factorial = $self->new(3);
+ $x->bone(); delete $x->{_a}; delete $x->{_p};
+
+ my $limit = $self->new("1E-". ($scale-1));
+ #my $steps = 0;
+ while (3 < 5)
{
- # exact result
- $x->{_m} = $gs; $x->{_e} = $MBI->bzero(); $x->bnorm();
- # shortcut to not run trough _find_round_parameters again
- if (defined $params[1])
+ # 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:
+ my $next = $over->copy()->bdiv($below,$scale);
+ last if $next->bacmp($limit) <= 0;
+
+ if ($sign == 0)
{
- $x->bround($params[1],$params[3]); # then round accordingly
+ $x->badd($next);
}
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;
+ $x->bsub($next);
}
- # re-enable A and P, upgrade is taken care of by "local"
- ${"$self\::accuracy"} = $ab; ${"$self\::precision"} = $pb;
- return $x;
+ $sign = 1-$sign; # alternate
+ # calculate things for the next term
+ $over->bmul($x2); # $x*$x
+ $below->bmul($factorial); $factorial->binc(); # n*(n+1)
+ $below->bmul($factorial); $factorial->binc(); # n*(n+1)
}
- $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
-
- 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');
- my $rem;
- while ($diff->bacmp($e) >= 0)
+ # shortcut to not run through _find_round_parameters again
+ if (defined $params[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};
-
- # shortcut to not run trough _find_round_parameters again
- if (defined $params[1])
- {
- $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
+sub bsin
{
- # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT
- # compute factorial numbers
- # modifies first argument
- my ($self,$x,@r) = objectify(1,@_);
-
- return $x->bnan()
- if (($x->{sign} ne '+') || # inf, NaN, <0 etc => NaN
- ($x->{_e}->{sign} ne '+')); # digits after dot?
+ # Calculate a sinus of x.
+ my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
- 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);
- }
+ # taylor: x^3 x^5 x^7 x^9
+ # sin = x - --- + --- - --- + --- ...
+ # 3! 5! 7! 9!
-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);
+ my ($scale,@params);
+ ($x,@params) = $x->_find_round_parameters(@r);
+
+ # constant object or error in _find_round_parameters?
+ return $x if $x->modify('bsin') || $x->is_nan();
+
+ return $x->bzero(@r) if $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; # disable P
+ $scale = $params[0]+4; # at least four more for proper round
+ $params[2] = $r[2]; # 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;
# 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)
+ my $last = 0;
+ my $over = $x * $x; # X ^ 2
+ my $x2 = $over->copy(); # X ^ 2; difference between terms
+ $over->bmul($x); # X ^ 3 as starting value
+ my $sign = 1; # start with -=
+ my $below = $self->new(6); my $factorial = $self->new(4);
+ delete $x->{_a}; delete $x->{_p};
+
+ my $limit = $self->new("1E-". ($scale-1));
+ #my $steps = 0;
+ while (3 < 5)
{
- $d->badd($d); # * 2
- last if $d->is_one(); # == 1
- $x->bsqrt(); # 0
- if ($d > 1)
+ # 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:
+ my $next = $over->copy()->bdiv($below,$scale);
+ last if $next->bacmp($limit) <= 0;
+
+ if ($sign == 0)
{
- $x->bsqrt(); $x->bmul($xc); $d->bdec(); # 1
+ $x->badd($next);
}
- print "at $x\n";
- }
- # assume fraction ends in 1
- $x->bsqrt(); # 1
- if (!$c->is_one())
- {
- $x->bmul( $xc->bpow($c) );
- }
- elsif (!$c->is_zero())
- {
- $x->bmul( $xc );
+ else
+ {
+ $x->bsub($next);
+ }
+ $sign = 1-$sign; # alternate
+ # calculate things for the next term
+ $over->bmul($x2); # $x*$x
+ $below->bmul($factorial); $factorial->binc(); # n*(n+1)
+ $below->bmul($factorial); $factorial->binc(); # n*(n+1)
}
- # done
- # 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 _pow
+sub batan2
+ {
+ # calculate arcus tangens of ($y/$x)
+
+ # set up parameters
+ my ($self,$y,$x,@r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$y,$x,@r) = objectify(2,@_);
+ }
+
+ return $y if $y->modify('batan2');
+
+ return $y->bnan() if ($y->{sign} eq $nan) || ($x->{sign} eq $nan);
+
+ return $upgrade->new($y)->batan2($upgrade->new($x),@r) if defined $upgrade;
+
+ # Y X
+ # 0 0 result is 0
+ # 0 +x result is 0
+ return $y->bzero(@r) if $y->is_zero() && $x->{sign} eq '+';
+
+ # Y X
+ # 0 -x result is PI
+ if ($y->is_zero())
+ {
+ # calculate PI
+ my $pi = $self->bpi(@r);
+ # modify $x in place
+ $y->{_m} = $pi->{_m};
+ $y->{_e} = $pi->{_e};
+ $y->{_es} = $pi->{_es};
+ $y->{sign} = '+';
+ return $y;
+ }
+
+ # Y X
+ # +y 0 result is PI/2
+ # -y 0 result is -PI/2
+ if ($y->is_inf() || $x->is_zero())
+ {
+ # calculate PI/2
+ my $pi = $self->bpi(@r);
+ # modify $x in place
+ $y->{_m} = $pi->{_m};
+ $y->{_e} = $pi->{_e};
+ $y->{_es} = $pi->{_es};
+ # -y => -PI/2, +y => PI/2
+ $y->{sign} = substr($y->{sign},0,1); # +inf => +
+ $MBI->_div($y->{_m}, $MBI->_new(2));
+ return $y;
+ }
+
+ # we need to limit the accuracy to protect against overflow
+ my $fallback = 0;
+ my ($scale,@params);
+ ($y,@params) = $y->_find_round_parameters(@r);
+
+ # error in _find_round_parameters?
+ return $y if $y->is_nan();
+
+ # 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
+ $params[1] = undef; # disable P
+ $scale = $params[0]+4; # at least four more for proper round
+ $params[2] = $r[2]; # 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[0] || $params[1]) + 4; # take whatever is defined
+ }
+
+ # inlined is_one() && is_one('-')
+ if ($MBI->_is_one($y->{_m}) && $MBI->_is_zero($y->{_e}))
+ {
+ # shortcut: 1 1 result is PI/4
+ # inlined is_one() && is_one('-')
+ if ($MBI->_is_one($x->{_m}) && $MBI->_is_zero($x->{_e}))
+ {
+ # 1,1 => PI/4
+ my $pi_4 = $self->bpi( $scale - 3);
+ # modify $x in place
+ $y->{_m} = $pi_4->{_m};
+ $y->{_e} = $pi_4->{_e};
+ $y->{_es} = $pi_4->{_es};
+ # 1 1 => +
+ # -1 1 => -
+ # 1 -1 => -
+ # -1 -1 => +
+ $y->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-';
+ $MBI->_div($y->{_m}, $MBI->_new(4));
+ return $y;
+ }
+ # shortcut: 1 int(X) result is _atan_inv(X)
+
+ # is integer
+ if ($x->{_es} eq '+')
+ {
+ my $x1 = $MBI->_copy($x->{_m});
+ $MBI->_lsft($x1, $x->{_e},10) unless $MBI->_is_zero($x->{_e});
+
+ my ($a,$b) = $self->_atan_inv($x1, $scale);
+ my $y_sign = $y->{sign};
+ # calculate A/B
+ $y->bone(); $y->{_m} = $a; my $y_d = $self->bone(); $y_d->{_m} = $b;
+ $y->bdiv($y_d, @r);
+ $y->{sign} = $y_sign;
+ return $y;
+ }
+ }
+
+ # handle all other cases
+ # X Y
+ # +x +y 0 to PI/2
+ # -x +y PI/2 to PI
+ # +x -y 0 to -PI/2
+ # -x -y -PI/2 to -PI
+
+ my $y_sign = $y->{sign};
+
+ # divide $x by $y
+ $y->bdiv($x, $scale) unless $x->is_one();
+ $y->batan(@r);
+
+ # restore sign
+ $y->{sign} = $y_sign;
+
+ $y;
+ }
+
+sub batan
{
- # Calculate a power where $y is a non-integer, like 2 ** 0.5
- my ($x,$y,$a,$p,$r) = @_;
+ # Calculate a arcus tangens of x.
+ my ($x,@r) = @_;
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;
-
- # u = y * ln x
- # _ _
- # Taylor: | u u^2 u^3 |
- # x ** y = 1 + | --- + --- + * ----- + ... |
- # |_ 1 1*2 1*2*3 _|
+ # taylor: x^3 x^5 x^7 x^9
+ # atan = x - --- + --- - --- + --- ...
+ # 3 5 7 9
# 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(@r);
+
+ # constant object or error in _find_round_parameters?
+ return $x if $x->modify('batan') || $x->is_nan();
+
+ if ($x->{sign} =~ /^[+-]inf\z/)
+ {
+ # +inf result is PI/2
+ # -inf result is -PI/2
+ # calculate PI/2
+ my $pi = $self->bpi(@r);
+ # modify $x in place
+ $x->{_m} = $pi->{_m};
+ $x->{_e} = $pi->{_e};
+ $x->{_es} = $pi->{_es};
+ # -y => -PI/2, +y => PI/2
+ $x->{sign} = substr($x->{sign},0,1); # +inf => +
+ $MBI->_div($x->{_m}, $MBI->_new(2));
+ return $x;
+ }
+
+ return $x->bzero(@r) if $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; # disable P
+ $scale = $params[0]+4; # at least four more for proper round
+ $params[2] = $r[2]; # 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
+ }
+
+ # 1 or -1 => PI/4
+ # inlined is_one() && is_one('-')
+ if ($MBI->_is_one($x->{_m}) && $MBI->_is_zero($x->{_e}))
+ {
+ my $pi = $self->bpi($scale - 3);
+ # modify $x in place
+ $x->{_m} = $pi->{_m};
+ $x->{_e} = $pi->{_e};
+ $x->{_es} = $pi->{_es};
+ # leave the sign of $x alone (+1 => +PI/4, -1 => -PI/4)
+ $MBI->_div($x->{_m}, $MBI->_new(4));
+ return $x;
+ }
+
+ # This series is only valid if -1 < x < 1, so for other x we need to
+ # to calculate PI/2 - atan(1/x):
+ my $one = $MBI->_new(1);
+ my $pi = undef;
+ if ($x->{_es} eq '+' && ($MBI->_acmp($x->{_m},$one) >= 0))
+ {
+ # calculate PI/2
+ $pi = $self->bpi($scale - 3);
+ $MBI->_div($pi->{_m}, $MBI->_new(2));
+ # calculate 1/$x:
+ my $x_copy = $x->copy();
+ # modify $x in place
+ $x->bone(); $x->bdiv($x_copy,$scale);
}
# 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;
# 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;
-
- my ($limit,$v,$u,$below,$factor,$next,$over);
-
- $u = $x->copy()->blog($scale)->bmul($y);
- $v = $self->bone(); # 1
- $factor = $self->new(2); # 2
- $x->bone(); # first term: 1
-
- $below = $v->copy();
- $over = $u->copy();
+ local $Math::BigInt::upgrade = undef;
- $limit = $self->new("1E-". ($scale-1));
+ my $last = 0;
+ my $over = $x * $x; # X ^ 2
+ my $x2 = $over->copy(); # X ^ 2; difference between terms
+ $over->bmul($x); # X ^ 3 as starting value
+ my $sign = 1; # start with -=
+ my $below = $self->new(3);
+ my $two = $self->new(2);
+ delete $x->{_a}; delete $x->{_p};
+
+ my $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,$scale);
- last if $next->bcmp($limit) <= 0;
- $x->badd($next);
-# print "at $x\n";
+ # anymore, so we stop:
+ my $next = $over->copy()->bdiv($below,$scale);
+ last if $next->bacmp($limit) <= 0;
+
+ if ($sign == 0)
+ {
+ $x->badd($next);
+ }
+ else
+ {
+ $x->bsub($next);
+ }
+ $sign = 1-$sign; # alternate
# calculate things for the next term
- $over *= $u; $below *= $factor; $factor->binc();
- #$steps++;
+ $over->bmul($x2); # $x*$x
+ $below->badd($two); # n += 2
}
-
- # shortcut to not run trough _find_round_parameters again
- if (defined $params[1])
+
+ if (defined $pi)
+ {
+ my $x_copy = $x->copy();
+ # modify $x in place
+ $x->{_m} = $pi->{_m};
+ $x->{_e} = $pi->{_e};
+ $x->{_es} = $pi->{_es};
+ # PI/2 - $x
+ $x->bsub($x_copy);
+ }
+
+ # 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;
$x;
}
-sub bpow
- {
- # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT
- # compute power of two numbers, second arg is used as integer
- # modifies first argument
-
- my ($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->is_one() || $y->is_one();
-
- return $x->_pow2($y,$a,$p,$r) if !$y->is_int(); # non-integer power
-
- my $y1 = $y->as_number(); # make bigint
- # if ($x == -1)
- if ($x->{sign} eq '-' && $x->{_m}->is_one() && $x->{_e}->is_zero())
- {
- # if $x == -1 and odd/even y => +1/-1 because +-1 ^ (+-1) => +-1
- return $y1->is_odd() ? $x : $x->babs(1);
- }
- if ($x->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();
- }
-
- # 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;
- $x->bnorm();
- if ($y->{sign} eq '-')
- {
- # modify $x in place!
- my $z = $x->copy(); $x->bzero()->binc();
- return $x->bdiv($z,$a,$p,$r); # round in one go (might ignore y's A!)
- }
- $x->round($a,$p,$r,$y);
- }
-
###############################################################################
# rounding functions
# 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);
+
+ # negative amount?
+ return $x->blsft($y->copy()->babs(),$n) if $y->{sign} =~ /^-/;
+
+ # the following call to bdiv() will return either quo or (quo,reminder):
+ $x->bdiv($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);
+
+ # negative amount?
+ return $x->brsft($y->copy()->babs(),$n) if $y->{sign} =~ /^-/;
+
+ $x->bmul($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 || $class;
no strict 'refs';
- if (!method_alias($name))
+ $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($name))
+ if (!_method_hand_up($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}) ));
}
##############################################################################
my $self = shift;
my $l = scalar @_;
my $lib = ''; my @a;
+ my $lib_kind = 'try';
+ $IMPORT=1;
for ( my $i = 0; $i < $l ; $i++)
{
-# print "at $_[$i] (",$_[$i+1]||'undef',")\n";
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 { $self->new(shift); };
}
elsif ($_[$i] eq 'upgrade')
$downgrade = $_[$i+1]; # or undef to disable
$i++;
}
- elsif ($_[$i] eq 'lib')
+ elsif ($_[$i] =~ /^(lib|try|only)\z/)
{
+ # alternative library
$lib = $_[$i+1] || ''; # default Calc
+ $lib_kind = $1; # lib, try or only
$i++;
}
elsif ($_[$i] eq 'with')
{
- $MBI = $_[$i+1] || 'Math::BigInt'; # default Math::BigInt
+ # alternative class for our private parts()
+ # XXX: no longer supported
+ # $MBI = $_[$i+1] || 'Math::BigInt';
$i++;
}
else
push @a, $_[$i];
}
}
-# print "mbf @a\n";
+ $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} };
- if ((defined $mbilib) && ($MBI eq 'Math::BigInt'))
+ if ((defined $mbilib) && ($MBI eq 'Math::BigInt::Calc'))
{
# MBI already loaded
- $MBI->import('lib',"$lib,$mbilib", 'objectify');
+ Math::BigInt->import( $lib_kind, "$lib,$mbilib", 'objectify');
}
else
{
- # MBI not loaded, or with ne "Math::BigInt"
+ # MBI not loaded, or with ne "Math::BigInt::Calc"
$lib .= ",$mbilib" if defined $mbilib;
-
-# my @parts = split /::/, $MBI; # Math::BigInt => Math BigInt
-# my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
-# $file = File::Spec->catfile (@parts, $file);
-
- if ($] < 5.006)
- {
- # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
- # used in the same script, or eval inside import().
- my @parts = split /::/, $MBI; # Math::BigInt => Math BigInt
- my $file = pop @parts; $file .= '.pm'; # BigInt => BigInt.pm
- $file = File::Spec->catfile (@parts, $file);
- eval { require $file; $MBI->import( lib => '$lib', 'objectify' ); }
- }
- else
- {
- my $rc = "use $MBI lib => '$lib', 'objectify';";
- eval $rc;
- }
+ $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_kind => $lib, 'objectify' );
+ }
+ if ($@)
+ {
+ require Carp; Carp::croak ("Couldn't load $lib: $! $@");
}
- die ("Couldn't load $MBI: $! $@") if $@;
+ # 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
- $self->export_to_level(1,$self,@a); # need this, too
+ $self->export_to_level(1,$self,@a); # export wanted functions
}
sub bnorm
{
# 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));
+ $z->as_bin();
+ }
+
+sub as_oct
+ {
+ # return number as octal digit string (only for integers defined)
+ my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+
+ return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
+ return '0' 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));
+ $z->as_oct();
+ }
sub as_number
{
# return copy as a bigint representation of this BigFloat number
my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
- my $z = $x->{_m}->copy();
- if ($x->{_e}->{sign} eq '-') # < 0
+ return $x if $x->modify('as_number');
+
+ 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;
}
use Math::BigFloat;
# Number creation
- $x = Math::BigFloat->new($str); # defaults to 0
- $nan = Math::BigFloat->bnan(); # create a NotANumber
- $zero = Math::BigFloat->bzero(); # create a +0
- $inf = Math::BigFloat->binf(); # create a +inf
- $inf = Math::BigFloat->binf('-'); # create a -inf
- $one = Math::BigFloat->bone(); # create a +1
- $one = Math::BigFloat->bone('-'); # create a -1
+ my $x = Math::BigFloat->new($str); # defaults to 0
+ my $y = $x->copy(); # make a true copy
+ my $nan = Math::BigFloat->bnan(); # create a NotANumber
+ my $zero = Math::BigFloat->bzero(); # create a +0
+ my $inf = Math::BigFloat->binf(); # create a +inf
+ my $inf = Math::BigFloat->binf('-'); # create a -inf
+ my $one = Math::BigFloat->bone(); # create a +1
+ my $mone = Math::BigFloat->bone('-'); # create a -1
+
+ my $pi = Math::BigFloat->bpi(100); # PI to 100 digits
+
+ # the following examples compute their result to 100 digits accuracy:
+ my $cos = Math::BigFloat->new(1)->bcos(100); # cosinus(1)
+ my $sin = Math::BigFloat->new(1)->bsin(100); # sinus(1)
+ my $atan = Math::BigFloat->new(1)->batan(100); # arcus tangens(1)
+
+ my $atan2 = Math::BigFloat->new( 1 )->batan2( 1 ,100); # batan(1)
+ my $atan2 = Math::BigFloat->new( 1 )->batan2( 8 ,100); # batan(1/8)
+ my $atan2 = Math::BigFloat->new( -2 )->batan2( 1 ,100); # batan(-2)
# Testing
$x->is_zero(); # true if arg is +0
$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)
- $x->blsft($y); # left shift
- $x->brsft($y); # right shift
- # return (quo,rem) or quo if scalar
+ $x->bmod($y); # modulus ($x % $y)
+ $x->bpow($y); # power of arguments ($x ** $y)
+ $x->bmodpow($exp,$mod); # modular exponentation (($num**$exp) % $mod))
+ $x->blsft($y, $n); # left shift by $y places in base $n
+ $x->brsft($y, $n); # right shift by $y places in base $n
+ # returns (quo,rem) or quo if in scalar context
- $x->blog($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->bexp(); # calculate e ** $x where e is Euler's number
$x->band($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 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.
+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.
-In case the result of one operation has more precision than specified,
+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
+
+The default value for C<div_scale> is 40.
+
+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 ( -$scale ) and fround ( 0 )
-These are effetively no-ops.
+These are effectively no-ops.
=back
All rounding functions take as a second parameter a rounding mode from one of
-the following: 'even', 'odd', '+inf', '-inf', 'zero' or 'trunc'.
+the following: 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'.
The default rounding mode is 'even'. By using
-C<< Math::BigFloat::round_mode($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
+
+Math::BigFloat supports all methods that Math::BigInt supports, except it
+calculates non-integer results when possible. Please see L<Math::BigInt>
+for a full description of each method. Below are just the most important
+differences:
+
+=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!
+
+=head2 bexp()
+
+ $x->bexp($accuracy); # calculate e ** X
+
+Calculates the expression C<e ** $x> where C<e> is Euler's number.
+
+This method was added in v1.82 of Math::BigInt (April 2007).
+
+=head2 bnok()
+
+ $x->bnok($y); # x over y (binomial coefficient n over k)
+
+Calculates the binomial coefficient n over k, also called the "choose"
+function. The result is equivalent to:
+
+ ( n ) n!
+ | - | = -------
+ ( k ) k!(n-k)!
+
+This method was added in v1.84 of Math::BigInt (April 2007).
+
+=head2 bpi()
+
+ print Math::BigFloat->bpi(100), "\n";
+
+Calculate PI to N digits (including the 3 before the dot). The result is
+rounded according to the current rounding mode, which defaults to "even".
+
+This method was added in v1.87 of Math::BigInt (June 2007).
+
+=head2 bcos()
+
+ my $x = Math::BigFloat->new(1);
+ print $x->bcos(100), "\n";
+
+Calculate the cosinus of $x, modifying $x in place.
+
+This method was added in v1.87 of Math::BigInt (June 2007).
+
+=head2 bsin()
+
+ my $x = Math::BigFloat->new(1);
+ print $x->bsin(100), "\n";
+
+Calculate the sinus of $x, modifying $x in place.
+
+This method was added in v1.87 of Math::BigInt (June 2007).
+
+=head2 batan2()
+
+ my $y = Math::BigFloat->new(2);
+ my $x = Math::BigFloat->new(3);
+ print $y->batan2($x), "\n";
+
+Calculate the arcus tanges of C<$y> divided by C<$x>, modifying $y in place.
+See also L<batan()>.
+
+This method was added in v1.87 of Math::BigInt (June 2007).
+
+=head2 batan()
+
+ my $x = Math::BigFloat->new(1);
+ print $x->batan(100), "\n";
+
+Calculate the arcus tanges of $x, modifying $x in place. See also L<batan2()>.
+
+This method was added in v1.87 of Math::BigInt (June 2007).
+
+=head2 bmuladd()
+
+ $x->bmuladd($y,$z);
+
+Multiply $x by $y, and then add $z to the result.
+
+This method was added in v1.87 of Math::BigInt (June 2007).
=head1 Autocreating constants
You can change this by using:
- use Math::BigFloat lib => 'BitVect';
+ use Math::BigFloat lib => 'GMP';
+
+Note: The keyword 'lib' will warn when the requested library could not be
+loaded. To suppress the warning use 'try' instead:
+
+ use Math::BigFloat try => 'GMP';
+
+To turn the warning into a die(), use 'only' instead:
+
+ use Math::BigFloat only => 'GMP';
The following would first try to find Math::BigInt::Foo, then
Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
use Math::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
-significant digit first, while BitVect.pm uses a bit vector of base 2, most
-significant bit first. Other modules might use even different means of
-representing the numbers. See the respective module documentation for further
-details.
+See the respective low-level library documentation for further details.
Please note that Math::BigFloat does B<not> use the denoted library itself,
but it merely passes the lib argument to Math::BigInt. So, instead of the need
use Math::BigFloat lib => 'GMP';
-Use the lib, Luke! And see L<Using Math::BigInt::Lite> for more details.
-
-=head2 Using Math::BigInt::Lite
-
-It is possible to use L<Math::BigInt::Lite> with Math::BigFloat:
-
- # 1
- use Math::BigFloat with => 'Math::BigInt::Lite';
+It is also possible to just require Math::BigFloat:
-There is no need to "use Math::BigInt" or "use Math::BigInt::Lite", but you
-can combine these if you want. For instance, you may want to use
-Math::BigInt objects in your main script, too.
+ require Math::BigFloat;
- # 2
- use Math::BigInt;
- use Math::BigFloat with => 'Math::BigInt::Lite';
-
-Of course, you can combine this with the C<lib> parameter.
+This will load the necessary things (like BigInt) when they are needed, and
+automatically.
- # 3
- use Math::BigFloat with => 'Math::BigInt::Lite', lib => 'GMP,Pari';
+See L<Math::BigInt> for more details than you ever wanted to know about using
+a different low-level library.
-If you want to use Math::BigInt's, too, simple add a Math::BigInt B<before>:
-
- # 4
- use Math::BigInt;
- use Math::BigFloat with => 'Math::BigInt::Lite', lib => 'GMP,Pari';
+=head2 Using Math::BigInt::Lite
-Notice that the module with the last C<lib> will "win" and thus
-it's lib will be used if the lib is available:
+For backwards compatibility reasons it is still possible to
+request a different storage class for use with Math::BigFloat:
- # 5
- use Math::BigInt lib => 'Bar,Baz';
- use Math::BigFloat with => 'Math::BigInt::Lite', lib => 'Foo';
+ use Math::BigFloat with => 'Math::BigInt::Lite';
-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.
+However, this request is ignored, as the current code now uses the low-level
+math libary for directly storing the number parts.
-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
+=head1 EXPORTS
-The old way still works though:
+C<Math::BigFloat> exports nothing by default, but can export the C<bpi()> method:
- # 6
- use Math::BigInt lib => 'Bar,Baz';
- use Math::BigFloat;
+ use Math::BigFloat qw/bpi/;
-But B<examples #3 and #4 are recommended> for usage.
+ print bpi(10), "\n";
=head1 BUGS
-=over 2
-
-=item *
-
-The following does not work yet:
+Please see the file BUGS in the CPAN distribution Math::BigInt for known bugs.
- $m = $x->mantissa();
- $e = $x->exponent();
- $y = $m * ( 10 ** $e );
- print "ok\n" if $x == $y;
+=head1 CAVEATS
-=item *
+Do not try to be clever to insert some operations in between switching
+libraries:
-There is no fmod() function yet.
+ require Math::BigFloat;
+ my $matter = Math::BigFloat->bone() + 4; # load BigInt and Calc
+ Math::BigFloat->import( lib => 'Pari' ); # load Pari, too
+ my $anti_matter = Math::BigFloat->bone()+4; # now use Pari
-=back
+This will create objects with numbers stored in two different backend libraries,
+and B<VERY BAD THINGS> will happen when you use these together:
-=head1 CAVEAT
+ my $flash_and_bang = $matter + $anti_matter; # Don't do this!
=over 1
=item bdiv
-The following will probably not do what you expect:
+The following will probably not print what you expect:
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
instead.
+=item brsft
+
+The following will probably not print what you expect:
+
+ my $c = Math::BigFloat->new('3.14159');
+ print $c->brsft(3,10),"\n"; # prints 0.00314153.1415
+
+It prints both quotient and remainder, since print calls C<brsft()> in list
+context. Also, C<< $c->brsft() >> will modify $c, so be careful.
+You probably want to use
+
+ print scalar $c->copy()->brsft(3,10),"\n";
+ # or if you really want to modify $c
+ print scalar $c->brsft(3,10),"\n";
+
+instead.
+
=item Modifying and =
Beware of:
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/~tels/Math-BigInt> 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 - 2006, and still
+at it in 2007.
=cut
projects / p5sagit/p5-mst-13.2.git / blobdiff