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-rwxr-xr-xMaster/tlpkg/tlperl0/lib/Math/BigFloat.pm4402
1 files changed, 4402 insertions, 0 deletions
diff --git a/Master/tlpkg/tlperl0/lib/Math/BigFloat.pm b/Master/tlpkg/tlperl0/lib/Math/BigFloat.pm
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+++ b/Master/tlpkg/tlperl0/lib/Math/BigFloat.pm
@@ -0,0 +1,4402 @@
+package Math::BigFloat;
+
+#
+# Mike grinned. 'Two down, infinity to go' - Mike Nostrus in 'Before and After'
+#
+
+# The following hash values are internally used:
+# _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 = '1.60';
+require 5.006;
+
+require Exporter;
+@ISA = qw/Math::BigInt/;
+@EXPORT_OK = qw/bpi/;
+
+use strict;
+# $_trap_inf/$_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 { my $rc = $_[2] ?
+ ref($_[0])->bcmp($_[1],$_[0]) :
+ 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 { $_[0]->as_number() }, # 'trunc' to bigint
+;
+
+##############################################################################
+# global constants, flags and assorted stuff
+
+# the following are public, but their usage is not recommended. Use the
+# accessor methods instead.
+
+# class constants, use Class->constant_name() to access
+# one of 'even', 'odd', '+inf', '-inf', 'zero', 'trunc' or 'common'
+$round_mode = 'even';
+$accuracy = undef;
+$precision = undef;
+$div_scale = 40;
+
+$upgrade = undef;
+$downgrade = undef;
+# 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 $rnd_mode, so we need to support it, too
+
+sub TIESCALAR { my ($class) = @_; bless \$round_mode, $class; }
+sub FETCH { return $round_mode; }
+sub STORE { $rnd_mode = $_[0]->round_mode($_[1]); }
+
+BEGIN
+ {
+ # 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;
+ }
+
+##############################################################################
+
+{
+ # 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 ffac fneg
+ fceil ffloor frsft flsft fone flog froot fexp
+ /;
+ # valid methods that can be handed up (for AUTOLOAD)
+ my %hand_ups = map { $_ => 1 }
+ 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 { exists $methods{$_[0]||''}; }
+ sub _method_hand_up { exists $hand_ups{$_[0]||''}; }
+}
+
+##############################################################################
+# constructors
+
+sub new
+ {
+ # create a new BigFloat object from a string or another bigfloat object.
+ # _e: exponent
+ # _m: mantissa
+ # sign => sign (+/-), or "NaN"
+
+ my ($class,$wanted,@r) = @_;
+
+ # avoid numify-calls by not using || on $wanted!
+ 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)) && UNIVERSAL::can( $wanted, "as_number"))
+ {
+ $self->{_m} = $wanted->as_number()->{value}; # get us a bigint copy
+ $self->{_e} = $MBI->_zero();
+ $self->{_es} = '+';
+ $self->{sign} = $wanted->sign();
+ return $self->bnorm();
+ }
+ # else: got a string or something maskerading as number (with overload)
+
+ # handle '+inf', '-inf' first
+ if ($wanted =~ /^[+-]?inf\z/)
+ {
+ return $downgrade->new($wanted) if $downgrade;
+
+ $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;
+ }
+
+ my ($mis,$miv,$mfv,$es,$ev) = Math::BigInt::_split($wanted);
+ if (!ref $mis)
+ {
+ if ($_trap_nan)
+ {
+ require Carp;
+ Carp::croak ("$wanted is not a number initialized to $class");
+ }
+
+ return $downgrade->bnan() if $downgrade;
+
+ $self->{_e} = $MBI->_zero();
+ $self->{_es} = '+';
+ $self->{_m} = $MBI->_zero();
+ $self->{sign} = $nan;
+ }
+ else
+ {
+ # 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
+ 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 && $self->{_es} eq '+')
+ {
+ if ($MBI->_is_zero( $self->{_e} ))
+ {
+ return $downgrade->new($$mis . $MBI->_str( $self->{_m} ));
+ }
+ 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;
+ }
+
+ 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 NaN
+ my $self = shift;
+
+ 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 +-inf
+ my $self = shift;
+
+ 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;
+ $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 0
+ my $self = shift;
+ $IMPORT=1; # call our import only once
+ $self->{_m} = $MBI->_zero();
+ $self->{_e} = $MBI->_one();
+ $self->{_es} = '+';
+ }
+
+sub isa
+ {
+ my ($self,$class) = @_;
+ return if $class =~ /^Math::BigInt/; # we aren't one of these
+ UNIVERSAL::isa($self,$class);
+ }
+
+sub config
+ {
+ # return (later set?) configuration data as hash ref
+ my $class = shift || 'Math::BigFloat';
+
+ if (@_ == 1 && ref($_[0]) ne 'HASH')
+ {
+ my $cfg = $class->SUPER::config();
+ return $cfg->{$_[0]};
+ }
+
+ my $cfg = $class->SUPER::config(@_);
+
+ # now we need only to override the ones that are different from our parent
+ $cfg->{class} = $class;
+ $cfg->{with} = $MBI;
+ $cfg;
+ }
+
+##############################################################################
+# string conversation
+
+sub bstr
+ {
+ # (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]) ? (undef,$_[0]) : objectify(1,@_);
+
+ 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 = '.';
+
+ # $x is zero?
+ my $not_zero = !($x->{sign} eq '+' && $MBI->_is_zero($x->{_m}));
+ if ($not_zero)
+ {
+ $es = $MBI->_str($x->{_m});
+ $len = CORE::length($es);
+ my $e = $MBI->_num($x->{_e});
+ $e = -$e if $x->{_es} eq '-';
+ if ($e < 0)
+ {
+ $dot = '';
+ # if _e is bigger than a scalar, the following will blow your memory
+ if ($e <= -$len)
+ {
+ my $r = abs($e) - $len;
+ $es = '0.'. ('0' x $r) . $es; $cad = -($len+$r);
+ }
+ else
+ {
+ 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 = '-'.$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
+ my $zeros = $x->{_a} - $cad; # cad == 0 => 12340
+ $zeros = $x->{_a} - $len if $cad != $len;
+ $es .= $dot.'0' x $zeros if $zeros > 0;
+ }
+ elsif ((($x->{_p} || 0) < 0))
+ {
+ # 123400 => 6, 0.1234 => 4, 0.001234 => 6
+ my $zeros = -$x->{_p} + $cad;
+ $es .= $dot.'0' x $zeros if $zeros > 0;
+ }
+ $es;
+ }
+
+sub bsstr
+ {
+ # (ref to BFLOAT or num_str ) return num_str
+ # Convert number from internal format to scientific string format.
+ # internal format is always normalized (no leading zeros, "-0E0" => "+0E0")
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ if ($x->{sign} !~ /^[+-]$/)
+ {
+ return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
+ return 'inf'; # +inf
+ }
+ my $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 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
+# XXX TODO this must be overwritten and return NaN for non-integer values
+# band(), bior(), bxor(), too
+#sub bnot
+# {
+# $class->SUPER::bnot($class,@_);
+# }
+
+sub bcmp
+ {
+ # Compares 2 values. Returns one of undef, <0, =0, >0. (suitable for sort)
+
+ # set up parameters
+ my ($self,$x,$y) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y) = objectify(2,@_);
+ }
+
+ return $upgrade->bcmp($x,$y) if defined $upgrade &&
+ ((!$x->isa($self)) || (!$y->isa($self)));
+
+ if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/))
+ {
+ # handle +-inf and NaN
+ return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
+ return 0 if ($x->{sign} eq $y->{sign}) && ($x->{sign} =~ /^[+-]inf$/);
+ return +1 if $x->{sign} eq '+inf';
+ return -1 if $x->{sign} eq '-inf';
+ return -1 if $y->{sign} eq '+inf';
+ return +1;
+ }
+
+ # check sign for speed first
+ return 1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # does also 0 <=> -y
+ return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # does also -x <=> 0
+
+ # shortcut
+ my $xz = $x->is_zero();
+ my $yz = $y->is_zero();
+ return 0 if $xz && $yz; # 0 <=> 0
+ return -1 if $xz && $y->{sign} eq '+'; # 0 <=> +y
+ return 1 if $yz && $x->{sign} eq '+'; # +x <=> 0
+
+ # adjust so that exponents are equal
+ my $lxm = $MBI->_len($x->{_m});
+ my $lym = $MBI->_len($y->{_m});
+ # the numify somewhat limits our length, but makes it much faster
+ 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;
+
+ # lengths (corrected by exponent) are equal
+ # so make mantissa equal length by padding with zero (shift left)
+ my $diff = $lxm - $lym;
+ my $xm = $x->{_m}; # not yet copy it
+ my $ym = $y->{_m};
+ if ($diff > 0)
+ {
+ $ym = $MBI->_copy($y->{_m});
+ $ym = $MBI->_lsft($ym, $MBI->_new($diff), 10);
+ }
+ elsif ($diff < 0)
+ {
+ $xm = $MBI->_copy($x->{_m});
+ $xm = $MBI->_lsft($xm, $MBI->_new(-$diff), 10);
+ }
+ my $rc = $MBI->_acmp($xm,$ym);
+ $rc = -$rc if $x->{sign} eq '-'; # -124 < -123
+ $rc <=> 0;
+ }
+
+sub bacmp
+ {
+ # Compares 2 values, ignoring their signs.
+ # Returns one of undef, <0, =0, >0. (suitable for sort)
+
+ # 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 undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
+ return 0 if ($x->is_inf() && $y->is_inf());
+ return 1 if ($x->is_inf() && !$y->is_inf());
+ return -1;
+ }
+
+ # shortcut
+ my $xz = $x->is_zero();
+ my $yz = $y->is_zero();
+ return 0 if $xz && $yz; # 0 <=> 0
+ return -1 if $xz && !$yz; # 0 <=> +y
+ return 1 if $yz && !$xz; # +x <=> 0
+
+ # adjust so that exponents are equal
+ 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 + $xes * $MBI->_num($x->{_e});
+ my $ly = $lym + $yes * $MBI->_num($y->{_e});
+ my $l = $lx - $ly;
+ return $l <=> 0 if $l != 0;
+
+ # lengths (corrected by exponent) are equal
+ # so make mantissa equal-length by padding with zero (shift left)
+ my $diff = $lxm - $lym;
+ my $xm = $x->{_m}; # not yet copy it
+ my $ym = $y->{_m};
+ if ($diff > 0)
+ {
+ $ym = $MBI->_copy($y->{_m});
+ $ym = $MBI->_lsft($ym, $MBI->_new($diff), 10);
+ }
+ elsif ($diff < 0)
+ {
+ $xm = $MBI->_copy($x->{_m});
+ $xm = $MBI->_lsft($xm, $MBI->_new(-$diff), 10);
+ }
+ $MBI->_acmp($xm,$ym);
+ }
+
+sub badd
+ {
+ # add second arg (BFLOAT or string) to first (BFLOAT) (modifies first)
+ # return result as BFLOAT
+
+ # 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} !~ /^[+-]$/))
+ {
+ # NaN first
+ return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
+ # inf handling
+ if (($x->{sign} =~ /^[+-]inf$/) && ($y->{sign} =~ /^[+-]inf$/))
+ {
+ # +inf++inf or -inf+-inf => same, rest is NaN
+ return $x if $x->{sign} eq $y->{sign};
+ return $x->bnan();
+ }
+ # +-inf + something => +inf; something +-inf => +-inf
+ $x->{sign} = $y->{sign}, return $x if $y->{sign} =~ /^[+-]inf$/;
+ return $x;
+ }
+
+ 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(@r) if $y->is_zero(); # x+0
+ if ($x->is_zero()) # 0+y
+ {
+ # make copy, clobbering up x (modify in place!)
+ $x->{_e} = $MBI->_copy($y->{_e});
+ $x->{_es} = $y->{_es};
+ $x->{_m} = $MBI->_copy($y->{_m});
+ $x->{sign} = $y->{sign} || $nan;
+ 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->_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
+ {
+ $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 $y->{sign})
+ {
+ # add
+ $x->{_m} = $MBI->_add($x->{_m}, $add);
+ }
+ 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,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+
+ return $x if $x->modify('binc');
+
+ if ($x->{_es} eq '-')
+ {
+ return $x->badd($self->bone(),@r); # digits after dot
+ }
+
+ if (!$MBI->_is_zero($x->{_e})) # _e == 0 for NaN, inf, -inf
+ {
+ # 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 '+')
+ {
+ $MBI->_inc($x->{_m});
+ return $x->bnorm()->bround(@r);
+ }
+ elsif ($x->{sign} eq '-')
+ {
+ $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->bone(),@r); # badd() does round
+ }
+
+sub bdec
+ {
+ # decrement arg by one
+ my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+
+ return $x if $x->modify('bdec');
+
+ if ($x->{_es} eq '-')
+ {
+ return $x->badd($self->bone('-'),@r); # digits after dot
+ }
+
+ if (!$MBI->_is_zero($x->{_e}))
+ {
+ $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)
+ {
+ $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 '+')
+ {
+ $MBI->_dec($x->{_m});
+ return $x->bnorm()->round(@r);
+ }
+ # inf, nan handling etc
+ $x->badd($self->bone('-'),@r); # does round
+ }
+
+sub DEBUG () { 0; }
+
+sub blog
+ {
+ my ($self,$x,$base,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+
+ return $x if $x->modify('blog');
+
+ # $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,@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 == 0)
+ {
+ # simulate old behaviour
+ $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[0] || $params[1]) + 4; # take whatever is defined
+ }
+
+ return $x->bzero(@params) if $x->is_one();
+ # 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 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;
+
+ # upgrade $x if $x is not a BigFloat (handle BigInt input)
+ # XXX TODO: rebless!
+ if (!$x->isa('Math::BigFloat'))
+ {
+ $x = Math::BigFloat->new($x);
+ $self = ref($x);
+ }
+
+ my $done = 0;
+
+ # If the base is defined and an integer, try to calculate integer result
+ # first. This is very fast, and in case the real result was found, we can
+ # stop right here.
+ if (defined $base && $base->is_int() && $x->is_int())
+ {
+ my $i = $MBI->_copy( $x->{_m} );
+ $MBI->_lsft( $i, $x->{_e}, 10 ) unless $MBI->_is_zero($x->{_e});
+ my $int = Math::BigInt->bzero();
+ $int->{value} = $i;
+ $int->blog($base->as_number());
+ # if ($exact)
+ if ($base->as_number()->bpow($int) == $x)
+ {
+ # found result, return it
+ $x->{_m} = $int->{value};
+ $x->{_e} = $MBI->_zero();
+ $x->{_es} = '+';
+ $x->bnorm();
+ $done = 1;
+ }
+ }
+
+ if ($done == 0)
+ {
+ # 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[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 _len_to_steps
+ {
+ # 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;
+
+ # two constants for the Ramanujan estimate of ln(N!)
+ my $lg2 = log(2 * 3.14159265) / 2;
+ my $lg10 = log(10);
+
+ # D = 50 => N => 42, so L = 40 and R = 50
+ my $l = 40; my $r = $d;
+
+ # 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);
+
+ # 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 bnok
+ {
+ # Calculate n over k (binomial coefficient or "choose" function) as integer.
+ # set up parameters
+ my ($self,$x,$y,@r) = (ref($_[0]),@_);
+
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$x,$y,@r) = objectify(2,@_);
+ }
+
+ return $x if $x->modify('bnok');
+
+ return $x->bnan() if $x->is_nan() || $y->is_nan();
+ return $x->binf() if $x->is_inf();
+
+ my $u = $x->as_int();
+ $u->bnok($y->as_int());
+
+ $x->{_m} = $u->{value};
+ $x->{_e} = $MBI->_zero();
+ $x->{_es} = '+';
+ $x->{sign} = '+';
+ $x->bnorm(@r);
+ }
+
+sub bexp
+ {
+ # Calculate e ** X (Euler's number to the power of X)
+ my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+
+ return $x if $x->modify('bexp');
+
+ 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,@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 == 0)
+ {
+ # simulate old behaviour
+ $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's not enough...
+ $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
+ }
+
+ return $x->bone(@params) if $x->is_zero();
+
+ if (!$x->isa('Math::BigFloat'))
+ {
+ $x = Math::BigFloat->new($x);
+ $self = ref($x);
+ }
+
+ # 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;
+
+ # This happens if the input had enough digits
+ # (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
+ 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; $$pbr = undef;
+ # we also need to disable any set A or P on $x (_find_round_parameters took
+ # them already into account), since these would interfere, too
+ delete $x->{_a}; delete $x->{_p};
+ # need to disable $upgrade in BigInt, to avoid deep recursion
+ local $Math::BigInt::upgrade = undef;
+
+ my ($limit,$v,$u,$below,$factor,$next,$over);
+
+ $u = $x->copy()->blog(undef,$scale)->bmul($y);
+ $v = $self->bone(); # 1
+ $factor = $self->new(2); # 2
+ $x->bone(); # first term: 1
+
+ $below = $v->copy();
+ $over = $u->copy();
+
+ $limit = $self->new("1E-". ($scale-1));
+ #my $steps = 0;
+ while (3 < 5)
+ {
+ # we calculate the next term, and add it to the last
+ # when the next term is below our limit, it won't affect the outcome
+ # anymore, so we stop:
+ $next = $over->copy()->bdiv($below,$scale);
+ last if $next->bacmp($limit) <= 0;
+ $x->badd($next);
+ # calculate things for the next term
+ $over *= $u; $below *= $factor; $factor->binc();
+
+ last if $x->{sign} !~ /^[-+]$/;
+
+ #$steps++;
+ }
+
+ # shortcut to not run through _find_round_parameters again
+ if (defined $params[0])
+ {
+ $x->bround($params[0],$params[2]); # then round accordingly
+ }
+ else
+ {
+ $x->bfround($params[1],$params[2]); # then round accordingly
+ }
+ if ($fallback)
+ {
+ # clear a/p after round, since user did not request it
+ delete $x->{_a}; delete $x->{_p};
+ }
+ # restore globals
+ $$abr = $ab; $$pbr = $pb;
+ $x;
+ }
+
+sub 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
+
+ # 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('bpow');
+
+ return $x->bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;
+ return $x if $x->{sign} =~ /^[+-]inf$/;
+
+ # 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}))
+ {
+ # 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();
+ }
+
+ 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 '-')
+ {
+ # 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!)
+ }
+ $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/)
+ {
+ # i.e., if it's NaN, +inf, or -inf...
+ return $num->bnan();
+ }
+
+ $num->bmodinv ($mod) if ($exp->{sign} eq '-');
+
+ # check num for valid values (also NaN if there was no inverse but $exp < 0)
+ return $num->bnan() if $num->{sign} !~ /^[+-]$/;
+
+ # $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)
+ {
+# $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;
+
+ }
+
+# 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 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;
+ }
+
+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,@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 == 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; $$pbr = undef;
+ # we also need to disable any set A or P on $x (_find_round_parameters took
+ # them already into account), since these would interfere, too
+ delete $x->{_a}; delete $x->{_p};
+ # need to disable $upgrade in BigInt, to avoid deep recursion
+ local $Math::BigInt::upgrade = undef;
+
+ my $last = 0;
+ my $over = $x * $x; # X ^ 2
+ my $x2 = $over->copy(); # X ^ 2; difference between terms
+ my $sign = 1; # start with -=
+ my $below = $self->new(2); my $factorial = $self->new(3);
+ $x->bone(); delete $x->{_a}; delete $x->{_p};
+
+ my $limit = $self->new("1E-". ($scale-1));
+ #my $steps = 0;
+ while (3 < 5)
+ {
+ # we calculate the next term, and add it to the last
+ # when the next term is below our limit, it won't affect the outcome
+ # anymore, so we stop:
+ my $next = $over->copy()->bdiv($below,$scale);
+ last if $next->bacmp($limit) <= 0;
+
+ if ($sign == 0)
+ {
+ $x->badd($next);
+ }
+ else
+ {
+ $x->bsub($next);
+ }
+ $sign = 1-$sign; # alternate
+ # calculate things for the next term
+ $over->bmul($x2); # $x*$x
+ $below->bmul($factorial); $factorial->binc(); # n*(n+1)
+ $below->bmul($factorial); $factorial->binc(); # n*(n+1)
+ }
+
+ # shortcut to not run through _find_round_parameters again
+ if (defined $params[0])
+ {
+ $x->bround($params[0],$params[2]); # then round accordingly
+ }
+ else
+ {
+ $x->bfround($params[1],$params[2]); # then round accordingly
+ }
+ if ($fallback)
+ {
+ # clear a/p after round, since user did not request it
+ delete $x->{_a}; delete $x->{_p};
+ }
+ # restore globals
+ $$abr = $ab; $$pbr = $pb;
+ $x;
+ }
+
+sub bsin
+ {
+ # Calculate a sinus of x.
+ my ($self,$x,@r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+
+ # taylor: x^3 x^5 x^7 x^9
+ # sin = x - --- + --- - --- + --- ...
+ # 3! 5! 7! 9!
+
+ # we need to limit the accuracy to protect against overflow
+ my $fallback = 0;
+ my ($scale,@params);
+ ($x,@params) = $x->_find_round_parameters(@r);
+
+ # constant object or error in _find_round_parameters?
+ return $x if $x->modify('bsin') || $x->is_nan();
+
+ return $x->bzero(@r) if $x->is_zero();
+
+ # no rounding at all, so must use fallback
+ if (scalar @params == 0)
+ {
+ # simulate old behaviour
+ $params[0] = $self->div_scale(); # and round to it as accuracy
+ $params[1] = undef; # disable P
+ $scale = $params[0]+4; # at least four more for proper round
+ $params[2] = $r[2]; # round mode by caller or undef
+ $fallback = 1; # to clear a/p afterwards
+ }
+ else
+ {
+ # the 4 below is empirical, and there might be cases where it is not
+ # enough...
+ $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
+ }
+
+ # when user set globals, they would interfere with our calculation, so
+ # disable them and later re-enable them
+ no strict 'refs';
+ my $abr = "$self\::accuracy"; my $ab = $$abr; $$abr = undef;
+ my $pbr = "$self\::precision"; my $pb = $$pbr; $$pbr = undef;
+ # we also need to disable any set A or P on $x (_find_round_parameters took
+ # them already into account), since these would interfere, too
+ delete $x->{_a}; delete $x->{_p};
+ # need to disable $upgrade in BigInt, to avoid deep recursion
+ local $Math::BigInt::upgrade = undef;
+
+ my $last = 0;
+ my $over = $x * $x; # X ^ 2
+ my $x2 = $over->copy(); # X ^ 2; difference between terms
+ $over->bmul($x); # X ^ 3 as starting value
+ my $sign = 1; # start with -=
+ my $below = $self->new(6); my $factorial = $self->new(4);
+ delete $x->{_a}; delete $x->{_p};
+
+ my $limit = $self->new("1E-". ($scale-1));
+ #my $steps = 0;
+ while (3 < 5)
+ {
+ # we calculate the next term, and add it to the last
+ # when the next term is below our limit, it won't affect the outcome
+ # anymore, so we stop:
+ my $next = $over->copy()->bdiv($below,$scale);
+ last if $next->bacmp($limit) <= 0;
+
+ if ($sign == 0)
+ {
+ $x->badd($next);
+ }
+ else
+ {
+ $x->bsub($next);
+ }
+ $sign = 1-$sign; # alternate
+ # calculate things for the next term
+ $over->bmul($x2); # $x*$x
+ $below->bmul($factorial); $factorial->binc(); # n*(n+1)
+ $below->bmul($factorial); $factorial->binc(); # n*(n+1)
+ }
+
+ # shortcut to not run through _find_round_parameters again
+ if (defined $params[0])
+ {
+ $x->bround($params[0],$params[2]); # then round accordingly
+ }
+ else
+ {
+ $x->bfround($params[1],$params[2]); # then round accordingly
+ }
+ if ($fallback)
+ {
+ # clear a/p after round, since user did not request it
+ delete $x->{_a}; delete $x->{_p};
+ }
+ # restore globals
+ $$abr = $ab; $$pbr = $pb;
+ $x;
+ }
+
+sub batan2
+ {
+ # calculate arcus tangens of ($y/$x)
+
+ # set up parameters
+ my ($self,$y,$x,@r) = (ref($_[0]),@_);
+ # objectify is costly, so avoid it
+ if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1])))
+ {
+ ($self,$y,$x,@r) = objectify(2,@_);
+ }
+
+ return $y if $y->modify('batan2');
+
+ return $y->bnan() if ($y->{sign} eq $nan) || ($x->{sign} eq $nan);
+
+ # Y X
+ # 0 0 result is 0
+ # 0 +x result is 0
+ # ? inf result is 0
+ return $y->bzero(@r) if ($x->is_inf('+') && !$y->is_inf()) || ($y->is_zero() && $x->{sign} eq '+');
+
+ # Y X
+ # != 0 -inf result is +- pi
+ if ($x->is_inf() || $y->is_inf())
+ {
+ # calculate PI
+ my $pi = $self->bpi(@r);
+ if ($y->is_inf())
+ {
+ # upgrade to BigRat etc.
+ return $upgrade->new($y)->batan2($upgrade->new($x),@r) if defined $upgrade;
+ if ($x->{sign} eq '-inf')
+ {
+ # calculate 3 pi/4
+ $MBI->_mul($pi->{_m}, $MBI->_new(3));
+ $MBI->_div($pi->{_m}, $MBI->_new(4));
+ }
+ elsif ($x->{sign} eq '+inf')
+ {
+ # calculate pi/4
+ $MBI->_div($pi->{_m}, $MBI->_new(4));
+ }
+ else
+ {
+ # calculate pi/2
+ $MBI->_div($pi->{_m}, $MBI->_new(2));
+ }
+ $y->{sign} = substr($y->{sign},0,1); # keep +/-
+ }
+ # modify $y in place
+ $y->{_m} = $pi->{_m};
+ $y->{_e} = $pi->{_e};
+ $y->{_es} = $pi->{_es};
+ # keep the sign of $y
+ return $y;
+ }
+
+ return $upgrade->new($y)->batan2($upgrade->new($x),@r) if defined $upgrade;
+
+ # Y X
+ # 0 -x result is PI
+ if ($y->is_zero())
+ {
+ # calculate PI
+ my $pi = $self->bpi(@r);
+ # modify $y 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 ($x->is_zero())
+ {
+ # calculate PI/2
+ my $pi = $self->bpi(@r);
+ # modify $y in place
+ $y->{_m} = $pi->{_m};
+ $y->{_e} = $pi->{_e};
+ $y->{_es} = $pi->{_es};
+ # -y => -PI/2, +y => PI/2
+ $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 $y 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 arcus tangens of x.
+ my ($x,@r) = @_;
+ my $self = ref($x);
+
+ # 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,@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 == 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
+ }
+
+ # 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 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; $$pbr = undef;
+ # we also need to disable any set A or P on $x (_find_round_parameters took
+ # them already into account), since these would interfere, too
+ delete $x->{_a}; delete $x->{_p};
+ # need to disable $upgrade in BigInt, to avoid deep recursion
+ local $Math::BigInt::upgrade = undef;
+
+ my $last = 0;
+ my $over = $x * $x; # X ^ 2
+ my $x2 = $over->copy(); # X ^ 2; difference between terms
+ $over->bmul($x); # X ^ 3 as starting value
+ my $sign = 1; # start with -=
+ my $below = $self->new(3);
+ my $two = $self->new(2);
+ delete $x->{_a}; delete $x->{_p};
+
+ my $limit = $self->new("1E-". ($scale-1));
+ #my $steps = 0;
+ while (3 < 5)
+ {
+ # we calculate the next term, and add it to the last
+ # when the next term is below our limit, it won't affect the outcome
+ # anymore, so we stop:
+ my $next = $over->copy()->bdiv($below,$scale);
+ last if $next->bacmp($limit) <= 0;
+
+ if ($sign == 0)
+ {
+ $x->badd($next);
+ }
+ else
+ {
+ $x->bsub($next);
+ }
+ $sign = 1-$sign; # alternate
+ # calculate things for the next term
+ $over->bmul($x2); # $x*$x
+ $below->badd($two); # n += 2
+ }
+
+ if (defined $pi)
+ {
+ my $x_copy = $x->copy();
+ # modify $x in place
+ $x->{_m} = $pi->{_m};
+ $x->{_e} = $pi->{_e};
+ $x->{_es} = $pi->{_es};
+ # PI/2 - $x
+ $x->bsub($x_copy);
+ }
+
+ # shortcut to not run through _find_round_parameters again
+ if (defined $params[0])
+ {
+ $x->bround($params[0],$params[2]); # then round accordingly
+ }
+ else
+ {
+ $x->bfround($params[1],$params[2]); # then round accordingly
+ }
+ if ($fallback)
+ {
+ # clear a/p after round, since user did not request it
+ delete $x->{_a}; delete $x->{_p};
+ }
+ # restore globals
+ $$abr = $ab; $$pbr = $pb;
+ $x;
+ }
+
+###############################################################################
+# rounding functions
+
+sub bfround
+ {
+ # precision: round to the $Nth digit left (+$n) or right (-$n) from the '.'
+ # $n == 0 means round to integer
+ # expects and returns normalized numbers!
+ my $x = shift; my $self = ref($x) || $x; $x = $self->new(shift) if !ref($x);
+
+ 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())
+ {
+ $x->{_p} = $scale if !defined $x->{_p} || $x->{_p} < $scale; # -3 < -2
+ return $x;
+ }
+ return $x if $x->{sign} !~ /^[+-]$/;
+
+ # 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
+ delete $x->{_a}; # and clear A
+ if ($scale < 0)
+ {
+ # round right from the '.'
+
+ return $x if $x->{_es} eq '+'; # e >= 0 => nothing to round
+
+ $scale = -$scale; # positive for simplicity
+ 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
+ # 0.001 1e-3 1 2 3
+ # 1.23 123e-2 3 0 2
+ # 1.2345 12345e-4 5 0 4
+
+ # do not round after/right of the $dad
+ return $x if $scale > $dad; # 0.123, scale >= 3 => exit
+
+ # round to zero if rounding inside the $zad, but not for last zero like:
+ # 0.0065, scale -2, round last '0' with following '65' (scale == zad case)
+ return $x->bzero() if $scale < $zad;
+ if ($scale == $zad) # for 0.006, scale -3 and trunc
+ {
+ $scale = -$len;
+ }
+ else
+ {
+ # adjust round-point to be inside mantissa
+ if ($zad != 0)
+ {
+ $scale = $scale-$zad;
+ }
+ else
+ {
+ my $dbd = $len - $dad; $dbd = 0 if $dbd < 0; # digits before dot
+ $scale = $dbd+$scale;
+ }
+ }
+ }
+ else
+ {
+ # round left from the '.'
+
+ # 123 => 100 means length(123) = 3 - $scale (2) => 1
+
+ my $dbt = $MBI->_len($x->{_m});
+ # digits before dot
+ 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
+ return $x if $scale == 1 && $dbt <= $dbd;
+ # maximum digits before dot
+ ++$dbd;
+
+ if ($scale > $dbd)
+ {
+ # not enough digits before dot, so round to zero
+ return $x->bzero;
+ }
+ elsif ( $scale == $dbd )
+ {
+ # maximum
+ $scale = -$dbt;
+ }
+ else
+ {
+ $scale = $dbd - $scale;
+ }
+ }
+ # pass sign to bround for rounding modes '+inf' and '-inf'
+ 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();
+ }
+
+sub bround
+ {
+ # 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);
+
+ if (($_[0] || 0) < 0)
+ {
+ require Carp; Carp::croak ('bround() needs positive accuracy');
+ }
+
+ my ($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 $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) || $x->{sign} !~ /^[+-]$/;
+
+ # 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
+ 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
+ delete $x->{_p}; # and clear P
+ $x->bnorm(); # del trailing zeros gen. by bround()
+ }
+
+sub bfloor
+ {
+ # return integer less or equal then $x
+ my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+
+ return $x if $x->modify('bfloor');
+
+ return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
+
+ # if $x has digits after dot
+ if ($x->{_es} eq '-')
+ {
+ $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);
+ }
+
+sub bceil
+ {
+ # return integer greater or equal then $x
+ my ($self,$x,$a,$p,$r) = ref($_[0]) ? (ref($_[0]),@_) : objectify(1,@_);
+
+ return $x if $x->modify('bceil');
+ return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
+
+ # if $x has digits after dot
+ if ($x->{_es} eq '-')
+ {
+ $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 $n)
+
+ # set up parameters
+ my ($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 = $self->new($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 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('blsft');
+ return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf
+
+ $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, avoid it by empty sub
+ }
+
+sub AUTOLOAD
+ {
+ # make fxxx and bxxx both work by selectively mapping fxxx() to MBF::bxxx()
+ # or falling back to MBI::bxxx()
+ my $name = $AUTOLOAD;
+
+ $name =~ s/(.*):://; # split package
+ my $c = $1 || $class;
+ no strict 'refs';
+ $c->import() if $IMPORT == 0;
+ if (!_method_alias($name))
+ {
+ if (!defined $name)
+ {
+ # delayed load of Carp and avoid recursion
+ require Carp;
+ Carp::croak ("$c: Can't call a method without name");
+ }
+ if (!_method_hand_up($name))
+ {
+ # delayed load of Carp and avoid recursion
+ require Carp;
+ Carp::croak ("Can't call $c\-\>$name, not a valid method");
+ }
+ # try one level up, but subst. bxxx() for fxxx() since MBI only got bxxx()
+ $name =~ s/^f/b/;
+ return &{"Math::BigInt"."::$name"}(@_);
+ }
+ my $bname = $name; $bname =~ s/^f/b/;
+ $c .= "::$name";
+ *{$c} = \&{$bname};
+ &{$c}; # uses @_
+ }
+
+sub exponent
+ {
+ # return a copy of the exponent
+ my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+
+ if ($x->{sign} !~ /^[+-]$/)
+ {
+ my $s = $x->{sign}; $s =~ s/^[+-]//;
+ return Math::BigInt->new($s); # -inf, +inf => +inf
+ }
+ Math::BigInt->new( $x->{_es} . $MBI->_str($x->{_e}));
+ }
+
+sub mantissa
+ {
+ # return a copy of the mantissa
+ my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+
+ if ($x->{sign} !~ /^[+-]$/)
+ {
+ my $s = $x->{sign}; $s =~ s/^[+]//;
+ return Math::BigInt->new($s); # -inf, +inf => +inf
+ }
+ my $m = Math::BigInt->new( $MBI->_str($x->{_m}));
+ $m->bneg() if $x->{sign} eq '-';
+
+ $m;
+ }
+
+sub parts
+ {
+ # return a copy of both the exponent and the mantissa
+ my ($self,$x) = ref($_[0]) ? (ref($_[0]),$_[0]) : objectify(1,@_);
+
+ if ($x->{sign} !~ /^[+-]$/)
+ {
+ 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 = Math::BigInt->bzero();
+ $m->{value} = $MBI->_copy($x->{_m});
+ $m->bneg() if $x->{sign} eq '-';
+ ($m, Math::BigInt->new( $x->{_es} . $MBI->_num($x->{_e}) ));
+ }
+
+##############################################################################
+# private stuff (internal use only)
+
+sub import
+ {
+ my $self = shift;
+ my $l = scalar @_;
+ my $lib = ''; my @a;
+ my $lib_kind = 'try';
+ $IMPORT=1;
+ for ( my $i = 0; $i < $l ; $i++)
+ {
+ if ( $_[$i] eq ':constant' )
+ {
+ # 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')
+ {
+ # this causes upgrading
+ $upgrade = $_[$i+1]; # or undef to disable
+ $i++;
+ }
+ elsif ($_[$i] eq 'downgrade')
+ {
+ # this causes downgrading
+ $downgrade = $_[$i+1]; # or undef to disable
+ $i++;
+ }
+ elsif ($_[$i] =~ /^(lib|try|only)\z/)
+ {
+ # alternative library
+ $lib = $_[$i+1] || ''; # default Calc
+ $lib_kind = $1; # lib, try or only
+ $i++;
+ }
+ elsif ($_[$i] eq 'with')
+ {
+ # alternative class for our private parts()
+ # XXX: no longer supported
+ # $MBI = $_[$i+1] || 'Math::BigInt';
+ $i++;
+ }
+ else
+ {
+ push @a, $_[$i];
+ }
+ }
+
+ $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::Calc'))
+ {
+ # MBI already loaded
+ Math::BigInt->import( $lib_kind, "$lib,$mbilib", 'objectify');
+ }
+ else
+ {
+ # MBI not loaded, or with ne "Math::BigInt::Calc"
+ $lib .= ",$mbilib" if defined $mbilib;
+ $lib =~ s/^,//; # don't leave empty
+
+ # replacement library can handle lib statement, but also could ignore it
+
+ # Perl < 5.6.0 dies with "out of memory!" when eval() and ':constant' is
+ # used in the same script, or eval inside import(). So we require MBI:
+ require Math::BigInt;
+ Math::BigInt->import( $lib_kind => $lib, 'objectify' );
+ }
+ if ($@)
+ {
+ require Carp; Carp::croak ("Couldn't load $lib: $! $@");
+ }
+ # find out which one was actually loaded
+ $MBI = Math::BigInt->config()->{lib};
+
+ # register us with MBI to get notified of future lib changes
+ Math::BigInt::_register_callback( $self, sub { $MBI = $_[0]; } );
+
+ $self->export_to_level(1,$self,@a); # export wanted functions
+ }
+
+sub bnorm
+ {
+ # adjust m and e so that m is smallest possible
+ my ($self,$x) = ref($_[0]) ? (undef,$_[0]) : objectify(1,@_);
+
+ return $x if $x->{sign} !~ /^[+-]$/; # inf, nan etc
+
+ 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 '-')
+ {
+ 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} = '+';
+ }
+ }
+ 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});
+ }
+
+ $x; # MBI bnorm is no-op, so dont call it
+ }
+
+##############################################################################
+
+sub as_hex
+ {
+ # return number as hexadecimal string (only for integers defined)
+ my ($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,@_);
+
+ return $x if $x->modify('as_number');
+
+ if (!$x->isa('Math::BigFloat'))
+ {
+ # if the object can as_number(), use it
+ return $x->as_number() if $x->can('as_number');
+ # otherwise, get us a float and then a number
+ $x = $x->can('as_float') ? $x->as_float() : $self->new(0+"$x");
+ }
+
+ my $z = $MBI->_copy($x->{_m});
+ if ($x->{_es} eq '-') # < 0
+ {
+ $MBI->_rsft( $z, $x->{_e},10);
+ }
+ elsif (! $MBI->_is_zero($x->{_e})) # > 0
+ {
+ $MBI->_lsft( $z, $x->{_e},10);
+ }
+ $z = Math::BigInt->new( $x->{sign} . $MBI->_num($z));
+ $z;
+ }
+
+sub length
+ {
+ my $x = shift;
+ my $class = ref($x) || $x;
+ $x = $class->new(shift) unless ref($x);
+
+ 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 = 0;
+ $t = $MBI->_num($x->{_e}) if $x->{_es} eq '-';
+ return ($len, $t);
+ }
+ $len;
+ }
+
+1;
+__END__
+
+=head1 NAME
+
+Math::BigFloat - Arbitrary size floating point math package
+
+=head1 SYNOPSIS
+
+ use Math::BigFloat;
+
+ # Number creation
+ 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_nan(); # true if arg is NaN
+ $x->is_one(); # true if arg is +1
+ $x->is_one('-'); # true if arg is -1
+ $x->is_odd(); # true if odd, false for even
+ $x->is_even(); # true if even, false for odd
+ $x->is_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)
+ $x->bacmp($y); # compare absolutely (undef,<0,=0,>0)
+ $x->sign(); # return the sign, either +,- or NaN
+ $x->digit($n); # return the nth digit, counting from right
+ $x->digit(-$n); # return the nth digit, counting from left
+
+ # The following all modify their first argument. 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->bone(); # set $x to +1
+ $x->bone('-'); # set $x to -1
+ $x->binf(); # set $x to inf
+ $x->binf('-'); # set $x to -inf
+
+ $x->bneg(); # negation
+ $x->babs(); # absolute value
+ $x->bnorm(); # normalize (no-op)
+ $x->bnot(); # two's complement (bit wise not)
+ $x->binc(); # increment x by 1
+ $x->bdec(); # decrement x by 1
+
+ $x->badd($y); # addition (add $y to $x)
+ $x->bsub($y); # subtraction (subtract $y from $x)
+ $x->bmul($y); # multiplication (multiply $x by $y)
+ $x->bdiv($y); # divide, set $x to quotient
+ # return (quo,rem) or quo if scalar
+
+ $x->bmod($y); # modulus ($x % $y)
+ $x->bpow($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(); # 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
+ $x->bxor($y); # bit-wise exclusive 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: 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->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 (including basic math operations) are overloaded if you
+declare your big floating point numbers as
+
+ $i = new Math::BigFloat '12_3.456_789_123_456_789E-2';
+
+Operations with overloaded operators preserve the arguments, which is
+exactly what you expect.
+
+=head2 Canonical notation
+
+Input to these routines are either BigFloat objects, or strings of the
+following four forms:
+
+=over 2
+
+=item *
+
+C</^[+-]\d+$/>
+
+=item *
+
+C</^[+-]\d+\.\d*$/>
+
+=item *
+
+C</^[+-]\d+E[+-]?\d+$/>
+
+=item *
+
+C</^[+-]\d*\.\d+E[+-]?\d+$/>
+
+=back
+
+all with optional leading and trailing zeros and/or spaces. Additionally,
+numbers are allowed to have an underscore between any two digits.
+
+Empty strings as well as other illegal numbers results in 'NaN'.
+
+bnorm() on a BigFloat object is now effectively a no-op, since the numbers
+are always stored in normalized form. On a string, it creates a BigFloat
+object.
+
+=head2 Output
+
+Output values are BigFloat objects (normalized), except for bstr() and bsstr().
+
+The string output will always have leading and trailing zeros stripped and drop
+a plus sign. C<bstr()> will give you always the form with a decimal point,
+while C<bsstr()> (s for scientific) gives you the scientific notation.
+
+ Input bstr() bsstr()
+ '-0' '0' '0E1'
+ ' -123 123 123' '-123123123' '-123123123E0'
+ '00.0123' '0.0123' '123E-4'
+ '123.45E-2' '1.2345' '12345E-4'
+ '10E+3' '10000' '1E4'
+
+Some routines (C<is_odd()>, C<is_even()>, C<is_zero()>, C<is_one()>,
+C<is_nan()>) return true or false, while others (C<bcmp()>, C<bacmp()>)
+return either undef, <0, 0 or >0 and are suited for sort.
+
+Actual math is done by using the class defined with C<with => Class;> (which
+defaults to BigInts) to represent the mantissa and exponent.
+
+The sign C</^[+-]$/> 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.
+
+=head2 C<mantissa()>, C<exponent()> and C<parts()>
+
+C<mantissa()> and C<exponent()> return the said parts of the BigFloat
+as BigInts such that:
+
+ $m = $x->mantissa();
+ $e = $x->exponent();
+ $y = $m * ( 10 ** $e );
+ print "ok\n" if $x == $y;
+
+C<< ($m,$e) = $x->parts(); >> is just a shortcut giving you both of them.
+
+A zero is represented and returned as C<0E1>, B<not> C<0E0> (after Knuth).
+
+Currently the mantissa is reduced as much as possible, favouring higher
+exponents over lower ones (e.g. returning 1e7 instead of 10e6 or 10000000e0).
+This might change in the future, so do not depend on it.
+
+=head2 Accuracy vs. Precision
+
+See also: L<Rounding|Rounding>.
+
+Math::BigFloat supports both precision (rounding to a certain place before or
+after the dot) and accuracy (rounding to a certain number of digits). For a
+full documentation, examples and tips on these topics please see the large
+section about rounding in L<Math::BigInt>.
+
+Since things like C<sqrt(2)> or C<1 / 3> must presented with a limited
+accuracy lest a operation consumes all resources, each operation produces
+no more than the requested number of digits.
+
+If there is no gloabl precision or accuracy set, B<and> the operation in
+question was not called with a requested precision or accuracy, B<and> the
+input $x has no accuracy or precision set, then a fallback parameter will
+be used. For historical reasons, it is called C<div_scale> and can be accessed
+via:
+
+ $d = Math::BigFloat->div_scale(); # query
+ Math::BigFloat->div_scale($n); # set to $n digits
+
+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->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
+
+=over 2
+
+=item ffround ( +$scale )
+
+Rounds to the $scale'th place left from the '.', counting from the dot.
+The first digit is numbered 1.
+
+=item ffround ( -$scale )
+
+Rounds to the $scale'th place right from the '.', counting from the dot.
+
+=item ffround ( 0 )
+
+Rounds to an integer.
+
+=item fround ( +$scale )
+
+Preserves accuracy to $scale digits from the left (aka significant digits)
+and pads the rest with zeros. If the number is between 1 and -1, the
+significant digits count from the first non-zero after the '.'
+
+=item fround ( -$scale ) and fround ( 0 )
+
+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', 'trunc' or 'common'.
+
+The default rounding mode is 'even'. By using
+C<< Math::BigFloat->round_mode($round_mode); >> you can get and set the default
+mode for subsequent rounding. The usage of C<$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
+'trunc' as rounding mode to make it equivalent to:
+
+ $x = 2.5;
+ $y = int($x) + 2;
+
+You can override this by passing the desired rounding mode as parameter to
+C<as_number()>:
+
+ $x = Math::BigFloat->new(2.5);
+ $y = $x->as_number('odd'); # $y = 3
+
+=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
+
+After C<use Math::BigFloat ':constant'> all the floating point constants
+in the given scope are converted to C<Math::BigFloat>. This conversion
+happens at compile time.
+
+In particular
+
+ perl -MMath::BigFloat=:constant -e 'print 2E-100,"\n"'
+
+prints the value of C<2E-100>. Note that without conversion of
+constants the expression 2E-100 will be calculated as normal floating point
+number.
+
+Please note that ':constant' does not affect integer constants, nor binary
+nor hexadecimal constants. Use L<bignum> or L<Math::BigInt> to get this to
+work.
+
+=head2 Math library
+
+Math with the numbers is done (by default) by a module called
+Math::BigInt::Calc. This is equivalent to saying:
+
+ use Math::BigFloat lib => 'Calc';
+
+You can change this by using:
+
+ use Math::BigFloat lib => 'GMP';
+
+B<Note>: General purpose packages should not be explicit about the library
+to use; let the script author decide which is best.
+
+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';
+
+If your script works with huge numbers and Calc is too slow for them,
+you can also for the loading of one of these libraries and if none
+of them can be used, the code will die:
+
+ use Math::BigFloat only => 'GMP,Pari';
+
+The following would first try to find Math::BigInt::Foo, then
+Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:
+
+ use Math::BigFloat lib => 'Foo,Math::BigInt::Bar';
+
+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
+to do:
+
+ use Math::BigInt lib => 'GMP';
+ use Math::BigFloat;
+
+you can roll it all into one line:
+
+ use Math::BigFloat lib => 'GMP';
+
+It is also possible to just require Math::BigFloat:
+
+ require Math::BigFloat;
+
+This will load the necessary things (like BigInt) when they are needed, and
+automatically.
+
+See L<Math::BigInt> for more details than you ever wanted to know about using
+a different low-level library.
+
+=head2 Using Math::BigInt::Lite
+
+For backwards compatibility reasons it is still possible to
+request a different storage class for use with Math::BigFloat:
+
+ use Math::BigFloat with => 'Math::BigInt::Lite';
+
+However, this request is ignored, as the current code now uses the low-level
+math libary for directly storing the number parts.
+
+=head1 EXPORTS
+
+C<Math::BigFloat> exports nothing by default, but can export the C<bpi()> method:
+
+ use Math::BigFloat qw/bpi/;
+
+ print bpi(10), "\n";
+
+=head1 BUGS
+
+Please see the file BUGS in the CPAN distribution Math::BigInt for known bugs.
+
+=head1 CAVEATS
+
+Do not try to be clever to insert some operations in between switching
+libraries:
+
+ 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
+
+This will create objects with numbers stored in two different backend libraries,
+and B<VERY BAD THINGS> will happen when you use these together:
+
+ my $flash_and_bang = $matter + $anti_matter; # Don't do this!
+
+=over 1
+
+=item stringify, bstr()
+
+Both stringify and bstr() now drop the leading '+'. The old code would return
+'+1.23', the new returns '1.23'. See the documentation in L<Math::BigInt> for
+reasoning and details.
+
+=item bdiv
+
+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 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:
+
+ $x = Math::BigFloat->new(5);
+ $y = $x;
+
+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 (except overloaded math operators), and vice
+versa. See L<Math::BigInt> for details and how to avoid that.
+
+=item bpow
+
+C<bpow()> now modifies the first argument, unlike the old code which left
+it alone and only returned the result. This is to be consistent with
+C<badd()> etc. The first will modify $x, the second one won't:
+
+ print bpow($x,$i),"\n"; # modify $x
+ 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
+the same terms as Perl itself.
+
+=head1 AUTHORS
+
+Mark Biggar, overloaded interface by Ilya Zakharevich.
+Completely rewritten by Tels L<http://bloodgate.com> in 2001 - 2006, and still
+at it in 2007.
+
+=cut