#acupoftea - www.loginstore.com

File BigFloat.pm

package Math::BigFloat;

#
# Mike grinned. 'Two down, infinity to go' - Mike Nostrus in 'Before and After'
#

# The following hash values are used internally:
# sign  : "+", "-", "+inf", "-inf", or "NaN" if not a number
#   _m  : mantissa ($LIB thingy)
#   _es : sign of _e
#   _e  : exponent ($LIB thingy)
#   _a  : accuracy
#   _p  : precision

use 5.006001;
use strict;
use warnings;

use Carp qw< carp croak >;
use Math::BigInt ();

our $VERSION = '1.999818';

require Exporter;
our @ISA        = qw/Math::BigInt/;
our @EXPORT_OK  = qw/bpi/;

# $_trap_inf/$_trap_nan are internal and should never be accessed from outside
our ($AUTOLOAD, $accuracy, $precision, $div_scale, $round_mode, $rnd_mode,
     $upgrade, $downgrade, $_trap_nan, $_trap_inf);

use overload

# overload key: with_assign

'+'     =>      sub { $_[0] -> copy() -> badd($_[1]); },

'-'     =>      sub { my $c = $_[0] -> copy();
                        $_[2] ? $c -> bneg() -> badd($_[1])
                              : $c -> bsub($_[1]); },

'*'     =>      sub { $_[0] -> copy() -> bmul($_[1]); },

'/'     =>      sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bdiv($_[0])
                              : $_[0] -> copy() -> bdiv($_[1]); },

'%'     =>      sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bmod($_[0])
                              : $_[0] -> copy() -> bmod($_[1]); },

'**'    =>      sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bpow($_[0])
                              : $_[0] -> copy() -> bpow($_[1]); },

'<<'    =>      sub { $_[2] ? ref($_[0]) -> new($_[1]) -> blsft($_[0])
                              : $_[0] -> copy() -> blsft($_[1]); },

'>>'    =>      sub { $_[2] ? ref($_[0]) -> new($_[1]) -> brsft($_[0])
                              : $_[0] -> copy() -> brsft($_[1]); },

# overload key: assign

'+='    =>      sub { $_[0]->badd($_[1]); },

'-='    =>      sub { $_[0]->bsub($_[1]); },

'*='    =>      sub { $_[0]->bmul($_[1]); },

'/='    =>      sub { scalar $_[0]->bdiv($_[1]); },

'%='    =>      sub { $_[0]->bmod($_[1]); },

'**='   =>      sub { $_[0]->bpow($_[1]); },

'<<='   =>      sub { $_[0]->blsft($_[1]); },

'>>='   =>      sub { $_[0]->brsft($_[1]); },

#  'x='    =>      sub { },

#  '.='    =>      sub { },

# overload key: num_comparison

'<'     =>      sub { $_[2] ? ref($_[0]) -> new($_[1]) -> blt($_[0])
                              : $_[0] -> blt($_[1]); },

'<='    =>      sub { $_[2] ? ref($_[0]) -> new($_[1]) -> ble($_[0])
                              : $_[0] -> ble($_[1]); },

'>'     =>      sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bgt($_[0])
                              : $_[0] -> bgt($_[1]); },

'>='    =>      sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bge($_[0])
                              : $_[0] -> bge($_[1]); },

'=='    =>      sub { $_[0] -> beq($_[1]); },

'!='    =>      sub { $_[0] -> bne($_[1]); },

# overload key: 3way_comparison

'<=>'   =>      sub { my $cmp = $_[0] -> bcmp($_[1]);
                        defined($cmp) && $_[2] ? -$cmp : $cmp; },

'cmp'   =>      sub { $_[2] ? "$_[1]" cmp $_[0] -> bstr()
                              : $_[0] -> bstr() cmp "$_[1]"; },

# overload key: str_comparison

#  'lt'    =>      sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bstrlt($_[0])
#                              : $_[0] -> bstrlt($_[1]); },
#
#  'le'    =>      sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bstrle($_[0])
#                              : $_[0] -> bstrle($_[1]); },
#
#  'gt'    =>      sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bstrgt($_[0])
#                              : $_[0] -> bstrgt($_[1]); },
#
#  'ge'    =>      sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bstrge($_[0])
#                              : $_[0] -> bstrge($_[1]); },
#
#  'eq'    =>      sub { $_[0] -> bstreq($_[1]); },
#
#  'ne'    =>      sub { $_[0] -> bstrne($_[1]); },

# overload key: binary

'&'     =>      sub { $_[2] ? ref($_[0]) -> new($_[1]) -> band($_[0])
                              : $_[0] -> copy() -> band($_[1]); },

'&='    =>      sub { $_[0] -> band($_[1]); },

'|'     =>      sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bior($_[0])
                              : $_[0] -> copy() -> bior($_[1]); },

'|='    =>      sub { $_[0] -> bior($_[1]); },

'^'     =>      sub { $_[2] ? ref($_[0]) -> new($_[1]) -> bxor($_[0])
                              : $_[0] -> copy() -> bxor($_[1]); },

'^='    =>      sub { $_[0] -> bxor($_[1]); },

#  '&.'    =>      sub { },

#  '&.='   =>      sub { },

#  '|.'    =>      sub { },

#  '|.='   =>      sub { },

#  '^.'    =>      sub { },

#  '^.='   =>      sub { },

# overload key: unary

'neg'   =>      sub { $_[0] -> copy() -> bneg(); },

#  '!'     =>      sub { },

'~'     =>      sub { $_[0] -> copy() -> bnot(); },

#  '~.'    =>      sub { },

# overload key: mutators

'++'    =>      sub { $_[0] -> binc() },

'--'    =>      sub { $_[0] -> bdec() },

# overload key: func

'atan2' =>      sub { $_[2] ? ref($_[0]) -> new($_[1]) -> batan2($_[0])
                              : $_[0] -> copy() -> batan2($_[1]); },

'cos'   =>      sub { $_[0] -> copy() -> bcos(); },

'sin'   =>      sub { $_[0] -> copy() -> bsin(); },

'exp'   =>      sub { $_[0] -> copy() -> bexp($_[1]); },

'abs'   =>      sub { $_[0] -> copy() -> babs(); },

'log'   =>      sub { $_[0] -> copy() -> blog(); },

'sqrt'  =>      sub { $_[0] -> copy() -> bsqrt(); },

'int'   =>      sub { $_[0] -> copy() -> bint(); },

# overload key: conversion

'bool'  =>      sub { $_[0] -> is_zero() ? '' : 1; },

'""'    =>      sub { $_[0] -> bstr(); },

'0+'    =>      sub { $_[0] -> numify(); },

'='     =>      sub { $_[0]->copy(); },

;

##############################################################################
# global constants, flags and 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 $LIB = 'Math::BigInt::Calc';

# 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;
}

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()
    my $name = $AUTOLOAD;

$name =~ s/(.*):://;        # split package
    my $c = $1 || __PACKAGE__;
    no strict 'refs';
    $c->import() if $IMPORT == 0;
    if (!_method_alias($name)) {
        if (!defined $name) {
            # delayed load of Carp and avoid recursion
            croak("$c: Can't call a method without name");
        }
        if (!_method_hand_up($name)) {
            # delayed load of Carp and avoid recursion
            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 @_
}

##############################################################################

{
    # 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]||''}; }
}

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';

# Getter/accessor.

if (@_ == 1 && ref($_[0]) ne 'HASH') {
        my $param = shift;
        return $class if $param eq 'class';
        return $LIB   if $param eq 'with';
        return $class->SUPER::config($param);
    }

# Setter.

my $cfg = $class->SUPER::config(@_);

# now we need only to override the ones that are different from our parent
    $cfg->{class} = $class;
    $cfg->{with} = $LIB;
    $cfg;
}

###############################################################################
# Constructor methods
###############################################################################

sub new {
    # Create a new Math::BigFloat object from a string or another bigfloat object.
    # _e: exponent
    # _m: mantissa
    # sign  => ("+", "-", "+inf", "-inf", or "NaN")

my $self    = shift;
    my $selfref = ref $self;
    my $class   = $selfref || $self;

my ($wanted, @r) = @_;

# avoid numify-calls by not using || on $wanted!

unless (defined $wanted) {
        #carp("Use of uninitialized value in new");
        return $self->bzero(@r);
    }

# Using $wanted->isa("Math::BigFloat") here causes a 'Deep recursion on
    # subroutine "Math::BigFloat::as_number"' in some tests. Fixme!

if (UNIVERSAL::isa($wanted, 'Math::BigFloat')) {
        my $copy = $wanted -> copy();
        if ($selfref) {                 # if new() called as instance method
            %$self = %$copy;
        } else {                        # if new() called as class method
            $self = $copy;
        }
        return $copy;
    }

$class->import() if $IMPORT == 0;             # make require work

# If called as a class method, initialize a new object.

$self = bless {}, $class unless $selfref;

# shortcut for bigints and its subclasses
    if ((ref($wanted)) && $wanted -> can("as_number")) {
        $self->{_m} = $wanted->as_number()->{value};  # get us a bigint copy
        $self->{_e} = $LIB->_zero();
        $self->{_es} = '+';
        $self->{sign} = $wanted->sign();
        return $self->bnorm();
    }

# else: got a string or something masquerading as number (with overload)

# Handle Infs.

if ($wanted =~ /^\s*([+-]?)inf(inity)?\s*\z/i) {
        return $downgrade->new($wanted) if $downgrade;
        my $sgn = $1 || '+';
        $self->{sign} = $sgn . 'inf';   # set a default sign for bstr()
        return $self->binf($sgn);
    }

# Handle explicit NaNs (not the ones returned due to invalid input).

if ($wanted =~ /^\s*([+-]?)nan\s*\z/i) {
        return $downgrade->new($wanted) if $downgrade;
        $self = $class -> bnan();
        $self->round(@r) unless @r >= 2 && !defined $r[0] && !defined $r[1];
        return $self;
    }

# Handle hexadecimal numbers. We auto-detect hexadecimal numbers if they
    # have a "0x" or "0X" prefix.

if ($wanted =~ /^\s*[+-]?0[Xx]/) {
        $self = $class -> from_hex($wanted);
        $self->round(@r) unless @r >= 2 && !defined $r[0] && !defined $r[1];
        return $self;
    }

# Handle octal numbers. We auto-detect octal numbers if they have a "0"
    # prefix and a binary exponent.

if ($wanted =~ /
                       ^
                       \s*

# sign
                       [+-]?

# prefix
                       0

# significand using the octal digits 0..7
                       [0-7]+ (?: _ [0-7]+ )*
                       (?:
                           \.
                           (?: [0-7]+ (?: _ [0-7]+ )* )?
                       )?

# exponent (power of 2) using decimal digits
                       [Pp]
                       [+-]?
                       \d+ (?: _ \d+ )*

\s*
                       $
                 /x)
    {
        $self = $class -> from_oct($wanted);
        $self->round(@r) unless @r >= 2 && !defined $r[0] && !defined $r[1];
        return $self;
    }

# Handle binary numbers. We auto-detect binary numbers if they have a "0b"
    # or "0B" prefix.

if ($wanted =~ /^\s*[+-]?0[Bb]/) {
        $self = $class -> from_bin($wanted);
        $self->round(@r) unless @r >= 2 && !defined $r[0] && !defined $r[1];
        return $self;
    }

# Shortcut for simple forms like '12' that have no trailing zeros.
    if ($wanted =~ /^([+-]?)0*([1-9][0-9]*[1-9])$/) {
        $self->{_e}   = $LIB -> _zero();
        $self->{_es}  = '+';
        $self->{sign} = $1 || '+';
        $self->{_m}   = $LIB -> _new($2);
        if (!$downgrade) {
            $self->round(@r) unless @r >= 2 && !defined $r[0] && !defined $r[1];
            return $self;
        }
    }

my ($mis, $miv, $mfv, $es, $ev) = Math::BigInt::_split($wanted);
    if (!ref $mis) {
        if ($_trap_nan) {
            croak("$wanted is not a number initialized to $class");
        }

return $downgrade->bnan() if $downgrade;

$self->{_e} = $LIB->_zero();
        $self->{_es} = '+';
        $self->{_m} = $LIB->_zero();
        $self->{sign} = $nan;
    } else {
        # make integer from mantissa by adjusting exp, then convert to int
        $self->{_e} = $LIB->_new($$ev); # exponent
        $self->{_es} = $$es || '+';
        my $mantissa = "$$miv$$mfv";     # create mant.
        $mantissa =~ s/^0+(\d)/$1/;      # strip leading zeros
        $self->{_m} = $LIB->_new($mantissa); # create mant.

# 3.123E0 = 3123E-3, and 3.123E-2 => 3123E-5
        if (CORE::length($$mfv) != 0) {
            my $len = $LIB->_new(CORE::length($$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 $$miv =~ /[1-9](0*)$/;
            if ($zeros != 0) {
                my $z = $LIB->_new($zeros);
                # turn '120e2' into '12e3'
                $self->{_m} = $LIB->_rsft($self->{_m}, $z, 10);
                ($self->{_e}, $self->{_es}) =
                  _e_add($self->{_e}, $z, $self->{_es}, '+');
            }
        }
        $self->{sign} = $$mis;

# for something like 0Ey, set y to 0, 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 $LIB->_is_zero().
        $self->{sign} = '+', $self->{_e} = $LIB->_zero()
          if $$miv eq '0' and $$mfv eq '';

if (!$downgrade) {
            $self->round(@r) unless @r >= 2 && !defined $r[0] && !defined $r[1];
            return $self;
        }
    }

# if downgrade, inf, NaN or integers go down

if ($downgrade && $self->{_es} eq '+') {
        if ($LIB->_is_zero($self->{_e})) {
            return $downgrade->new($$mis . $LIB->_str($self->{_m}));
        }
        return $downgrade->new($self->bsstr());
    }
    $self->bnorm();
    $self->round(@r) unless @r >= 2 && !defined $r[0] && !defined $r[1];
    return $self;
}

sub from_hex {
    my $self    = shift;
    my $selfref = ref $self;
    my $class   = $selfref || $self;

# Don't modify constant (read-only) objects.

return if $selfref && $self->modify('from_hex');

my $str = shift;

# If called as a class method, initialize a new object.

$self = $class -> bzero() unless $selfref;

if ($str =~ s/
                     ^
                     \s*

# sign
                     ( [+-]? )

# optional "hex marker"
                     (?: 0? x )?

# significand using the hex digits 0..9 and a..f
                     (
                         [0-9a-fA-F]+ (?: _ [0-9a-fA-F]+ )*
                         (?:
                             \.
                             (?: [0-9a-fA-F]+ (?: _ [0-9a-fA-F]+ )* )?
                         )?
                     |
                         \.
                         [0-9a-fA-F]+ (?: _ [0-9a-fA-F]+ )*
                     )

# exponent (power of 2) using decimal digits
                     (?:
                         [Pp]
                         ( [+-]? )
                         ( \d+ (?: _ \d+ )* )
                     )?

\s*
                     $
                 //x)
    {
        my $s_sign  = $1 || '+';
        my $s_value = $2;
        my $e_sign  = $3 || '+';
        my $e_value = $4 || '0';
        $s_value =~ tr/_//d;
        $e_value =~ tr/_//d;

# The significand must be multiplied by 2 raised to this exponent.

my $two_expon = $class -> new($e_value);
        $two_expon -> bneg() if $e_sign eq '-';

# If there is a dot in the significand, remove it and adjust the
        # exponent according to the number of digits in the fraction part of
        # the significand. Since the digits in the significand are in base 16,
        # but the exponent is only in base 2, multiply the exponent adjustment
        # value by log(16) / log(2) = 4.

my $idx = index($s_value, '.');
        if ($idx >= 0) {
            substr($s_value, $idx, 1) = '';
            $two_expon -= $class -> new(CORE::length($s_value))
                                 -> bsub($idx)
                                 -> bmul("4");
        }

$self -> {sign} = $s_sign;
        $self -> {_m}   = $LIB -> _from_hex('0x' . $s_value);

if ($two_expon > 0) {
            my $factor = $class -> new("2") -> bpow($two_expon);
            $self -> bmul($factor);
        } elsif ($two_expon < 0) {
            my $factor = $class -> new("0.5") -> bpow(-$two_expon);
            $self -> bmul($factor);
        }

return $self;
    }

return $self->bnan();
}

sub from_oct {
    my $self    = shift;
    my $selfref = ref $self;
    my $class   = $selfref || $self;

# Don't modify constant (read-only) objects.

return if $selfref && $self->modify('from_oct');

my $str = shift;

# If called as a class method, initialize a new object.

$self = $class -> bzero() unless $selfref;

if ($str =~ s/
                     ^
                     \s*

# sign
                     ( [+-]? )

# significand using the octal digits 0..7
                     (
                         [0-7]+ (?: _ [0-7]+ )*
                         (?:
                             \.
                             (?: [0-7]+ (?: _ [0-7]+ )* )?
                         )?
                     |
                         \.
                         [0-7]+ (?: _ [0-7]+ )*
                     )

# exponent (power of 2) using decimal digits
                     (?:
                         [Pp]
                         ( [+-]? )
                         ( \d+ (?: _ \d+ )* )
                     )?

\s*
                     $
                 //x)
    {
        my $s_sign  = $1 || '+';
        my $s_value = $2;
        my $e_sign  = $3 || '+';
        my $e_value = $4 || '0';
        $s_value =~ tr/_//d;
        $e_value =~ tr/_//d;

# The significand must be multiplied by 2 raised to this exponent.

my $two_expon = $class -> new($e_value);
        $two_expon -> bneg() if $e_sign eq '-';

# If there is a dot in the significand, remove it and adjust the
        # exponent according to the number of digits in the fraction part of
        # the significand. Since the digits in the significand are in base 8,
        # but the exponent is only in base 2, multiply the exponent adjustment
        # value by log(8) / log(2) = 3.

my $idx = index($s_value, '.');
        if ($idx >= 0) {
            substr($s_value, $idx, 1) = '';
            $two_expon -= $class -> new(CORE::length($s_value))
                                 -> bsub($idx)
                                 -> bmul("3");
        }

$self -> {sign} = $s_sign;
        $self -> {_m}   = $LIB -> _from_oct($s_value);

if ($two_expon > 0) {
            my $factor = $class -> new("2") -> bpow($two_expon);
            $self -> bmul($factor);
        } elsif ($two_expon < 0) {
            my $factor = $class -> new("0.5") -> bpow(-$two_expon);
            $self -> bmul($factor);
        }

return $self;
    }

return $self->bnan();
}

sub from_bin {
    my $self    = shift;
    my $selfref = ref $self;
    my $class   = $selfref || $self;

# Don't modify constant (read-only) objects.

return if $selfref && $self->modify('from_bin');

my $str = shift;

# If called as a class method, initialize a new object.

$self = $class -> bzero() unless $selfref;

if ($str =~ s/
                     ^
                     \s*

# sign
                     ( [+-]? )

# optional "bin marker"
                     (?: 0? b )?

# significand using the binary digits 0 and 1
                     (
                         [01]+ (?: _ [01]+ )*
                         (?:
                             \.
                             (?: [01]+ (?: _ [01]+ )* )?
                         )?
                     |
                         \.
                         [01]+ (?: _ [01]+ )*
                     )

# exponent (power of 2) using decimal digits
                     (?:
                         [Pp]
                         ( [+-]? )
                         ( \d+ (?: _ \d+ )* )
                     )?

\s*
                     $
                 //x)
    {
        my $s_sign  = $1 || '+';
        my $s_value = $2;
        my $e_sign  = $3 || '+';
        my $e_value = $4 || '0';
        $s_value =~ tr/_//d;
        $e_value =~ tr/_//d;

# The significand must be multiplied by 2 raised to this exponent.

my $two_expon = $class -> new($e_value);
        $two_expon -> bneg() if $e_sign eq '-';

# If there is a dot in the significand, remove it and adjust the
        # exponent according to the number of digits in the fraction part of
        # the significand.

my $idx = index($s_value, '.');
        if ($idx >= 0) {
            substr($s_value, $idx, 1) = '';
            $two_expon -= $class -> new(CORE::length($s_value))
                                 -> bsub($idx);
        }

$self -> {sign} = $s_sign;
        $self -> {_m}   = $LIB -> _from_bin('0b' . $s_value);

if ($two_expon > 0) {
            my $factor = $class -> new("2") -> bpow($two_expon);
            $self -> bmul($factor);
        } elsif ($two_expon < 0) {
            my $factor = $class -> new("0.5") -> bpow(-$two_expon);
            $self -> bmul($factor);
        }

return $self;
    }

return $self->bnan();
}

sub from_ieee754 {
    my $self    = shift;
    my $selfref = ref $self;
    my $class   = $selfref || $self;

# Don't modify constant (read-only) objects.

return if $selfref && $self->modify('from_ieee754');

my $in     = shift;     # input string (or raw bytes)
    my $format = shift;     # format ("binary32", "decimal64" etc.)
    my $enc;                # significand encoding (applies only to decimal)
    my $k;                  # storage width in bits
    my $b;                  # base

if ($format =~ /^binary(\d+)\z/) {
        $k = $1;
        $b = 2;
    } elsif ($format =~ /^decimal(\d+)(dpd|bcd)?\z/) {
        $k = $1;
        $b = 10;
        $enc = $2 || 'dpd';     # default is dencely-packed decimals (DPD)
    } elsif ($format eq 'half') {
        $k = 16;
        $b = 2;
    } elsif ($format eq 'single') {
        $k = 32;
        $b = 2;
    } elsif ($format eq 'double') {
        $k = 64;
        $b = 2;
    } elsif ($format eq 'quadruple') {
        $k = 128;
        $b = 2;
    } elsif ($format eq 'octuple') {
        $k = 256;
        $b = 2;
    } elsif ($format eq 'sexdecuple') {
        $k = 512;
        $b = 2;
    }

if ($b == 2) {

# Get the parameters for this format.

my $p;                      # precision (in bits)
        my $t;                      # number of bits in significand
        my $w;                      # number of bits in exponent

if ($k == 16) {             # binary16 (half-precision)
            $p = 11;
            $t = 10;
            $w =  5;
        } elsif ($k == 32) {        # binary32 (single-precision)
            $p = 24;
            $t = 23;
            $w =  8;
        } elsif ($k == 64) {        # binary64 (double-precision)
            $p = 53;
            $t = 52;
            $w = 11;
        } else {                    # binaryN (quadruple-precision and above)
            if ($k < 128 || $k != 32 * sprintf('%.0f', $k / 32)) {
                croak "Number of bits must be 16, 32, 64, or >= 128 and",
                  " a multiple of 32";
            }
            $p = $k - sprintf('%.0f', 4 * log($k) / log(2)) + 13;
            $t = $p - 1;
            $w = $k - $t - 1;
        }

# The maximum exponent, minimum exponent, and exponent bias.

my $emax = Math::BigInt -> new(2) -> bpow($w - 1) -> bdec();
        my $emin = 1 - $emax;
        my $bias = $emax;

# Undefined input.

unless (defined $in) {
            carp("Input is undefined");
            return $self -> bzero();
        }

# Make sure input string is a string of zeros and ones.

my $len = CORE::length $in;
        if (8 * $len == $k) {                   # bytes
            $in = unpack "B*", $in;
        } elsif (4 * $len == $k) {              # hexadecimal
            if ($in =~ /([^\da-f])/i) {
                croak "Illegal hexadecimal digit '$1'";
            }
            $in = unpack "B*", pack "H*", $in;
        } elsif ($len == $k) {                  # bits
            if ($in =~ /([^01])/) {
                croak "Illegal binary digit '$1'";
            }
        } else {
            croak "Unknown input -- $in";
        }

# Split bit string into sign, exponent, and mantissa/significand.

my $sign = substr($in, 0, 1) eq '1' ? '-' : '+';
        my $expo = $class -> from_bin(substr($in, 1, $w));
        my $mant = $class -> from_bin(substr($in, $w + 1));

my $x;

$expo -> bsub($bias);                   # subtract bias

if ($expo < $emin) {                    # zero and subnormals
            if ($mant == 0) {                   # zero
                $x = $class -> bzero();
            } else {                            # subnormals
                # compute (1/$b)**(N) rather than ($b)**(-N)
                $x = $class -> new("0.5");      # 1/$b
                $x -> bpow($bias + $t - 1) -> bmul($mant);
                $x -> bneg() if $sign eq '-';
            }
        }

elsif ($expo > $emax) {                 # inf and nan
            if ($mant == 0) {                   # inf
                $x = $class -> binf($sign);
            } else {                            # nan
                $x = $class -> bnan();
            }
        }

else {                                  # normals
            $mant = $class -> new(2) -> bpow($t) -> badd($mant);
            if ($expo < $t) {
                # compute (1/$b)**(N) rather than ($b)**(-N)
                $x = $class -> new("0.5");      # 1/$b
                $x -> bpow($t - $expo) -> bmul($mant);
            } else {
                $x = $class -> new(2);
                $x -> bpow($expo - $t) -> bmul($mant);
            }
            $x -> bneg() if $sign eq '-';
        }

if ($selfref) {
            $self -> {sign} = $x -> {sign};
            $self -> {_m}   = $x -> {_m};
            $self -> {_es}  = $x -> {_es};
            $self -> {_e}   = $x -> {_e};
        } else {
            $self = $x;
        }
        return $self;
    }

croak("The format '$format' is not yet supported.");
}

sub bzero {
    # create/assign '+0'

if (@_ == 0) {
        #carp("Using bone() as a function is deprecated;",
        #           " use bone() as a method instead");
        unshift @_, __PACKAGE__;
    }

my $self    = shift;
    my $selfref = ref $self;
    my $class   = $selfref || $self;

$self->import() if $IMPORT == 0;            # make require work
    return if $selfref && $self->modify('bzero');

$self = bless {}, $class unless $selfref;

$self -> {sign} = '+';
    $self -> {_m}   = $LIB -> _zero();
    $self -> {_es}  = '+';
    $self -> {_e}   = $LIB -> _zero();

# If rounding parameters are given as arguments, use them. If no rounding
    # parameters are given, and if called as a class method initialize the new
    # instance with the class variables.

if (@_) {
        croak "can't specify both accuracy and precision"
          if @_ >= 2 && defined $_[0] && defined $_[1];
        $self->{_a} = $_[0];
        $self->{_p} = $_[1];
    } else {
        unless($selfref) {
            $self->{_a} = $class -> accuracy();
            $self->{_p} = $class -> precision();
        }
    }

return $self;
}

sub bone {
    # Create or assign '+1' (or -1 if given sign '-').

if (@_ == 0 || (defined($_[0]) && ($_[0] eq '+' || $_[0] eq '-'))) {
        #carp("Using bone() as a function is deprecated;",
        #           " use bone() as a method instead");
        unshift @_, __PACKAGE__;
    }

my $self    = shift;
    my $selfref = ref $self;
    my $class   = $selfref || $self;

$self->import() if $IMPORT == 0;            # make require work
    return if $selfref && $self->modify('bone');

my $sign = shift;
    $sign = defined $sign && $sign =~ /^\s*-/ ? "-" : "+";

$self = bless {}, $class unless $selfref;

$self -> {sign} = $sign;
    $self -> {_m}   = $LIB -> _one();
    $self -> {_es}  = '+';
    $self -> {_e}   = $LIB -> _zero();

# If rounding parameters are given as arguments, use them. If no rounding
    # parameters are given, and if called as a class method initialize the new
    # instance with the class variables.

if (@_) {
        croak "can't specify both accuracy and precision"
          if @_ >= 2 && defined $_[0] && defined $_[1];
        $self->{_a} = $_[0];
        $self->{_p} = $_[1];
    } else {
        unless($selfref) {
            $self->{_a} = $class -> accuracy();
            $self->{_p} = $class -> precision();
        }
    }

return $self;
}

sub binf {
    # create/assign a '+inf' or '-inf'

if (@_ == 0 || (defined($_[0]) && !ref($_[0]) &&
                    $_[0] =~ /^\s*[+-](inf(inity)?)?\s*$/))
    {
        #carp("Using binf() as a function is deprecated;",
        #           " use binf() as a method instead");
        unshift @_, __PACKAGE__;
    }

my $self    = shift;
    my $selfref = ref $self;
    my $class   = $selfref || $self;

{
        no strict 'refs';
        if (${"${class}::_trap_inf"}) {
            croak("Tried to create +-inf in $class->binf()");
        }
    }

$self->import() if $IMPORT == 0;            # make require work
    return if $selfref && $self->modify('binf');

my $sign = shift;
    $sign = defined $sign && $sign =~ /^\s*-/ ? "-" : "+";

$self = bless {}, $class unless $selfref;

$self -> {sign} = $sign . 'inf';
    $self -> {_m}   = $LIB -> _zero();
    $self -> {_es}  = '+';
    $self -> {_e}   = $LIB -> _zero();

# If rounding parameters are given as arguments, use them. If no rounding
    # parameters are given, and if called as a class method initialize the new
    # instance with the class variables.

if (@_) {
        croak "can't specify both accuracy and precision"
          if @_ >= 2 && defined $_[0] && defined $_[1];
        $self->{_a} = $_[0];
        $self->{_p} = $_[1];
    } else {
        unless($selfref) {
            $self->{_a} = $class -> accuracy();
            $self->{_p} = $class -> precision();
        }
    }

return $self;
}

sub bnan {
    # create/assign a 'NaN'

if (@_ == 0) {
        #carp("Using bnan() as a function is deprecated;",
        #           " use bnan() as a method instead");
        unshift @_, __PACKAGE__;
    }

my $self    = shift;
    my $selfref = ref $self;
    my $class   = $selfref || $self;

{
        no strict 'refs';
        if (${"${class}::_trap_nan"}) {
            croak("Tried to create NaN in $class->bnan()");
        }
    }

$self->import() if $IMPORT == 0;            # make require work
    return if $selfref && $self->modify('bnan');

$self = bless {}, $class unless $selfref;

$self -> {sign} = $nan;
    $self -> {_m}   = $LIB -> _zero();
    $self -> {_es}  = '+';
    $self -> {_e}   = $LIB -> _zero();

# If rounding parameters are given as arguments, use them. If no rounding
    # parameters are given, and if called as a class method initialize the new
    # instance with the class variables.

if (@_) {
        croak "can't specify both accuracy and precision"
          if @_ >= 2 && defined $_[0] && defined $_[1];
        $self->{_a} = $_[0];
        $self->{_p} = $_[1];
    } else {
        unless($selfref) {
            $self->{_a} = $class -> accuracy();
            $self->{_p} = $class -> precision();
        }
    }

return $self;
}

sub bpi {

# Called as                 Argument list
    # ---------                 -------------
    # Math::BigFloat->bpi()     ("Math::BigFloat")
    # Math::BigFloat->bpi(10)   ("Math::BigFloat", 10)
    # $x->bpi()                 ($x)
    # $x->bpi(10)               ($x, 10)
    # Math::BigFloat::bpi()     ()
    # Math::BigFloat::bpi(10)   (10)
    #
    # In ambiguous cases, we favour the OO-style, so the following case
    #
    #   $n = Math::BigFloat->new("10");
    #   $x = Math::BigFloat->bpi($n);
    #
    # which gives an argument list with the single element $n, is resolved as
    #
    #   $n->bpi();

my $self    = shift;
    my $selfref = ref $self;
    my $class   = $selfref || $self;

my @r;                      # rounding paramters

# If bpi() is called as a function ...
    #
    # This cludge is necessary because we still support bpi() as a function. If
    # bpi() is called with either no argument or one argument, and that one
    # argument is either undefined or a scalar that looks like a number, then
    # we assume bpi() is called as a function.

if (@_ == 0 &&
        (defined($self) && !ref($self) && $self =~ /^\s*[+-]?\d/i)
          ||
        !defined($self))
    {
        $r[0]  = $self;
        $class = __PACKAGE__;
        $self  = $class -> bzero(@r);       # initialize
    }

# ... or if bpi() is called as a method ...

else {
        @r = @_;
        if ($selfref) {                     # bpi() called as instance method
            return $self if $self -> modify('bpi');
        } else {                            # bpi() called as class method
            $self = $class -> bzero(@r);    # initialize
        }
    }

($self, @r) = $self -> _find_round_parameters(@r);

# The accuracy, i.e., the number of digits. Pi has one digit before the
    # dot, so a precision of 4 digits is equivalent to an accuracy of 5 digits.

my $n = defined $r[0] ? $r[0]
          : defined $r[1] ? 1 - $r[1]
          : $self -> div_scale();

my $rmode = defined $r[2] ? $r[2] : $self -> round_mode();

my $pi;

if ($n <= 1000) {

# 75 x 14 = 1050 digits

my $all_digits = <<EOF;
314159265358979323846264338327950288419716939937510582097494459230781640628
620899862803482534211706798214808651328230664709384460955058223172535940812
848111745028410270193852110555964462294895493038196442881097566593344612847
564823378678316527120190914564856692346034861045432664821339360726024914127
372458700660631558817488152092096282925409171536436789259036001133053054882
046652138414695194151160943305727036575959195309218611738193261179310511854
807446237996274956735188575272489122793818301194912983367336244065664308602
139494639522473719070217986094370277053921717629317675238467481846766940513
200056812714526356082778577134275778960917363717872146844090122495343014654
958537105079227968925892354201995611212902196086403441815981362977477130996
051870721134999999837297804995105973173281609631859502445945534690830264252
230825334468503526193118817101000313783875288658753320838142061717766914730
359825349042875546873115956286388235378759375195778185778053217122680661300
192787661119590921642019893809525720106548586327886593615338182796823030195
EOF

# Should we round up?

my $round_up;

# From the string above, we need to extract the number of digits we
        # want plus extra characters for the newlines.

my $nchrs = $n + int($n / 75);

# Extract the digits we want.

my $digits = substr($all_digits, 0, $nchrs);

# Find out whether we should round up or down. Rounding is easy, since
        # pi is trancendental. With directed rounding, it doesn't matter what
        # the following digits are. With rounding to nearest, we only have to
        # look at one extra digit.

if ($rmode eq 'trunc') {
            $round_up = 0;
        } else {
            my $next_digit = substr($all_digits, $nchrs, 1);
            $round_up = $next_digit lt '5' ? 0 : 1;
        }

# Remove the newlines.

$digits =~ tr/0-9//cd;

# Now do the rounding. We could easily make the regex substitution
        # handle all cases, but we avoid using the regex engine when it is
        # simple to avoid it.

if ($round_up) {
            my $last_digit = substr($digits, -1, 1);
            if ($last_digit lt '9') {
                substr($digits, -1, 1) = ++$last_digit;
            } else {
                $digits =~ s/([0-8])(9+)$/ ($1 + 1) . ("0" x CORE::length($2)) /e;
            }
        }

# Append the exponent and convert to an object.

$pi = Math::BigFloat -> new($digits . 'e-' . ($n - 1));

} else {

# For large accuracy, the arctan formulas become very inefficient with
        # Math::BigFloat, so use Brent-Salamin (aka AGM or Gauss-Legendre).

# Use a few more digits in the intermediate computations.
        $n += 8;

$HALF = $class -> new($HALF) unless ref($HALF);
        my ($an, $bn, $tn, $pn) = ($class -> bone, $HALF -> copy() -> bsqrt($n),
                                   $HALF -> copy() -> bmul($HALF), $class -> bone);
        while ($pn < $n) {
            my $prev_an = $an -> copy();
            $an -> badd($bn) -> bmul($HALF, $n);
            $bn -> bmul($prev_an) -> bsqrt($n);
            $prev_an -> bsub($an);
            $tn -> bsub($pn * $prev_an * $prev_an);
            $pn -> badd($pn);
        }
        $an -> badd($bn);
        $an -> bmul($an, $n) -> bdiv(4 * $tn, $n);

$an -> round(@r);
        $pi = $an;
    }

if (defined $r[0]) {
        $pi -> accuracy($r[0]);
    } elsif (defined $r[1]) {
        $pi -> precision($r[1]);
    }

for my $key (qw/ sign _m _es _e _a _p /) {
        $self -> {$key} = $pi -> {$key};
    }

return $self;
}

sub copy {
    my $self    = shift;
    my $selfref = ref $self;
    my $class   = $selfref || $self;

# If called as a class method, the object to copy is the next argument.

$self = shift() unless $selfref;

my $copy = bless {}, $class;

$copy->{sign} = $self->{sign};
    $copy->{_es}  = $self->{_es};
    $copy->{_m}   = $LIB->_copy($self->{_m});
    $copy->{_e}   = $LIB->_copy($self->{_e});
    $copy->{_a}   = $self->{_a} if exists $self->{_a};
    $copy->{_p}   = $self->{_p} if exists $self->{_p};

return $copy;
}

sub as_number {
    # return copy as a bigint representation of this Math::BigFloat number
    my ($class, $x) = ref($_[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() : $class->new(0+"$x");
    }

return Math::BigInt->binf($x->sign()) if $x->is_inf();
    return Math::BigInt->bnan()           if $x->is_nan();

my $z = $LIB->_copy($x->{_m});
    if ($x->{_es} eq '-') {                     # < 0
        $z = $LIB->_rsft($z, $x->{_e}, 10);
    } elsif (! $LIB->_is_zero($x->{_e})) {      # > 0
        $z = $LIB->_lsft($z, $x->{_e}, 10);
    }
    $z = Math::BigInt->new($x->{sign} . $LIB->_str($z));
    $z;
}

###############################################################################
# Boolean methods
###############################################################################

sub is_zero {
    # return true if arg (BFLOAT or num_str) is zero
    my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);

($x->{sign} eq '+' && $LIB->_is_zero($x->{_m})) ? 1 : 0;
}

sub is_one {
    # return true if arg (BFLOAT or num_str) is +1 or -1 if signis given
    my ($class, $x, $sign) = ref($_[0]) ? (undef, @_) : objectify(1, @_);

$sign = '+' if !defined $sign || $sign ne '-';

($x->{sign} eq $sign &&
     $LIB->_is_zero($x->{_e}) &&
     $LIB->_is_one($x->{_m})) ? 1 : 0;
}

sub is_odd {
    # return true if arg (BFLOAT or num_str) is odd or false if even
    my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);

(($x->{sign} =~ /^[+-]$/) && # NaN & +-inf aren't
     ($LIB->_is_zero($x->{_e})) &&
     ($LIB->_is_odd($x->{_m}))) ? 1 : 0;
}

sub is_even {
    # return true if arg (BINT or num_str) is even or false if odd
    my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);

(($x->{sign} =~ /^[+-]$/) &&        # NaN & +-inf aren't
     ($x->{_es} eq '+') &&              # 123.45 isn't
     ($LIB->_is_even($x->{_m}))) ? 1 : 0; # but 1200 is
}

sub is_int {
    # return true if arg (BFLOAT or num_str) is an integer
    my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);

(($x->{sign} =~ /^[+-]$/) && # NaN and +-inf aren't
     ($x->{_es} eq '+')) ? 1 : 0; # 1e-1 => no integer
}

###############################################################################
# Comparison methods
###############################################################################

sub bcmp {
    # Compares 2 values.  Returns one of undef, <0, =0, >0. (suitable for sort)

# set up parameters
    my ($class, $x, $y) = (ref($_[0]), @_);

# objectify is costly, so avoid it
    if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
        ($class, $x, $y) = objectify(2, @_);
    }

return $upgrade->bcmp($x, $y) if defined $upgrade &&
      ((!$x->isa($class)) || (!$y->isa($class)));

# Handle all 'nan' cases.

return undef if ($x->{sign} eq $nan) || ($y->{sign} eq $nan);

# Handle all '+inf' and '-inf' cases.

return  0 if ($x->{sign} eq '+inf' && $y->{sign} eq '+inf' ||
                  $x->{sign} eq '-inf' && $y->{sign} eq '-inf');
    return +1 if $x->{sign} eq '+inf'; # x = +inf and y < +inf
    return -1 if $x->{sign} eq '-inf'; # x = -inf and y > -inf
    return -1 if $y->{sign} eq '+inf'; # x < +inf and y = +inf
    return +1 if $y->{sign} eq '-inf'; # x > -inf and y = -inf

# Handle all cases with opposite signs.

return +1 if $x->{sign} eq '+' && $y->{sign} eq '-'; # also does 0 <=> -y
    return -1 if $x->{sign} eq '-' && $y->{sign} eq '+'; # also does -x <=> 0

# Handle all remaining zero cases.

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

# Both arguments are now finite, non-zero numbers with the same sign.

my $cmp;

# The next step is to compare the exponents, but since each mantissa is an
    # integer of arbitrary value, the exponents must be normalized by the length
    # of the mantissas before we can compare them.

my $mxl = $LIB->_len($x->{_m});
    my $myl = $LIB->_len($y->{_m});

# If the mantissas have the same length, there is no point in normalizing the
    # exponents by the length of the mantissas, so treat that as a special case.

if ($mxl == $myl) {

# First handle the two cases where the exponents have different signs.

if ($x->{_es} eq '+' && $y->{_es} eq '-') {
            $cmp = +1;
        } elsif ($x->{_es} eq '-' && $y->{_es} eq '+') {
            $cmp = -1;
        }

# Then handle the case where the exponents have the same sign.

else {
            $cmp = $LIB->_acmp($x->{_e}, $y->{_e});
            $cmp = -$cmp if $x->{_es} eq '-';
        }

# Adjust for the sign, which is the same for x and y, and bail out if
        # we're done.

$cmp = -$cmp if $x->{sign} eq '-'; # 124 > 123, but -124 < -123
        return $cmp if $cmp;

}

# We must normalize each exponent by the length of the corresponding
    # mantissa. Life is a lot easier if we first make both exponents
    # non-negative. We do this by adding the same positive value to both
    # exponent. This is safe, because when comparing the exponents, only the
    # relative difference is important.

my $ex;
    my $ey;

if ($x->{_es} eq '+') {

# If the exponent of x is >= 0 and the exponent of y is >= 0, there is no
        # need to do anything special.

if ($y->{_es} eq '+') {
            $ex = $LIB->_copy($x->{_e});
            $ey = $LIB->_copy($y->{_e});
        }

# If the exponent of x is >= 0 and the exponent of y is < 0, add the
        # absolute value of the exponent of y to both.

else {
            $ex = $LIB->_copy($x->{_e});
            $ex = $LIB->_add($ex, $y->{_e}); # ex + |ey|
            $ey = $LIB->_zero();             # -ex + |ey| = 0
        }

} else {

# If the exponent of x is < 0 and the exponent of y is >= 0, add the
        # absolute value of the exponent of x to both.

if ($y->{_es} eq '+') {
            $ex = $LIB->_zero(); # -ex + |ex| = 0
            $ey = $LIB->_copy($y->{_e});
            $ey = $LIB->_add($ey, $x->{_e}); # ey + |ex|
        }

# If the exponent of x is < 0 and the exponent of y is < 0, add the
        # absolute values of both exponents to both exponents.

else {
            $ex = $LIB->_copy($y->{_e}); # -ex + |ey| + |ex| = |ey|
            $ey = $LIB->_copy($x->{_e}); # -ey + |ex| + |ey| = |ex|
        }

}

# Now we can normalize the exponents by adding lengths of the mantissas.

$ex = $LIB->_add($ex, $LIB->_new($mxl));
    $ey = $LIB->_add($ey, $LIB->_new($myl));

# We're done if the exponents are different.

$cmp = $LIB->_acmp($ex, $ey);
    $cmp = -$cmp if $x->{sign} eq '-'; # 124 > 123, but -124 < -123
    return $cmp if $cmp;

# Compare the mantissas, but first normalize them by padding the shorter
    # mantissa with zeros (shift left) until it has the same length as the longer
    # mantissa.

my $mx = $x->{_m};
    my $my = $y->{_m};

if ($mxl > $myl) {
        $my = $LIB->_lsft($LIB->_copy($my), $LIB->_new($mxl - $myl), 10);
    } elsif ($mxl < $myl) {
        $mx = $LIB->_lsft($LIB->_copy($mx), $LIB->_new($myl - $mxl), 10);
    }

$cmp = $LIB->_acmp($mx, $my);
    $cmp = -$cmp if $x->{sign} eq '-'; # 124 > 123, but -124 < -123
    return $cmp;

}

sub bacmp {
    # Compares 2 values, ignoring their signs.
    # Returns one of undef, <0, =0, >0. (suitable for sort)

# set up parameters
    my ($class, $x, $y) = (ref($_[0]), @_);
    # objectify is costly, so avoid it
    if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
        ($class, $x, $y) = objectify(2, @_);
    }

return $upgrade->bacmp($x, $y) if defined $upgrade &&
      ((!$x->isa($class)) || (!$y->isa($class)));

# 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 = $LIB->_len($x->{_m});
    my $lym = $LIB->_len($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 * $LIB->_num($x->{_e});
    my $ly = $lym + $yes * $LIB->_num($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 = $LIB->_copy($y->{_m});
        $ym = $LIB->_lsft($ym, $LIB->_new($diff), 10);
    } elsif ($diff < 0) {
        $xm = $LIB->_copy($x->{_m});
        $xm = $LIB->_lsft($xm, $LIB->_new(-$diff), 10);
    }
    $LIB->_acmp($xm, $ym);
}

###############################################################################
# Arithmetic methods
###############################################################################

sub bneg {
    # (BINT or num_str) return BINT
    # negate number or make a negated number from string
    my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);

return $x if $x->modify('bneg');

# for +0 do not negate (to have always normalized +0). Does nothing for 'NaN'
    $x->{sign} =~ tr/+-/-+/ unless ($x->{sign} eq '+' && $LIB->_is_zero($x->{_m}));
    $x;
}

sub bnorm {
    # adjust m and e so that m is smallest possible
    my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);

return $x if $x->{sign} !~ /^[+-]$/; # inf, nan etc

my $zeros = $LIB->_zeros($x->{_m}); # correct for trailing zeros
    if ($zeros != 0) {
        my $z = $LIB->_new($zeros);
        $x->{_m} = $LIB->_rsft($x->{_m}, $z, 10);
        if ($x->{_es} eq '-') {
            if ($LIB->_acmp($x->{_e}, $z) >= 0) {
                $x->{_e} = $LIB->_sub($x->{_e}, $z);
                $x->{_es} = '+' if $LIB->_is_zero($x->{_e});
            } else {
                $x->{_e} = $LIB->_sub($LIB->_copy($z), $x->{_e});
                $x->{_es} = '+';
            }
        } else {
            $x->{_e} = $LIB->_add($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} = $LIB->_one()
          if $LIB->_is_zero($x->{_m});
    }

$x;
}

sub binc {
    # increment arg by one
    my ($class, $x, @r) = ref($_[0]) ? (ref($_[0]), @_) : objectify(1, @_);

return $x if $x->modify('binc');

if ($x->{_es} eq '-') {
        return $x->badd($class->bone(), @r); #  digits after dot
    }

if (!$LIB->_is_zero($x->{_e})) # _e == 0 for NaN, inf, -inf
    {
        # 1e2 => 100, so after the shift below _m has a '0' as last digit
        $x->{_m} = $LIB->_lsft($x->{_m}, $x->{_e}, 10); # 1e2 => 100
        $x->{_e} = $LIB->_zero();                      # normalize
        $x->{_es} = '+';
        # we know that the last digit of $x will be '1' or '9', depending on the
        # sign
    }
    # now $x->{_e} == 0
    if ($x->{sign} eq '+') {
        $x->{_m} = $LIB->_inc($x->{_m});
        return $x->bnorm()->bround(@r);
    } elsif ($x->{sign} eq '-') {
        $x->{_m} = $LIB->_dec($x->{_m});
        $x->{sign} = '+' if $LIB->_is_zero($x->{_m}); # -1 +1 => -0 => +0
        return $x->bnorm()->bround(@r);
    }
    # inf, nan handling etc
    $x->badd($class->bone(), @r); # badd() does round
}

sub bdec {
    # decrement arg by one
    my ($class, $x, @r) = ref($_[0]) ? (ref($_[0]), @_) : objectify(1, @_);

return $x if $x->modify('bdec');

if ($x->{_es} eq '-') {
        return $x->badd($class->bone('-'), @r); #  digits after dot
    }

if (!$LIB->_is_zero($x->{_e})) {
        $x->{_m} = $LIB->_lsft($x->{_m}, $x->{_e}, 10); # 1e2 => 100
        $x->{_e} = $LIB->_zero();                      # normalize
        $x->{_es} = '+';
    }
    # now $x->{_e} == 0
    my $zero = $x->is_zero();
    # <= 0
    if (($x->{sign} eq '-') || $zero) {
        $x->{_m} = $LIB->_inc($x->{_m});
        $x->{sign} = '-' if $zero;                # 0 => 1 => -1
        $x->{sign} = '+' if $LIB->_is_zero($x->{_m}); # -1 +1 => -0 => +0
        return $x->bnorm()->round(@r);
    }
    # > 0
    elsif ($x->{sign} eq '+') {
        $x->{_m} = $LIB->_dec($x->{_m});
        return $x->bnorm()->round(@r);
    }
    # inf, nan handling etc
    $x->badd($class->bone('-'), @r); # does round
}

sub badd {
    # add second arg (BFLOAT or string) to first (BFLOAT) (modifies first)
    # return result as BFLOAT

# set up parameters
    my ($class, $x, $y, @r) = (ref($_[0]), @_);
    # objectify is costly, so avoid it
    if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
        ($class, $x, $y, @r) = objectify(2, @_);
    }

return $x if $x->modify('badd');

# 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($class)) || (!$y->isa($class)));

$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} = $LIB->_copy($y->{_e});
        $x->{_es} = $y->{_es};
        $x->{_m} = $LIB->_copy($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 = $LIB->_zero() if !defined $e; # if no BFLOAT?
    $e = $LIB->_copy($e);              # make copy (didn't do it yet)

my $es;

($e, $es) = _e_sub($e, $x->{_e}, $y->{_es} || '+', $x->{_es});

my $add = $LIB->_copy($y->{_m});

if ($es eq '-')             # < 0
    {
        $x->{_m} = $LIB->_lsft($x->{_m}, $e, 10);
        ($x->{_e}, $x->{_es}) = _e_add($x->{_e}, $e, $x->{_es}, $es);
    } elsif (!$LIB->_is_zero($e)) # > 0
    {
        $add = $LIB->_lsft($add, $e, 10);
    }
    # else: both e are the same, so just leave them

if ($x->{sign} eq $y->{sign}) {
        # add
        $x->{_m} = $LIB->_add($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 {
    # (BINT or num_str, BINT or num_str) return BINT
    # subtract second arg from first, modify first

# set up parameters
    my ($class, $x, $y, @r) = (ref($_[0]), @_);

# objectify is costly, so avoid it
    if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
        ($class, $x, $y, @r) = objectify(2, @_);
    }

return $x if $x -> modify('bsub');

return $upgrade -> new($x) -> bsub($upgrade -> new($y), @r)
      if defined $upgrade && (!$x -> isa($class) || !$y -> isa($class));

return $x -> round(@r) if $y -> is_zero();

# To correctly handle the lone special case $x -> bsub($x), we note the
    # sign of $x, then flip the sign from $y, and if the sign of $x did change,
    # too, then we caught the special case:

my $xsign = $x -> {sign};
    $y -> {sign} =~ tr/+-/-+/;  # does nothing for NaN
    if ($xsign ne $x -> {sign}) {
        # special case of $x -> bsub($x) results in 0
        return $x -> bzero(@r) if $xsign =~ /^[+-]$/;
        return $x -> bnan();    # NaN, -inf, +inf
    }
    $x -> badd($y, @r);         # badd does not leave internal zeros
    $y -> {sign} =~ tr/+-/-+/;  # refix $y (does nothing for NaN)
    $x;                         # already rounded by badd() or no rounding
}

sub bmul {
    # multiply two numbers

# set up parameters
    my ($class, $x, $y, @r) = (ref($_[0]), @_);
    # objectify is costly, so avoid it
    if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
        ($class, $x, $y, @r) = objectify(2, @_);
    }

return $x if $x->modify('bmul');

return $x->bnan() if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));

# inf handling
    if (($x->{sign} =~ /^[+-]inf$/) || ($y->{sign} =~ /^[+-]inf$/)) {
        return $x->bnan() if $x->is_zero() || $y->is_zero();
        # result will always be +-inf:
        # +inf * +/+inf => +inf, -inf * -/-inf => +inf
        # +inf * -/-inf => -inf, -inf * +/+inf => -inf
        return $x->binf() if ($x->{sign} =~ /^\+/ && $y->{sign} =~ /^\+/);
        return $x->binf() if ($x->{sign} =~ /^-/ && $y->{sign} =~ /^-/);
        return $x->binf('-');
    }

return $upgrade->bmul($x, $y, @r) if defined $upgrade &&
      ((!$x->isa($class)) || (!$y->isa($class)));

# aEb * cEd = (a*c)E(b+d)
    $x->{_m} = $LIB->_mul($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 ($class, $x, $y, $z, @r) = objectify(3, @_);

return $x if $x->modify('bmuladd');

return $x->bnan() if (($x->{sign} eq $nan) ||
                          ($y->{sign} eq $nan) ||
                          ($z->{sign} eq $nan));

# inf handling
    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($class)) || (!$y->isa($class)));

# aEb * cEd = (a*c)E(b+d)
    $x->{_m} = $LIB->_mul($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 = $LIB->_zero() if !defined $e; # if no BFLOAT?
    $e = $LIB->_copy($e);              # make copy (didn't do it yet)

my $es;

($e, $es) = _e_sub($e, $x->{_e}, $z->{_es} || '+', $x->{_es});

my $add = $LIB->_copy($z->{_m});

if ($es eq '-')             # < 0
    {
        $x->{_m} = $LIB->_lsft($x->{_m}, $e, 10);
        ($x->{_e}, $x->{_es}) = _e_add($x->{_e}, $e, $x->{_es}, $es);
    } elsif (!$LIB->_is_zero($e)) # > 0
    {
        $add = $LIB->_lsft($add, $e, 10);
    }
    # else: both e are the same, so just leave them

if ($x->{sign} eq $z->{sign}) {
        # add
        $x->{_m} = $LIB->_add($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 quo)

# set up parameters
    my ($class, $x, $y, $a, $p, $r) = (ref($_[0]), @_);
    # objectify is costly, so avoid it
    if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
        ($class, $x, $y, $a, $p, $r) = objectify(2, @_);
    }

return $x if $x->modify('bdiv');

my $wantarray = wantarray;  # call only once

# At least one argument is NaN. This is handled the same way as in
    # Math::BigInt -> bdiv().

if ($x -> is_nan() || $y -> is_nan()) {
        return $wantarray ? ($x -> bnan(), $class -> bnan()) : $x -> bnan();
    }

# Divide by zero and modulo zero. This is handled the same way as in
    # Math::BigInt -> bdiv(). See the comment in the code for Math::BigInt ->
    # bdiv() for further details.

if ($y -> is_zero()) {
        my ($quo, $rem);
        if ($wantarray) {
            $rem = $x -> copy();
        }
        if ($x -> is_zero()) {
            $quo = $x -> bnan();
        } else {
            $quo = $x -> binf($x -> {sign});
        }
        return $wantarray ? ($quo, $rem) : $quo;
    }

# Numerator (dividend) is +/-inf. This is handled the same way as in
    # Math::BigInt -> bdiv(). See the comment in the code for Math::BigInt ->
    # bdiv() for further details.

if ($x -> is_inf()) {
        my ($quo, $rem);
        $rem = $class -> bnan() if $wantarray;
        if ($y -> is_inf()) {
            $quo = $x -> bnan();
        } else {
            my $sign = $x -> bcmp(0) == $y -> bcmp(0) ? '+' : '-';
            $quo = $x -> binf($sign);
        }
        return $wantarray ? ($quo, $rem) : $quo;
    }

# Denominator (divisor) is +/-inf. This is handled the same way as in
    # Math::BigInt -> bdiv(), with one exception: In scalar context,
    # Math::BigFloat does true division (although rounded), not floored division
    # (F-division), so a finite number divided by +/-inf is always zero. See the
    # comment in the code for Math::BigInt -> bdiv() for further details.

if ($y -> is_inf()) {
        my ($quo, $rem);
        if ($wantarray) {
            if ($x -> is_zero() || $x -> bcmp(0) == $y -> bcmp(0)) {
                $rem = $x -> copy();
                $quo = $x -> bzero();
            } else {
                $rem = $class -> binf($y -> {sign});
                $quo = $x -> bone('-');
            }
            return ($quo, $rem);
        } else {
            if ($y -> is_inf()) {
                if ($x -> is_nan() || $x -> is_inf()) {
                    return $x -> bnan();
                } else {
                    return $x -> bzero();
                }
            }
        }
    }

# At this point, both the numerator and denominator are finite numbers, and
    # the denominator (divisor) is non-zero.

# x == 0?
    return wantarray ? ($x, $class->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] = $class->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 = $class -> bzero() if wantarray;

$y = $class->new($y) unless $y->isa('Math::BigFloat');

my $lx = $LIB -> _len($x->{_m}); my $ly = $LIB -> _len($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!

# check that $y is not 1 nor -1 and cache the result:
    my $y_not_one = !($LIB->_is_zero($y->{_e}) && $LIB->_is_one($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 remainder 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 Math::BigFloat)
            $y = $class->new($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)
            $x->{_m} = $LIB->_lsft($x->{_m}, $LIB->_new($scale), 10);
            $x->{_m} = $LIB->_div($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}, $LIB->_new($scale), $x->{_es}, '+');
            $x->bnorm();        # remove trailing 0's
        }
    }                           # end 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) {
            $x -> bfloor();
            $rem->bmod($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 remainder

# set up parameters
    my ($class, $x, $y, $a, $p, $r) = (ref($_[0]), @_);
    # objectify is costly, so avoid it
    if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
        ($class, $x, $y, $a, $p, $r) = objectify(2, @_);
    }

return $x if $x->modify('bmod');

# At least one argument is NaN. This is handled the same way as in
    # Math::BigInt -> bmod().

if ($x -> is_nan() || $y -> is_nan()) {
        return $x -> bnan();
    }

# Modulo zero. This is handled the same way as in Math::BigInt -> bmod().

if ($y -> is_zero()) {
        return $x;
    }

# Numerator (dividend) is +/-inf. This is handled the same way as in
    # Math::BigInt -> bmod().

if ($x -> is_inf()) {
        return $x -> bnan();
    }

# Denominator (divisor) is +/-inf. This is handled the same way as in
    # Math::BigInt -> bmod().

if ($y -> is_inf()) {
        if ($x -> is_zero() || $x -> bcmp(0) == $y -> bcmp(0)) {
            return $x;
        } else {
            return $x -> binf($y -> sign());
        }
    }

return $x->bzero() if $x->is_zero()
      || ($x->is_int() &&
          # check that $y == +1 or $y == -1:
          ($LIB->_is_zero($y->{_e}) && $LIB->_is_one($y->{_m})));

my $cmp = $x->bacmp($y);    # equal or $x < $y?
    if ($cmp == 0) {            # $x == $y => result 0
        return $x -> bzero($a, $p);
    }

# only $y of the operands negative?
    my $neg = $x->{sign} ne $y->{sign} ? 1 : 0;

$x->{sign} = $y->{sign};     # calc sign first
    if ($cmp < 0 && $neg == 0) { # $x < $y => result $x
        return $x -> round($a, $p, $r);
    }

my $ym = $LIB->_copy($y->{_m});

# 2e1 => 20
    $ym = $LIB->_lsft($ym, $y->{_e}, 10)
      if $y->{_es} eq '+' && !$LIB->_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 = $LIB->_num($y->{_e});  # no more digits after dot
        $x->{_m} = $LIB->_lsft($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 = $LIB->_num($x->{_e}); # no more digits after dot
        $ym = $LIB->_lsft($ym, $x->{_e}, 10); # 123 => 1230
    }
    # 123e1 % 20 => 1230 % 20
    if ($x->{_es} eq '+' && !$LIB->_is_zero($x->{_e})) {
        $x->{_m} = $LIB->_lsft($x->{_m}, $x->{_e}, 10); # es => '+' here
    }

$x->{_e} = $LIB->_new($shiftx);
    $x->{_es} = '+';
    $x->{_es} = '-' if $shiftx != 0 || $shifty != 0;
    $x->{_e} = $LIB->_add($x->{_e}, $LIB->_new($shifty)) if $shifty != 0;

# now mantissas are equalized, exponent of $x is adjusted, so calc result

$x->{_m} = $LIB->_mod($x->{_m}, $ym);

$x->{sign} = '+' if $LIB->_is_zero($x->{_m}); # fix sign for -0
    $x->bnorm();

if ($neg != 0 && ! $x -> is_zero()) # 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 $LIB->_is_zero($x->{_m}); # fix sign for -0
        $x->bnorm();
    }

$x->round($a, $p, $r, $y);     # round and return
}

sub bmodpow {
    # takes a very large number to a very large exponent in a given very
    # large modulus, quickly, thanks to binary exponentiation. Supports
    # negative exponents.
    my ($class, $num, $exp, $mod) = 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);
}

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 ($class, $x, $y, $a, $p, $r) = (ref($_[0]), @_);
    # objectify is costly, so avoid it
    if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
        ($class, $x, $y, $a, $p, $r) = objectify(2, @_);
    }

return $x if $x->modify('bpow');

# $x and/or $y is a NaN
    return $x->bnan() if $x->is_nan() || $y->is_nan();

# $x and/or $y is a +/-Inf
    if ($x->is_inf("-")) {
        return $x->bzero()   if $y->is_negative();
        return $x->bnan()    if $y->is_zero();
        return $x            if $y->is_odd();
        return $x->bneg();
    } elsif ($x->is_inf("+")) {
        return $x->bzero()   if $y->is_negative();
        return $x->bnan()    if $y->is_zero();
        return $x;
    } elsif ($y->is_inf("-")) {
        return $x->bnan()    if $x -> is_one("-");
        return $x->binf("+") if $x > -1 && $x < 1;
        return $x->bone()    if $x -> is_one("+");
        return $x->bzero();
    } elsif ($y->is_inf("+")) {
        return $x->bnan()    if $x -> is_one("-");
        return $x->bzero()   if $x > -1 && $x < 1;
        return $x->bone()    if $x -> is_one("+");
        return $x->binf("+");
    }

# we don't support complex numbers, so return NaN
    return $x->bnan() if $x->is_negative() && !$y->is_int();

# 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();

my $y1 = $y->as_number()->{value}; # make MBI part

if ($x->is_one("-")) {
        # if $x == -1 and odd/even y => +1/-1  because +-1 ^ (+-1) => +-1
        return $LIB->_is_odd($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 = $LIB->_is_odd($y1) ? '-' : '+' if $x->{sign} ne '+';

# calculate $x->{_m} ** $y and $x->{_e} * $y separately (faster)
    $x->{_m} = $LIB->_pow($x->{_m}, $y1);
    $x->{_e} = $LIB->_mul ($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 blog {
    # Return the logarithm of the operand. If a second operand is defined, that
    # value is used as the base, otherwise the base is assumed to be Euler's
    # constant.

my ($class, $x, $base, $a, $p, $r);

# Don't objectify the base, since an undefined base, as in $x->blog() or
    # $x->blog(undef) signals that the base is Euler's number.

if (!ref($_[0]) && $_[0] =~ /^[A-Za-z]|::/) {
        # E.g., Math::BigFloat->blog(256, 2)
        ($class, $x, $base, $a, $p, $r) =
          defined $_[2] ? objectify(2, @_) : objectify(1, @_);
    } else {
        # E.g., Math::BigFloat::blog(256, 2) or $x->blog(2)
        ($class, $x, $base, $a, $p, $r) =
          defined $_[1] ? objectify(2, @_) : objectify(1, @_);
    }

return $x if $x->modify('blog');

return $x -> bnan() if $x -> is_nan();

# 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);

# no rounding at all, so must use fallback
    if (scalar @params == 0) {
        # simulate old behaviour
        $params[0] = $class->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
    }

my $done = 0;
    if (defined $base) {
        $base = $class -> new($base) unless ref $base;
        if ($base -> is_nan() || $base -> is_one()) {
            $x -> bnan();
            $done = 1;
        } elsif ($base -> is_inf() || $base -> is_zero()) {
            if ($x -> is_inf() || $x -> is_zero()) {
                $x -> bnan();
            } else {
                $x -> bzero(@params);
            }
            $done = 1;
        } elsif ($base -> is_negative()) { # -inf < base < 0
            if ($x -> is_one()) {          #     x = 1
                $x -> bzero(@params);
            } elsif ($x == $base) {
                $x -> bone('+', @params); #     x = base
            } else {
                $x -> bnan();   #     otherwise
            }
            $done = 1;
        } elsif ($x == $base) {
            $x -> bone('+', @params); # 0 < base && 0 < x < inf
            $done = 1;
        }
    }

# We now know that the base is either undefined or positive and finite.

unless ($done) {
        if ($x -> is_inf()) {   #   x = +/-inf
            my $sign = defined $base && $base < 1 ? '-' : '+';
            $x -> binf($sign);
            $done = 1;
        } elsif ($x -> is_neg()) { #   -inf < x < 0
            $x -> bnan();
            $done = 1;
        } elsif ($x -> is_one()) { #   x = 1
            $x -> bzero(@params);
            $done = 1;
        } elsif ($x -> is_zero()) { #   x = 0
            my $sign = defined $base && $base < 1 ? '+' : '-';
            $x -> binf($sign);
            $done = 1;
        }
    }

if ($done) {
        if ($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 $abr = "$class\::accuracy"; my $ab = $$abr; $$abr = undef;
    my $pbr = "$class\::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 $x is not a Math::BigFloat (handle BigInt input)
    # XXX TODO: rebless!
    if (!$x->isa('Math::BigFloat')) {
        $x = Math::BigFloat->new($x);
        $class = ref($x);
    }

$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 $xint = Math::BigInt -> new($x    -> bdstr());
        my $bint = Math::BigInt -> new($base -> bdstr());
        $xint->blog($bint);

# if we found the exact result, we're done
        if ($bint -> bpow($xint) == $x) {
            my $xflt = Math::BigFloat -> new($xint -> bdstr());
            $x->{sign} = $xflt->{sign};
            $x->{_m}   = $xflt->{_m};
            $x->{_es}  = $xflt->{_es};
            $x->{_e}   = $xflt->{_e};
            $done = 1;
        }
    }

if ($done == 0) {
        # First calculate the log to base e (using reduction by 10 and possibly
        # also by 2):
        $x->_log_10($scale);

# and if a different base was requested, convert it
        if (defined $base) {
            $base = Math::BigFloat->new($base)
              unless $base->isa('Math::BigFloat');
            # log_b(x) = ln(x) / ln(b), so compute ln(b)
            my $base_log_e = $base->copy()->_log_10($scale);
            $x->bdiv($base_log_e, $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 bexp {
    # Calculate e ** X (Euler's number to the power of X)
    my ($class, $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] = $class->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);
        $class = ref($x);
    }

# when user set globals, they would interfere with our calculation, so
    # disable them and later re-enable them
    no strict 'refs';
    my $abr = "$class\::accuracy"; my $ab = $$abr; $$abr = undef;
    my $pbr = "$class\::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 simply reduce 13700/5040 to 685/252, but must keep
    # the numerator and the denominator!

if ($scale <= 75) {
        # set $x directly from a cached string form
        $x->{_m} = $LIB->_new("2718281828459045235360287471352662497757" .
                              "2470936999595749669676277240766303535476");
        $x->{sign} = '+';
        $x->{_es} = '-';
        $x->{_e} = $LIB->_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 = $LIB->_new("9093339520860578540197197" .
                           "0164779391644753259799242");
        my $F = $LIB->_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 = $LIB->_mul($A, $F);
            $A = $LIB->_inc($A);
            # increment f
            $F = $LIB->_inc($F);
        }
        # compute $B as factorial of $steps (this is faster than doing it manually)
        my $B = $LIB->_fac($LIB->_new($steps));

#  print "A ", $LIB->_str($A), "\nB ", $LIB->_str($B), "\n";

# compute A/B with $scale digits in the result (truncate, not round)
        $A = $LIB->_lsft($A, $LIB->_new($scale), 10);
        $A = $LIB->_div($A, $B);

$x->{_m} = $A;
        $x->{sign} = '+';
        $x->{_es} = '-';
        $x->{_e} = $LIB->_new($scale);
    }

# $x contains now an estimate of e, with some surplus digits, so we can round
    if (!$x_org->is_one()) {
        # Reduce size of fractional part, followup with integer power of two.
        my $lshift = 0;
        while ($lshift < 30 && $x_org->bacmp(2 << $lshift) > 0) {
            $lshift++;
        }
        # Raise $x to the wanted power and round it.
        if ($lshift == 0) {
            $x->bpow($x_org, @params);
        } else {
            my($mul, $rescale) = (1 << $lshift, $scale+1+$lshift);
            $x->bpow(scalar $x_org->bdiv($mul, $rescale), $rescale)->bpow($mul, @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 bnok {
    # Calculate n over k (binomial coefficient or "choose" function) as integer.
    # set up parameters
    my ($class, $x, $y, @r) = (ref($_[0]), @_);

# objectify is costly, so avoid it
    if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
        ($class, $x, $y, @r) = objectify(2, @_);
    }

return $x if $x->modify('bnok');

return $x->bnan() if $x->is_nan() || $y->is_nan();
    return $x->bnan() if (($x->is_finite() && !$x->is_int()) ||
                          ($y->is_finite() && !$y->is_int()));

my $xint = Math::BigInt -> new($x -> bsstr());
    my $yint = Math::BigInt -> new($y -> bsstr());
    $xint -> bnok($yint);
    my $xflt = Math::BigFloat -> new($xint);

$x->{_m}   = $xflt->{_m};
    $x->{_e}   = $xflt->{_e};
    $x->{_es}  = $xflt->{_es};
    $x->{sign} = $xflt->{sign};

return $x;
}

sub bsin {
    # Calculate a sinus of x.
    my ($class, $x, @r) = ref($_[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] = $class->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 $abr = "$class\::accuracy"; my $ab = $$abr; $$abr = undef;
    my $pbr = "$class\::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 $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 = $class->new(6); my $factorial = $class->new(4);
    delete $x->{_a};
    delete $x->{_p};

my $limit = $class->new("1E-". ($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 bcos {
    # Calculate a cosinus of x.
    my ($class, $x, @r) = ref($_[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] = $class->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 $abr = "$class\::accuracy"; my $ab = $$abr; $$abr = undef;
    my $pbr = "$class\::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 $over = $x * $x;         # X ^ 2
    my $x2 = $over->copy();     # X ^ 2; difference between terms
    my $sign = 1;               # start with -=
    my $below = $class->new(2);
    my $factorial = $class->new(3);
    $x->bone();
    delete $x->{_a};
    delete $x->{_p};

my $limit = $class->new("1E-". ($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 batan {
    # Calculate a arcus tangens of x.

my $self    = shift;
    my $selfref = ref $self;
    my $class   = $selfref || $self;

my (@r) = @_;

# 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);
    ($self, @params) = $self->_find_round_parameters(@r);

# Constant object or error in _find_round_parameters?

return $self if $self->modify('batan') || $self->is_nan();

if ($self->{sign} =~ /^[+-]inf\z/) {
        # +inf result is PI/2
        # -inf result is -PI/2
        # calculate PI/2
        my $pi = $class->bpi(@r);
        # modify $self in place
        $self->{_m} = $pi->{_m};
        $self->{_e} = $pi->{_e};
        $self->{_es} = $pi->{_es};
        # -y => -PI/2, +y => PI/2
        $self->{sign} = substr($self->{sign}, 0, 1); # "+inf" => "+"
        $self -> {_m} = $LIB->_div($self->{_m}, $LIB->_new(2));
        return $self;
    }

return $self->bzero(@r) if $self->is_zero();

# no rounding at all, so must use fallback
    if (scalar @params == 0) {
        # simulate old behaviour
        $params[0] = $class->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 ($LIB->_is_one($self->{_m}) && $LIB->_is_zero($self->{_e})) {
        my $pi = $class->bpi($scale - 3);
        # modify $self in place
        $self->{_m} = $pi->{_m};
        $self->{_e} = $pi->{_e};
        $self->{_es} = $pi->{_es};
        # leave the sign of $self alone (+1 => +PI/4, -1 => -PI/4)
        $self->{_m} = $LIB->_div($self->{_m}, $LIB->_new(4));
        return $self;
    }

# This series is only valid if -1 < x < 1, so for other x we need to
    # calculate PI/2 - atan(1/x):
    my $pi = undef;
    if ($self->bacmp($self->copy()->bone) >= 0) {
        # calculate PI/2
        $pi = $class->bpi($scale - 3);
        $pi->{_m} = $LIB->_div($pi->{_m}, $LIB->_new(2));
        # calculate 1/$self:
        my $self_copy = $self->copy();
        # modify $self in place
        $self->bone();
        $self->bdiv($self_copy, $scale);
    }

my $fmul = 1;
    foreach (0 .. int($scale / 20)) {
        $fmul *= 2;
        $self->bdiv($self->copy()->bmul($self)->binc->bsqrt($scale + 4)->binc, $scale + 4);
    }

# When user set globals, they would interfere with our calculation, so
    # disable them and later re-enable them.
    no strict 'refs';
    my $abr = "$class\::accuracy";  my $ab = $$abr; $$abr = undef;
    my $pbr = "$class\::precision"; my $pb = $$pbr; $$pbr = undef;
    # We also need to disable any set A or P on $self (_find_round_parameters
    # took them already into account), since these would interfere, too
    delete $self->{_a};
    delete $self->{_p};
    # Need to disable $upgrade in BigInt, to avoid deep recursion.
    local $Math::BigInt::upgrade = undef;

my $over = $self * $self;   # X ^ 2
    my $self2 = $over->copy();  # X ^ 2; difference between terms
    $over->bmul($self);         # X ^ 3 as starting value
    my $sign = 1;               # start with -=
    my $below = $class->new(3);
    my $two = $class->new(2);
    delete $self->{_a};
    delete $self->{_p};

my $limit = $class->new("1E-". ($scale-1));
    #my $steps = 0;
    while (1) {
        # We calculate the next term, and add it to the last. When the next
        # term is below our limit, it won't affect the outcome anymore, so we
        # stop:
        my $next = $over->copy()->bdiv($below, $scale);
        last if $next->bacmp($limit) <= 0;

if ($sign == 0) {
            $self->badd($next);
        } else {
            $self->bsub($next);
        }
        $sign = 1-$sign;        # alternatex
        # calculate things for the next term
        $over->bmul($self2);    # $self*$self
        $below->badd($two);     # n += 2
    }
    $self->bmul($fmul);

if (defined $pi) {
        my $self_copy = $self->copy();
        # modify $self in place
        $self->{_m} = $pi->{_m};
        $self->{_e} = $pi->{_e};
        $self->{_es} = $pi->{_es};
        # PI/2 - $self
        $self->bsub($self_copy);
    }

# Shortcut to not run through _find_round_parameters again.
    if (defined $params[0]) {
        $self->bround($params[0], $params[2]); # then round accordingly
    } else {
        $self->bfround($params[1], $params[2]); # then round accordingly
    }
    if ($fallback) {
        # Clear a/p after round, since user did not request it.
        delete $self->{_a};
        delete $self->{_p};
    }

# restore globals
    $$abr = $ab;
    $$pbr = $pb;
    $self;
}

sub batan2 {
    # $y -> batan2($x) returns the arcus tangens of $y / $x.

# Set up parameters.
    my ($class, $y, $x, @r) = (ref($_[0]), @_);

# Objectify is costly, so avoid it if we can.
    if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
        ($class, $y, $x, @r) = objectify(2, @_);
    }

# Quick exit if $y is read-only.
    return $y if $y -> modify('batan2');

# Handle all NaN cases.
    return $y -> bnan() if $x->{sign} eq $nan || $y->{sign} eq $nan;

# 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] = $class -> 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
    }

if ($x -> is_inf("+")) {                    # x = inf
        if ($y -> is_inf("+")) {                #    y = inf
            $y -> bpi($scale) -> bmul("0.25");  #       pi/4
        } elsif ($y -> is_inf("-")) {           #    y = -inf
            $y -> bpi($scale) -> bmul("-0.25"); #       -pi/4
        } else {                                #    -inf < y < inf
            return $y -> bzero(@r);             #       0
        }
    } elsif ($x -> is_inf("-")) {               # x = -inf
        if ($y -> is_inf("+")) {                #    y = inf
            $y -> bpi($scale) -> bmul("0.75");  #       3/4 pi
        } elsif ($y -> is_inf("-")) {           #    y = -inf
            $y -> bpi($scale) -> bmul("-0.75"); #       -3/4 pi
        } elsif ($y >= 0) {                     #    y >= 0
            $y -> bpi($scale);                  #       pi
        } else {                                #    y < 0
            $y -> bpi($scale) -> bneg();        #       -pi
        }
    } elsif ($x > 0) {                               # 0 < x < inf
        if ($y -> is_inf("+")) {                     #    y = inf
            $y -> bpi($scale) -> bmul("0.5");        #       pi/2
        } elsif ($y -> is_inf("-")) {                #    y = -inf
            $y -> bpi($scale) -> bmul("-0.5");       #       -pi/2
        } else {                                     #   -inf < y < inf
            $y -> bdiv($x, $scale) -> batan($scale); #       atan(y/x)
        }
    } elsif ($x < 0) {                        # -inf < x < 0
        my $pi = $class -> bpi($scale);
        if ($y >= 0) {                        #    y >= 0
            $y -> bdiv($x, $scale) -> batan() #       atan(y/x) + pi
               -> badd($pi);
        } else {                              #    y < 0
            $y -> bdiv($x, $scale) -> batan() #       atan(y/x) - pi
               -> bsub($pi);
        }
    } else {                                   # x = 0
        if ($y > 0) {                          #    y > 0
            $y -> bpi($scale) -> bmul("0.5");  #       pi/2
        } elsif ($y < 0) {                     #    y < 0
            $y -> bpi($scale) -> bmul("-0.5"); #       -pi/2
        } else {                               #    y = 0
            return $y -> bzero(@r);            #       0
        }
    }

$y -> round(@r);

if ($fallback) {
        delete $y->{_a};
        delete $y->{_p};
    }

return $y;
}
##############################################################################

sub bsqrt {
    # calculate square root
    my ($class, $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] = $class->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 $abr = "$class\::accuracy"; my $ab = $$abr; $$abr = undef;
    my $pbr = "$class\::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 = $LIB->_copy($x->{_m});
    $i = $LIB->_lsft($i, $x->{_e}, 10) unless $LIB->_is_zero($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} = $LIB->_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"
        ${"$class\::accuracy"} = $ab;
        ${"$class\::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 multiplying 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 = $LIB->_copy($x->{_m});

my $length = $LIB->_len($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 $LIB->_is_odd($x->{_e});

$y1 = $LIB->_lsft($y1, $LIB->_new($s2), 10);

# now take the square root and truncate to integer
    $y1 = $LIB->_sqrt($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 = $LIB->_num($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 -= $LIB->_len($y1);
    if ($dat < 0) {
        $dat = abs($dat);
        $x->{_e} = $LIB->_new($dat);
        $x->{_es} = '-';
    } else {
        $x->{_e} = $LIB->_new($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 broot {
    # calculate $y'th root of $x

# set up parameters
    my ($class, $x, $y, $a, $p, $r) = (ref($_[0]), @_);
    # objectify is costly, so avoid it
    if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
        ($class, $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] = $class->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 $abr = "$class\::accuracy"; my $ab = $$abr; $$abr = undef;
    my $pbr = "$class\::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 '+' && $LIB->_is_two($y->{_m}) && $LIB->_is_zero($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 = $class->bone()->bdiv($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 = $LIB->_copy($x->{_m});
            $i = $LIB->_lsft($i, $x->{_e}, 10) unless $LIB->_is_zero($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} = $LIB->_zero();
                $x->{_es} = '+';
                $x->bnorm();
                $done = 1;
            }
        }
        if ($done == 0) {
            my $u = $class->bone()->bdiv($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 bfac {
    # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT
    # compute factorial number, modifies first argument

# set up parameters
    my ($class, $x, @r) = (ref($_[0]), @_);
    # objectify is costly, so avoid it
    ($class, $x, @r) = objectify(1, @_) if !ref($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?

if (! $LIB->_is_zero($x->{_e})) {
        $x->{_m} = $LIB->_lsft($x->{_m}, $x->{_e}, 10); # change 12e1 to 120e0
        $x->{_e} = $LIB->_zero();           # normalize
        $x->{_es} = '+';
    }
    $x->{_m} = $LIB->_fac($x->{_m});       # calculate factorial
    $x->bnorm()->round(@r);     # norm again and round result
}

sub bdfac {
    # compute double factorial

# set up parameters
    my ($class, $x, @r) = (ref($_[0]), @_);
    # objectify is costly, so avoid it
    ($class, $x, @r) = objectify(1, @_) if !ref($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?

croak("bdfac() requires a newer version of the $LIB library.")
        unless $LIB->can('_dfac');

if (! $LIB->_is_zero($x->{_e})) {
        $x->{_m} = $LIB->_lsft($x->{_m}, $x->{_e}, 10); # change 12e1 to 120e0
        $x->{_e} = $LIB->_zero();           # normalize
        $x->{_es} = '+';
    }
    $x->{_m} = $LIB->_dfac($x->{_m});       # calculate factorial
    $x->bnorm()->round(@r);     # norm again and round result
}

sub blsft {
    # shift left by $y (multiply by $b ** $y)

# set up parameters
    my ($class, $x, $y, $b, $a, $p, $r) = (ref($_[0]), @_);

# objectify is costly, so avoid it
    if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
        ($class, $x, $y, $b, $a, $p, $r) = objectify(2, @_);
    }

return $x if $x -> modify('blsft');
    return $x if $x -> {sign} !~ /^[+-]$/; # nan, +inf, -inf

$b = 2 if !defined $b;
    $b = $class -> new($b) unless ref($b) && $b -> isa($class);

return $x -> bnan() if $x -> is_nan() || $y -> is_nan() || $b -> is_nan();

# shift by a negative amount?
    return $x -> brsft($y -> copy() -> babs(), $b) if $y -> {sign} =~ /^-/;

$x -> bmul($b -> bpow($y), $a, $p, $r, $y);
}

sub brsft {
    # shift right by $y (divide $b ** $y)

# set up parameters
    my ($class, $x, $y, $b, $a, $p, $r) = (ref($_[0]), @_);

# objectify is costly, so avoid it
    if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
        ($class, $x, $y, $b, $a, $p, $r) = objectify(2, @_);
    }

return $x if $x -> modify('brsft');
    return $x if $x -> {sign} !~ /^[+-]$/; # nan, +inf, -inf

$b = 2 if !defined $b;
    $b = $class -> new($b) unless ref($b) && $b -> isa($class);

return $x -> bnan() if $x -> is_nan() || $y -> is_nan() || $b -> is_nan();

# shift by a negative amount?
    return $x -> blsft($y -> copy() -> babs(), $b) if $y -> {sign} =~ /^-/;

# the following call to bdiv() will return either quotient (scalar context)
    # or quotient and remainder (list context).
    $x -> bdiv($b -> bpow($y), $a, $p, $r, $y);
}

###############################################################################
# Bitwise methods
###############################################################################

sub band {
    my $x     = shift;
    my $xref  = ref($x);
    my $class = $xref || $x;

croak 'band() is an instance method, not a class method' unless $xref;
    croak 'Not enough arguments for band()' if @_ < 1;

return if $x -> modify('band');

my $y = shift;
    $y = $class -> new($y) unless ref($y);

my @r = @_;

my $xtmp = Math::BigInt -> new($x -> bint());   # to Math::BigInt
    $xtmp -> band($y);
    $xtmp = $class -> new($xtmp);                   # back to Math::BigFloat

$x -> {sign} = $xtmp -> {sign};
    $x -> {_m}   = $xtmp -> {_m};
    $x -> {_es}  = $xtmp -> {_es};
    $x -> {_e}   = $xtmp -> {_e};

return $x -> round(@r);
}

sub bior {
    my $x     = shift;
    my $xref  = ref($x);
    my $class = $xref || $x;

croak 'bior() is an instance method, not a class method' unless $xref;
    croak 'Not enough arguments for bior()' if @_ < 1;

return if $x -> modify('bior');

my $y = shift;
    $y = $class -> new($y) unless ref($y);

my @r = @_;

my $xtmp = Math::BigInt -> new($x -> bint());   # to Math::BigInt
    $xtmp -> bior($y);
    $xtmp = $class -> new($xtmp);                   # back to Math::BigFloat

$x -> {sign} = $xtmp -> {sign};
    $x -> {_m}   = $xtmp -> {_m};
    $x -> {_es}  = $xtmp -> {_es};
    $x -> {_e}   = $xtmp -> {_e};

return $x -> round(@r);
}

sub bxor {
    my $x     = shift;
    my $xref  = ref($x);
    my $class = $xref || $x;

croak 'bxor() is an instance method, not a class method' unless $xref;
    croak 'Not enough arguments for bxor()' if @_ < 1;

return if $x -> modify('bxor');

my $y = shift;
    $y = $class -> new($y) unless ref($y);

my @r = @_;

my $xtmp = Math::BigInt -> new($x -> bint());   # to Math::BigInt
    $xtmp -> bxor($y);
    $xtmp = $class -> new($xtmp);                   # back to Math::BigFloat

$x -> {sign} = $xtmp -> {sign};
    $x -> {_m}   = $xtmp -> {_m};
    $x -> {_es}  = $xtmp -> {_es};
    $x -> {_e}   = $xtmp -> {_e};

return $x -> round(@r);
}

sub bnot {
    my $x     = shift;
    my $xref  = ref($x);
    my $class = $xref || $x;

croak 'bnot() is an instance method, not a class method' unless $xref;

return if $x -> modify('bnot');

my @r = @_;

my $xtmp = Math::BigInt -> new($x -> bint());   # to Math::BigInt
    $xtmp -> bnot();
    $xtmp = $class -> new($xtmp);                   # back to Math::BigFloat

$x -> {sign} = $xtmp -> {sign};
    $x -> {_m}   = $xtmp -> {_m};
    $x -> {_es}  = $xtmp -> {_es};
    $x -> {_e}   = $xtmp -> {_e};

return $x -> round(@r);
}

###############################################################################
# Rounding methods
###############################################################################

sub bround {
    # accuracy: preserve $N digits, and overwrite the rest with 0's
    my $x = shift;
    my $class = ref($x) || $x;
    $x = $class->new(shift) if !ref($x);

if (($_[0] || 0) < 0) {
        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() || $LIB->_len($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 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 $class = ref($x) || $x;
    $x = $class->new(shift) if !ref($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 = $LIB->_len($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}.$LIB->_num($x->{_e}))); # digits after dot
        my $zad = 0;                                      # zeros after dot
        $zad = $dad - $len if (-$dad < -$len); # for 0.00..00xxx style

# print "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 = $LIB->_len($x->{_m});
        # digits before dot
        my $dbd = $dbt + ($x->{_es} . $LIB->_num($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 bfloor {
    # round towards minus infinity
    my ($class, $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} = $LIB->_rsft($x->{_m}, $x->{_e}, 10); # cut off digits after dot
        $x->{_e} = $LIB->_zero();                     # trunc/norm
        $x->{_es} = '+';                              # abs e
        $x->{_m} = $LIB->_inc($x->{_m}) if $x->{sign} eq '-';    # increment if negative
    }
    $x->round($a, $p, $r);
}

sub bceil {
    # round towards plus infinity
    my ($class, $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} = $LIB->_rsft($x->{_m}, $x->{_e}, 10); # cut off digits after dot
        $x->{_e} = $LIB->_zero();                     # trunc/norm
        $x->{_es} = '+';                              # abs e
        if ($x->{sign} eq '+') {
            $x->{_m} = $LIB->_inc($x->{_m}); # increment if positive
        } else {
            $x->{sign} = '+' if $LIB->_is_zero($x->{_m}); # avoid -0
        }
    }
    $x->round($a, $p, $r);
}

sub bint {
    # round towards zero
    my ($class, $x, $a, $p, $r) = ref($_[0]) ? (ref($_[0]), @_) : objectify(1, @_);

return $x if $x->modify('bint');
    return $x if $x->{sign} !~ /^[+-]$/; # nan, +inf, -inf

# if $x has digits after the decimal point
    if ($x->{_es} eq '-') {
        $x->{_m} = $LIB->_rsft($x->{_m}, $x->{_e}, 10); # cut off digits after dot
        $x->{_e} = $LIB->_zero();                     # truncate/normalize
        $x->{_es} = '+';                              # abs e
        $x->{sign} = '+' if $LIB->_is_zero($x->{_m}); # avoid -0
    }
    $x->round($a, $p, $r);
}

###############################################################################
# Other mathematical methods
###############################################################################

sub bgcd {
    # (BINT or num_str, BINT or num_str) return BINT
    # does not modify arguments, but returns new object

unshift @_, __PACKAGE__
      unless ref($_[0]) || $_[0] =~ /^[a-z]\w*(?:::[a-z]\w*)*$/i;

my ($class, @args) = objectify(0, @_);

my $x = shift @args;
    $x = ref($x) && $x -> isa($class) ? $x -> copy() : $class -> new($x);
    return $class->bnan() unless $x -> is_int();

while (@args) {
        my $y = shift @args;
        $y = $class->new($y) unless ref($y) && $y -> isa($class);
        return $class->bnan() unless $y -> is_int();

# greatest common divisor
        while (! $y->is_zero()) {
            ($x, $y) = ($y->copy(), $x->copy()->bmod($y));
        }

last if $x -> is_one();
    }
    return $x -> babs();
}

sub blcm {
    # (BFLOAT or num_str, BFLOAT or num_str) return BFLOAT
    # does not modify arguments, but returns new object
    # Least Common Multiple

unshift @_, __PACKAGE__
      unless ref($_[0]) || $_[0] =~ /^[a-z]\w*(?:::[a-z]\w*)*$/i;

my ($class, @args) = objectify(0, @_);

my $x = shift @args;
    $x = ref($x) && $x -> isa($class) ? $x -> copy() : $class -> new($x);
    return $class->bnan() if $x->{sign} !~ /^[+-]$/;    # x NaN?

while (@args) {
        my $y = shift @args;
        $y = $class -> new($y) unless ref($y) && $y -> isa($class);
        return $x->bnan() unless $y -> is_int();
        my $gcd = $x -> bgcd($y);
        $x -> bdiv($gcd) -> bmul($y);
    }

return $x -> babs();
}

###############################################################################
# Object property methods
###############################################################################

sub length {
    my $x = shift;
    my $class = ref($x) || $x;
    $x = $class->new(shift) unless ref($x);

return 1 if $LIB->_is_zero($x->{_m});

my $len = $LIB->_len($x->{_m});
    $len += $LIB->_num($x->{_e}) if $x->{_es} eq '+';
    if (wantarray()) {
        my $t = 0;
        $t = $LIB->_num($x->{_e}) if $x->{_es} eq '-';
        return ($len, $t);
    }
    $len;
}

sub mantissa {
    # return a copy of the mantissa
    my ($class, $x) = ref($_[0]) ? (ref($_[0]), $_[0]) : objectify(1, @_);

if ($x->{sign} !~ /^[+-]$/) {
        my $s = $x->{sign};
        $s =~ s/^\+//;
        return Math::BigInt->new($s, undef, undef); # -inf, +inf => +inf
    }
    my $m = Math::BigInt->new($LIB->_str($x->{_m}), undef, undef);
    $m->bneg() if $x->{sign} eq '-';

$m;
}

sub exponent {
    # return a copy of the exponent
    my ($class, $x) = ref($_[0]) ? (ref($_[0]), $_[0]) : objectify(1, @_);

if ($x->{sign} !~ /^[+-]$/) {
        my $s = $x->{sign};
        $s =~ s/^[+-]//;
        return Math::BigInt->new($s, undef, undef); # -inf, +inf => +inf
    }
    Math::BigInt->new($x->{_es} . $LIB->_str($x->{_e}), undef, undef);
}

sub parts {
    # return a copy of both the exponent and the mantissa
    my ($class, $x) = ref($_[0]) ? (ref($_[0]), $_[0]) : objectify(1, @_);

if ($x->{sign} !~ /^[+-]$/) {
        my $s = $x->{sign};
        $s =~ s/^\+//;
        my $se = $s;
        $se =~ s/^-//;
        return ($class->new($s), $class->new($se)); # +inf => inf and -inf, +inf => inf
    }
    my $m = Math::BigInt->bzero();
    $m->{value} = $LIB->_copy($x->{_m});
    $m->bneg() if $x->{sign} eq '-';
    ($m, Math::BigInt->new($x->{_es} . $LIB->_num($x->{_e})));
}

sub sparts {
    my $self  = shift;
    my $class = ref $self;

croak("sparts() is an instance method, not a class method")
        unless $class;

# Not-a-number.

if ($self -> is_nan()) {
        my $mant = $self -> copy();             # mantissa
        return $mant unless wantarray;          # scalar context
        my $expo = $class -> bnan();            # exponent
        return ($mant, $expo);                  # list context
    }

# Infinity.

if ($self -> is_inf()) {
        my $mant = $self -> copy();             # mantissa
        return $mant unless wantarray;          # scalar context
        my $expo = $class -> binf('+');         # exponent
        return ($mant, $expo);                  # list context
    }

# Finite number.

my $mant = $class -> bzero();
    $mant -> {sign} = $self -> {sign};
    $mant -> {_m}   = $LIB->_copy($self -> {_m});
    return $mant unless wantarray;

my $expo = $class -> bzero();
    $expo -> {sign} = $self -> {_es};
    $expo -> {_m}   = $LIB->_copy($self -> {_e});

return ($mant, $expo);
}

sub nparts {
    my $self  = shift;
    my $class = ref $self;

croak("nparts() is an instance method, not a class method")
        unless $class;

# Not-a-number.

if ($self -> is_nan()) {
        my $mant = $self -> copy();             # mantissa
        return $mant unless wantarray;          # scalar context
        my $expo = $class -> bnan();            # exponent
        return ($mant, $expo);                  # list context
    }

# Infinity.

if ($self -> is_inf()) {
        my $mant = $self -> copy();             # mantissa
        return $mant unless wantarray;          # scalar context
        my $expo = $class -> binf('+');         # exponent
        return ($mant, $expo);                  # list context
    }

# Finite number.

my ($mant, $expo) = $self -> sparts();

if ($mant -> bcmp(0)) {
        my ($ndigtot, $ndigfrac) = $mant -> length();
        my $expo10adj = $ndigtot - $ndigfrac - 1;

if ($expo10adj != 0) {
            my $factor  = "1e" . -$expo10adj;
            $mant -> bmul($factor);
            return $mant unless wantarray;
            $expo -> badd($expo10adj);
            return ($mant, $expo);
        }
    }

return $mant unless wantarray;
    return ($mant, $expo);
}

sub eparts {
    my $self  = shift;
    my $class = ref $self;

croak("eparts() is an instance method, not a class method")
        unless $class;

# Not-a-number and Infinity.

return $self -> sparts() if $self -> is_nan() || $self -> is_inf();

# Finite number.

my ($mant, $expo) = $self -> nparts();

my $c = $expo -> copy() -> bmod(3);
    $mant -> blsft($c, 10);
    return $mant unless wantarray;

$expo -> bsub($c);
    return ($mant, $expo);
}

sub dparts {
    my $self  = shift;
    my $class = ref $self;

croak("dparts() is an instance method, not a class method")
        unless $class;

# Not-a-number and Infinity.

if ($self -> is_nan() || $self -> is_inf()) {
        my $int = $self -> copy();
        return $int unless wantarray;
        my $frc = $class -> bzero();
        return ($int, $frc);
    }

my $int = $self  -> copy();
    my $frc = $class -> bzero();

# If the input has a fraction part.

if ($int->{_es} eq '-') {
        $int->{_m} = $LIB -> _rsft($int->{_m}, $int->{_e}, 10);
        $int->{_e} = $LIB -> _zero();
        $int->{_es} = '+';
        $int->{sign} = '+' if $LIB->_is_zero($int->{_m});   # avoid -0

return $int unless wantarray;
        $frc = $self -> copy() -> bsub($int);
        return ($int, $frc);
    }

return $int unless wantarray;
    return ($int, $frc);
}

###############################################################################
# String conversion methods
###############################################################################

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 ($class, $x) = ref($_[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 '+' && $LIB->_is_zero($x->{_m}));
    if ($not_zero) {
        $es = $LIB->_str($x->{_m});
        $len = CORE::length($es);
        my $e = $LIB->_num($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 = $LIB->_num($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;
}

# Decimal notation, e.g., "12345.6789".

sub bdstr {
    my $x = shift;

if ($x->{sign} ne '+' && $x->{sign} ne '-') {
        return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
        return 'inf';                                  # +inf
    }

my $mant = $LIB->_str($x->{_m});
    my $expo = $x -> exponent();

my $str = $mant;
    if ($expo >= 0) {
        $str .= "0" x $expo;
    } else {
        my $mantlen = CORE::length($mant);
        my $c = $mantlen + $expo;
        $str = "0" x (1 - $c) . $str if $c <= 0;
        substr($str, $expo, 0) = '.';
    }

return $x->{sign} eq '-' ? "-$str" : $str;
}

# Scientific notation with significand/mantissa as an integer, e.g., "12345.6789"
# is written as "123456789e-4".

sub bsstr {
    my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);

if ($x->{sign} ne '+' && $x->{sign} ne '-') {
        return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
        return 'inf';                                  # +inf
    }

my $str = $LIB->_str($x->{_m}) . 'e' . $x->{_es}. $LIB->_str($x->{_e});
    return $x->{sign} eq '-' ? "-$str" : $str;
}

# Normalized notation, e.g., "12345.6789" is written as "1.23456789e+4".

sub bnstr {
    my $x = shift;

if ($x->{sign} ne '+' && $x->{sign} ne '-') {
        return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
        return 'inf';                                  # +inf
    }

my ($mant, $expo) = $x -> nparts();

my $esgn = $expo < 0 ? '-' : '+';
    my $eabs = $expo -> babs() -> bfround(0) -> bstr();
    #$eabs = '0' . $eabs if length($eabs) < 2;

return $mant . 'e' . $esgn . $eabs;
}

# Engineering notation, e.g., "12345.6789" is written as "12.3456789e+3".

sub bestr {
    my $x = shift;

if ($x->{sign} ne '+' && $x->{sign} ne '-') {
        return $x->{sign} unless $x->{sign} eq '+inf'; # -inf, NaN
        return 'inf';                                  # +inf
    }

my ($mant, $expo) = $x -> eparts();

my $esgn = $expo < 0 ? '-' : '+';
    my $eabs = $expo -> babs() -> bfround(0) -> bstr();
    #$eabs = '0' . $eabs if length($eabs) < 2;

return $mant . 'e' . $esgn . $eabs;
}

sub to_hex {
    # return number as hexadecimal string (only for integers defined)

my ($class, $x) = ref($_[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 = $LIB->_copy($x->{_m});
    if (! $LIB->_is_zero($x->{_e})) {   # > 0
        $z = $LIB->_lsft($z, $x->{_e}, 10);
    }
    my $str = $LIB->_to_hex($z);
    return $x->{sign} eq '-' ? "-$str" : $str;
}

sub to_oct {
    # return number as octal digit string (only for integers defined)

my ($class, $x) = ref($_[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 octal?

my $z = $LIB->_copy($x->{_m});
    if (! $LIB->_is_zero($x->{_e})) {   # > 0
        $z = $LIB->_lsft($z, $x->{_e}, 10);
    }
    my $str = $LIB->_to_oct($z);
    return $x->{sign} eq '-' ? "-$str" : $str;
}

sub to_bin {
    # return number as binary digit string (only for integers defined)

my ($class, $x) = ref($_[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 binary?

my $z = $LIB->_copy($x->{_m});
    if (! $LIB->_is_zero($x->{_e})) {   # > 0
        $z = $LIB->_lsft($z, $x->{_e}, 10);
    }
    my $str = $LIB->_to_bin($z);
    return $x->{sign} eq '-' ? "-$str" : $str;
}

sub to_ieee754 {
    my $x = shift;
    my $format = shift;
    my $class = ref $x;

my $enc;            # significand encoding (applies only to decimal)
    my $k;              # storage width in bits
    my $b;              # base

if ($format =~ /^binary(\d+)\z/) {
        $k = $1;
        $b = 2;
    } elsif ($format =~ /^decimal(\d+)(dpd|bcd)?\z/) {
        $k = $1;
        $b = 10;
        $enc = $2 || 'dpd';     # default is dencely-packed decimals (DPD)
    } elsif ($format eq 'half') {
        $k = 16;
        $b = 2;
    } elsif ($format eq 'single') {
        $k = 32;
        $b = 2;
    } elsif ($format eq 'double') {
        $k = 64;
        $b = 2;
    } elsif ($format eq 'quadruple') {
        $k = 128;
        $b = 2;
    } elsif ($format eq 'octuple') {
        $k = 256;
        $b = 2;
    } elsif ($format eq 'sexdecuple') {
        $k = 512;
        $b = 2;
    }

if ($b == 2) {

# Get the parameters for this format.

my $p;                      # precision (in bits)
        my $t;                      # number of bits in significand
        my $w;                      # number of bits in exponent

if ($k == 16) {             # binary16 (half-precision)
            $p = 11;
            $t = 10;
            $w =  5;
        } elsif ($k == 32) {        # binary32 (single-precision)
            $p = 24;
            $t = 23;
            $w =  8;
        } elsif ($k == 64) {        # binary64 (double-precision)
            $p = 53;
            $t = 52;
            $w = 11;
        } else {                    # binaryN (quadruple-precition and above)
            if ($k < 128 || $k != 32 * sprintf('%.0f', $k / 32)) {
                croak "Number of bits must be 16, 32, 64, or >= 128 and",
                  " a multiple of 32";
            }
            $p = $k - sprintf('%.0f', 4 * log($k) / log(2)) + 13;
            $t = $p - 1;
            $w = $k - $t - 1;
        }

# The maximum exponent, minimum exponent, and exponent bias.

my $emax = $class -> new(2) -> bpow($w - 1) -> bdec();
        my $emin = 1 - $emax;
        my $bias = $emax;

# Get numerical sign, exponent, and mantissa/significand for bit
        # string.

my $sign = 0;
        my $expo;
        my $mant;

if ($x -> is_nan()) {                   # nan
            $sign = 1;
            $expo = $emax -> copy() -> binc();
            $mant = $class -> new(2) -> bpow($t - 1);
        } elsif ($x -> is_inf()) {              # inf
            $sign = 1 if $x -> is_neg();
            $expo = $emax -> copy() -> binc();
            $mant = $class -> bzero();
        } elsif ($x -> is_zero()) {             # zero
            $expo = $emin -> copy() -> bdec();
            $mant = $class -> bzero();
        } else {                                # normal and subnormal

$sign = 1 if $x -> is_neg();

# Now we need to compute the mantissa and exponent in base $b.

my $binv = $class -> new("0.5");
            my $b    = $class -> new(2);
            my $one  = $class -> bone();

# We start off by initializing the exponent to zero and the
            # mantissa to the input value. Then we increase the mantissa and
            # decrease the exponent, or vice versa, until the mantissa is in
            # the desired range or we hit one of the limits for the exponent.

$mant = $x -> copy() -> babs();

# We need to find the base 2 exponent. First make an estimate of
            # the base 2 exponent, before adjusting it below. We could skip
            # this estimation and go straight to the while-loops below, but the
            # loops are slow, especially when the final exponent is far from
            # zero and even more so if the number of digits is large. This
            # initial estimation speeds up the computation dramatically.
            #
            #   log2($m * 10**$e) = log10($m + 10**$e) * log(10)/log(2)
            #                     = (log10($m) + $e) * log(10)/log(2)
            #                     = (log($m)/log(10) + $e) * log(10)/log(2)

my ($m, $e) = $x -> nparts();
            my $ms = $m -> numify();
            my $es = $e -> numify();

my $expo_est = (log(abs($ms))/log(10) + $es) * log(10)/log(2);
            $expo_est = int($expo_est);

# Limit the exponent.

if ($expo_est > $emax) {
                $expo_est = $emax;
            } elsif ($expo_est < $emin) {
                $expo_est = $emin;
            }

# Don't multiply by a number raised to a negative exponent. This
            # will cause a division, whose result is truncated to some fixed
            # number of digits. Instead, multiply by the inverse number raised
            # to a positive exponent.

$expo = $class -> new($expo_est);
            if ($expo_est > 0) {
                $mant -> bmul($binv -> copy() -> bpow($expo));
            } elsif ($expo_est < 0) {
                my $expo_abs = $expo -> copy() -> bneg();
                $mant -> bmul($b -> copy() -> bpow($expo_abs));
            }

# Final adjustment.

while ($mant >= $b && $expo <= $emax) {
                $mant -> bmul($binv);
                $expo -> binc();
            }

while ($mant < $one && $expo >= $emin) {
                $mant -> bmul($b);
                $expo -> bdec();
            }

# Encode as infinity, normal number or subnormal number?

if ($expo > $emax) {                # overflow => infinity
                $expo = $emax -> copy() -> binc();
                $mant = $class -> bzero();
            } elsif ($expo < $emin) {           # subnormal number
                my $const = $class -> new(2) -> bpow($t - 1);
                $mant -> bmul($const);
                $mant -> bfround(0);
            } else {                            # normal number
                $mant -> bdec();                # remove implicit leading bit
                my $const = $class -> new(2) -> bpow($t);
                $mant -> bmul($const) -> bfround(0);
            }
        }

$expo -> badd($bias);                   # add bias

my $signbit = "$sign";

my $mantbits = $mant -> to_bin();
        $mantbits = ("0" x ($t - CORE::length($mantbits))) . $mantbits;

my $expobits = $expo -> to_bin();
        $expobits = ("0" x ($w - CORE::length($expobits))) . $expobits;

my $bin = $signbit . $expobits . $mantbits;
        return pack "B*", $bin;
    }

croak("The format '$format' is not yet supported.");
}

sub as_hex {
    # return number as hexadecimal string (only for integers defined)

my ($class, $x) = ref($_[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 = $LIB->_copy($x->{_m});
    if (! $LIB->_is_zero($x->{_e})) {   # > 0
        $z = $LIB->_lsft($z, $x->{_e}, 10);
    }
    my $str = $LIB->_as_hex($z);
    return $x->{sign} eq '-' ? "-$str" : $str;
}

sub as_oct {
    # return number as octal digit string (only for integers defined)

my ($class, $x) = ref($_[0]) ? (ref($_[0]), $_[0]) : objectify(1, @_);

return $x->bstr() if $x->{sign} !~ /^[+-]$/; # inf, nan etc
    return '00' if $x->is_zero();

return $nan if $x->{_es} ne '+';    # how to do 1e-1 in octal?

my $z = $LIB->_copy($x->{_m});
    if (! $LIB->_is_zero($x->{_e})) {   # > 0
        $z = $LIB->_lsft($z, $x->{_e}, 10);
    }
    my $str = $LIB->_as_oct($z);
    return $x->{sign} eq '-' ? "-$str" : $str;
}

sub as_bin {
    # return number as binary digit string (only for integers defined)

my ($class, $x) = ref($_[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 binary?

my $z = $LIB->_copy($x->{_m});
    if (! $LIB->_is_zero($x->{_e})) {   # > 0
        $z = $LIB->_lsft($z, $x->{_e}, 10);
    }
    my $str = $LIB->_as_bin($z);
    return $x->{sign} eq '-' ? "-$str" : $str;
}

sub numify {
    # Make a Perl scalar number from a Math::BigFloat object.
    my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);

if ($x -> is_nan()) {
        require Math::Complex;
        my $inf = Math::Complex::Inf();
        return $inf - $inf;
    }

if ($x -> is_inf()) {
        require Math::Complex;
        my $inf = Math::Complex::Inf();
        return $x -> is_negative() ? -$inf : $inf;
    }

# Create a string and let Perl's atoi()/atof() handle the rest.
    return 0 + $x -> bnstr();
}

###############################################################################
# Private methods and functions.
###############################################################################

sub import {
    my $class = 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 { $class->new(shift); };
        } elsif ($_[$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
            # $LIB = $_[$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 $mbilib) && ($LIB eq 'Math::BigInt::Calc')) {
        # $LIB already loaded
        Math::BigInt->import($lib_kind, "$lib, $mbilib", 'objectify');
    } else {
        # $LIB 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 => $lib, 'objectify');
    }
    if ($@) {
        croak("Couldn't load $lib: $! $@");
    }
    # find out which one was actually loaded
    $LIB = Math::BigInt->config('lib');

# register us with MBI to get notified of future lib changes
    Math::BigInt::_register_callback($class, sub { $LIB = $_[0]; });

$class->export_to_level(1, $class, @a); # export wanted functions
}

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 _log {
    # internal log function to calculate ln() based on Taylor series.
    # Modifies $x in place.
    my ($x, $scale) = @_;
    my $class = ref $x;

# in case of $x == 1, result is 0
    return $x->bzero() if $x->is_one();

# XXX TODO: rewrite this in a similar 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, $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 = $class->new(3); $f = $class->new(2);

$limit = $class->new("1E-". ($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);
    }
    $x->bmul($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 $x in place.
    my ($x, $scale) = @_;
    my $class = ref $x;

# 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. x = 123, dbd = 3; x = 1.23, dbd = 1;
    # x = 0.0123, dbd = -1; x = 0.000123, dbd = -3, etc.

my $dbd = $LIB->_num($x->{_e});
    $dbd = -$dbd if $x->{_es} eq '-';
    $dbd += $LIB->_len($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 '+' &&                     # $x == 10
        ($LIB->_is_one($x->{_e}) &&
         $LIB->_is_one($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
            $calc = 0;                      # no need to calc, but round
        }
        # if we can't use the shortcut, we continue normally
    } else {
        # disable the shortcut for 2, since we maybe have it cached
        if (($LIB->_is_zero($x->{_e}) &&        # $x == 2
             $LIB->_is_two($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 '-' &&                    # $x == 0.1
         ($LIB->_is_one($x->{_e}) &&
          $LIB->_is_one($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 $x 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 = $class->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 = $class->new($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 = $class->new($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();
                $l_2->_log($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 = $class->new('1.25');
            $l_10->_log($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
        $l_10->bmul($class->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}, $LIB->_new($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 = $class->new($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
    }
    $x->bround($scale+4);
    # $twos > 0 => did mul 2, < 0 => did div 2 (but we never did both)
    # So calculate correction factor based on ln(2):
    if ($twos != 0) {
        $LOG_2 = $class->new($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();
            $l_2->_log($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
    } else {
        undef $l_2;
    }

$x->_log($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 _e_add {
    # Internal helper sub to take two positive integers and their signs and
    # then add them. Input ($LIB, $LIB, ('+'|'-'), ('+'|'-')), output
    # ($LIB, ('+'|'-')).

my ($x, $y, $xs, $ys) = @_;

# if the signs are equal we can add them (-5 + -3 => -(5 + 3) => -8)
    if ($xs eq $ys) {
        $x = $LIB->_add($x, $y); # +a + +b or -a + -b
    } else {
        my $a = $LIB->_acmp($x, $y);
        if ($a == 0) {
            # This does NOT modify $x in-place. TODO: Fix this?
            $x = $LIB->_zero(); # result is 0
            $xs = '+';
            return ($x, $xs);
        }
        if ($a > 0) {
            $x = $LIB->_sub($x, $y);     # abs sub
        } else {                         # a < 0
            $x = $LIB->_sub ($y, $x, 1); # abs sub
            $xs = $ys;
        }
    }

$xs = '+' if $xs eq '-' && $LIB->_is_zero($x); # no "-0"

return ($x, $xs);
}

sub _e_sub {
    # Internal helper sub to take two positive integers and their signs and
    # then subtract them. Input ($LIB, $LIB, ('+'|'-'), ('+'|'-')),
    # output ($LIB, ('+'|'-'))
    my ($x, $y, $xs, $ys) = @_;

# flip sign
    $ys = $ys eq '+' ? '-' : '+'; # swap sign of second operand ...
    _e_add($x, $y, $xs, $ys);     # ... and let _e_add() do the job
}

sub _pow {
    # Calculate a power where $y is a non-integer, like 2 ** 0.3
    my ($x, $y, @r) = @_;
    my $class = ref($x);

# if $y == 0.5, it is sqrt($x)
    $HALF = $class->new($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] = $class->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 $abr = "$class\::accuracy"; my $ab = $$abr; $$abr = undef;
    my $pbr = "$class\::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);
    my $do_invert = ($u->{sign} eq '-');
    $u->bneg()  if $do_invert;
    $v = $class->bone();        # 1
    $factor = $class->new(2);   # 2
    $x->bone();                 # first term: 1

$below = $v->copy();
    $over = $u->copy();

$limit = $class->new("1E-". ($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:
        $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 $x->{sign} !~ /^[-+]$/;
    }

if ($do_invert) {
        my $x_copy = $x->copy();
        $x->bone->bdiv($x_copy, $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;
}

1;

__END__

=pod

=head1 NAME

Math::BigFloat - Arbitrary size floating point math package

=head1 SYNOPSIS

use Math::BigFloat;

# Configuration methods (may be used as class methods and instance methods)

Math::BigFloat->accuracy();     # get class accuracy
  Math::BigFloat->accuracy($n);   # set class accuracy
  Math::BigFloat->precision();    # get class precision
  Math::BigFloat->precision($n);  # set class precision
  Math::BigFloat->round_mode();   # get class rounding mode
  Math::BigFloat->round_mode($m); # set global round mode, must be one of
                                  # 'even', 'odd', '+inf', '-inf', 'zero',
                                  # 'trunc', or 'common'
  Math::BigFloat->config("lib");  # name of backend math library

# Constructor methods (when the class methods below are used as instance
  # methods, the value is assigned the invocand)

$x = Math::BigFloat->new($str);               # defaults to 0
  $x = Math::BigFloat->new('0x123');            # from hexadecimal
  $x = Math::BigFloat->new('0b101');            # from binary
  $x = Math::BigFloat->from_hex('0xc.afep+3');  # from hex
  $x = Math::BigFloat->from_hex('cafe');        # ditto
  $x = Math::BigFloat->from_oct('1.3267p-4');   # from octal
  $x = Math::BigFloat->from_oct('0377');        # ditto
  $x = Math::BigFloat->from_bin('0b1.1001p-4'); # from binary
  $x = Math::BigFloat->from_bin('0101');        # ditto
  $x = Math::BigFloat->from_ieee754($b, "binary64");  # from IEEE-754 bytes
  $x = Math::BigFloat->bzero();                 # create a +0
  $x = Math::BigFloat->bone();                  # create a +1
  $x = Math::BigFloat->bone('-');               # create a -1
  $x = Math::BigFloat->binf();                  # create a +inf
  $x = Math::BigFloat->binf('-');               # create a -inf
  $x = Math::BigFloat->bnan();                  # create a Not-A-Number
  $x = Math::BigFloat->bpi();                   # returns pi

$y = $x->copy();        # make a copy (unlike $y = $x)
  $y = $x->as_int();      # return as BigInt

# Boolean methods (these don't modify the invocand)

$x->is_zero();          # if $x is 0
  $x->is_one();           # if $x is +1
  $x->is_one("+");        # ditto
  $x->is_one("-");        # if $x is -1
  $x->is_inf();           # if $x is +inf or -inf
  $x->is_inf("+");        # if $x is +inf
  $x->is_inf("-");        # if $x is -inf
  $x->is_nan();           # if $x is NaN

$x->is_positive();      # if $x > 0
  $x->is_pos();           # ditto
  $x->is_negative();      # if $x < 0
  $x->is_neg();           # ditto

$x->is_odd();           # if $x is odd
  $x->is_even();          # if $x is even
  $x->is_int();           # if $x is an integer

# Comparison methods

$x->bcmp($y);           # compare numbers (undef, < 0, == 0, > 0)
  $x->bacmp($y);          # compare absolutely (undef, < 0, == 0, > 0)
  $x->beq($y);            # true if and only if $x == $y
  $x->bne($y);            # true if and only if $x != $y
  $x->blt($y);            # true if and only if $x < $y
  $x->ble($y);            # true if and only if $x <= $y
  $x->bgt($y);            # true if and only if $x > $y
  $x->bge($y);            # true if and only if $x >= $y

# Arithmetic methods

$x->bneg();             # negation
  $x->babs();             # absolute value
  $x->bsgn();             # sign function (-1, 0, 1, or NaN)
  $x->bnorm();            # normalize (no-op)
  $x->binc();             # increment $x by 1
  $x->bdec();             # decrement $x by 1
  $x->badd($y);           # addition (add $y to $x)
  $x->bsub($y);           # subtraction (subtract $y from $x)
  $x->bmul($y);           # multiplication (multiply $x by $y)
  $x->bmuladd($y,$z);     # $x = $x * $y + $z
  $x->bdiv($y);           # division (floored), set $x to quotient
                          # return (quo,rem) or quo if scalar
  $x->btdiv($y);          # division (truncated), set $x to quotient
                          # return (quo,rem) or quo if scalar
  $x->bmod($y);           # modulus (x % y)
  $x->btmod($y);          # modulus (truncated)
  $x->bmodinv($mod);      # modular multiplicative inverse
  $x->bmodpow($y,$mod);   # modular exponentiation (($x ** $y) % $mod)
  $x->bpow($y);           # power of arguments (x ** y)
  $x->blog();             # logarithm of $x to base e (Euler's number)
  $x->blog($base);        # logarithm of $x to base $base (e.g., base 2)
  $x->bexp();             # calculate e ** $x where e is Euler's number
  $x->bnok($y);           # x over y (binomial coefficient n over k)
  $x->bsin();             # sine
  $x->bcos();             # cosine
  $x->batan();            # inverse tangent
  $x->batan2($y);         # two-argument inverse tangent
  $x->bsqrt();            # calculate square root
  $x->broot($y);          # $y'th root of $x (e.g. $y == 3 => cubic root)
  $x->bfac();             # factorial of $x (1*2*3*4*..$x)

$x->blsft($n);          # left shift $n places in base 2
  $x->blsft($n,$b);       # left shift $n places in base $b
                          # returns (quo,rem) or quo (scalar context)
  $x->brsft($n);          # right shift $n places in base 2
  $x->brsft($n,$b);       # right shift $n places in base $b
                          # returns (quo,rem) or quo (scalar context)

# Bitwise methods

$x->band($y);           # bitwise and
  $x->bior($y);           # bitwise inclusive or
  $x->bxor($y);           # bitwise exclusive or
  $x->bnot();             # bitwise not (two's complement)

# Rounding methods
  $x->round($A,$P,$mode); # round to accuracy or precision using
                          # rounding mode $mode
  $x->bround($n);         # accuracy: preserve $n digits
  $x->bfround($n);        # $n > 0: round to $nth digit left of dec. point
                          # $n < 0: round to $nth digit right of dec. point
  $x->bfloor();           # round towards minus infinity
  $x->bceil();            # round towards plus infinity
  $x->bint();             # round towards zero

# Other mathematical methods

$x->bgcd($y);            # greatest common divisor
  $x->blcm($y);            # least common multiple

# Object property methods (do not modify the invocand)

$x->sign();              # the sign, either +, - or NaN
  $x->digit($n);           # the nth digit, counting from the right
  $x->digit(-$n);          # the nth digit, counting from the left
  $x->length();            # return number of digits in number
  ($xl,$f) = $x->length(); # length of number and length of fraction
                           # part, latter is always 0 digits long
                           # for Math::BigInt objects
  $x->mantissa();          # return (signed) mantissa as BigInt
  $x->exponent();          # return exponent as BigInt
  $x->parts();             # return (mantissa,exponent) as BigInt
  $x->sparts();            # mantissa and exponent (as integers)
  $x->nparts();            # mantissa and exponent (normalised)
  $x->eparts();            # mantissa and exponent (engineering notation)
  $x->dparts();            # integer and fraction part

# Conversion methods (do not modify the invocand)

$x->bstr();         # decimal notation, possibly zero padded
  $x->bsstr();        # string in scientific notation with integers
  $x->bnstr();        # string in normalized notation
  $x->bestr();        # string in engineering notation
  $x->bdstr();        # string in decimal notation
  $x->as_hex();       # as signed hexadecimal string with prefixed 0x
  $x->as_bin();       # as signed binary string with prefixed 0b
  $x->as_oct();       # as signed octal string with prefixed 0
  $x->to_ieee754($format); # to bytes encoded according to IEEE 754-2008

# Other conversion methods

$x->numify();           # return as scalar (might overflow or underflow)

=head1 DESCRIPTION

Math::BigFloat provides support for arbitrary precision floating point.
Overloading is also provided for Perl operators.

All operators (including basic math operations) are overloaded if you
declare your big floating point numbers as

$x = Math::BigFloat -> new('12_3.456_789_123_456_789E-2');

Operations with overloaded operators preserve the arguments, which is
exactly what you expect.

=head2 Input

Input values to these routines may be any scalar number or string that looks
like a number and represents a floating point number.

=over

=item *

Leading and trailing whitespace is ignored.

=item *

Leading and trailing zeros are ignored.

=item *

If the string has a "0x" prefix, it is interpreted as a hexadecimal number.

=item *

If the string has a "0b" prefix, it is interpreted as a binary number.

=item *

For hexadecimal and binary numbers, the exponent must be separated from the
significand (mantissa) by the letter "p" or "P", not "e" or "E" as with decimal
numbers.

=item *

One underline is allowed between any two digits, including hexadecimal and
binary digits.

=item *

If the string can not be interpreted, NaN is returned.

=back

Octal numbers are typically prefixed by "0", but since leading zeros are
stripped, these methods can not automatically recognize octal numbers, so use
the constructor from_oct() to interpret octal strings.

Some examples of valid string input

Input string                Resulting value
    123                         123
    1.23e2                      123
    12300e-2                    123
    0xcafe                      51966
    0b1101                      13
    67_538_754                  67538754
    -4_5_6.7_8_9e+0_1_0         -4567890000000
    0x1.921fb5p+1               3.14159262180328369140625e+0
    0b1.1001p-4                 9.765625e-2

=head2 Output

Output values are usually Math::BigFloat objects.

Boolean operators C<is_zero()>, C<is_one()>, C<is_inf()>, etc. return true or
false.

Comparison operators C<bcmp()> and C<bacmp()>) return -1, 0, 1, or
undef.

=head1 METHODS

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 Configuration methods

=over

=item 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<Math::BigInt/round()>, L<Math::BigInt/bround()> or L<Math::BigInt/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

=item 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!

=back

=head2 Constructor methods

=over

=item from_hex()

$x -> from_hex("0x1.921fb54442d18p+1");
    $x = Math::BigFloat -> from_hex("0x1.921fb54442d18p+1");

Interpret input as a hexadecimal string.A prefix ("0x", "x", ignoring case) is
optional. A single underscore character ("_") may be placed between any two
digits. If the input is invalid, a NaN is returned. The exponent is in base 2
using decimal digits.

If called as an instance method, the value is assigned to the invocand.

=item from_oct()

$x -> from_oct("1.3267p-4");
    $x = Math::BigFloat -> from_oct("1.3267p-4");

Interpret input as an octal string. A single underscore character ("_") may be
placed between any two digits. If the input is invalid, a NaN is returned. The
exponent is in base 2 using decimal digits.

If called as an instance method, the value is assigned to the invocand.

=item from_bin()

$x -> from_bin("0b1.1001p-4");
    $x = Math::BigFloat -> from_bin("0b1.1001p-4");

Interpret input as a hexadecimal string. A prefix ("0b" or "b", ignoring case)
is optional. A single underscore character ("_") may be placed between any two
digits. If the input is invalid, a NaN is returned. The exponent is in base 2
using decimal digits.

If called as an instance method, the value is assigned to the invocand.

=item from_ieee754()

Interpret the input as a value encoded as described in IEEE754-2008.  The input
can be given as a byte string, hex string or binary string. The input is
assumed to be in big-endian byte-order.

# both $dbl and $mbf are 3.141592...
        $bytes = "\x40\x09\x21\xfb\x54\x44\x2d\x18";
        $dbl = unpack "d>", $bytes;
        $mbf = Math::BigFloat -> from_ieee754($bytes, "binary64");

=item 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).

=back

=head2 Arithmetic methods

=over

=item bmuladd()

$x->bmuladd($y,$z);

Multiply $x by $y, and then add $z to the result.

This method was added in v1.87 of Math::BigInt (June 2007).

=item bdiv()

$q = $x->bdiv($y);
    ($q, $r) = $x->bdiv($y);

In scalar context, divides $x by $y and returns the result to the given or
default accuracy/precision. In list context, does floored division
(F-division), returning an integer $q and a remainder $r so that $x = $q * $y +
$r. The remainer (modulo) is equal to what is returned by C<< $x->bmod($y) >>.

=item bmod()

$x->bmod($y);

Returns $x modulo $y. When $x is finite, and $y is finite and non-zero, the
result is identical to the remainder after floored division (F-division). If,
in addition, both $x and $y are integers, the result is identical to the result
from Perl's % operator.

=item bexp()

$x->bexp($accuracy);            # calculate e ** X

Calculates the expression C<e ** $x> where C<e> is Euler's number.

This method was added in v1.82 of Math::BigInt (April 2007).

=item bnok()

$x->bnok($y);   # x over y (binomial coefficient n over k)

Calculates the binomial coefficient n over k, also called the "choose"
function. The result is equivalent to:

( n )      n!
    | - |  = -------
    ( k )    k!(n-k)!

This method was added in v1.84 of Math::BigInt (April 2007).

=item bsin()

my $x = Math::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).

=item 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).

=item 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).

=item 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).

=item as_float()

This method is called when Math::BigFloat encounters an object it doesn't know
how to handle. For instance, assume $x is a Math::BigFloat, or subclass
thereof, and $y is defined, but not a Math::BigFloat, or subclass thereof. If
you do

$x -> badd($y);

$y needs to be converted into an object that $x can deal with. This is done by
first checking if $y is something that $x might be upgraded to. If that is the
case, no further attempts are made. The next is to see if $y supports the
method C<as_float()>. The method C<as_float()> is expected to return either an
object that has the same class as $x, a subclass thereof, or a string that
C<ref($x)-E<gt>new()> can parse to create an object.

In Math::BigFloat, C<as_float()> has the same effect as C<copy()>.

=item to_ieee754()

Encodes the invocand as a byte string in the given format as specified in IEEE
754-2008. Note that the encoded value is the nearest possible representation of
the value. This value might not be exactly the same as the value in the
invocand.

# $x = 3.1415926535897932385
    $x = Math::BigFloat -> bpi(30);

$b = $x -> to_ieee754("binary64");  # encode as 8 bytes
    $h = unpack "H*", $b;               # "400921fb54442d18"

# 3.141592653589793115997963...
    $y = Math::BigFloat -> from_ieee754($h, "binary64");

All binary formats in IEEE 754-2008 are accepted. For convenience, som aliases
are recognized: "half" for "binary16", "single" for "binary32", "double" for
"binary64", "quadruple" for "binary128", "octuple" for "binary256", and
"sexdecuple" for "binary512".

See also L<https://en.wikipedia.org/wiki/IEEE_754>.

=back

=head2 ACCURACY AND PRECISION

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.	DIR	-	drwxr-xr-x	2026-06-24 01:59:38
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BigFloat	DIR	-	drwxr-xr-x	2026-06-24 01:59:38
BigInt	DIR	-	drwxr-xr-x	2026-06-24 01:59:38
BigFloat.pm	text/plain	194.67 KB	-rw-r--r--	2026-06-23 08:11:00
BigInt.pm	text/plain	212.67 KB	-rw-r--r--	2026-06-23 08:11:00
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Complex.pm	text/plain	48.89 KB	-rw-r--r--	2026-06-23 08:11:00
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