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BigRat.pm

#
# "Tax the rat farms." - Lord Vetinari
#

# The following hash values are used:
#   sign : +,-,NaN,+inf,-inf
#   _d   : denominator
#   _n   : numerator (value = _n/_d)
#   _a   : accuracy
#   _p   : precision
# You should not look at the innards of a BigRat - use the methods for this.

package Math::BigRat;

use 5.006;
use strict;
use warnings;

use Carp qw< carp croak >;

use Math::BigFloat 1.999718;

our $VERSION = '0.2614';

our @ISA = qw(Math::BigFloat);

our ($accuracy, $precision, $round_mode, $div_scale,
     $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(); },

;

BEGIN {
    *objectify = \&Math::BigInt::objectify;  # inherit this from BigInt
    *AUTOLOAD  = \&Math::BigFloat::AUTOLOAD; # can't inherit AUTOLOAD
    # We inherit these from BigFloat because currently it is not possible that
    # Math::BigFloat has a different $LIB variable than we, because
    # Math::BigFloat also uses Math::BigInt::config->('lib') (there is always
    # only one library loaded)
    *_e_add = \&Math::BigFloat::_e_add;
    *_e_sub = \&Math::BigFloat::_e_sub;
    *as_number = \&as_int;
    *is_pos = \&is_positive;
    *is_neg = \&is_negative;
}

##############################################################################
# Global constants and flags. Access these only via the accessor methods!

$accuracy   = $precision = undef;
$round_mode = 'even';
$div_scale  = 40;
$upgrade    = undef;
$downgrade  = undef;

# These are internally, and not to be used from the outside at all!

$_trap_nan = 0;                         # are NaNs ok? set w/ config()
$_trap_inf = 0;                         # are infs ok? set w/ config()

# the math backend library

my $LIB = 'Math::BigInt::Calc';

my $nan   = 'NaN';
#my $class = 'Math::BigRat';

sub isa {
    return 0 if $_[1] =~ /^Math::Big(Int|Float)/;       # we aren't
    UNIVERSAL::isa(@_);
}

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

sub new {
    my $proto    = shift;
    my $protoref = ref $proto;
    my $class    = $protoref || $proto;

# Check the way we are called.

if ($protoref) {
        croak("new() is a class method, not an instance method");
    }

if (@_ < 1) {
        #carp("Using new() with no argument is deprecated;",
        #           " use bzero() or new(0) instead");
        return $class -> bzero();
    }

if (@_ > 2) {
        carp("Superfluous arguments to new() ignored.");
    }

# Get numerator and denominator. If any of the arguments is undefined,
    # return zero.

my ($n, $d) = @_;

if (@_ == 1 && !defined $n ||
        @_ == 2 && (!defined $n || !defined $d))
    {
        #carp("Use of uninitialized value in new()");
        return $class -> bzero();
    }

# Initialize a new object.

my $self = bless {}, $class;

# One or two input arguments may be given. First handle the numerator $n.

if (ref($n)) {
        $n = Math::BigFloat -> new($n, undef, undef)
          unless ($n -> isa('Math::BigRat') ||
                  $n -> isa('Math::BigInt') ||
                  $n -> isa('Math::BigFloat'));
    } else {
        if (defined $d) {
            # If the denominator is defined, the numerator is not a string
            # fraction, e.g., "355/113".
            $n = Math::BigFloat -> new($n, undef, undef);
        } else {
            # If the denominator is undefined, the numerator might be a string
            # fraction, e.g., "355/113".
            if ($n =~ m| ^ \s* (\S+) \s* / \s* (\S+) \s* $ |x) {
                $n = Math::BigFloat -> new($1, undef, undef);
                $d = Math::BigFloat -> new($2, undef, undef);
            } else {
                $n = Math::BigFloat -> new($n, undef, undef);
            }
        }
    }

# At this point $n is an object and $d is either an object or undefined. An
    # undefined $d means that $d was not specified by the caller (not that $d
    # was specified as an undefined value).

unless (defined $d) {
        #return $n -> copy($n)               if $n -> isa('Math::BigRat');
        return $class -> copy($n)           if $n -> isa('Math::BigRat');
        return $class -> bnan()             if $n -> is_nan();
        return $class -> binf($n -> sign()) if $n -> is_inf();

if ($n -> isa('Math::BigInt')) {
            $self -> {_n}   = $LIB -> _new($n -> copy() -> babs() -> bstr());
            $self -> {_d}   = $LIB -> _one();
            $self -> {sign} = $n -> sign();
            return $self;
        }

if ($n -> isa('Math::BigFloat')) {
            my $m = $n -> mantissa() -> babs();
            my $e = $n -> exponent();
            $self -> {_n} = $LIB -> _new($m -> bstr());
            $self -> {_d} = $LIB -> _one();

if ($e > 0) {
                $self -> {_n} = $LIB -> _lsft($self -> {_n},
                                              $LIB -> _new($e -> bstr()), 10);
            } elsif ($e < 0) {
                $self -> {_d} = $LIB -> _lsft($self -> {_d},
                                              $LIB -> _new(-$e -> bstr()), 10);

my $gcd = $LIB -> _gcd($LIB -> _copy($self -> {_n}), $self -> {_d});
                if (!$LIB -> _is_one($gcd)) {
                    $self -> {_n} = $LIB -> _div($self->{_n}, $gcd);
                    $self -> {_d} = $LIB -> _div($self->{_d}, $gcd);
                }
            }

$self -> {sign} = $n -> sign();
            return $self;
        }

die "I don't know how to handle this";  # should never get here
    }

# At the point we know that both $n and $d are defined. We know that $n is
    # an object, but $d might still be a scalar. Now handle $d.

$d = Math::BigFloat -> new($d, undef, undef)
      unless ref($d) && ($d -> isa('Math::BigRat') ||
                         $d -> isa('Math::BigInt') ||
                         $d -> isa('Math::BigFloat'));

# At this point both $n and $d are objects.

return $class -> bnan() if $n -> is_nan() || $d -> is_nan();

# At this point neither $n nor $d is a NaN.

if ($n -> is_zero()) {
        return $class -> bnan() if $d -> is_zero();     # 0/0 = NaN
        return $class -> bzero();
    }

return $class -> binf($d -> sign()) if $d -> is_zero();

# At this point, neither $n nor $d is a NaN or a zero.

if ($d < 0) {               # make sure denominator is positive
        $n -> bneg();
        $d -> bneg();
    }

if ($n -> is_inf()) {
        return $class -> bnan() if $d -> is_inf();      # Inf/Inf = NaN
        return $class -> binf($n -> sign());
    }

# At this point $n is finite.

return $class -> bzero()            if $d -> is_inf();
    return $class -> binf($d -> sign()) if $d -> is_zero();

# At this point both $n and $d are finite and non-zero.

if ($n < 0) {
        $n -> bneg();
        $self -> {sign} = '-';
    } else {
        $self -> {sign} = '+';
    }

if ($n -> isa('Math::BigRat')) {

if ($d -> isa('Math::BigRat')) {

# At this point both $n and $d is a Math::BigRat.

# p   r    p * s    (p / gcd(p, r)) * (s / gcd(s, q))
            # - / -  = ----- =  ---------------------------------
            # q   s    q * r    (q / gcd(s, q)) * (r / gcd(p, r))

my $p = $n -> {_n};
            my $q = $n -> {_d};
            my $r = $d -> {_n};
            my $s = $d -> {_d};
            my $gcd_pr = $LIB -> _gcd($LIB -> _copy($p), $r);
            my $gcd_sq = $LIB -> _gcd($LIB -> _copy($s), $q);
            $self -> {_n} = $LIB -> _mul($LIB -> _div($LIB -> _copy($p), $gcd_pr),
                                         $LIB -> _div($LIB -> _copy($s), $gcd_sq));
            $self -> {_d} = $LIB -> _mul($LIB -> _div($LIB -> _copy($q), $gcd_sq),
                                         $LIB -> _div($LIB -> _copy($r), $gcd_pr));

return $self;       # no need for $self -> bnorm() here
        }

# At this point, $n is a Math::BigRat and $d is a Math::Big(Int|Float).

my $p = $n -> {_n};
        my $q = $n -> {_d};
        my $m = $d -> mantissa();
        my $e = $d -> exponent();

#                   /      p
        #                  |  ------------  if e > 0
        #                  |  q * m * 10^e
        #                  |
        # p                |    p
        # - / (m * 10^e) = |  -----         if e == 0
        # q                |  q * m
        #                  |
        #                  |  p * 10^-e
        #                  |  --------      if e < 0
        #                   \  q * m

$self -> {_n} = $LIB -> _copy($p);
        $self -> {_d} = $LIB -> _mul($LIB -> _copy($q), $m);
        if ($e > 0) {
            $self -> {_d} = $LIB -> _lsft($self -> {_d}, $e, 10);
        } elsif ($e < 0) {
            $self -> {_n} = $LIB -> _lsft($self -> {_n}, -$e, 10);
        }

return $self -> bnorm();

} else {

if ($d -> isa('Math::BigRat')) {

# At this point $n is a Math::Big(Int|Float) and $d is a
            # Math::BigRat.

my $m = $n -> mantissa();
            my $e = $n -> exponent();
            my $p = $d -> {_n};
            my $q = $d -> {_d};

#                   /  q * m * 10^e
            #                  |   ------------  if e > 0
            #                  |        p
            #                  |
            #              p   |   m * q
            # (m * 10^e) / - = |   -----         if e == 0
            #              q   |     p
            #                  |
            #                  |     q * m
            #                  |   ---------     if e < 0
            #                   \  p * 10^-e

$self -> {_n} = $LIB -> _mul($LIB -> _copy($q), $m);
            $self -> {_d} = $LIB -> _copy($p);
            if ($e > 0) {
                $self -> {_n} = $LIB -> _lsft($self -> {_n}, $e, 10);
            } elsif ($e < 0) {
                $self -> {_d} = $LIB -> _lsft($self -> {_d}, -$e, 10);
            }
            return $self -> bnorm();

} else {

# At this point $n and $d are both a Math::Big(Int|Float)

my $m1 = $n -> mantissa();
            my $e1 = $n -> exponent();
            my $m2 = $d -> mantissa();
            my $e2 = $d -> exponent();

#               /
            #              |  m1 * 10^(e1 - e2)
            #              |  -----------------  if e1 > e2
            #              |         m2
            #              |
            # m1 * 10^e1   |  m1
            # ---------- = |  --                 if e1 = e2
            # m2 * 10^e2   |  m2
            #              |
            #              |         m1
            #              |  -----------------  if e1 < e2
            #              |  m2 * 10^(e2 - e1)
            #               \

$self -> {_n} = $LIB -> _new($m1 -> bstr());
            $self -> {_d} = $LIB -> _new($m2 -> bstr());
            my $ediff = $e1 - $e2;
            if ($ediff > 0) {
                $self -> {_n} = $LIB -> _lsft($self -> {_n},
                                              $LIB -> _new($ediff -> bstr()),
                                              10);
            } elsif ($ediff < 0) {
                $self -> {_d} = $LIB -> _lsft($self -> {_d},
                                              $LIB -> _new(-$ediff -> bstr()),
                                              10);
            }

return $self -> bnorm();
        }
    }

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->{_d} = $LIB->_copy($self->{_d});
    $copy->{_n} = $LIB->_copy($self->{_n});
    $copy->{_a} = $self->{_a} if defined $self->{_a};
    $copy->{_p} = $self->{_p} if defined $self->{_p};

#($copy, $copy->{_a}, $copy->{_p})
    #  = $copy->_find_round_parameters(@_);

return $copy;
}

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

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

if ($_trap_nan) {
        croak ("Tried to set a variable to NaN in $class->bnan()");
    }

$self -> {sign} = $nan;
    $self -> {_n}   = $LIB -> _zero();
    $self -> {_d}   = $LIB -> _one();

($self, $self->{_a}, $self->{_p})
      = $self->_find_round_parameters(@_);

return $self;
}

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

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

my $sign = shift();
    $sign = defined($sign) && substr($sign, 0, 1) eq '-' ? '-inf' : '+inf';

if ($_trap_inf) {
        croak ("Tried to set a variable to +-inf in $class->binf()");
    }

$self -> {sign} = $sign;
    $self -> {_n}   = $LIB -> _zero();
    $self -> {_d}   = $LIB -> _one();

($self, $self->{_a}, $self->{_p})
      = $self->_find_round_parameters(@_);

return $self;
}

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

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

my $sign = shift();
    $sign = '+' unless defined($sign) && $sign eq '-';

$self -> {sign} = $sign;
    $self -> {_n}   = $LIB -> _one();
    $self -> {_d}   = $LIB -> _one();

($self, $self->{_a}, $self->{_p})
      = $self->_find_round_parameters(@_);

return $self;
}

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

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

$self -> {sign} = '+';
    $self -> {_n}   = $LIB -> _zero();
    $self -> {_d}   = $LIB -> _one();

($self, $self->{_a}, $self->{_p})
      = $self->_find_round_parameters(@_);

return $self;
}

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

sub config {
    # return (later set?) configuration data as hash ref
    my $class = shift() || 'Math::BigRat';

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

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

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

$cfg;
}

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

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

if ($x->{sign} !~ /^[+-]$/) {               # inf, NaN etc
        my $s = $x->{sign};
        $s =~ s/^\+//;                          # +inf => inf
        return $s;
    }

my $s = '';
    $s = $x->{sign} if $x->{sign} ne '+';       # '+3/2' => '3/2'

return $s . $LIB->_str($x->{_n}) if $LIB->_is_one($x->{_d});
    $s . $LIB->_str($x->{_n}) . '/' . $LIB->_str($x->{_d});
}

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

if ($x->{sign} !~ /^[+-]$/) {               # inf, NaN etc
        my $s = $x->{sign};
        $s =~ s/^\+//;                          # +inf => inf
        return $s;
    }

my $s = '';
    $s = $x->{sign} if $x->{sign} ne '+';       # +3 vs 3
    $s . $LIB->_str($x->{_n}) . '/' . $LIB->_str($x->{_d});
}

sub bnorm {
    # reduce the number to the shortest form
    my ($class, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);

# Both parts must be objects of whatever we are using today.
    if (my $c = $LIB->_check($x->{_n})) {
        croak("n did not pass the self-check ($c) in bnorm()");
    }
    if (my $c = $LIB->_check($x->{_d})) {
        croak("d did not pass the self-check ($c) in bnorm()");
    }

# no normalize for NaN, inf etc.
    return $x if $x->{sign} !~ /^[+-]$/;

# normalize zeros to 0/1
    if ($LIB->_is_zero($x->{_n})) {
        $x->{sign} = '+';                               # never leave a -0
        $x->{_d} = $LIB->_one() unless $LIB->_is_one($x->{_d});
        return $x;
    }

return $x if $LIB->_is_one($x->{_d});               # no need to reduce

# Compute the GCD.
    my $gcd = $LIB->_gcd($LIB->_copy($x->{_n}), $x->{_d});
    if (!$LIB->_is_one($gcd)) {
        $x->{_n} = $LIB->_div($x->{_n}, $gcd);
        $x->{_d} = $LIB->_div($x->{_d}, $gcd);
    }

$x;
}

##############################################################################
# sign manipulation

sub bneg {
    # (BRAT or num_str) return BRAT
    # 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->{_n}));
    $x;
}

##############################################################################
# special values

sub _bnan {
    # used by parent class bnan() to initialize number to NaN
    my $self = shift;

if ($_trap_nan) {
        my $class = ref($self);
        # "$self" below will stringify the object, this blows up if $self is a
        # partial object (happens under trap_nan), so fix it beforehand
        $self->{_d} = $LIB->_zero() unless defined $self->{_d};
        $self->{_n} = $LIB->_zero() unless defined $self->{_n};
        croak ("Tried to set $self to NaN in $class\::_bnan()");
    }
    $self->{_n} = $LIB->_zero();
    $self->{_d} = $LIB->_zero();
}

sub _binf {
    # used by parent class bone() to initialize number to +inf/-inf
    my $self = shift;

if ($_trap_inf) {
        my $class = ref($self);
        # "$self" below will stringify the object, this blows up if $self is a
        # partial object (happens under trap_nan), so fix it beforehand
        $self->{_d} = $LIB->_zero() unless defined $self->{_d};
        $self->{_n} = $LIB->_zero() unless defined $self->{_n};
        croak ("Tried to set $self to inf in $class\::_binf()");
    }
    $self->{_n} = $LIB->_zero();
    $self->{_d} = $LIB->_zero();
}

sub _bone {
    # used by parent class bone() to initialize number to +1/-1
    my $self = shift;
    $self->{_n} = $LIB->_one();
    $self->{_d} = $LIB->_one();
}

sub _bzero {
    # used by parent class bzero() to initialize number to 0
    my $self = shift;
    $self->{_n} = $LIB->_zero();
    $self->{_d} = $LIB->_one();
}

##############################################################################
# mul/add/div etc

sub badd {
    # add two rational 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, @_);
    }

# +inf + +inf => +inf, -inf + -inf => -inf
    return $x->binf(substr($x->{sign}, 0, 1))
      if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;

# +inf + -inf or -inf + +inf => NaN
    return $x->bnan() if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/);

#  1   1    gcd(3, 4) = 1    1*3 + 1*4    7
    #  - + -                  = --------- = --
    #  4   3                      4*3       12

# we do not compute the gcd() here, but simple do:
    #  5   7    5*3 + 7*4   43
    #  - + -  = --------- = --
    #  4   3       4*3      12

# and bnorm() will then take care of the rest

# 5 * 3
    $x->{_n} = $LIB->_mul($x->{_n}, $y->{_d});

# 7 * 4
    my $m = $LIB->_mul($LIB->_copy($y->{_n}), $x->{_d});

# 5 * 3 + 7 * 4
    ($x->{_n}, $x->{sign}) = _e_add($x->{_n}, $m, $x->{sign}, $y->{sign});

# 4 * 3
    $x->{_d} = $LIB->_mul($x->{_d}, $y->{_d});

# normalize result, and possible round
    $x->bnorm()->round(@r);
}

sub bsub {
    # subtract two rational numbers

# flip sign of $x, call badd(), then flip sign of result
    $x->{sign} =~ tr/+-/-+/
      unless $x->{sign} eq '+' && $LIB->_is_zero($x->{_n}); # not -0
    $x->badd($y, @r);           # does norm and round
    $x->{sign} =~ tr/+-/-+/
      unless $x->{sign} eq '+' && $LIB->_is_zero($x->{_n}); # not -0

$x;
}

sub bmul {
    # multiply two rational numbers

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('-');
    }

# x == 0  # also: or y == 1 or y == -1
    return wantarray ? ($x, $class->bzero()) : $x if $x -> is_zero();

if ($y -> is_zero()) {
        $x -> bzero();
        return wantarray ? ($x, $class->bzero()) : $x;
    }

# According to Knuth, this can be optimized by doing gcd twice (for d
    # and n) and reducing in one step. This saves us a bnorm() at the end.
    #
    # p   s    p * s    (p / gcd(p, r)) * (s / gcd(s, q))
    # - * -  = ----- =  ---------------------------------
    # q   r    q * r    (q / gcd(s, q)) * (r / gcd(p, r))

my $gcd_pr = $LIB -> _gcd($LIB -> _copy($x->{_n}), $y->{_d});
    my $gcd_sq = $LIB -> _gcd($LIB -> _copy($y->{_n}), $x->{_d});

$x->{_n} = $LIB -> _mul(scalar $LIB -> _div($x->{_n}, $gcd_pr),
                            scalar $LIB -> _div($LIB -> _copy($y->{_n}),
                                                $gcd_sq));
    $x->{_d} = $LIB -> _mul(scalar $LIB -> _div($x->{_d}, $gcd_sq),
                            scalar $LIB -> _div($LIB -> _copy($y->{_d}),
                                                $gcd_pr));

# compute new sign
    $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-';

$x->round(@r);
}

sub bdiv {
    # (dividend: BRAT or num_str, divisor: BRAT or num_str) return
    # (BRAT, BRAT) (quo, rem) or BRAT (only rem)

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(). See the comments in the code implementing that
    # method.

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 comments in the code implementing that
    # method.

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 comments in the code implementing that
    # method.

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::BigFloat -> bdiv(). See the comments in the code implementing that
    # method.

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();

# XXX TODO: list context, upgrade
    # According to Knuth, this can be optimized by doing gcd twice (for d and n)
    # and reducing in one step. This would save us the bnorm() at the end.
    #
    # p   r    p * s    (p / gcd(p, r)) * (s / gcd(s, q))
    # - / -  = ----- =  ---------------------------------
    # q   s    q * r    (q / gcd(s, q)) * (r / gcd(p, r))

$x->{_n} = $LIB->_mul($x->{_n}, $y->{_d});
    $x->{_d} = $LIB->_mul($x->{_d}, $y->{_n});

# compute new sign
    $x->{sign} = $x->{sign} eq $y->{sign} ? '+' : '-';

$x -> bnorm();
    if (wantarray) {
        my $rem = $x -> copy();
        $x -> bfloor();
        $x -> round(@r);
        $rem -> bsub($x -> copy()) -> bmul($y);
        return $x, $rem;
    } else {
        $x -> round(@r);
        return $x;
    }
}

sub bmod {
    # compute "remainder" (in Perl way) of $x / $y

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());
        }
    }

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

return $x if $x->is_zero(); # 0 / 7 = 0, mod 0

# Compute $x - $y * floor($x/$y). This can probably be optimized by working
    # on a lower level.

$x -> bsub($x -> copy() -> bdiv($y) -> bfloor() -> bmul($y));
    return $x -> round(@r);
}

##############################################################################
# bdec/binc

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

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

if ($x->{sign} eq '-') {
        $x->{_n} = $LIB->_add($x->{_n}, $x->{_d}); # -5/2 => -7/2
    } else {
        if ($LIB->_acmp($x->{_n}, $x->{_d}) < 0) # n < d?
        {
            # 1/3 -- => -2/3
            $x->{_n} = $LIB->_sub($LIB->_copy($x->{_d}), $x->{_n});
            $x->{sign} = '-';
        } else {
            $x->{_n} = $LIB->_sub($x->{_n}, $x->{_d}); # 5/2 => 3/2
        }
    }
    $x->bnorm()->round(@r);
}

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

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

if ($x->{sign} eq '-') {
        if ($LIB->_acmp($x->{_n}, $x->{_d}) < 0) {
            # -1/3 ++ => 2/3 (overflow at 0)
            $x->{_n} = $LIB->_sub($LIB->_copy($x->{_d}), $x->{_n});
            $x->{sign} = '+';
        } else {
            $x->{_n} = $LIB->_sub($x->{_n}, $x->{_d}); # -5/2 => -3/2
        }
    } else {
        $x->{_n} = $LIB->_add($x->{_n}, $x->{_d}); # 5/2 => 7/2
    }
    $x->bnorm()->round(@r);
}

##############################################################################
# is_foo methods (the rest is inherited)

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

return 1 if ($x->{sign} =~ /^[+-]$/) && # NaN and +-inf aren't
      $LIB->_is_one($x->{_d});              # x/y && y != 1 => no integer
    0;
}

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

return 1 if $x->{sign} eq '+' && $LIB->_is_zero($x->{_n});
    0;
}

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

croak "too many arguments for is_one()" if @_ > 2;
    my $sign = $_[1] || '';
    $sign = '+' if $sign ne '-';
    return 1 if ($x->{sign} eq $sign &&
                 $LIB->_is_one($x->{_n}) && $LIB->_is_one($x->{_d}));
    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, @_);

return 1 if ($x->{sign} =~ /^[+-]$/) &&               # NaN & +-inf aren't
      ($LIB->_is_one($x->{_d}) && $LIB->_is_odd($x->{_n})); # x/2 is not, but 3/1
    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, @_);

return 0 if $x->{sign} !~ /^[+-]$/; # NaN & +-inf aren't
    return 1 if ($LIB->_is_one($x->{_d}) # x/3 is never
                 && $LIB->_is_even($x->{_n})); # but 4/1 is
    0;
}

##############################################################################
# parts() and friends

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

# NaN, inf, -inf
    return Math::BigInt->new($x->{sign}) if ($x->{sign} !~ /^[+-]$/);

my $n = Math::BigInt->new($LIB->_str($x->{_n}));
    $n->{sign} = $x->{sign};
    $n;
}

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

# NaN
    return Math::BigInt->new($x->{sign}) if $x->{sign} eq 'NaN';
    # inf, -inf
    return Math::BigInt->bone() if $x->{sign} !~ /^[+-]$/;

Math::BigInt->new($LIB->_str($x->{_d}));
}

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

my $c = 'Math::BigInt';

return ($c->bnan(), $c->bnan()) if $x->{sign} eq 'NaN';
    return ($c->binf(), $c->binf()) if $x->{sign} eq '+inf';
    return ($c->binf('-'), $c->binf()) if $x->{sign} eq '-inf';

my $n = $c->new($LIB->_str($x->{_n}));
    $n->{sign} = $x->{sign};
    my $d = $c->new($LIB->_str($x->{_d}));
    ($n, $d);
}

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

return $nan unless $x->is_int();
    $LIB->_len($x->{_n});       # length(-123/1) => length(123)
}

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

return $nan unless $x->is_int();
    $LIB->_digit($x->{_n}, $n || 0); # digit(-123/1, 2) => digit(123, 2)
}

##############################################################################
# special calc routines

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

return $x if ($x->{sign} !~ /^[+-]$/ ||     # not for NaN, inf
                  $LIB->_is_one($x->{_d}));     # 22/1 => 22, 0/1 => 0

$x->{_n} = $LIB->_div($x->{_n}, $x->{_d});  # 22/7 => 3/1 w/ truncate
    $x->{_d} = $LIB->_one();                    # d => 1
    $x->{_n} = $LIB->_inc($x->{_n}) if $x->{sign} eq '+';   # +22/7 => 4/1
    $x->{sign} = '+' if $x->{sign} eq '-' && $LIB->_is_zero($x->{_n}); # -0 => 0
    $x;
}

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

return $x if ($x->{sign} !~ /^[+-]$/ ||     # not for NaN, inf
                  $LIB->_is_one($x->{_d}));     # 22/1 => 22, 0/1 => 0

$x->{_n} = $LIB->_div($x->{_n}, $x->{_d});  # 22/7 => 3/1 w/ truncate
    $x->{_d} = $LIB->_one();                    # d => 1
    $x->{_n} = $LIB->_inc($x->{_n}) if $x->{sign} eq '-';   # -22/7 => -4/1
    $x;
}

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

return $x if ($x->{sign} !~ /^[+-]$/ ||     # +/-inf or NaN
                  $LIB -> _is_one($x->{_d}));   # already an integer

$x->{_n} = $LIB->_div($x->{_n}, $x->{_d});  # 22/7 => 3/1 w/ truncate
    $x->{_d} = $LIB->_one();                    # d => 1
    $x->{sign} = '+' if $x->{sign} eq '-' && $LIB -> _is_zero($x->{_n});
    return $x;
}

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

# if $x is not an integer
    if (($x->{sign} ne '+') || (!$LIB->_is_one($x->{_d}))) {
        return $x->bnan();
    }

$x->{_n} = $LIB->_fac($x->{_n});
    # since _d is 1, we don't need to reduce/norm the result
    $x->round(@r);
}

sub bpow {
    # power ($x ** $y)

# 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, @_);
    }

# $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("+");
    }

if ($x->is_zero()) {
        return $x->binf()    if $y->is_negative();
        return $x->bone("+") if $y->is_zero();
        return $x;
    } elsif ($x->is_one()) {
        return $x->round(@r) if $y->is_odd();   # x is -1, y is odd => -1
        return $x->babs()->round(@r);           # x is -1, y is even => 1
    } elsif ($y->is_zero()) {
        return $x->bone(@r);                    # x^0 and x != 0 => 1
    } elsif ($y->is_one()) {
        return $x->round(@r);                   # x^1 => x
    }

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

# (a/b)^-(c/d) = (b/a)^(c/d)
    ($x->{_n}, $x->{_d}) = ($x->{_d}, $x->{_n}) if $y->is_negative();

unless ($LIB->_is_one($y->{_n})) {
        $x->{_n} = $LIB->_pow($x->{_n}, $y->{_n});
        $x->{_d} = $LIB->_pow($x->{_d}, $y->{_n});
        $x->{sign} = '+' if $x->{sign} eq '-' && $LIB->_is_even($y->{_n});
    }

unless ($LIB->_is_one($y->{_d})) {
        return $x->bsqrt(@r) if $LIB->_is_two($y->{_d}); # 1/2 => sqrt
        return $x->broot($LIB->_str($y->{_d}), @r);      # 1/N => root(N)
    }

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

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, @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, @r) =
          defined $_[2] ? objectify(2, @_) : objectify(1, @_);
    } else {
        # E.g., Math::BigFloat::blog(256, 2) or $x->blog(2)
        ($class, $x, $base, @r) =
          defined $_[1] ? objectify(2, @_) : objectify(1, @_);
    }

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

# Handle all exception cases and all trivial cases. I have used Wolfram Alpha
    # (http://www.wolframalpha.com) as the reference for these cases.

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

if (defined $base) {
        $base = $class -> new($base) unless ref $base;
        if ($base -> is_nan() || $base -> is_one()) {
            return $x -> bnan();
        } elsif ($base -> is_inf() || $base -> is_zero()) {
            return $x -> bnan() if $x -> is_inf() || $x -> is_zero();
            return $x -> bzero();
        } elsif ($base -> is_negative()) {        # -inf < base < 0
            return $x -> bzero() if $x -> is_one(); #     x = 1
            return $x -> bone()  if $x == $base;    #     x = base
            return $x -> bnan();                    #     otherwise
        }
        return $x -> bone() if $x == $base; # 0 < base && 0 < x < inf
    }

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

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

# At this point we are done handling all exception cases and trivial cases.

$base = Math::BigFloat -> new($base) if defined $base;

my $xn = Math::BigFloat -> new($LIB -> _str($x->{_n}));
    my $xd = Math::BigFloat -> new($LIB -> _str($x->{_d}));

my $xtmp = Math::BigRat -> new($xn -> bdiv($xd) -> blog($base, @r) -> bsstr());

$x -> {sign} = $xtmp -> {sign};
    $x -> {_n}   = $xtmp -> {_n};
    $x -> {_d}   = $xtmp -> {_d};

return $x;
}

sub bexp {
    # 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(1, @_);
    }

return $x->binf(@r)  if $x->{sign} eq '+inf';
    return $x->bzero(@r) 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(@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[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's not enough...
        $scale = abs($params[0] || $params[1]) + 4; # take whatever is defined
    }

return $x->bone(@params) if $x->is_zero();

# See the comments in Math::BigFloat on how this algorithm works.
    # Basically we calculate A and B (where B is faculty(N)) so that A/B = e

my $x_org = $x->copy();
    if ($scale <= 75) {
        # set $x directly from a cached string form
        $x->{_n} =
          $LIB->_new("90933395208605785401971970164779391644753259799242");
        $x->{_d} =
          $LIB->_new("33452526613163807108170062053440751665152000000000");
        $x->{sign} = '+';
    } 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("90933395208605785401971970164779391644753259799242");
        my $F = $LIB->_new(42); my $step = 42;

# Compute how many steps we need to take to get $A and $B sufficiently big
        my $steps = Math::BigFloat::_len_to_steps($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";

$x->{_n} = $A;
        $x->{_d} = $B;
        $x->{sign} = '+';
    }

# $x contains now an estimate of e, with some surplus digits, so we can round
    if (!$x_org->is_one()) {
        # raise $x to the wanted power and round it in one step:
        $x->bpow($x_org, @params);
    } else {
        # else just round the already computed result
        delete $x->{_a}; delete $x->{_p};
        # shortcut to not run through _find_round_parameters again
        if (defined $params[0]) {
            $x->bround($params[0], $params[2]); # then round accordingly
        } else {
            $x->bfround($params[1], $params[2]); # then round accordingly
        }
    }
    if ($fallback) {
        # clear a/p after round, since user did not request it
        delete $x->{_a}; delete $x->{_p};
    }

$x;
}

sub bnok {
    # 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, @_);
    }

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

$x -> {sign} = $xint -> {sign};
    $x -> {_n}   = $xint -> {_n};
    $x -> {_d}   = $xint -> {_d};

return $x;
}

sub broot {
    # 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, @_);
    }

# Convert $x into a Math::BigFloat.

my $xd   = Math::BigFloat -> new($LIB -> _str($x->{_d}));
    my $xflt = Math::BigFloat -> new($LIB -> _str($x->{_n})) -> bdiv($xd);
    $xflt -> {sign} = $x -> {sign};

# Convert $y into a Math::BigFloat.

my $yd   = Math::BigFloat -> new($LIB -> _str($y->{_d}));
    my $yflt = Math::BigFloat -> new($LIB -> _str($y->{_n})) -> bdiv($yd);
    $yflt -> {sign} = $y -> {sign};

# Compute the root and convert back to a Math::BigRat.

$xflt -> broot($yflt, @r);
    my $xtmp = Math::BigRat -> new($xflt -> bsstr());

$x -> {sign} = $xtmp -> {sign};
    $x -> {_n}   = $xtmp -> {_n};
    $x -> {_d}   = $xtmp -> {_d};

return $x;
}

sub bmodpow {
    # set up parameters
    my ($class, $x, $y, $m, @r) = (ref($_[0]), @_);
    # objectify is costly, so avoid it
    if ((!ref($_[0])) || (ref($_[0]) ne ref($_[1]))) {
        ($class, $x, $y, $m, @r) = objectify(3, @_);
    }

# Convert $x, $y, and $m into Math::BigInt objects.

my $xint = Math::BigInt -> new($x -> copy() -> bint());
    my $yint = Math::BigInt -> new($y -> copy() -> bint());
    my $mint = Math::BigInt -> new($m -> copy() -> bint());

$xint -> bmodpow($y, $m, @r);
    my $xtmp = Math::BigRat -> new($xint -> bsstr());

$x -> {sign} = $xtmp -> {sign};
    $x -> {_n}   = $xtmp -> {_n};
    $x -> {_d}   = $xtmp -> {_d};
    return $x;
}

sub bmodinv {
    # 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, @_);
    }

# Convert $x and $y into Math::BigInt objects.

my $xint = Math::BigInt -> new($x -> copy() -> bint());
    my $yint = Math::BigInt -> new($y -> copy() -> bint());

$xint -> bmodinv($y, @r);
    my $xtmp = Math::BigRat -> new($xint -> bsstr());

$x -> {sign} = $xtmp -> {sign};
    $x -> {_n}   = $xtmp -> {_n};
    $x -> {_d}   = $xtmp -> {_d};
    return $x;
}

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

return $x->bnan() if $x->{sign} !~ /^[+]/; # NaN, -inf or < 0
    return $x if $x->{sign} eq '+inf';         # sqrt(inf) == inf
    return $x->round(@r) if $x->is_zero() || $x->is_one();

my $n = $x -> {_n};
    my $d = $x -> {_d};

# Look for an exact solution. For the numerator and the denominator, take
    # the square root and square it and see if we got the original value. If we
    # did, for both the numerator and the denominator, we have an exact
    # solution.

{
        my $nsqrt = $LIB -> _sqrt($LIB -> _copy($n));
        my $n2    = $LIB -> _mul($LIB -> _copy($nsqrt), $nsqrt);
        if ($LIB -> _acmp($n, $n2) == 0) {
            my $dsqrt = $LIB -> _sqrt($LIB -> _copy($d));
            my $d2    = $LIB -> _mul($LIB -> _copy($dsqrt), $dsqrt);
            if ($LIB -> _acmp($d, $d2) == 0) {
                $x -> {_n} = $nsqrt;
                $x -> {_d} = $dsqrt;
                return $x->round(@r);
            }
        }
    }

local $Math::BigFloat::upgrade   = undef;
    local $Math::BigFloat::downgrade = undef;
    local $Math::BigFloat::precision = undef;
    local $Math::BigFloat::accuracy  = undef;
    local $Math::BigInt::upgrade     = undef;
    local $Math::BigInt::precision   = undef;
    local $Math::BigInt::accuracy    = undef;

my $xn = Math::BigFloat -> new($LIB -> _str($n));
    my $xd = Math::BigFloat -> new($LIB -> _str($d));

my $xtmp = Math::BigRat -> new($xn -> bdiv($xd) -> bsqrt() -> bsstr());

$x -> {sign} = $xtmp -> {sign};
    $x -> {_n}   = $xtmp -> {_n};
    $x -> {_d}   = $xtmp -> {_d};

$x->round(@r);
}

sub blsft {
    my ($class, $x, $y, $b, @r) = objectify(2, @_);

$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));
}

sub brsft {
    my ($class, $x, $y, $b, @r) = objectify(2, @_);

$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));
}

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;

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::BigRat

$x -> {sign} = $xtmp -> {sign};
    $x -> {_n}   = $xtmp -> {_n};
    $x -> {_d}   = $xtmp -> {_d};

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;

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::BigRat

$x -> {sign} = $xtmp -> {sign};
    $x -> {_n}   = $xtmp -> {_n};
    $x -> {_d}   = $xtmp -> {_d};

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;

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::BigRat

$x -> {sign} = $xtmp -> {sign};
    $x -> {_n}   = $xtmp -> {_n};
    $x -> {_d}   = $xtmp -> {_d};

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;

my @r = @_;

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

$x -> {sign} = $xtmp -> {sign};
    $x -> {_n}   = $xtmp -> {_n};
    $x -> {_d}   = $xtmp -> {_d};

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

##############################################################################
# round

sub round {
    $_[0];
}

sub bround {
    $_[0];
}

sub bfround {
    $_[0];
}

##############################################################################
# comparing

sub bcmp {
    # compare two signed numbers

# 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, @_);
    }

if ($x->{sign} !~ /^[+-]$/ || $y->{sign} !~ /^[+-]$/) {
        # $x is NaN and/or $y is NaN
        return undef if $x->{sign} eq $nan || $y->{sign} eq $nan;
        # $x and $y are both either +inf or -inf
        return 0     if $x->{sign} eq $y->{sign} && $x->{sign} =~ /^[+-]inf$/;
        # $x = +inf and $y < +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;
    }

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

# At this point, we know that $x and $y have the same sign.

# shortcut
    my $xz = $LIB->_is_zero($x->{_n});
    my $yz = $LIB->_is_zero($y->{_n});
    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

my $t = $LIB->_mul($LIB->_copy($x->{_n}), $y->{_d});
    my $u = $LIB->_mul($LIB->_copy($y->{_n}), $x->{_d});

my $cmp = $LIB->_acmp($t, $u);     # signs are equal
    $cmp = -$cmp if $x->{sign} eq '-'; # both are '-' => reverse
    $cmp;
}

sub bacmp {
    # compare two numbers (as unsigned)

# 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, @_);
    }

if (($x->{sign} !~ /^[+-]$/) || ($y->{sign} !~ /^[+-]$/)) {
        # handle +-inf and NaN
        return undef if (($x->{sign} eq $nan) || ($y->{sign} eq $nan));
        return 0 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} =~ /^[+-]inf$/;
        return 1 if $x->{sign} =~ /^[+-]inf$/ && $y->{sign} !~ /^[+-]inf$/;
        return -1;
    }

my $t = $LIB->_mul($LIB->_copy($x->{_n}), $y->{_d});
    my $u = $LIB->_mul($LIB->_copy($y->{_n}), $x->{_d});
    $LIB->_acmp($t, $u);        # ignore signs
}

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

croak 'beq() is an instance method, not a class method' unless $selfref;
    croak 'Wrong number of arguments for beq()' unless @_ == 1;

my $cmp = $self -> bcmp(shift);
    return defined($cmp) && ! $cmp;
}

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

croak 'bne() is an instance method, not a class method' unless $selfref;
    croak 'Wrong number of arguments for bne()' unless @_ == 1;

my $cmp = $self -> bcmp(shift);
    return defined($cmp) && ! $cmp ? '' : 1;
}

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

croak 'blt() is an instance method, not a class method' unless $selfref;
    croak 'Wrong number of arguments for blt()' unless @_ == 1;

my $cmp = $self -> bcmp(shift);
    return defined($cmp) && $cmp < 0;
}

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

croak 'ble() is an instance method, not a class method' unless $selfref;
    croak 'Wrong number of arguments for ble()' unless @_ == 1;

my $cmp = $self -> bcmp(shift);
    return defined($cmp) && $cmp <= 0;
}

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

croak 'bgt() is an instance method, not a class method' unless $selfref;
    croak 'Wrong number of arguments for bgt()' unless @_ == 1;

my $cmp = $self -> bcmp(shift);
    return defined($cmp) && $cmp > 0;
}

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

croak 'bge() is an instance method, not a class method'
        unless $selfref;
    croak 'Wrong number of arguments for bge()' unless @_ == 1;

my $cmp = $self -> bcmp(shift);
    return defined($cmp) && $cmp >= 0;
}

##############################################################################
# output conversion

sub numify {
    # convert 17/8 => float (aka 2.125)
    my ($self, $x) = ref($_[0]) ? (undef, $_[0]) : objectify(1, @_);

# Non-finite number.

return $x->bstr() if $x->{sign} !~ /^[+-]$/;

# Finite number.

my $abs = $LIB->_is_one($x->{_d})
            ? $LIB->_num($x->{_n})
            : Math::BigFloat -> new($LIB->_str($x->{_n}))
                             -> bdiv($LIB->_str($x->{_d}))
                             -> bstr();
    return $x->{sign} eq '-' ? 0 - $abs : 0 + $abs;
}

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

# NaN, inf etc
    return Math::BigInt->new($x->{sign}) if $x->{sign} !~ /^[+-]$/;

my $u = Math::BigInt->bzero();
    $u->{value} = $LIB->_div($LIB->_copy($x->{_n}), $x->{_d}); # 22/7 => 3
    $u->bneg if $x->{sign} eq '-'; # no negative zero
    $u;
}

sub as_float {
    # return N/D as Math::BigFloat

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

# NaN, inf etc
    return Math::BigFloat->new($x->{sign}) if $x->{sign} !~ /^[+-]$/;

my $xd   = Math::BigFloat -> new($LIB -> _str($x->{_d}));
    my $xflt = Math::BigFloat -> new($LIB -> _str($x->{_n}));
    $xflt -> {sign} = $x -> {sign};
    $xflt -> bdiv($xd, @r);

return $xflt;
}

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

return $x unless $x->is_int();

my $s = $x->{sign};
    $s = '' if $s eq '+';
    $s . $LIB->_as_bin($x->{_n});
}

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

return $x unless $x->is_int();

my $s = $x->{sign}; $s = '' if $s eq '+';
    $s . $LIB->_as_hex($x->{_n});
}

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

return $x unless $x->is_int();

my $s = $x->{sign}; $s = '' if $s eq '+';
    $s . $LIB->_as_oct($x->{_n});
}

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

sub from_hex {
    my $class = shift;

$class->new(@_);
}

sub from_bin {
    my $class = shift;

$class->new(@_);
}

sub from_oct {
    my $class = shift;

my @parts;
    for my $c (@_) {
        push @parts, Math::BigInt->from_oct($c);
    }
    $class->new (@parts);
}

##############################################################################
# import

sub import {
    my $class = shift;
    my $l = scalar @_;
    my $lib = ''; my @a;
    my $try = 'try';

for (my $i = 0; $i < $l ; $i++) {
        if ($_[$i] eq ':constant') {
            # this rest causes overlord er load to step in
            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/) {
            $lib = $_[$i+1] || ''; # default Calc
            $try = $1;             # lib, try or only
            $i++;
        } elsif ($_[$i] eq 'with') {
            # this argument is no longer used
            #$LIB = $_[$i+1] || 'Math::BigInt::Calc'; # default Math::BigInt::Calc
            $i++;
        } else {
            push @a, $_[$i];
        }
    }
    require Math::BigInt;

# let use Math::BigInt lib => 'GMP'; use Math::BigRat; still have GMP
    if ($lib ne '') {
        my @c = split /\s*,\s*/, $lib;
        foreach (@c) {
            $_ =~ tr/a-zA-Z0-9://cd; # limit to sane characters
        }
        $lib = join(",", @c);
    }
    my @import = ('objectify');
    push @import, $try => $lib if $lib ne '';

# LIB already loaded, so feed it our lib arguments
    Math::BigInt->import(@import);

$LIB = Math::BigFloat->config("lib");

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

# any non :constant stuff is handled by Exporter (loaded by parent class)
    # even if @_ is empty, to give it a chance
    $class->SUPER::import(@a);           # for subclasses
    $class->export_to_level(1, $class, @a); # need this, too
}

__END__

=pod

=head1 NAME

Math::BigRat - Arbitrary big rational numbers

=head1 SYNOPSIS

use Math::BigRat;

my $x = Math::BigRat->new('3/7'); $x += '5/9';

print $x->bstr(), "\n";
    print $x ** 2, "\n";

my $y = Math::BigRat->new('inf');
    print "$y ", ($y->is_inf ? 'is' : 'is not'), " infinity\n";

my $z = Math::BigRat->new(144); $z->bsqrt();

=head1 DESCRIPTION

Math::BigRat complements Math::BigInt and Math::BigFloat by providing support
for arbitrary big rational numbers.

=head2 MATH LIBRARY

You can change the underlying module that does the low-level
math operations by using:

use Math::BigRat try => 'GMP';

Note: This needs Math::BigInt::GMP installed.

The following would first try to find Math::BigInt::Foo, then
Math::BigInt::Bar, and when this also fails, revert to Math::BigInt::Calc:

use Math::BigRat try => 'Foo,Math::BigInt::Bar';

If you want to get warned when the fallback occurs, replace "try" with "lib":

use Math::BigRat lib => 'Foo,Math::BigInt::Bar';

If you want the code to die instead, replace "try" with "only":

use Math::BigRat only => 'Foo,Math::BigInt::Bar';

=head1 METHODS

Any methods not listed here are derived from Math::BigFloat (or
Math::BigInt), so make sure you check these two modules for further
information.

=over

=item new()

$x = Math::BigRat->new('1/3');

Create a new Math::BigRat object. Input can come in various forms:

$x = Math::BigRat->new(123);                            # scalars
    $x = Math::BigRat->new('inf');                          # infinity
    $x = Math::BigRat->new('123.3');                        # float
    $x = Math::BigRat->new('1/3');                          # simple string
    $x = Math::BigRat->new('1 / 3');                        # spaced
    $x = Math::BigRat->new('1 / 0.1');                      # w/ floats
    $x = Math::BigRat->new(Math::BigInt->new(3));           # BigInt
    $x = Math::BigRat->new(Math::BigFloat->new('3.1'));     # BigFloat
    $x = Math::BigRat->new(Math::BigInt::Lite->new('2'));   # BigLite

# You can also give D and N as different objects:
    $x = Math::BigRat->new(
            Math::BigInt->new(-123),
            Math::BigInt->new(7),
         );                      # => -123/7

=item numerator()

$n = $x->numerator();

Returns a copy of the numerator (the part above the line) as signed BigInt.

=item denominator()

$d = $x->denominator();

Returns a copy of the denominator (the part under the line) as positive BigInt.

=item parts()

($n, $d) = $x->parts();

Return a list consisting of (signed) numerator and (unsigned) denominator as
BigInts.

=item numify()

my $y = $x->numify();

Returns the object as a scalar. This will lose some data if the object
cannot be represented by a normal Perl scalar (integer or float), so
use L</as_int()> or L</as_float()> instead.

This routine is automatically used whenever a scalar is required:

my $x = Math::BigRat->new('3/1');
    @array = (0, 1, 2, 3);
    $y = $array[$x];                # set $y to 3

=item as_int()

=item as_number()

$x = Math::BigRat->new('13/7');
    print $x->as_int(), "\n";               # '1'

Returns a copy of the object as BigInt, truncated to an integer.

C<as_number()> is an alias for C<as_int()>.

=item as_float()

$x = Math::BigRat->new('13/7');
    print $x->as_float(), "\n";             # '1'

$x = Math::BigRat->new('2/3');
    print $x->as_float(5), "\n";            # '0.66667'

Returns a copy of the object as BigFloat, preserving the
accuracy as wanted, or the default of 40 digits.

This method was added in v0.22 of Math::BigRat (April 2008).

=item as_hex()

$x = Math::BigRat->new('13');
    print $x->as_hex(), "\n";               # '0xd'

Returns the BigRat as hexadecimal string. Works only for integers.

=item as_bin()

$x = Math::BigRat->new('13');
    print $x->as_bin(), "\n";               # '0x1101'

Returns the BigRat as binary string. Works only for integers.

=item as_oct()

$x = Math::BigRat->new('13');
    print $x->as_oct(), "\n";               # '015'

Returns the BigRat as octal string. Works only for integers.

=item from_hex()

my $h = Math::BigRat->from_hex('0x10');

Create a BigRat from a hexadecimal number in string form.

=item from_oct()

my $o = Math::BigRat->from_oct('020');

Create a BigRat from an octal number in string form.

=item from_bin()

my $b = Math::BigRat->from_bin('0b10000000');

Create a BigRat from an binary number in string form.

=item bnan()

$x = Math::BigRat->bnan();

Creates a new BigRat object representing NaN (Not A Number).
If used on an object, it will set it to NaN:

$x->bnan();

=item bzero()

$x = Math::BigRat->bzero();

Creates a new BigRat object representing zero.
If used on an object, it will set it to zero:

$x->bzero();

=item binf()

$x = Math::BigRat->binf($sign);

Creates a new BigRat object representing infinity. The optional argument is
either '-' or '+', indicating whether you want infinity or minus infinity.
If used on an object, it will set it to infinity:

$x->binf();
    $x->binf('-');

=item bone()

$x = Math::BigRat->bone($sign);

Creates a new BigRat object representing one. The optional argument is
either '-' or '+', indicating whether you want one or minus one.
If used on an object, it will set it to one:

$x->bone();                 # +1
    $x->bone('-');              # -1

=item length()

$len = $x->length();

Return the length of $x in digits for integer values.

=item digit()

print Math::BigRat->new('123/1')->digit(1);     # 1
    print Math::BigRat->new('123/1')->digit(-1);    # 3

Return the N'ths digit from X when X is an integer value.

=item bnorm()

$x->bnorm();

Reduce the number to the shortest form. This routine is called
automatically whenever it is needed.

=item bfac()

$x->bfac();

Calculates the factorial of $x. For instance:

print Math::BigRat->new('3/1')->bfac(), "\n";   # 1*2*3
    print Math::BigRat->new('5/1')->bfac(), "\n";   # 1*2*3*4*5

Works currently only for integers.

=item bround()/round()/bfround()

Are not yet implemented.

=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 bmodinv()

$x->bmodinv($mod);          # modular multiplicative inverse

Returns the multiplicative inverse of C<$x> modulo C<$mod>. If

$y = $x -> copy() -> bmodinv($mod)

then C<$y> is the number closest to zero, and with the same sign as C<$mod>,
satisfying

($x * $y) % $mod = 1 % $mod

If C<$x> and C<$y> are non-zero, they must be relative primes, i.e.,
C<bgcd($y, $mod)==1>. 'C<NaN>' is returned when no modular multiplicative
inverse exists.

=item bmodpow()

$num->bmodpow($exp,$mod);           # modular exponentiation
                                        # ($num**$exp % $mod)

Returns the value of C<$num> taken to the power C<$exp> in the modulus
C<$mod> using binary exponentiation.  C<bmodpow> is far superior to
writing

$num ** $exp % $mod

because it is much faster - it reduces internal variables into
the modulus whenever possible, so it operates on smaller numbers.

C<bmodpow> also supports negative exponents.

bmodpow($num, -1, $mod)

is exactly equivalent to

bmodinv($num, $mod)

=item bneg()

$x->bneg();

Used to negate the object in-place.

=item is_one()

print "$x is 1\n" if $x->is_one();

Return true if $x is exactly one, otherwise false.

=item is_zero()

print "$x is 0\n" if $x->is_zero();

Return true if $x is exactly zero, otherwise false.

=item is_pos()/is_positive()

print "$x is >= 0\n" if $x->is_positive();

Return true if $x is positive (greater than or equal to zero), otherwise
false. Please note that '+inf' is also positive, while 'NaN' and '-inf' aren't.

C<is_positive()> is an alias for C<is_pos()>.

=item is_neg()/is_negative()

print "$x is < 0\n" if $x->is_negative();

Return true if $x is negative (smaller than zero), otherwise false. Please
note that '-inf' is also negative, while 'NaN' and '+inf' aren't.

C<is_negative()> is an alias for C<is_neg()>.

=item is_int()

print "$x is an integer\n" if $x->is_int();

Return true if $x has a denominator of 1 (e.g. no fraction parts), otherwise
false. Please note that '-inf', 'inf' and 'NaN' aren't integer.

=item is_odd()

print "$x is odd\n" if $x->is_odd();

Return true if $x is odd, otherwise false.

=item is_even()

print "$x is even\n" if $x->is_even();

Return true if $x is even, otherwise false.

=item bceil()

$x->bceil();

Set $x to the next bigger integer value (e.g. truncate the number to integer
and then increment it by one).

=item bfloor()

$x->bfloor();

Truncate $x to an integer value.

=item bint()

$x->bint();

Round $x towards zero.

=item bsqrt()

$x->bsqrt();

Calculate the square root of $x.

=item broot()

$x->broot($n);

Calculate the N'th root of $x.

=item badd()

$x->badd($y);

Adds $y to $x and returns the result.

=item bmul()

$x->bmul($y);

Multiplies $y to $x and returns the result.

=item bsub()

$x->bsub($y);

Subtracts $y from $x and returns the result.

=item bdiv()

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

In scalar context, divides $x by $y and returns the result. 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 bdec()

$x->bdec();

Decrements $x by 1 and returns the result.

=item binc()

$x->binc();

Increments $x by 1 and returns the result.

=item copy()

my $z = $x->copy();

Makes a deep copy of the object.

Please see the documentation in L<Math::BigInt> for further details.

=item bstr()/bsstr()

my $x = Math::BigRat->new('8/4');
    print $x->bstr(), "\n";             # prints 1/2
    print $x->bsstr(), "\n";            # prints 1/2

Return a string representing this object.

=item bcmp()

$x->bcmp($y);

Compares $x with $y and takes the sign into account.
Returns -1, 0, 1 or undef.

=item bacmp()

$x->bacmp($y);

Compares $x with $y while ignoring their sign. Returns -1, 0, 1 or undef.

=item beq()

$x -> beq($y);

Returns true if and only if $x is equal to $y, and false otherwise.

=item bne()

$x -> bne($y);

Returns true if and only if $x is not equal to $y, and false otherwise.

=item blt()

$x -> blt($y);

Returns true if and only if $x is equal to $y, and false otherwise.

=item ble()

$x -> ble($y);

Returns true if and only if $x is less than or equal to $y, and false
otherwise.

=item bgt()

$x -> bgt($y);

Returns true if and only if $x is greater than $y, and false otherwise.

=item bge()

$x -> bge($y);

Returns true if and only if $x is greater than or equal to $y, and false
otherwise.

=item blsft()/brsft()

Used to shift numbers left/right.

Please see the documentation in L<Math::BigInt> for further details.

=item band()

$x->band($y);               # bitwise and

=item bior()

$x->bior($y);               # bitwise inclusive or

=item bxor()

$x->bxor($y);               # bitwise exclusive or

=item bnot()

$x->bnot();                 # bitwise not (two's complement)

=item bpow()

$x->bpow($y);

Compute $x ** $y.

Please see the documentation in L<Math::BigInt> for further details.

=item blog()

$x->blog($base, $accuracy);         # logarithm of x to the base $base

If C<$base> is not defined, Euler's number (e) is used:

print $x->blog(undef, 100);         # log(x) to 100 digits

=item bexp()

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

Calculates two integers A and B so that A/B is equal to C<e ** $x>, where C<e> is
Euler's number.

This method was added in v0.20 of Math::BigRat (May 2007).